Finite Group Databasebeta

← #30550591

Subgroups — #30550591

Order 15616.

2394 conjugacy classes of subgroups.

The subgroup lattice has 700 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840275C212yesmaximal, complemented
2#21456613C214yesmaximal, complemented
3#21456603C214yesmaximal, complemented
4#21456589C214yesmaximal, complemented
5#21456572C212yesmaximal, complemented
6#21456570C211yesmaximal, characteristic, complemented
7#21455735C214yesmaximal, complemented
8#21455402C214yesmaximal, complemented
9#21455042C214yesmaximal, complemented
10C2×C22⋊C8×D61C211yesmaximal, characteristic, complemented
11C61×#3189C211yesmaximal, characteristic, complemented
12C22×#11758226C2212yescomplemented
13C23×C4×D61C2211yescharacteristic
14C2×#14519800C2214yescomplemented
15C22×C8×D61C2212yescomplemented
16#16840275—22no—
17#16840275—22no—
18#16840275—22no—
19#16840275—22no—
20#16840275C2214yes—
21#16840275C2214yes—
22#16840275C2214yescomplemented
23#16840275C2214yescomplemented
24C2×#14519762—22no—
25C2×#14519753C2212yescomplemented
26C2×#14519752C2212yescomplemented
27C2×#14519751C2212yes—
28C2×#14519747—22no—
29C2×#14519739—22no—
30C2×#14519737—22no—
31C2×#14519735C2212yescomplemented
32#16840102C2218yescomplemented
33#16840090C2214yescomplemented
34#16840089C2218yescomplemented
35#16840080C2214yescomplemented
36#16840076C2214yescomplemented
37#16840000C2214yescomplemented
38#16839999C2214yescomplemented
39#16839998C2214yescomplemented
40#16839994C2218yescomplemented
41#16839993C2218yescomplemented
42#16839923C2218yescomplemented
43#16839912C2214yescomplemented
44#16839910C2218yescomplemented
45#16839901C2214yescomplemented
46#16839898C2214yescomplemented
47#16839799C2214yescomplemented
48#16839757C2218yescomplemented
49C2×C22⋊#92418C2211yescharacteristic
50C2×#14519689C2212yescomplemented
51C22⋊C8×D61C2214yescomplemented
52C22⋊C8×D61C2214yescomplemented
53C2×C22⋊Q8×C61C2212yes—
54C61×#484C2214yescomplemented
55C61×#467C2214yescomplemented
56C61×#451C2214yescomplemented
57C2×C22⋊C8×C61C2211yescharacteristic
58C24×D61D411yescharacteristic
59C2×#11758226—22no—
60C2×#11758226—22no—
61C2×#11758226—48no—
62C2×#11758226C2314yes—
63C2×#11758226C2314yes—
64C22×D244—22no—
65C22×D244—22no—
66C22×D244C2312yes—
67C22×C22⋊D61—22no—
68C23×Dic61D411yescharacteristic
69C22×C4×D61—22no—
70C22×C4×D61—44no—
71C22×C4×D61D412yes—
72C22×C4×D61D412yes—
73C22×C4×D61D411yescharacteristic
74C22×C4×D61D411yescharacteristic
75C22×C4×D61C2312yes—
76C22×C4×D61C2311yescharacteristic
77C22×C4×D61C2312yes—
78C22×C4×D61C2311yescharacteristic
79#14519800—48no—
80C2×#11758201C2318yescomplemented
81C2×D488C2314yescomplemented
82C22×#92418C2312yes—
83C2×C8×D61—44no—
84C2×C8×D61C2314yes—
85C2×C8×D61C2314yes—
86#14519762—24no—
