Finite Group Databasebeta

← #29603499

Subgroups — #29603499

Order 15616.

2358 conjugacy classes of subgroups.

The subgroup lattice has 640 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14519766C211yesmaximal, characteristic
2C22×Q8×Dic61C211yesmaximal, characteristic, complemented
3C2×#16839664C211yesmaximal, characteristic
4C2×#16839663C211yesmaximal, characteristic
5#21444065C214yesmaximal, complemented
6#21444055C214yesmaximal, complemented
7#21444048C211yesmaximal, characteristic, complemented
8C4×#14519684C211yesmaximal, characteristic, complemented
9C2×C61×#391C211yesmaximal, characteristic, complemented
10C23×D244C2211yescharacteristic
11C2×#14519766—24no—
12C2×#14519766—24no—
13C2×#14519766C418yescomplemented
14C2×Q8×Dic61—24no—
15C2×Q8×Dic61—24no—
16C2×Q8×Dic61—24no—
17C22×C4⋊Dic61C2212yes—
18C22×C4⋊Dic61C2211yescharacteristic
19C22×#11758198C2212yes—
20C22×#11758198C2214yes—
21C22×#11758198C2214yes—
22C22×#11758198C2212yes—
23C22×C4×Dic61C2212yes—
24C22×C4×Dic61C2211yescharacteristic
25C22×C4×Dic61C2211yescharacteristic
26#16839664—24no—
27#16839663—24no—
28C2×#14519684C2212yes—
29C2×#14519684C2212yes—
30C2×#14519684C2214yes—
31C23×Q8×C61C2211yescharacteristic
32C22×C4⋊C4×C61C2211yescharacteristic
33C22×C42×C61C2211yescharacteristic
34C61×#391—24no—
35C2×#95×C61C2214yes—
36C24×D61C2312yes—
37C22×D244—22no—
38C22×D244—22no—
39C22×D244—22no—
40C22×D244—22no—
41C22×D244—22no—
42C22×D244—22no—
43C22×D244C2×C414yes—
44C23×Dic61C2312yes—
45C23×Dic61C2311yescharacteristic
46C23×Dic61C2312yes—
47C23×Dic61C2311yescharacteristic
48#14519766—232no—
49Q8×Dic61—232no—
50C2×C4⋊Dic61—24no—
51C2×C4⋊Dic61—22no—
52C2×C4⋊Dic61—22no—
53C2×C4⋊Dic61—24no—
54C2×C4⋊Dic61—24no—
55C2×C4⋊Dic61D414yes—
56C2×#11758198—28no—
57C2×#11758198—28no—
58C2×#11758198—28no—
59C2×#11758198—24no—
60C2×#11758198—24no—
61C2×#11758198—28no—
62C2×#11758198—28no—
63C2×#11758198—44no—
64C2×#11758198—44no—
65C2×#11758198C2×C4116yes—
66C2×C4×Dic61—24no—
67C2×C4×Dic61—22no—
68C2×C4×Dic61—22no—
69C2×C4×Dic61—22no—
70C2×C4×Dic61—22no—
71C2×C4×Dic61—24no—
72C2×C4×Dic61—24no—
73C2×C4×Dic61C2×C414yes—
74C2×C4×Dic61D414yes—
75#14519684—24no—
76#14519684—24no—
77#14519684—44no—
78C22×Q8×C61—22no—
79C22×Q8×C61—22no—
80C22×Q8×C61—22no—
81C22×Q8×C61—22no—
82C22×Q8×C61C2×C414yes—
83C22×Q8×C61D414yes—
84C22×Q8×C61D414yes—
85C23×C4×C61C2312yes—
86C23×C4×C61C2312yes—
87C23×C4×C61C2312yes—
88C23×C4×C61C2311yescharacteristic
89C2×C4⋊C4×C61—22no—
90C2×C4⋊C4×C61—22no—
91C2×C4⋊C4×C61—22no—
92C2×C42×C61—22no—
93C2×C42×C61—22no—
94C2×C42×C61—22no—
95#95×C61—28no—
96C24×C61C2411yescharacteristic
97C23×D61—22no—
98C23×D61—22no—
99C23×D61—22no—
100C23×D61—22no—
101C23×D61—22no—
102C23×D61—22no—
103C23×D61—24no—
104C23×D61—24no—
105C23×D61—24no—
106C23×D61—24no—
107C23×D61C22×C414yes—
108C2×Q8×C61—28no—
109C2×Q8×C61—28no—
110C2×Q8×C61—28no—
111C2×Q8×C61—28no—
