Finite Group Databasebeta

← #29572603

Subgroups — #29572603

Order 15616.

820 conjugacy classes of subgroups.

The subgroup lattice has 616 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21446288C211yesmaximal, characteristic, complemented
2#21446283C211yesmaximal, characteristic, complemented
3#21446117C211yesmaximal, characteristic, complemented
4#21444183C211yesmaximal, characteristic, complemented
5#21444179C211yesmaximal, characteristic, complemented
6#21444157C211yesmaximal, characteristic, complemented
7C61×#3505C211yesmaximal, characteristic
8C22×#11758194C2211yescharacteristic
9C2×#14519696—21no—
10C2×#14519694—21no—
11#16839726C2211yescharacteristic
12#16839726C2211yescharacteristic
13C2×C22⋊#92418—21no—
14C2×C22⋊#92418C2211yescharacteristic
15#16839531—21no—
16#16839531—21no—
17#16839517—21no—
18#16839488—21no—
19#16839488—22no—
20#16839486—21no—
21#16839486—22no—
22C61×#553—22no—
23C61×#553—21no—
24C61×#552—21no—
25C61×#551—22no—
26C61×#548—21no—
27C61×#546—21no—
28C61×#539—21no—
29C61×#538—21no—
30C61×#534C412yes—
31C61×#529—21no—
32C61×#525—21no—
33C2×C22⋊Q8×C61C2211yescharacteristic
34C2×#121×C61C2211yescharacteristic
35C61×#518C412yes—
36C2×C4×D4×C61C2211yescharacteristic
37C2×#11758194—21no—
38C2×#11758194—21no—
39C2×#11758194—21no—
40C2×#11758194—21no—
41C2×#11758194—42no—
42C22×#92418—21no—
43C22×#92418—21no—
44#14519696—42no—
45#14519694—42no—
46C22⋊#92418—21no—
47C22⋊#92418—21no—
48C22⋊#92418—21no—
49C22⋊#92418—21no—
50C22⋊#92418—21no—
51C22⋊#92418—21no—
52C22⋊#92418—42no—
53C2×#41×C61—21no—
54C22×Q8×C61D411yescharacteristic
55C22×D4×C61C2×C411yescharacteristic
56C22×D4×C61C2×C412yes—
57C22×D4×C61D411yescharacteristic
58C23×C4×C61D411yescharacteristic
59C23×C4×C61C2311yescharacteristic
60#126×C61—22no—
61#126×C61—22no—
62#126×C61—22no—
63#125×C61—22no—
64#125×C61—21no—
65#124×C61—22no—
66#124×C61—21no—
67#123×C61—22no—
68#123×C61—22no—
69#123×C61—22no—
70#123×C61—22no—
71#123×C61—22no—
72C22⋊Q8×C61—22no—
73C22⋊Q8×C61—21no—
74C22⋊Q8×C61—21no—
75C22⋊Q8×C61—21no—
76C22⋊Q8×C61—21no—
77C22⋊Q8×C61—21no—
78#121×C61—22no—
79#121×C61—21no—
80#121×C61—22no—
81#121×C61—22no—
82#121×C61—21no—
83#121×C61—21no—
84#121×C61—21no—
85#121×C61—21no—
86#120×C61—22no—
87#120×C61—22no—
88C4×Q8×C61—21no—
89C4×D4×C61—21no—
90C4×D4×C61—22no—
91C4×D4×C61—22no—
92C4×D4×C61—22no—
93C4×D4×C61—22no—
94C4×D4×C61—22no—
95C4×D4×C61—21no—
96C4×D4×C61—21no—
97C4×D4×C61—21no—
98C4×D4×C61—21no—
99#117×C61—22no—
100#117×C61—21no—
101#117×C61—22no—
102C2×C4⋊C4×C61D411yescharacteristic
103C2×C4⋊C4×C61D411yescharacteristic