87#14519762—24no—
88#14519762—24no—
89#14519762—24no—
90#14519753—22no—
91#14519753—22no—
92#14519753C2314yes—
93#14519753C2314yes—
94#14519752—22no—
95#14519752—22no—
96#14519752C2314yes—
97#14519752C2314yes—
98#14519751—22no—
99#14519751—22no—
100#14519751C2314yes—
101#14519751C2314yes—
102#14519747—24no—
103#14519747—24no—
104#14519747—24no—
105#14519747—24no—
106#14519739—24no—
107#14519739—24no—
108#14519739—24no—
109#14519739—24no—
110#14519737—24no—
111#14519737—24no—
112#14519737—24no—
113#14519737—24no—
114#14519735—22no—
115#14519735—22no—
116#14519735C2314yes—
117#14519735C2314yes—
118C2×C4⋊Dic61C2312yes—
119C2×#11758196—22no—
120C2×#11758198—22no—
121C2×#11758198—22no—
122C2×#11758198—22no—
123C2×#11758198—22no—
124C2×C4×Dic61—22no—
125#14519711C2314yescomplemented
126#14519708C2318yescomplemented
127#14519701C2318yescomplemented
128#14519699C2318yescomplemented
129#14519697C2318yescomplemented
130C22⋊#92418C2314yes—
131#14519689C2318yes—
132C22×Q8×C61C2312yes—
133C23×C4×C61C2311yescharacteristic
134C2×Q16×C61C2314yescomplemented
135C22×C8×C61C2312yes—
136C22⋊Q8×C61—22no—
137C22⋊Q8×C61—22no—
138C22⋊Q8×C61C2314yes—
139C22⋊Q8×C61C2314yes—
140C2×C4⋊C4×C61—22no—
141C2×C4⋊C4×C61C2312yes—
142C2×C22⋊C4×C61—22no—
143#107×C61C2314yescomplemented
144#103×C61C2318yescomplemented
145C22⋊C8×C61C2314yes—
146C24×C61C2×D411yescharacteristic
147C23×D61—24no—
148C23×D61—22no—
149C23×D61—42no—
150C23×D61—42no—
151C23×D61C2×D411yescharacteristic
152C23×D61C2×D412yes—
153C23×D61C2×D412yes—
154C23×D61C2×D411yescharacteristic
155C2×Q8×C61—22no—
156C2×Q8×C61—44no—
157C2×Q8×C61C2414yes—
158C22×C4×C61—22no—
159C22×C4×C61—22no—
160C22×C4×C61—42no—
161C22×C4×C61C2×D412yes—
162C22×C4×C61C2×D411yescharacteristic
163C22×C4×C61C2412yes—
164C22×C4×C61C2411yescharacteristic
165C2×C22⋊D61—24no—
166C2×C22⋊D61—24no—
167C2×C22⋊D61—48no—
168C22×Dic61—22no—
169C22×Dic61—42no—
170C22×Dic61C2×D412yes—
171C22×Dic61C2×D411yescharacteristic
172C22×Dic61C2×D412yes—
173C22×Dic61C2×D411yescharacteristic
174#11758226—216no—
175#11758226—48no—
176#11758226—48no—
177#11758226—48no—
178C2×D244—24no—
179C2×D244—24no—
180C2×D244—24no—
181C2×D244—24no—
182C2×D244—22no—
183C2×D244—44no—
184C2×D244—48no—
185C2×D244—48no—
186C2×D244C2414yes—
187C2×C4×D61—28no—
188C2×C4×D61—24no—
189C2×C4×D61—24no—
190C2×C4×D61—24no—
191C2×C4×D61—44no—
192C2×C4×D61—42no—
193C2×C4×D61—44no—
194C2×C4×D61—42no—
195C2×C4×D61—44no—
196C2×C4×D61—42no—
197C2×C4×D61—48no—
198C2×C4×D61C2×D412yes—
199C2×C4×D61C2×D412yes—
200C2×C4×D61C2×D411yescharacteristic
201C2×C4×D61C2×D411yescharacteristic
202C2×C4×D61C2×D412yes—
203C2×C4×D61C2×D412yes—
204C2×C4×D61C2×D412yes—
205C2×C4×D61C2412yes—
206C2×C4×D61C2411yescharacteristic
207C2×C4×D61C2411yescharacteristic
208Q16×C61—44no—