112C2×Q8×C61—28no—
113C2×Q8×C61—28no—
114C2×Q8×C61C22⋊C4116yescomplemented
115C22×C4×C61—22no—
116C22×C4×C61—22no—
117C22×C4×C61—22no—
118C22×C4×C61—22no—
119C22×C4×C61—22no—
120C22×C4×C61—22no—
121C22×C4×C61—22no—
122C22×C4×C61—22no—
123C22×C4×C61—22no—
124C22×C4×C61—22no—
125C22×C4×C61—22no—
126C22×C4×C61—22no—
127C22×C4×C61—22no—
128C22×C4×C61—24no—
129C22×C4×C61—24no—
130C22×C4×C61C22×C414yes—
131C22×C4×C61C22×C412yes—
132C22×C4×C61C2×D414yes—
133C22×C4×C61C2×D414yes—
134C22×C4×C61C2×D412yes—
135C22×C4×C61C2×D412yes—
136C22×C4×C61C2×D412yes—
137C22×C4×C61C2×D412yes—
138C22×C4×C61#4114yes—
139C22×C4×C61#4112yes—
140C22×C4×C61#4112yes—
141C22×Dic61—21no—
142C22×Dic61—21no—
143C22×Dic61—22no—
144C22×Dic61—22no—
145C22×Dic61—22no—
146C22×Dic61—22no—
147C22×Dic61—21no—
148C22×Dic61—22no—
149C22×Dic61—21no—
150C22×Dic61—22no—
151C22×Dic61—22no—
152C22×Dic61—22no—
153C22×Dic61—24no—
154C22×Dic61—22no—
155C22×Dic61—24no—
156C22×Dic61—22no—
157C22×Dic61—22no—
158C22×Dic61C22×C414yes—
159C22×Dic61C2×D414yes—
160C22×Dic61C2×D412yes—
161C22×Dic61C2×D412yes—
162C22×Dic61#4114yes—
163C2×D244—28no—
164C2×D244—28no—
165C2×D244—28no—
166C2×D244—44no—
167C2×D244—44no—
168C2×D244—44no—
169C2×D244—44no—
170C4⋊C4×C61—44no—
171C42×C61—44no—
172#11758198—232no—
173#11758198—416no—
174#11758198—48no—
175#11758198—48no—
176C4⋊Dic61—216no—
177C4⋊Dic61—28no—
178C4×Dic61—216no—
179C4×Dic61—28no—
180C4×Dic61—28no—
181C23×C61C23×C411yescharacteristic
182C23×C61C22×D411yescharacteristic
183C23×C61C22×D411yescharacteristic
184C23×C61C22×D411yescharacteristic
185C23×C61C22×D411yescharacteristic
186C23×C61C2×#4111yescharacteristic
187C23×C61C2×#4111yescharacteristic
188C23×C61C2×#4112yes—
189C23×C61C2×#4112yes—
190C23×C61H(5,𝔽2)12yes—
191C23×C612−1+412yes—
192C22×D61—24no—
193C22×D61—24no—
194C22×D61—24no—
195C22×D61—28no—
196C22×D61—28no—
197C22×D61—42no—
198C22×D61—42no—
199C22×D61—42no—
200C22×D61—42no—
201C22×D61—42no—
202C22×D61—44no—
203C22×D61—42no—
204C22×D61—42no—
205C22×D61—42no—
206C22×D61—44no—
207C22×D61—44no—
208C22×D61—44no—
209C22×D61—44no—
210C22×D61—44no—
211C22×D61—44no—
212C22×D61—44no—
213C22×D61—44no—
214C22×D61—44no—
215Q8×C61—232no—
216C2×C4×C61—24no—
217C2×C4×C61—24no—
218C2×C4×C61—24no—
219C2×C4×C61—24no—
220C2×C4×C61—24no—
221C2×C4×C61—24no—
222C2×C4×C61—24no—
223C2×C4×C61—24no—
224C2×C4×C61—24no—
225C2×C4×C61—42no—
226C2×C4×C61—42no—
227C2×C4×C61—42no—
228C2×C4×C61—42no—
229C2×C4×C61—42no—
230C2×C4×C61—42no—
231C2×C4×C61—42no—
232C2×C4×C61—44no—
233C2×C4×C61—44no—
234C2×C4×C61C2×C22⋊C418yes—
235C2×C4×C61C2×C22⋊C414yes—
236C2×C4×C61C4×D414yes—
237C2×C4×C61C4×D414yes—
238C2×C4×C61#12014yes—
239C2×C4×C61#12114yes—
240C2×C4×C61#12114yes—
241C2×C4×C61#12314yes—
242C2×Dic61—22no—
243C2×Dic61—22no—
244C2×Dic61—24no—
245C2×Dic61—24no—
246C2×Dic61—24no—
247C2×Dic61—24no—
248C2×Dic61—24no—
249C2×Dic61—24no—
