Finite Group Databasebeta

← #29689043

Subgroups — #29689043

Order 15616.

1498 conjugacy classes of subgroups.

The subgroup lattice has 622 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C22⋊Q8×D61C211yesmaximal, characteristic
2C22×#14519689C211yesmaximal, characteristic
3#21448201C211yesmaximal, characteristic
4#21446243C211yesmaximal, characteristic, complemented
5#21446242C211yesmaximal, characteristic, complemented
6#21446240C211yesmaximal, characteristic, complemented
7#21446130C212yesmaximal, complemented
8#21446128C212yesmaximal, complemented
9#21445904C212yesmaximal, complemented
10#21445752C212yesmaximal, complemented
11C61×#1892C211yesmaximal, characteristic, complemented
12C22×Q8×D61C2211yescharacteristic
13C23×C4×D61C2211yescharacteristic
14C23×#92418C2211yescharacteristic
15C2×#14519757C2211yescharacteristic
16C2×#14519756C2211yescharacteristic
17C2×#14519755—21no—
18C2×#14519754—21no—
19C2×#14519753C2211yescharacteristic
20C22⋊Q8×D61—24no—
21C22⋊Q8×D61C418yescomplemented
22C2×#14519735C2212yes—
23C2×C4⋊C4×D61—21no—
24C2×C4⋊C4×D61C2211yescharacteristic
25C2×C22⋊C4×D61—21no—
26C2×#14519702C2212yescomplemented
27C2×#14519701C2212yescomplemented
28C2×#14519700C2212yescomplemented
29C2×#14519689—24no—
30C2×#14519689C2212yes—
31C2×#14519689C2212yes—
32#16839624—21no—
33#16839622C2211yescharacteristic
34#16839619C2212yes—
35#16839617—21no—
36C22⋊C4×Dic61—21no—
37#16839609—21no—
38#16839525C2212yescomplemented
39#16839524C2212yescomplemented
40#16839524C2212yescomplemented
41C23×C8×C61C2211yescharacteristic
42C2×C22⋊Q8×C61C2211yescharacteristic
43C2×#103×C61C2212yescomplemented
44C61×#380C2211yescharacteristic
45C61×#340C2212yescomplemented
46C24×D61D411yescharacteristic
47C22×Dic122—21no—
48C22×Dic122C2311yescharacteristic
49C2×Q8×D61—22no—
50C2×Q8×D61—42no—
51C2×Q8×D61—42no—
52C2×Q8×D61C2×C414yes—
53C23×Dic61D411yescharacteristic
54C22×C4×D61—21no—
55C22×C4×D61—21no—
56C22×C4×D61—42no—
57C22×C4×D61—42no—
58C22×C4×D61C2×C414yes—
59C22×C4×D61D411yescharacteristic
60C22×C4×D61D411yescharacteristic
61C22×C4×D61D414yes—
62C22×C4×D61C2311yescharacteristic
63C22×C4×D61C2311yescharacteristic
64C22×#92418—22no—
65C22×#92418—42no—
66C22×#92418C2312yes—
67#14519757—22no—
68#14519757C2×C414yes—
69#14519756—22no—
70#14519756C2×C414yes—
71#14519755—28no—
72#14519754—28no—
73#14519753—22no—
74#14519753C2×C414yes—
75#14519735—42no—
76#14519735—42no—
77C2×C4⋊Dic61—21no—
78C2×C4⋊Dic61C2312yes—
79C2×C4⋊Dic61C2311yescharacteristic
80C2×#11758196—21no—
81C2×#11758196—21no—
82C2×#11758196—21no—
83C4⋊C4×D61—28no—
84C4⋊C4×D61—22no—
85C4⋊C4×D61C2×C414yes—
86C2×C22⋊Dic61—21no—
87C2×C22⋊Dic61—21no—
88C2×#11758198—21no—
89C2×#11758198—21no—
90C2×#11758198—22no—
91C2×#11758198—22no—
92C22⋊C4×D61—28no—
93C2×C4×Dic61—22no—
94#14519702—42no—
95#14519701—42no—