104C2×C4⋊C4×C61D412yes—
105C2×C4⋊C4×C61D411yescharacteristic
106C2×C22⋊C4×C61—22no—
107C2×C22⋊C4×C61C2×C412yes—
108C2×C22⋊C4×C61C2×C411yescharacteristic
109C2×C22⋊C4×C61D411yescharacteristic
110C2×C22⋊C4×C61D412yes—
111C2×C42×C61D411yescharacteristic
112C24×C61C22⋊C411yescharacteristic
113C24×C61C22⋊C412yes—
114C24×C61C22×C411yescharacteristic
115#41×C61—81no—
116#41×C61—81no—
117C2×Q8×C61—21no—
118C2×Q8×C61—21no—
119C2×Q8×C61—41no—
120C2×Q8×C61—41no—
121C2×D4×C61—21no—
122C2×D4×C61—22no—
123C2×D4×C61—22no—
124C2×D4×C61—21no—
125C2×D4×C61—22no—
126C2×D4×C61—22no—
127C2×D4×C61—21no—
128C2×D4×C61—21no—
129C2×D4×C61—41no—
130C2×D4×C61—41no—
131C2×D4×C61—41no—
132C2×D4×C61—41no—
133C2×D4×C61—42no—
134C2×D4×C61—42no—
135C22×C4×C61—22no—
136C22×C4×C61—22no—
137C22×C4×C61—21no—
138C22×C4×C61—21no—
139C22×C4×C61—41no—
140C22×C4×C61—41no—
141C22×C4×C61—41no—
142C22×C4×C61—41no—
143C22×C4×C61C22⋊C411yescharacteristic
144C22×C4×C61C22⋊C411yescharacteristic
145C22×C4×C61C22⋊C412yes—
146C22×C4×C61C22⋊C411yescharacteristic
147C22×C4×C61C22⋊C412yes—
148C22×C4×C61C22⋊C412yes—
149C22×C4×C61C2×D411yescharacteristic
150C22×C4×C61C2×D411yescharacteristic
151C22×C4×C61C2×D411yescharacteristic
152C22×C4×C61C2×D411yescharacteristic
153C22×C4×C61C2×D411yescharacteristic
154C22×C4×C61C2×D411yescharacteristic
155C4⋊C4×C61—22no—
156C4⋊C4×C61—22no—
157C4⋊C4×C61—22no—
158C4⋊C4×C61—22no—
159C4⋊C4×C61—21no—
160C4⋊C4×C61—22no—
161C4⋊C4×C61—21no—
162C4⋊C4×C61—21no—
163C4⋊C4×C61—21no—
164C4⋊C4×C61—21no—
165C4⋊C4×C61—22no—
166C4⋊C4×C61—21no—
167C22⋊C4×C61—22no—
168C22⋊C4×C61—22no—
169C22⋊C4×C61—22no—
170C22⋊C4×C61—22no—
171C22⋊C4×C61—22no—
172C22⋊C4×C61—22no—
173C22⋊C4×C61—21no—
174C22⋊C4×C61—22no—
175C22⋊C4×C61—21no—
176C22⋊C4×C61—22no—
177C22⋊C4×C61—22no—
178C22⋊C4×C61—21no—
179C22⋊C4×C61—21no—
180C22⋊C4×C61—22no—
181C42×C61—22no—
182C42×C61—22no—
183C42×C61—21no—
184C42×C61—21no—
185#11758194—22no—
186#11758194—22no—
187#11758194—41no—
188#11758194—41no—
189#11758194—41no—
190#11758194—41no—
191#11758194—41no—
192#11758194—41no—
193C2×#92418—22no—
194C2×#92418—41no—
195C2×#92418—41no—
196C2×#92418—41no—
197C2×#92418—41no—
198C2×#92418—41no—
199C2×#92418—41no—
200C2×#92418—41no—
201C23×C61—21no—
202C23×C61—21no—
203C23×C61—21no—
204C23×C61—22no—
205C23×C61—21no—
206C23×C61—21no—
207C23×C61—41no—
208C23×C61—41no—
209C23×C61—41no—
210C23×C61—41no—
211C23×C61—42no—
212C23×C61—42no—
213C23×C61—42no—
214C23×C61C2×C22⋊C411yescharacteristic
215C23×C61C2×C22⋊C411yescharacteristic
216C23×C61C2×C22⋊C411yescharacteristic
217Q8×C61—41no—
218Q8×C61—41no—
219Q8×C61—81no—
220D4×C61—42no—