250C2×Dic61—22no—
251C2×Dic61—22no—
252C2×Dic61—22no—
253C2×Dic61—28no—
254C2×Dic61—42no—
255C2×Dic61—42no—
256C2×Dic61—42no—
257C2×Dic61—42no—
258C2×Dic61—42no—
259C2×Dic61—42no—
260C2×Dic61—42no—
261C2×Dic61—42no—
262C2×Dic61—44no—
263C2×Dic61#12414yescomplemented
264D244—416no—
265#14026—611nomaximal, Sylow
266C22×C61C22×C22⋊C411yescharacteristic
267C22×C61C2×C4×D411yescharacteristic
268C22×C61C2×C4×D411yescharacteristic
269C22×C61C4×#4112yes—
270C22×C61#52012yes—
271C22×C61C2×#12011yescharacteristic
272C22×C61C2×#12111yescharacteristic
273C22×C61C2×#12111yescharacteristic
274C22×C61C2×#12311yescharacteristic
275C22×C61C2×#12411yescharacteristic
276C22×C61#52912yes—
277C22×C61#53211yescharacteristic
278C22×C61#53511yescharacteristic
279C22×C61#53611yescharacteristic
280C22×C61#54212yes—
281C22×C61#54612yes—
282C22×C61#54812yes—
283C22×C61#54812yes—
284C22×C61#54812yes—
285C22×C61#55012yes—
286C22×C61#55012yes—
287C22×C61#55612yes—
288C22×C61#55912yes—
289C2×D61—44no—
290C2×D61—48no—
291C2×D61—48no—
292C2×D61—44no—
293C2×D61—44no—
294C2×D61—84no—
295C2×D61—82no—
296C2×D61—82no—
297C2×D61—84no—
298C2×D61—82no—
299C2×D61—82no—
300C4×C61—28no—
301C4×C61—44no—
302C4×C61—82no—
303C4×C61#38618yes—
304Dic61—24no—
305Dic61—44no—
306Dic61—42no—
307Dic61—44no—
308C22×#124—611no—
309C22×C4×Q8—611no—
310C2×#391D6111yescharacteristic, complemented
311C2×#390—611no—
312C2×#390—611no—
313#1439—614no—
314#1437—614no—
315C4⋊#95—611no—
316C4×#95—611no—
317C2×C61C2×#38611yescharacteristic
318C2×C61C4×#12411yescharacteristic
319C2×C61#232611yescharacteristic
320C2×C61#234112yes—
321C2×C61#236712yes—
322C2×C61#242612yes—
323C2×C61#243111yescharacteristic
324C2×C61#243611yescharacteristic
325C2×C61#244812yes—
326C2×C61#254511yescharacteristic
327C2×C61#254811yescharacteristic
328D61—84no—
329C23×Q8C2×D6111yescharacteristic
330C23×D4—611no—
331C2×#124—618no—
332C2×#124—1224no—
333C2×#124—1224no—
334C2×C4×Q8—1224no—
335C2×C4×Q8—1224no—
336C2×C4×Q8—1224no—
337C22×C4⋊C4—611no—
338C22×C4⋊C4—612no—
339C22×C4⋊C4C2×D6111yescharacteristic
340C22×C22⋊C4—614no—
341C22×C22⋊C4—612no—
342C22×C22⋊C4—614no—
343C22×C22⋊C4—612no—
344C22×C42—611no—
345C22×C42—611no—
346C22×C42—612no—
347C22×C42C2×D6111yescharacteristic
348#391—24no—
349#390—1224no—
350#390—1224no—
351C2×#95—612no—
352C2×#95—612no—
353C2×#95—614no—
354C2×#95C2×D6114yes—
355C61#1402611yescharacteristic, complemented, Sylow
356C25—612no—
357C22×Q8—22no—
358C22×Q8—22no—
359C22×Q8—22no—
360C22×Q8—22no—
361C22×Q8C4×D6114yes—
362C22×Q8D24414yes—
363C22×Q8C22⋊D6114yes—
364C22×D4—614no—
365C22×D4—1222no—
366C22×D4—1222no—
367C22×D4—1222no—
368C22×D4—1222no—
369C22×D4—1222no—
370C22×D4—1222no—
371C23×C4—611no—
372C23×C4—611no—
373C23×C4—612no—
374C23×C4—612no—
375C23×C4C22×D6112yes—
376C23×C4C22×D6112yes—
377C23×C4C22×D6112yes—
378C23×C4C22×D6111yescharacteristic
379#124—12232no—