96#14519700—42no—
97#14519689—24no—
98#14519689—24no—
99#14519684—21no—
100#14519684—22no—
101C22×Q8×C61C2311yescharacteristic
102C23×C4×C61C2311yescharacteristic
103C22×C8×C61—22no—
104C22×C8×C61—42no—
105C22×C8×C61C2312yes—
106C22⋊Q8×C61—22no—
107C22⋊Q8×C61C2×C414yes—
108C2×C4⋊C4×C61—21no—
109C2×C4⋊C4×C61C2312yes—
110C2×C4⋊C4×C61C2311yescharacteristic
111C2×C22⋊C4×C61—21no—
112C2×C22⋊C4×C61—21no—
113#103×C61—42no—
114#95×C61—21no—
115C24×C61C2×D411yescharacteristic
116C23×D61—22no—
117C23×D61—22no—
118C23×D61—24no—
119C23×D61—42no—
120C23×D61C22⋊C414yes—
121C23×D61C2×D411yescharacteristic
122C23×D61C2×D411yescharacteristic
123C2×Q8×C61—21no—
124C2×Q8×C61—22no—
125C2×Q8×C61—22no—
126C2×Q8×C61C22×C412yes—
127C22×C4×C61—22no—
128C22×C4×C61—22no—
129C22×C4×C61—21no—
130C22×C4×C61—21no—
131C22×C4×C61—22no—
132C22×C4×C61C22×C412yes—
133C22×C4×C61C2×D411yescharacteristic
134C22×C4×C61C2×D412yes—
135C22×C4×C61C2411yescharacteristic
136C22×Dic61—21no—
137C22×Dic61—21no—
138C22×Dic61—22no—
139C22×Dic61—42no—
140C22×Dic61C22⋊C414yes—
141C22×Dic61C2×D411yescharacteristic
142C22×Dic61C2×D411yescharacteristic
143Q8×D61—48no—
144Q8×D61—44no—
145Q8×D61—44no—
146C2×C4×D61—22no—
147C2×C4×D61—22no—
148C2×C4×D61—22no—
149C2×C4×D61—22no—
150C2×C4×D61—24no—
151C2×C4×D61—24no—
152C2×C4×D61—48no—
153C2×C4×D61—44no—
154C2×C4×D61—44no—
155C2×C4×D61—42no—
156C2×C4×D61—42no—
157C2×C4×D61—44no—
158C2×C4×D61—44no—
159C2×C4×D61C22⋊C414yes—
160C2×C4×D61C22⋊C414yes—
161C2×C4×D61C22×C412yes—
162C2×C4×D61C22×C412yes—
163C2×C4×D61C2×D412yes—
164C2×C4×D61C2×D412yes—
165C2×Dic122—22no—
166C2×Dic122—22no—
167C2×Dic122—21no—
168C2×Dic122—24no—
169C2×Dic122—44no—
170C2×Dic122C22×C412yes—
171C2×C8×C61—22no—
172C2×C8×C61—22no—
173C2×C8×C61—22no—
174C2×C8×C61—44no—
175C4⋊C4×C61—24no—
176C4⋊C4×C61—21no—
177C4⋊C4×C61—22no—
178C4⋊C4×C61—22no—
179C4⋊C4×C61C22×C412yes—
180C22⋊C4×C61—24no—
181C22⋊C4×C61—42no—
182C22⋊Dic61—24no—
183C22⋊Dic61—42no—
184#11758198—24no—
185#11758198—24no—
186#11758198—44no—
187#11758198—44no—
188C4⋊Dic61—21no—
189C4⋊Dic61—24no—
190C4⋊Dic61—22no—
191C4⋊Dic61—22no—
192C4⋊Dic61C22×C412yes—
193#11758196—24no—
194#11758196—24no—
195#11758196—24no—
196C4×Dic61—44no—
197C2×#92418—22no—
198C2×#92418—22no—
199C2×#92418—22no—
200C2×#92418—44no—
201C23×C61—22no—
202C23×C61—22no—
203C23×C61C2×C22⋊C412yes—
204C23×C61#13712yes—
205C23×C61#13712yes—
206C23×C61C22×D411yescharacteristic
207C22×D61—22no—
208C22×D61—21no—
209C22×D61—21no—
210C22×D61—22no—
211C22×D61—22no—
212C22×D61—21no—
213C22×D61—21no—
214C22×D61—28no—
215C22×D61—28no—
216C22×D61—44no—
217C22×D61—48no—
218C22×D61—48no—
219C22×D61C2×C22⋊C412yes—
220C22×D61C2×C22⋊C412yes—
221C22×D61#13514yes—
222Q8×C61—24no—
223Q8×C61—41no—
224Q8×C61—41no—
225C2×C4×C61—22no—
226C2×C4×C61—24no—
227C2×C4×C61—22no—
228C2×C4×C61—22no—