209C2×C8×C61—42no—
210C2×C8×C61C2414yes—
211C4⋊C4×C61—24no—
212C4⋊C4×C61—24no—
213C4⋊C4×C61—22no—
214C4⋊C4×C61C2414yes—
215C22⋊C4×C61—24no—
216C22⋊C4×C61—24no—
217#11758201—48no—
218#11758198—24no—
219#11758198—24no—
220#11758198—24no—
221#11758198—24no—
222#11758198—24no—
223#11758198—24no—
224#11758198—24no—
225#11758198—24no—
226C4⋊Dic61—22no—
227C4⋊Dic61C2414yes—
228#11758196—24no—
229#11758196—24no—
230C4×Dic61—24no—
231C4×Dic61—24no—
232C2×#92418—42no—
233C2×#92418C2414yes—
234D488—44no—
235C8×D61—28no—
236C8×D61—44no—
237C23×C61—41no—
238C23×C61—41no—
239C23×C61#13514yescomplemented
240C23×C61C22×D411yescharacteristic
241C23×C61C22×D412yes—
242C22×D61—22no—
243C22×D61—22no—
244C22×D61—22no—
245C22×D61—24no—
246C22×D61—24no—
247C22×D61—22no—
248C22×D61—24no—
249C22×D61—24no—
250C22×D61—24no—
251C22×D61—42no—
252C22×D61—41no—
253C22×D61—42no—
254C22×D61—41no—
255C22×D61—416no—
256C22×D61—44no—
257C22×D61—44no—
258C22×D61—44no—
259C22×D61C22×D412yes—
260C22×D61C22×D411yescharacteristic
261C22×D61C22×D411yescharacteristic
262Q8×C61—28no—
263Q8×C61—44no—
264C2×C4×C61—22no—
265C2×C4×C61—24no—
266C2×C4×C61—24no—
267C2×C4×C61—24no—
268C2×C4×C61—41no—
269C2×C4×C61—42no—
270C2×C4×C61—44no—
271C2×C4×C61—44no—
272C2×C4×C61C22×D412yes—
273C2×C4×C61C22×D411yescharacteristic
274C2×C4×C61C2511yescharacteristic
275C22⋊D61—48no—
276C22⋊D61—48no—
277C2×Dic61—24no—
278C2×Dic61—22no—
279C2×Dic61—24no—
280C2×Dic61—41no—
281C2×Dic61—42no—
282C2×Dic61—44no—
283C2×Dic61C22×D412yes—
284C2×Dic61C22×D411yescharacteristic
285C2×Dic61C22×D411yescharacteristic
286D244—28no—
287D244—44no—
288D244—48no—
289D244—48no—
290D244—48no—
291D244—48no—
292C4×D61—22no—
293C4×D61—24no—
294C4×D61—22no—
295C4×D61—24no—
296C4×D61—22no—
297C4×D61—24no—
298C4×D61—42no—
299C4×D61—42no—
300C4×D61—42no—
301C4×D61—48no—
302C4×D61—48no—
303C4×D61H(5,𝔽2)14yes—
304C8×C61—24no—
305#92418—24no—
306#60734—611nomaximal, Sylow
307C22×C61—22no—
308C22×C61—21no—
309C22×C61—22no—
310C22×C61—21no—
311C22×C61—44no—
312C22×C61C2×#13514yes—
313C22×C61C2×#13512yes—
314C22×C61C23×D411yescharacteristic
315C2×D61—28no—
316C2×D61—21no—
317C2×D61—21no—
318C2×D61—21no—
319C2×D61—21no—
320C2×D61—22no—
321C2×D61—22no—
322C2×D61—22no—
323C2×D61—48no—
324C2×D61—48no—
325C2×D61—44no—
326C2×D61—44no—
327C2×D61—44no—
328C2×D61—44no—
329C2×D61—88no—
330C2×D61#53412yes—
331C2×D61#57512yes—
332C4×C61—22no—
333C4×C61—21no—
334C4×C61—44no—
335C4×C61—44no—
336C4×C61C2×H(5,𝔽2)12yes—
337Dic61—22no—
338Dic61—21no—
339Dic61—44no—
340Dic61#53412yes—
341C2×#525—612no—
342#3191—614no—
343#3190—614no—
344#3189D6111yescharacteristic, complemented
345#3188—614no—
346#3187—612no—
347#3186—611no—
348#3091—614no—
349#3050—614no—
350#3002—614no—
351C22×C22⋊C8—611no—
352C2×C61—22no—
