Finite Group Databasebeta

← #29572291

Subgroups — #29572291

Order 15616.

828 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
1#21446805C211yesmaximal, characteristic, complemented
2#21446288C211yesmaximal, characteristic, complemented
3#21446258C211yesmaximal, characteristic, complemented
4#21446122C211yesmaximal, characteristic, complemented
5#21444362C211yesmaximal, characteristic, complemented
6#21444335C211yesmaximal, characteristic, complemented
7C61×#3536C211yesmaximal, characteristic
8C22×#11758194C2211yescharacteristic
9C2×#14519704—21no—
10C2×#14519698—21no—
11C2×#14519696—21no—
12#16839726C2211yescharacteristic
13#16839622C2211yescharacteristic
14C4×C22⋊Dic61C2211yescharacteristic
15#16839531—21no—
16#16839529—22no—
17#16839524—21no—
18#16839521—21no—
19#16839499—21no—
20#16839496—21no—
21#16839496—22no—
22C61×#562—22no—
23C61×#554—21no—
24C61×#548—21no—
25C61×#547—21no—
26C61×#546—21no—
27C61×#546—22no—
28C61×#543—21no—
29C61×#542—21no—
30C61×#539C412yescomplemented
31C61×#538—21no—
32C61×#537—21no—
33C61×#537C412yescomplemented
34C61×#525C2211yescharacteristic
35C2×C22⋊Q8×C61C2211yescharacteristic
36C2×#121×C61C2211yescharacteristic
37C2×#11758194—21no—
38C2×#11758194—21no—
39C2×#11758194—21no—
40C2×#11758194—42no—
41C22×#92418—21no—
42C2×C4⋊Dic61—21no—
43C2×C22⋊Dic61—21no—
44C2×C22⋊Dic61—21no—
45C2×C4×Dic61—21no—
46#14519704—42no—
47#14519698—42no—
48#14519696—42no—
49C22⋊#92418—21no—
50C22⋊#92418—21no—
51#14519684—21no—
52#14519684—21no—
53C2×#41×C61—21no—
54C2×#41×C61—22no—
55C2×#41×C61D411yescharacteristic
56C22×Q8×C61D411yescharacteristic
57C22×D4×C61—21no—
58C22×D4×C61D411yescharacteristic
59C23×C4×C61C2311yescharacteristic
60C4⋊Q8×C61—21no—
61#126×C61—22no—
62#125×C61—22no—
63#124×C61—22no—
64#124×C61—21no—
65#124×C61—22no—
66#123×C61—22no—
67#123×C61—22no—
68#123×C61—22no—
69#123×C61—22no—
70#123×C61—21no—
71C22⋊Q8×C61—21no—
72C22⋊Q8×C61—22no—
73C22⋊Q8×C61—22no—
74C22⋊Q8×C61—22no—
75C22⋊Q8×C61—22no—
76C22⋊Q8×C61—21no—
77C22⋊Q8×C61—21no—
78C22⋊Q8×C61—21no—
79C22⋊Q8×C61—21no—
80C22⋊Q8×C61—22no—
81C22⋊Q8×C61—22no—
82C22⋊Q8×C61C2×C411yescharacteristic
83C22⋊Q8×C61D411yescharacteristic
84#121×C61—21no—
85#121×C61—22no—
86#121×C61—22no—
87#121×C61—22no—
88#121×C61—22no—
89#121×C61—22no—
90#121×C61—21no—
91#121×C61—21no—
92#121×C61C2×C411yescharacteristic
93#121×C61C2×C412yes—
94#121×C61D411yescharacteristic
95#121×C61D412yes—