229C2×C4×C61—22no—
230C2×C4×C61—22no—
231C2×C4×C61—42no—
232C2×C4×C61—42no—
233C2×C4×C61—42no—
234C2×C4×C61—42no—
235C2×C4×C61—42no—
236C2×C4×C61C2×C22⋊C412yes—
237C2×C4×C61C23×C411yescharacteristic
238C2×C4×C61C22×D411yescharacteristic
239C2×C4×C61H(5,𝔽2)12yes—
240C2×C4×C61H(5,𝔽2)12yes—
241C2×Dic61—22no—
242C2×Dic61—22no—
243C2×Dic61—22no—
244C2×Dic61—22no—
245C2×Dic61—44no—
246C2×Dic61—42no—
247C2×Dic61—42no—
248C2×Dic61—42no—
249C2×Dic61—42no—
250C2×Dic61—42no—
251C2×Dic61C2×C22⋊C412yes—
252C2×Dic61C2×C22⋊C412yes—
253C4×D61—24no—
254C4×D61—24no—
255C4×D61—22no—
256C4×D61—22no—
257C4×D61—22no—
258C4×D61—22no—
259C4×D61—44no—
260C4×D61—44no—
261C4×D61—48no—
262C4×D61—48no—
263Dic122—24no—
264Dic122—48no—
265Dic122—41no—
266Dic122—41no—
267C8×C61—42no—
268#92418—42no—
269#21762—611nomaximal, Sylow
270C22×C61—22no—
271C22×C61—22no—
272C22×C61—24no—
273C22×C61—24no—
274C22×C61#41914yes—
275C22×C61C22×C22⋊C411yescharacteristic
276C22×C61#53412yes—
277C22×C61C2×#13512yes—
278C22×C61C2×#13711yescharacteristic
279C22×C61C2×#13711yescharacteristic
280C2×D61—22no—
281C2×D61—22no—
282C2×D61—24no—
283C2×D61—24no—
284C2×D61—22no—
285C2×D61—22no—
286C2×D61—22no—
287C2×D61—416no—
288C2×D61—42no—
289C2×D61—42no—
290C2×D61#41614yes—
291C4×C61—22no—
292C4×C61—42no—
293C4×C61—42no—
294C4×C61—82no—
295C4×C61#51812yes—
296C4×C61#53411yescharacteristic
297C4×C61#53411yescharacteristic
298Dic61—22no—
299Dic61—22no—
300Dic61—42no—
301Dic61—42no—
302Dic61—82no—
303C22×C22⋊Q8—611no—
304C22×C22⋊C8—611no—
305#2280—611no—
306#1892—611no—
307#1892—611no—
308#1892—611no—
309#1892D6111yescharacteristic, complemented
310#1873—612no—
311#1873—612no—
312#1823—612no—
313#1787—612no—
314C2×C61—24no—
315C2×C61#230811yescharacteristic
316C2×C61C2×#41611yescharacteristic
317C2×C61C2×#41911yescharacteristic
318C2×C61#319012yes—
319C2×C61#319012yes—
320D61—24no—
321D61—42no—
322C23×Q8—611no—
323C24×C4—611no—
324C23×C8—611no—
325C23×C8C2×D6111yescharacteristic
326C2×C22⋊Q8—611no—
327C2×C22⋊Q8—611no—
328C2×C22⋊Q8—618no—
329C2×C22⋊Q8—611no—
330C2×C22⋊Q8—1221no—
331C2×C22⋊Q8—1221no—
332C2×C22⋊Q8—1224no—
333C2×C22⋊Q8C2×D6111yescharacteristic
334C2×#117—612no—
335C22×C4⋊C4—611no—
336C22×C4⋊C4—1221no—
337C22×C22⋊C4—1221no—
338C2×#103—612no—
339C2×#103—612no—
340C2×#103—612no—
341C2×#103C2×D6112yescomplemented
342C2×C22⋊C8—612no—
343C2×C22⋊C8—612no—
344C2×C22⋊C8—1224no—
345#381—1221no—
346#380—611no—
347#380—612no—
348#380C2×D6111yescharacteristic
349#379—1221no—
350C4×C22⋊C4—1221no—
351C4×C22⋊C4—1221no—
352#340—612no—
353#340—612no—
354#340—612no—
355#340C2×D6112yescomplemented
356C61#2176211yescharacteristic, complemented, Sylow
357C25—611no—
358C22×Q8—614no—
359C22×Q8—611no—
360C22×Q8—1221no—
361C22×Q8—1222no—
362C22×Q8—2442no—
363C22×Q8—2442no—
364C22×Q8C22×D6111yescharacteristic