380C4×Q8—12232no—
381C2×C4⋊C4—22no—
382C2×C4⋊C4—22no—
383C2×C4⋊C4—22no—
384C2×C4⋊C4—614no—
385C2×C4⋊C4—1224no—
386C2×C4⋊C4—1224no—
387C2×C4⋊C4—1224no—
388C2×C4⋊C4—1222no—
389C2×C4⋊C4—1222no—
390C2×C22⋊C4—6116no—
391C2×C22⋊C4—1228no—
392C2×C22⋊C4—1228no—
393C2×C22⋊C4—1228no—
394C2×C22⋊C4—1224no—
395C2×C22⋊C4—1224no—
396C2×C22⋊C4—1228no—
397C2×C22⋊C4—1228no—
398C2×C22⋊C4—2444no—
399C2×C22⋊C4—2444no—
400C2×C42—22no—
401C2×C42—22no—
402C2×C42—22no—
403C2×C42—614no—
404C2×C42—614no—
405C2×C42—1224no—
406C2×C42—1224no—
407C2×C42—1224no—
408C2×C42—1222no—
409C2×C42—1222no—
410C2×C42—1222no—
411C2×C42—1222no—
412#95—28no—
413#95—1224no—
414#95—1224no—
415#95—2444no—
416C24—614no—
417C24—1222no—
418C24—1222no—
419C24—1222no—
420C24—1222no—
421C24—1224no—
422C24—1222no—
423C24—1222no—
424C24—1224no—
425C24—1224no—
426C24—1224no—
427C24C23×D6111yescharacteristic
428C2×Q8—28no—
429C2×Q8—28no—
430C2×Q8—28no—
431C2×Q8—28no—
432C2×Q8—28no—
433C2×Q8—28no—
434C2×Q8#11758198116yescomplemented
435C2×D4—1228no—
436C2×D4—1228no—
437C2×D4—1228no—
438C2×D4—2444no—
439C2×D4—2444no—
440C2×D4—2444no—
441C2×D4—2444no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—24no—
455C22×C4—22no—
456C22×C4—24no—
457C22×C4—612no—
458C22×C4—614no—
459C22×C4—614no—
460C22×C4—612no—
461C22×C4—614no—
462C22×C4—1221no—
463C22×C4—1221no—
464C22×C4—1222no—
465C22×C4—1222no—
466C22×C4—1221no—
467C22×C4—1222no—
468C22×C4—1222no—
469C22×C4—1222no—
470C22×C4—1221no—
471C22×C4—1222no—
472C22×C4—1222no—
473C22×C4—1222no—
474C22×C4—1224no—
475C22×C4—1222no—
476C22×C4—1224no—
477C22×C4—1222no—
478C22×C4—1222no—
479C22×C4C2×C4×D6114yes—
480C22×C4C2×C4×D6112yes—
481C22×C4C2×D24414yes—
482C22×C4C2×D24412yes—
483C22×C4D4×D6112yes—
484C22×C4D4×D6112yes—
485C22×C4#1175822412yes—
486C22×C4#1175822412yes—
487C22×C4#1175822614yes—
488C22×C4C2×C22⋊D6114yes—
489C22×C4C2×C22⋊D6112yes—
490C4⋊C4—44no—
491C4⋊C4—12216no—
492C4⋊C4—1228no—
493C22⋊C4—12232no—
494C22⋊C4—24416no—
495C22⋊C4—2448no—
496C22⋊C4—2448no—
497C42—44no—
498C42—12216no—
499C42—1228no—
500C42—1228no—
501C23—1224no—
502C23—1224no—
503C23—1224no—
504C23—1228no—
505C23—1228no—
506C23—2444no—
507C23—2444no—
508C23—2442no—
509C23—2442no—
510C23—2442no—
511C23—2442no—
512C23—2442no—
513C23—2444no—
514C23—2442no—
515C23—2442no—
516C23—2442no—
517C23—2444no—
518C23—2444no—
519C23—2444no—
520C23—2444no—
521C23—2444no—
522C23—2444no—
523C23—2444no—
524C23C22×C4×D6111yescharacteristic
525C23C2×D4×D6111yescharacteristic
526C23C2×D4×D6111yescharacteristic
527C23C22×C22⋊D6111yescharacteristic
528C23C22×D24411yescharacteristic
529C23#41×D6112yes—
530C23C2×#1175822411yescharacteristic
531C23C2×#1175822411yescharacteristic
532C23C2×#1175822612yes—
533C23#1451982312yes—
534C23#1451982512yes—
535Q8—232no—
536D4—24416no—
537C2×C4—24no—
538C2×C4—24no—
539C2×C4—24no—
540C2×C4—24no—
541C2×C4—24no—
542C2×C4—24no—
543C2×C4—24no—