221D4×C61—42no—
222D4×C61—44no—
223D4×C61—42no—
224D4×C61—42no—
225D4×C61—41no—
226D4×C61—41no—
227D4×C61—81no—
228D4×C61—82no—
229D4×C61—82no—
230C2×C4×C61—21no—
231C2×C4×C61—21no—
232C2×C4×C61—21no—
233C2×C4×C61—21no—
234C2×C4×C61—21no—
235C2×C4×C61—21no—
236C2×C4×C61—21no—
237C2×C4×C61—21no—
238C2×C4×C61—22no—
239C2×C4×C61—21no—
240C2×C4×C61—22no—
241C2×C4×C61—22no—
242C2×C4×C61—21no—
243C2×C4×C61—21no—
244C2×C4×C61—41no—
245C2×C4×C61—41no—
246C2×C4×C61—41no—
247C2×C4×C61—41no—
248C2×C4×C61—41no—
249C2×C4×C61—41no—
250C2×C4×C61—41no—
251C2×C4×C61—42no—
252C2×C4×C61—42no—
253C2×C4×C61—42no—
254C2×C4×C61—81no—
255C2×C4×C61—82no—
256C2×C4×C61—82no—
257C2×C4×C61#12011yescharacteristic
258C2×C4×C61#12011yescharacteristic
259C2×C4×C61#12011yescharacteristic
260C2×C4×C61#12011yescharacteristic
261#92418—42no—
262#92418—81no—
263#92418—81no—
264#92418—81no—
265#8538—611nomaximal, Sylow
266C22×C61—21no—
267C22×C61—21no—
268C22×C61—21no—
269C22×C61—21no—
270C22×C61—21no—
271C22×C61—21no—
272C22×C61—21no—
273C22×C61—21no—
274C22×C61—42no—
275C22×C61—41no—
276C22×C61—42no—
277C22×C61—41no—
278C22×C61—42no—
279C22×C61—42no—
280C22×C61—41no—
281C22×C61—81no—
282C22×C61—81no—
283C22×C61—81no—
284C22×C61—162no—
285C22×C61#37911yescharacteristic
286C22×C61#41311yescharacteristic
287C22×C61#41311yescharacteristic
288C4×C61—21no—
289C4×C61—21no—
290C4×C61—21no—
291C4×C61—41no—
292C4×C61—42no—
293C4×C61—41no—
294C4×C61—42no—
295C4×C61—41no—
296C4×C61—42no—
297C4×C61B(4,𝔽2)11yescharacteristic
298C4×C61#45811yescharacteristic
299#3505D6111yescharacteristic
300#1901—611no—
301#1900—611no—
302#1870—611no—
303#1472—611no—
304#1471—611no—
305#1465—611no—
306C2×C61—21no—
307C2×C61—21no—
308C2×C61—41no—
309C2×C61—41no—
310C2×C61—81no—
311C2×C61—82no—
312C2×C61#178511yescharacteristic
313C2×C61#179011yescharacteristic
314C2×C61#179111yescharacteristic
315C22×M16—611no—
316#553—22no—
317#553—21no—
318#552—21no—
319#551—22no—
320#548—21no—
321#546—21no—
322#539—21no—
323#538—21no—
324#534Dic6112yes—
325#529—21no—
326#525—21no—
327C2×C22⋊Q8C2×D6111yescharacteristic
328C2×#121C2×D6111yescharacteristic
329#518Dic6112yes—
330C2×C4×D4C2×D6111yescharacteristic
331C2×#101—1221no—
332C2×#100—1221no—
333#407—611no—
334#407—611no—
335C2×C22⋊C8—611no—
336C2×C22⋊C8—1221no—
337#343—1221no—
338#343—1221no—
339#336—1221no—
340#324—1221no—
341#324—1222no—
342#323—1221no—
343#323—1222no—
344C61#853811yescharacteristic, complemented, Sylow
345C2×#41—21no—
346C22×Q8C22⋊D6111yescharacteristic
347C22×D4C2×Dic6111yescharacteristic
348C22×D4C2×Dic6112yes—