353C2×C61—24no—
354C2×C61C2×#53411yescharacteristic
355C2×C61C22×#13511yescharacteristic
356C2×C61C2×#57511yescharacteristic
357D61—22no—
358D61—22no—
359D61—42no—
360D61—84no—
361C22×#41—612no—
362C24×C4—611no—
363C2×#135—614no—
364C23×C8—612no—
365#525—614no—
366#525—614no—
367#525—614no—
368#525—614no—
369#525—1222no—
370#525—1222no—
371#525—1222no—
372#525—1222no—
373C2×#123—1222no—
374C2×C22⋊Q8—612no—
375C2×C22⋊Q8C2×D6112yes—
376C2×#121—612no—
377C2×#121—612no—
378C2×#120—1222no—
379C2×C4×D4—1222no—
380C2×C4×D4—1222no—
381C2×#117—612no—
382#485—618no—
383#484C2×D6114yescomplemented
384#483—614no—
385#482—618no—
386#481—614no—
387#480—614no—
388#467—614no—
389#467C2×D6114yescomplemented
390#466—614no—
391#466—614no—
392#465—618no—
393#465—618no—
394#452—618no—
395#451C2×D6114yescomplemented
396#450—614no—
397#449—618no—
398#448—614no—
399#447—614no—
400#427—614no—
401#416—618no—
402C2×C22⋊C8—611no—
403C2×C22⋊C8—612no—
404C2×C22⋊C8—614no—
405C2×C22⋊C8—614no—
406C2×C22⋊C8C2×D6111yescharacteristic
407C61#6073411yescharacteristic, complemented, Sylow
408C25—611no—
409C2×#41—614no—
410C2×#41—614no—
411C2×#41—1222no—
412C2×#41—1222no—
413C2×#41—2448no—
414C22×Q8C22×D6112yes—
415C22×D4—612no—
416C22×D4—1222no—
417C22×D4—1222no—
418C22×D4—1222no—
419C23×C4—611no—
420C23×C4—612no—
421C23×C4—611no—
422C23×C4—612no—
423C23×C4—612no—
424C23×C4—611no—
425C23×C4—611no—
426C23×C4—612no—
427C23×C4—611no—
428C23×C4—1222no—
429C23×C4—2444no—
430C23×C4C22×D6111yescharacteristic
431#135—2448no—
432C2×Q16C22×D6114yescomplemented
433C2×SD16—618no—
434C2×D8—614no—
435C22×C8—612no—
436C22×C8—614no—
437C22×C8—614no—
438C22×C8—2444no—
439C22×C8C22×D6112yes—
440#123—1224no—
441#123—1224no—
442#123—1224no—
443#123—1224no—
444C22⋊Q8—22no—
445C22⋊Q8—22no—
446C22⋊Q8—614no—
447C22⋊Q8—614no—
448C22⋊Q8—1222no—
449C22⋊Q8—1222no—
450C22⋊Q8C22×D6114yes—
451C22⋊Q8C22×D6114yes—
452#121—614no—
453#121—614no—
454#121—614no—
455#121—614no—
456#121—1222no—
457#121—1222no—
458#121—1222no—
459#121—1222no—
460#120—1224no—
461#120—1224no—
462#120—1224no—
463#120—1224no—
464C4×D4—1224no—
465C4×D4—1224no—
466C4×D4—1224no—
467C4×D4—1224no—
468C4×D4—1224no—
469C4×D4—1224no—
470C4×D4—1224no—
471C4×D4—1224no—
472#117—614no—
473#117—614no—
474#117—1222no—
475#117—1222no—
476C2×C4⋊C4—22no—
477C2×C4⋊C4—612no—
478C2×C4⋊C4—1222no—
479C2×C4⋊C4C22×D6112yes—
480C2×C22⋊C4—22no—
481C2×C22⋊C4—1222no—
482C2×C22⋊C4—1222no—
483C2×C22⋊C4—1222no—
484C2×C22⋊C4—1222no—
485C2×C42—1222no—
486#107—614no—
487#107C22×D6114yescomplemented
488#106—618no—
489#103—618no—
490#103C22×D6118yescomplemented
491#102—618no—
492#102—618no—
493C22⋊C8—614no—
494C22⋊C8—618no—
495C22⋊C8C22×D6114yes—
496C24—611no—
497C24—612no—
498C24—612no—
499C24—611no—
500C24—1224no—
501C24—1222no—