544C2×C4—24no—
545C2×C4—24no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—44no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—44no—
555C2×C4—614no—
556C2×C4—1222no—
557C2×C4—1222no—
558C2×C4—1224no—
559C2×C4—1224no—
560C2×C4—1224no—
561C2×C4—1224no—
562C2×C4—1224no—
563C2×C4—1224no—
564C2×C4—1222no—
565C2×C4—1222no—
566C2×C4—1222no—
567C2×C4—1228no—
568C2×C4—2442no—
569C2×C4—2442no—
570C2×C4—2442no—
571C2×C4—2442no—
572C2×C4—2442no—
573C2×C4—2442no—
574C2×C4—2442no—
575C2×C4—2442no—
576C2×C4—2444no—
577C2×C4C2×#1175819818yes—
578C2×C4C2×#1175819814yes—
579C2×C4#1451973714yes—
580C2×C4#1451973714yes—
581C2×C4#1451974714yes—
582C2×C4#1451974914yes—
583C2×C4#1451974914yes—
584C2×C4#1451976114yes—
585C22—4882no—
586C22—4882no—
587C22—4882no—
588C22—4884no—
589C22—4884no—
590C22—4882no—
591C22—2444no—
592C22—2444no—
593C22—2444no—
594C22—2448no—
595C22—2448no—
596C22C22×#1175819811yescharacteristic
597C22C2×#1451973711yescharacteristic
598C22C2×#1451973711yescharacteristic
599C22C4×#1175822612yes—
600C22#1684024412yes—
601C22C2×#1451974711yescharacteristic
602C22C2×#1451974911yescharacteristic
603C22C2×#1451974911yescharacteristic
604C22C2×#1451976111yescharacteristic
605C22#124×D6111yescharacteristic
606C22#1684029812yes—
607C22#1684031311yescharacteristic
608C22#1684033011yescharacteristic
609C22#1684033711yescharacteristic
610C22#1684038512yes—
611C22#1684040312yes—
612C22#1684042212yes—
613C22#1684042312yes—
614C22#1684042412yes—
615C22#1684043112yes—
616C22#1684043412yes—
617C22#1684048612yes—
618C22#1684049412yes—
619C4—28no—
620C4—44no—
621C4—82no—
622C4—1224no—
623C4—2444no—
624C4—2442no—
625C4—2444no—
626C4#1683963918yes—
627C2—4884no—
628C2C2×#1683963911yescharacteristic
629C2#2144836211yescharacteristic
630C2#2144853311yescharacteristic
631C2#2144865712yes—
632C2#2144889512yes—
633C2#2144938712yes—
634C2#2144944011yescharacteristic
635C2#2144947611yescharacteristic
636C2#2144961512yes—
637C2#2145079611yescharacteristic
638C2#2145081911yescharacteristic
subgroupG/H
center, FrattiniC24C23×D61
commutatorC23×C61C23×C4
FittingC2×C61×#391C2
socle, cosocleC24×C61C24
prime radicalC23×D244C22
hypercenterC2×#391D61
hypocenterC61#14026
LCSG ▷ C23×C61 ▷ C61
UCSC1 ◁ C24 ◁ C2×#391

Complements

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

NH ≅ G/Ntype
2. C22×Q8×Dic61C2semidirect
5. #21444065C2semidirect
6. #21444055C2semidirect
7. #21444048C2semidirect
8. C4×#14519684C2semidirect
9. C2×C61×#391C2semidirect
13. C2×#14519766C4semidirect
114. C2×Q8×C61C22⋊C4semidirect
263. C2×Dic61#124semidirect
310. C2×#391D61semidirect
355. C61#14026semidirect
434. C2×Q8#11758198semidirect

Maximal subgroups

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

subgroupindex
C22×#145197662
C22×Q8×Dic612
C2×#168396642
C2×#168396632
#214440652
#214440552
#214440482
C4×#145196842
C2×C61×#3912
#1402661