96#120×C61—22no—
97#120×C61—21no—
98C4×Q8×C61—21no—
99C4×D4×C61—22no—
100C4×D4×C61—21no—
101C4×D4×C61—22no—
102C4×D4×C61—21no—
103C4×D4×C61—21no—
104#117×C61D411yescharacteristic
105C2×C4⋊C4×C61—21no—
106C2×C4⋊C4×C61C2×C412yes—
107C2×C4⋊C4×C61D411yescharacteristic
108C2×C4⋊C4×C61D411yescharacteristic
109C2×C22⋊C4×C61—22no—
110C2×C22⋊C4×C61—21no—
111C2×C22⋊C4×C61D412yes—
112C24×C61—21no—
113C24×C61C2×D411yescharacteristic
114#41×C61—41no—
115#41×C61—41no—
116#41×C61—84no—
117#41×C61—81no—
118#41×C61—81no—
119C2×Q8×C61—22no—
120C2×Q8×C61—21no—
121C2×Q8×C61—21no—
122C2×Q8×C61—41no—
123C2×Q8×C61—41no—
124C2×Q8×C61C22⋊C411yescharacteristic
125C2×D4×C61—21no—
126C2×D4×C61—22no—
127C2×D4×C61—21no—
128C2×D4×C61—22no—
129C2×D4×C61—22no—
130C2×D4×C61—21no—
131C2×D4×C61—22no—
132C2×D4×C61—21no—
133C2×D4×C61—41no—
134C2×D4×C61—41no—
135C2×D4×C61—82no—
136C2×D4×C61C22⋊C411yescharacteristic
137C2×D4×C61C22⋊C412yes—
138C22×C4×C61—22no—
139C22×C4×C61—22no—
140C22×C4×C61—22no—
141C22×C4×C61—21no—
142C22×C4×C61—21no—
143C22×C4×C61—41no—
144C22×C4×C61—41no—
145C22×C4×C61C22⋊C412yes—
146C22×C4×C61C22⋊C412yes—
147C22×C4×C61C22×C411yescharacteristic
148C22×C4×C61C2×D411yescharacteristic
149C22×C4×C61C2×D411yescharacteristic
150C22×C4×C61C2×D411yescharacteristic
151C22×C4×C61C2×D411yescharacteristic
152C22×C4×C61C2×D411yescharacteristic
153C22×Dic61—41no—
154C22×Dic61—41no—
155C4⋊C4×C61—22no—
156C4⋊C4×C61—22no—
157C4⋊C4×C61—22no—
158C4⋊C4×C61—22no—
159C4⋊C4×C61—22no—
160C4⋊C4×C61—22no—
161C4⋊C4×C61—21no—
162C4⋊C4×C61—21no—
163C4⋊C4×C61—21no—
164C4⋊C4×C61—21no—
165C4⋊C4×C61—21no—
166C4⋊C4×C61C22⋊C412yes—
167C4⋊C4×C61C22⋊C411yescharacteristic
168C4⋊C4×C61C22⋊C411yescharacteristic
169C22⋊C4×C61—22no—
170C22⋊C4×C61—22no—
171C22⋊C4×C61—22no—
172C22⋊C4×C61—22no—
173C22⋊C4×C61—22no—
174C22⋊C4×C61—21no—
175C22⋊C4×C61—21no—
176C22⋊C4×C61—22no—
177C22⋊C4×C61—22no—
178C22⋊C4×C61—21no—
179C22⋊C4×C61—22no—
180C22⋊C4×C61—21no—
181C22⋊C4×C61—22no—
182C22⋊C4×C61—21no—
183C42×C61—22no—
184C42×C61—21no—
185C22⋊Dic61—22no—
186C22⋊Dic61—41no—
187C22⋊Dic61—42no—
188C4⋊Dic61—41no—
189C4⋊Dic61—41no—
190C4×Dic61—41no—
191C4×Dic61—41no—
192#11758194—22no—
193#11758194—22no—
194#11758194—41no—
195#11758194—41no—
196#11758194—41no—
197#11758194—41no—
198C2×#92418—22no—
199C2×#92418—41no—
200C2×#92418—41no—
201C2×#92418—41no—
202C23×C61—21no—
203C23×C61—21no—
204C23×C61—22no—
205C23×C61—21no—
206C23×C61—41no—