502C24—2442no—
503C24—2442no—
504C24D4×D6111yescharacteristic
505#41—12216no—
506#41—2448no—
507#41—2448no—
508#41—2448no—
509C2×Q8—22no—
510C2×Q8—44no—
511C2×Q8C23×D6114yes—
512C2×D4—614no—
513C2×D4—1224no—
514C2×D4—1224no—
515C2×D4—1224no—
516C2×D4—1224no—
517C2×D4—1224no—
518C2×D4—1224no—
519C2×D4—1222no—
520C2×D4—2444no—
521C2×D4—2448no—
522C2×D4—2448no—
523C2×D4—2448no—
524C22×C4—22no—
525C22×C4—22no—
526C22×C4—42no—
527C22×C4—611no—
528C22×C4—612no—
529C22×C4—611no—
530C22×C4—612no—
531C22×C4—611no—
532C22×C4—612no—
533C22×C4—612no—
534C22×C4—611no—
535C22×C4—611no—
536C22×C4—612no—
537C22×C4—611no—
538C22×C4—612no—
539C22×C4—612no—
540C22×C4—612no—
541C22×C4—1224no—
542C22×C4—1228no—
543C22×C4—1224no—
544C22×C4—1224no—
545C22×C4—1222no—
546C22×C4—2442no—
547C22×C4—2444no—
548C22×C4—2444no—
549C22×C4—2442no—
550C22×C4—2444no—
551C22×C4—2442no—
552C22×C4—2448no—
553C22×C4—2442no—
554C22×C4D4×D6112yes—
555C22×C4D4×D6111yescharacteristic
556C22×C4C23×D6112yes—
557C22×C4C23×D6111yescharacteristic
558Q16—44no—
559SD16—2448no—
560D8—2444no—
561C2×C8—42no—
562C2×C8—614no—
563C2×C8—1228no—
564C2×C8—2444no—
565C2×C8—2442no—
566C2×C8C23×D6114yes—
567C4⋊C4—24no—
568C4⋊C4—24no—
569C4⋊C4—22no—
570C4⋊C4—614no—
571C4⋊C4—1224no—
572C4⋊C4—1224no—
573C4⋊C4—1222no—
574C4⋊C4C23×D6114yes—
575C22⋊C4—24no—
576C22⋊C4—24no—
577C22⋊C4—1224no—
578C22⋊C4—1224no—
579C22⋊C4—1224no—
580C22⋊C4—1224no—
581C22⋊C4—1224no—
582C22⋊C4—1224no—
583C22⋊C4—1224no—
584C22⋊C4—1224no—
585C42—1224no—
586C42—1224no—
587C23—41no—
588C23—41no—
589C23—611no—
590C23—611no—
591C23—612no—
592C23—1222no—
593C23—1222no—
594C23—1224no—
595C23—1224no—
596C23—1222no—
597C23—1224no—
598C23—1224no—
599C23—1222no—
600C23—1224no—
601C23—2442no—
602C23—2441no—
603C23—2442no—
604C23—2441no—
605C23—2444no—
606C23—2444no—
607C23—24416no—
608C23—2444no—
609C23#1451980014yescomplemented
610C23C2×D4×D6111yescharacteristic
611C23C2×D4×D6112yes—
612Q8—28no—
613Q8—44no—
614D4—1228no—
615D4—2444no—
616D4—2448no—
617D4—2448no—
618D4—2448no—
619D4—2448no—
620D4—2448no—
621D4—2448no—
622C2×C4—22no—
623C2×C4—24no—
624C2×C4—24no—
625C2×C4—24no—
626C2×C4—42no—
627C2×C4—41no—
628C2×C4—44no—
629C2×C4—44no—
630C2×C4—611no—
631C2×C4—614no—
632C2×C4—612no—
633C2×C4—611no—
634C2×C4—1222no—
635C2×C4—1224no—
636C2×C4—1222no—
637C2×C4—1224no—
638C2×C4—1222no—
639C2×C4—1224no—
640C2×C4—1222no—
641C2×C4—1224no—
642C2×C4—1224no—
643C2×C4—2442no—
644C2×C4—2442no—
645C2×C4—2442no—
646C2×C4—2448no—
647C2×C4—2448no—
648C2×C4—2442no—
649C2×C4—2441no—
650C2×C4—2444no—
651C2×C4C2×D4×D6111yescharacteristic
652C2×C4C2×D4×D6112yes—
653C2×C4C24×D6111yescharacteristic
654C8—24no—
655C8—1224no—
656C22—22no—
657C22—21no—
658C22—22no—
659C22—21no—
660C22—44no—
661C22—612no—
662C22—612no—
663C22—1228no—