365C23×C4—611no—
366C23×C4—611no—
367C23×C4—611no—
368C23×C4—614no—
369C23×C4—611no—
370C23×C4—611no—
371C23×C4—614no—
372C23×C4—1221no—
373C23×C4—1221no—
374C23×C4—2442no—
375C23×C4—2442no—
376C23×C4C22×D6111yescharacteristic
377C22×C8—22no—
378C22×C8—42no—
379C22×C8—612no—
380C22×C8—1222no—
381C22×C8—2442no—
382C22×C8C22×D6112yes—
383C22⋊Q8—22no—
384C22⋊Q8—614no—
385C22⋊Q8—614no—
386C22⋊Q8—614no—
387C22⋊Q8—1228no—
388C22⋊Q8—1228no—
389C22⋊Q8—1222no—
390C22⋊Q8—1222no—
391C22⋊Q8—1222no—
392C22⋊Q8C4×D6114yes—
393#117—2442no—
394#117—2442no—
395C2×C4⋊C4—21no—
396C2×C4⋊C4—611no—
397C2×C4⋊C4—614no—
398C2×C4⋊C4—612no—
399C2×C4⋊C4—1222no—
400C2×C4⋊C4—1228no—
401C2×C4⋊C4—1221no—
402C2×C4⋊C4—1221no—
403C2×C4⋊C4—1221no—
404C2×C4⋊C4—1221no—
405C2×C4⋊C4C22×D6112yes—
406C2×C4⋊C4C22×D6111yescharacteristic
407C2×C22⋊C4—21no—
408C2×C22⋊C4—21no—
409C2×C22⋊C4—1228no—
410C2×C22⋊C4—1221no—
411C2×C22⋊C4—1221no—
412C2×C22⋊C4—1221no—
413C2×C22⋊C4—1222no—
414C2×C22⋊C4—1222no—
415C2×C22⋊C4—1221no—
416C2×C42—1222no—
417#103—42no—
418#103—2442no—
419#103—2442no—
420#103—2442no—
421C22⋊C8—1224no—
422C22⋊C8—1224no—
423#95—21no—
424#95—1222no—
425#95—1221no—
426C24—611no—
427C24—611no—
428C24—614no—
429C24—1224no—
430C24—1222no—
431C24—1222no—
432C24—2442no—
433C24D4×D6111yescharacteristic
434C2×Q8—21no—
435C2×Q8—22no—
436C2×Q8—22no—
437C2×Q8—612no—
438C2×Q8—1224no—
439C2×Q8—1221no—
440C2×Q8—1222no—
441C2×Q8—1222no—
442C2×Q8—2444no—
443C2×Q8—2448no—
444C2×Q8—2444no—
445C2×Q8—2444no—
446C2×Q8C2×C4×D6112yes—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—22no—
452C22×C4—611no—
453C22×C4—611no—
454C22×C4—614no—
455C22×C4—612no—
456C22×C4—612no—
457C22×C4—612no—
458C22×C4—612no—
459C22×C4—614no—
460C22×C4—614no—
461C22×C4—1224no—
462C22×C4—1224no—
463C22×C4—1222no—
464C22×C4—1222no—
465C22×C4—1222no—
466C22×C4—1222no—
467C22×C4—1221no—
468C22×C4—1221no—
469C22×C4—1222no—
470C22×C4—2448no—
471C22×C4—2444no—
472C22×C4—2444no—
473C22×C4—2442no—
474C22×C4—2442no—
475C22×C4—2444no—
476C22×C4—2444no—
477C22×C4—2442no—
478C22×C4C2×C4×D6112yes—
479C22×C4D4×D6112yes—
480C22×C4D4×D6111yescharacteristic
481C22×C4C23×D6111yescharacteristic
482C2×C8—22no—
483C2×C8—22no—
484C2×C8—22no—
485C2×C8—44no—
486C2×C8—1222no—
487C2×C8—1222no—
488C2×C8—1222no—
489C2×C8—2444no—
490C4⋊C4—21no—
491C4⋊C4—24no—
492C4⋊C4—22no—
493C4⋊C4—22no—
494C4⋊C4—612no—
495C4⋊C4—1224no—
496C4⋊C4—1224no—
497C4⋊C4—1224no—
498C4⋊C4—1221no—
499C4⋊C4—1224no—
500C4⋊C4—1222no—
501C4⋊C4—1222no—
502C4⋊C4C2×C4×D6112yes—
503C22⋊C4—24no—
504C22⋊C4—42no—
505C22⋊C4—1224no—
506C22⋊C4—1224no—
507C22⋊C4—1224no—
508C22⋊C4—2444no—
509C22⋊C4—2444no—
510C22⋊C4—2442no—
511C42—2444no—
512C23—22no—
513C23—22no—
514C23—612no—
515C23—614no—
516C23—612no—
517C23—1228no—
518C23—1221no—
519C23—1221no—
520C23—1222no—
521C23—1228no—