349C22×D4C22⋊D6111yescharacteristic
350C23×C4C22⋊D6111yescharacteristic
351C23×C4C22×D6111yescharacteristic
352C2×M16—1221no—
353C2×M16—1221no—
354C2×M16—1221no—
355C2×M16—1221no—
356C2×M16—2442no—
357C22×C8—1221no—
358C22×C8—1221no—
359#126—22no—
360#126—22no—
361#126—22no—
362#125—21no—
363#125—22no—
364#124—21no—
365#124—22no—
366#123—22no—
367#123—22no—
368#123—22no—
369#123—22no—
370#123—22no—
371C22⋊Q8—21no—
372C22⋊Q8—22no—
373C22⋊Q8—21no—
374C22⋊Q8—21no—
375C22⋊Q8—21no—
376C22⋊Q8—21no—
377#121—21no—
378#121—22no—
379#121—22no—
380#121—22no—
381#121—21no—
382#121—21no—
383#121—21no—
384#121—21no—
385#120—22no—
386#120—22no—
387C4×Q8—21no—
388C4×D4—22no—
389C4×D4—21no—
390C4×D4—22no—
391C4×D4—22no—
392C4×D4—22no—
393C4×D4—22no—
394C4×D4—21no—
395C4×D4—21no—
396C4×D4—21no—
397C4×D4—21no—
398#117—21no—
399#117—22no—
400#117—22no—
401C2×C4⋊C4C22⋊D6111yescharacteristic
402C2×C4⋊C4C22⋊D6111yescharacteristic
403C2×C4⋊C4C22⋊D6112yes—
404C2×C4⋊C4C22⋊D6111yescharacteristic
405C2×C22⋊C4—22no—
406C2×C22⋊C4C2×Dic6111yescharacteristic
407C2×C22⋊C4C2×Dic6112yes—
408C2×C22⋊C4C22⋊D6111yescharacteristic
409C2×C22⋊C4C22⋊D6112yes—
410C2×C42C22⋊D6111yescharacteristic
411#101—2442no—
412#100—2442no—
413C22⋊C8—1221no—
414C22⋊C8—1221no—
415C22⋊C8—1221no—
416C22⋊C8—1221no—
417C22⋊C8—1221no—
418C22⋊C8—1221no—
419C22⋊C8—2442no—
420C24C22⋊Dic6111yescharacteristic
421C24C22⋊Dic6112yes—
422C24C22×Dic6111yescharacteristic
423#41—81no—
424#41—81no—
425C2×Q8—21no—
426C2×Q8—21no—
427C2×Q8—41no—
428C2×Q8—41no—
429C2×D4—21no—
430C2×D4—22no—
431C2×D4—22no—
432C2×D4—21no—
433C2×D4—22no—
434C2×D4—22no—
435C2×D4—21no—
436C2×D4—21no—
437C2×D4—41no—
438C2×D4—41no—
439C2×D4—41no—
440C2×D4—41no—
441C2×D4—42no—
442C2×D4—42no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—41no—
448C22×C4—41no—
449C22×C4—41no—
450C22×C4—41no—
451C22×C4C22⋊Dic6111yescharacteristic
452C22×C4C22⋊Dic6111yescharacteristic
453C22×C4C22⋊Dic6112yes—
454C22×C4C22⋊Dic6111yescharacteristic
455C22×C4C22⋊Dic6112yes—
456C22×C4C22⋊Dic6112yes—
457C22×C4C2×C22⋊D6111yescharacteristic
458C22×C4C2×C22⋊D6111yescharacteristic
459C22×C4C2×C22⋊D6111yescharacteristic
460C22×C4C2×C22⋊D6111yescharacteristic
461C22×C4C2×C22⋊D6111yescharacteristic
462C22×C4C2×C22⋊D6111yescharacteristic
463M16—1222no—
464M16—1222no—
465M16—2441no—
466M16—2441no—
467M16—2441no—
468M16—2441no—
469M16—2441no—
470M16—2441no—
471C2×C8—1222no—
472C2×C8—2441no—
473C2×C8—2441no—
474C2×C8—2441no—
475C2×C8—2441no—
476C2×C8—2441no—
477C2×C8—2441no—
478C2×C8—2441no—
479C4⋊C4—22no—
480C4⋊C4—22no—
481C4⋊C4—22no—
482C4⋊C4—22no—
483C4⋊C4—21no—