664C22—1221no—
665C22—1221no—
666C22—1221no—
667C22—1221no—
668C22—1222no—
669C22—1222no—
670C22—1222no—
671C22—4888no—
672C22—2448no—
673C22—2448no—
674C22—2444no—
675C22—2444no—
676C22—2444no—
677C22—2444no—
678C22C2×#1451980014yes—
679C22C2×#1451980012yes—
680C22C22×D4×D6111yescharacteristic
681C4—21no—
682C4—22no—
683C4—44no—
684C4—44no—
685C4—612no—
686C4—1221no—
687C4—1222no—
688C4—2444no—
689C4H(5,𝔽2)×D6112yes—
690C2—22no—
691C2—24no—
692C2—1222no—
693C2—1222no—
694C2—4884no—
695C2—2442no—
696C2#534×D6111yescharacteristic
697C2C22×#1451980011yescharacteristic
698C2#575×D6111yescharacteristic
subgroupG/H
centerC22C22×D4×D61
commutator, cosocleC2×C4×C61C25
FrattiniC2×C4C24×D61
FittingC61×#3189C2
socleC22×C61C23×D4
hypercenter#3189D61
hypocenterC61#60734
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #3189

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C2×#16840275C2semidirect
2. #21456613C2semidirect
3. #21456603C2semidirect
4. #21456589C2semidirect
5. #21456572C2semidirect
6. #21456570C2semidirect
7. #21455735C2semidirect
8. #21455402C2semidirect
9. #21455042C2semidirect
10. C2×C22⋊C8×D61C2semidirect
11. C61×#3189C2semidirect
12. C22×#11758226C22semidirect
14. C2×#14519800C22semidirect
15. C22×C8×D61C22semidirect
22. #16840275C22semidirect
23. #16840275C22semidirect
25. C2×#14519753C22semidirect
26. C2×#14519752C22semidirect
31. C2×#14519735C22semidirect
32. #16840102C22semidirect
33. #16840090C22semidirect
34. #16840089C22semidirect
35. #16840080C22semidirect
36. #16840076C22semidirect
37. #16840000C22semidirect
38. #16839999C22semidirect
39. #16839998C22semidirect
40. #16839994C22semidirect
41. #16839993C22semidirect
42. #16839923C22semidirect
43. #16839912C22semidirect
44. #16839910C22semidirect
45. #16839901C22semidirect
46. #16839898C22semidirect
47. #16839799C22semidirect
48. #16839757C22semidirect
50. C2×#14519689C22semidirect
51. C22⋊C8×D61C22semidirect
52. C22⋊C8×D61C22semidirect
54. C61×#484C22semidirect
55. C61×#467C22semidirect
56. C61×#451C22semidirect
80. C2×#11758201C23semidirect
81. C2×D488C23semidirect
125. #14519711C23semidirect
126. #14519708C23semidirect
127. #14519701C23semidirect
128. #14519699C23semidirect
129. #14519697C23semidirect
134. C2×Q16×C61C23semidirect
143. #107×C61C23semidirect
144. #103×C61C23semidirect
239. C23×C61#135semidirect
344. #3189D61semidirect
383. #484C2×D61semidirect
389. #467C2×D61semidirect
395. #451C2×D61semidirect
407. C61#60734semidirect
432. C2×Q16C22×D61semidirect
487. #107C22×D61semidirect
490. #103C22×D61semidirect
609. C23#14519800semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#168402752
#214566132
#214566032
#214565892
#214565722
#214565702
#214557352
#214554022
#214550422
C2×C22⋊C8×D612
C61×#31892
#6073461