207C23×C61—41no—
208C23×C61—41no—
209C23×C61—41no—
210C23×C61—81no—
211C23×C61—81no—
212C23×C61C2×C22⋊C411yescharacteristic
213C23×C61#12011yescharacteristic
214C23×C61#12011yescharacteristic
215Q8×C61—41no—
216Q8×C61—41no—
217Q8×C61—41no—
218Q8×C61—81no—
219Q8×C61—82no—
220D4×C61—42no—
221D4×C61—41no—
222D4×C61—44no—
223D4×C61—42no—
224D4×C61—41no—
225D4×C61—41no—
226D4×C61—41no—
227D4×C61—41no—
228D4×C61—82no—
229D4×C61—81no—
230D4×C61—82no—
231D4×C61—82no—
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—21no—
239C2×C4×C61—22no—
240C2×C4×C61—21no—
241C2×C4×C61—22no—
242C2×C4×C61—21no—
243C2×C4×C61—22no—
244C2×C4×C61—41no—
245C2×C4×C61—41no—
246C2×C4×C61—41no—
247C2×C4×C61—41no—
248C2×C4×C61—41no—
249C2×C4×C61—42no—
250C2×C4×C61—42no—
251C2×C4×C61—81no—
252C2×C4×C61—82no—
253C2×C4×C61—82no—
254C2×C4×C61—82no—
255C2×C4×C61C2×C22⋊C411yescharacteristic
256C2×C4×C61C2×C22⋊C411yescharacteristic
257C2×C4×C61#12011yescharacteristic
258C2×C4×C61#12011yescharacteristic
259C2×Dic61—42no—
260C2×Dic61—81no—
261C2×Dic61—81no—
262C2×Dic61—81no—
263C2×Dic61—81no—
264C2×Dic61—81no—
265#92418—42no—
266#92418—81no—
267#8480—611nomaximal, Sylow
268C22×C61—21no—
269C22×C61—21no—
270C22×C61—21no—
271C22×C61—21no—
272C22×C61—21no—
273C22×C61—21no—
274C22×C61—21no—
275C22×C61—21no—
276C22×C61—41no—
277C22×C61—42no—
278C22×C61—41no—
279C22×C61—42no—
280C22×C61—41no—
281C22×C61—42no—
282C22×C61—81no—
283C22×C61—81no—
284C22×C61—161no—
285C22×C61—161no—
286C22×C61#37911yescharacteristic
287C22×C61#45411yescharacteristic
288C22×C61#45611yescharacteristic
289C4×C61—21no—
290C4×C61—21no—
291C4×C61—21no—
292C4×C61—41no—
293C4×C61—42no—
294C4×C61—42no—
295C4×C61—41no—
296C4×C61—41no—
297C4×C61—42no—
298C4×C61B(4,𝔽2)11yescharacteristic
299C4×C61#45811yescharacteristic
300Dic61—82no—
301Dic61—161no—
302#3536D6111yescharacteristic
303#1996—611no—
304#1901—611no—
305#1895—611no—
306#1871—611no—
307#1513—611no—
308#1507—611no—
309C2×C61—21no—
310C2×C61—21no—
311C2×C61—41no—
312C2×C61—41no—
313C2×C61—81no—
314C2×C61—82no—
315C2×C61#179011yescharacteristic
316C2×C61#179311yescharacteristic
317C2×C61#179511yescharacteristic
318C22×M16—611no—
319#562—22no—
320#554—21no—
321#548—21no—
322#547—21no—
323#546—21no—
324#546—22no—
325#543—21no—
326#542—21no—
327#539Dic6112yescomplemented
328#538—21no—
329#537—21no—
330#537Dic6112yescomplemented
331#525C2×D6111yescharacteristic