522C23—1222no—
523C23—1221no—
524C23—1221no—
525C23—1222no—
526C23—2444no—
527C23—2448no—
528C23—2448no—
529C23C22⋊C4×D6112yes—
530C23#1451980612yes—
531C23#1451980912yes—
532C23C2×D4×D6111yescharacteristic
533Q8—24no—
534Q8—41no—
535Q8—41no—
536Q8—1224no—
537Q8—2448no—
538Q8—2441no—
539Q8—2441no—
540C2×C4—22no—
541C2×C4—24no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—612no—
552C2×C4—612no—
553C2×C4—1222no—
554C2×C4—1222no—
555C2×C4—1224no—
556C2×C4—1224no—
557C2×C4—1222no—
558C2×C4—1222no—
559C2×C4—1222no—
560C2×C4—1222no—
561C2×C4—1222no—
562C2×C4—1222no—
563C2×C4—2444no—
564C2×C4—2444no—
565C2×C4—2448no—
566C2×C4—2448no—
567C2×C4—2444no—
568C2×C4—2442no—
569C2×C4—2442no—
570C2×C4—2442no—
571C2×C4—2442no—
572C2×C4—2442no—
573C2×C4C22⋊C4×D6112yes—
574C2×C4C22×C4×D6111yescharacteristic
575C2×C4C2×D4×D6111yescharacteristic
576C2×C4#1451982212yes—
577C2×C4#1451982312yes—
578C8—42no—
579C8—2442no—
580C22—22no—
581C22—24no—
582C22—24no—
583C22—22no—
584C22—614no—
585C22—1222no—
586C22—1224no—
587C22—1224no—
588C22—1222no—
589C22—1222no—
590C22—1222no—
591C22—1222no—
592C22—24416no—
593C22—2442no—
594C22—2442no—
595C22#1683976914yes—
596C22C2×C22⋊C4×D6111yescharacteristic
597C22#1684032012yes—
598C22#135×D6112yes—
599C22C2×#1451980611yescharacteristic
600C22C2×#1451980911yescharacteristic
601C4—22no—
602C4—42no—
603C4—42no—
604C4—82no—
605C4—1222no—
606C4—1222no—
607C4—4882no—
608C4—2442no—
609C4—2442no—
610C4#1684023412yes—
611C4#1684032211yescharacteristic
612C4#1684032311yescharacteristic
613C2—24no—
614C2—1224no—
615C2—2442no—
616C2#2144839311yescharacteristic
617C2#416×D6111yescharacteristic
618C2C2×#1683976911yescharacteristic
619C2#2145659512yes—
620C2#2145660112yes—
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1892C2
socleC23×C61C22×D4
prime radicalC24×D61D4
cosocleC22×C4×C61C24
hypercenter#1892D61
hypocenterC61#21762
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1892

Complements

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

NH ≅ G/Ntype
4. #21446243C2semidirect
5. #21446242C2semidirect
6. #21446240C2semidirect
7. #21446130C2semidirect
8. #21446128C2semidirect
9. #21445904C2semidirect
10. #21445752C2semidirect
11. C61×#1892C2semidirect
21. C22⋊Q8×D61C4semidirect
26. C2×#14519702C22semidirect
27. C2×#14519701C22semidirect
28. C2×#14519700C22semidirect
38. #16839525C22semidirect
39. #16839524C22semidirect
40. #16839524C22semidirect
43. C2×#103×C61C22semidirect
45. C61×#340C22semidirect
309. #1892D61semidirect
341. C2×#103C2×D61semidirect
355. #340C2×D61semidirect
356. C61#21762semidirect

Maximal subgroups

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

subgroupindex
C2×C22⋊Q8×D612
C22×#145196892
#214482012
#214462432
#214462422
#214462402
#214461302
#214461282
#214459042
#214457522
C61×#18922
#2176261