484C4⋊C4—22no—
485C4⋊C4—21no—
486C4⋊C4—21no—
487C4⋊C4—21no—
488C4⋊C4—21no—
489C4⋊C4—22no—
490C4⋊C4—21no—
491C22⋊C4—22no—
492C22⋊C4—22no—
493C22⋊C4—22no—
494C22⋊C4—22no—
495C22⋊C4—22no—
496C22⋊C4—22no—
497C22⋊C4—21no—
498C22⋊C4—22no—
499C22⋊C4—21no—
500C22⋊C4—22no—
501C22⋊C4—22no—
502C22⋊C4—21no—
503C22⋊C4—21no—
504C22⋊C4—22no—
505C42—22no—
506C42—22no—
507C42—21no—
508C42—21no—
509C23—21no—
510C23—21no—
511C23—21no—
512C23—21no—
513C23—22no—
514C23—21no—
515C23—41no—
516C23—41no—
517C23—41no—
518C23—41no—
519C23—42no—
520C23—42no—
521C23—42no—
522C23C2×C22⋊Dic6111yescharacteristic
523C23C2×C22⋊Dic6111yescharacteristic
524C23C2×C22⋊Dic6111yescharacteristic
525Q8—41no—
526Q8—41no—
527Q8—81no—
528D4—42no—
529D4—42no—
530D4—44no—
531D4—42no—
532D4—42no—
533D4—41no—
534D4—41no—
535D4—81no—
536D4—82no—
537D4—82no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—22no—
549C2×C4—21no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—81no—
563C2×C4—82no—
564C2×C4—82no—
565C2×C4#1451974611yescharacteristic
566C2×C4#1451974611yescharacteristic
567C2×C4#1451974611yescharacteristic
568C2×C4#1451974611yescharacteristic
569C8—4881no—
570C8—4881no—
571C8—4881no—
572C8—2442no—
573C22—21no—
574C22—21no—
575C22—21no—
576C22—21no—
577C22—21no—
578C22—21no—
579C22—21no—
580C22—21no—
581C22—42no—
582C22—41no—
583C22—42no—
584C22—41no—
585C22—42no—
586C22—42no—
587C22—41no—
588C22—81no—
589C22—81no—
590C22—81no—
591C22—162no—
592C22#1683961811yescharacteristic
593C22#1683974611yescharacteristic
594C22#1683974611yescharacteristic
595C4—21no—
596C4—21no—
597C4—21no—
598C4—41no—
599C4—42no—
600C4—41no—
601C4—42no—
602C4—41no—
603C4—42no—
604C4#1683994911yescharacteristic
605C4#1683995211yescharacteristic
606C2—21no—
607C2—21no—
608C2—41no—
609C2—41no—
610C2—81no—
611C2—82no—
612C2#2144574711yescharacteristic
613C2#2144576711yescharacteristic
614C2#2144577011yescharacteristic
subgroupG/H
centerC22#16839618
commutatorC24×C61C22×C4
FrattiniC23×C4C22×D61
Fitting, prime radicalC61×#3505C2
socleC22×C61#379
cosocleC23×C4×C61C23
hypercenter#3505D61
hypocenterC61#8538
LCSG ▷ C24×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #3505

Complements

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

NH ≅ G/Ntype
1. #21446288C2semidirect
2. #21446283C2semidirect
3. #21446117C2semidirect
4. #21444183C2semidirect
5. #21444179C2semidirect
6. #21444157C2semidirect
344. C61#8538semidirect

Maximal subgroups

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

subgroupindex
#214462882
#214462832
#214461172
#214441832
#214441792
#214441572
C61×#35052
#853861