332C2×C22⋊Q8C2×D6111yescharacteristic
333C2×#121C2×D6111yescharacteristic
334C2×#104—1221no—
335C2×#102—1221no—
336C2×#101—1221no—
337#407—611no—
338#380—611no—
339C4×C22⋊C4—611no—
340#343—1221no—
341#342—1222no—
342#340—1221no—
343#339—1221no—
344#328—1221no—
345#327—1222no—
346#327—1221no—
347C61#848011yescharacteristic, complemented, Sylow
348C2×#41—21no—
349C2×#41—22no—
350C2×#41C22⋊D6111yescharacteristic
351C22×Q8C22⋊D6111yescharacteristic
352C22×D4—21no—
353C22×D4C22⋊D6111yescharacteristic
354C23×C4C22×D6111yescharacteristic
355C2×M16—1221no—
356C2×M16—1221no—
357C2×M16—1221no—
358C2×M16—2442no—
359C22×C8—1221no—
360C4⋊Q8—21no—
361#126—22no—
362#125—22no—
363#124—22no—
364#124—21no—
365#124—22no—
366#123—22no—
367#123—22no—
368#123—22no—
369#123—22no—
370#123—21no—
371C22⋊Q8—21no—
372C22⋊Q8—22no—
373C22⋊Q8—22no—
374C22⋊Q8—22no—
375C22⋊Q8—22no—
376C22⋊Q8—21no—
377C22⋊Q8—21no—
378C22⋊Q8—21no—
379C22⋊Q8—21no—
380C22⋊Q8—22no—
381C22⋊Q8—22no—
382C22⋊Q8C2×Dic6111yescharacteristic
383C22⋊Q8C22⋊D6111yescharacteristic
384#121—21no—
385#121—22no—
386#121—22no—
387#121—22no—
388#121—22no—
389#121—22no—
390#121—21no—
391#121—21no—
392#121C2×Dic6112yes—
393#121C2×Dic6111yescharacteristic
394#121C22⋊D6112yes—
395#121C22⋊D6111yescharacteristic
396#120—22no—
397#120—21no—
398C4×Q8—21no—
399C4×D4—22no—
400C4×D4—21no—
401C4×D4—22no—
402C4×D4—21no—
403C4×D4—21no—
404#117C22⋊D6111yescharacteristic
405C2×C4⋊C4—21no—
406C2×C4⋊C4—1221no—
407C2×C4⋊C4C2×Dic6112yes—
408C2×C4⋊C4C22⋊D6111yescharacteristic
409C2×C4⋊C4C22⋊D6111yescharacteristic
410C2×C22⋊C4—22no—
411C2×C22⋊C4—21no—
412C2×C22⋊C4—1221no—
413C2×C22⋊C4—1221no—
414C2×C22⋊C4C22⋊D6112yes—
415C2×C42—1221no—
416#104—2442no—
417#102—2442no—
418#101—2442no—
419C22⋊C8—1221no—
420C22⋊C8—1221no—
421#95—1221no—
422#95—1221no—
423C24—21no—
424C24C2×C22⋊D6111yescharacteristic
425#41—41no—
426#41—41no—
427#41—84no—
428#41—81no—
429#41—81no—
430C2×Q8—22no—
431C2×Q8—21no—
432C2×Q8—21no—
433C2×Q8—41no—
434C2×Q8—41no—
435C2×Q8C22⋊Dic6111yescharacteristic
436C2×D4—21no—
437C2×D4—22no—
438C2×D4—21no—
439C2×D4—22no—
440C2×D4—22no—
441C2×D4—21no—
442C2×D4—22no—
443C2×D4—21no—
444C2×D4—41no—
445C2×D4—41no—
446C2×D4—82no—
447C2×D4C22⋊Dic6111yescharacteristic
448C2×D4C22⋊Dic6112yes—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—41no—
455C22×C4—41no—
456C22×C4—2441no—
457C22×C4—2441no—
458C22×C4C22⋊Dic6112yes—
459C22×C4C22⋊Dic6112yes—
460C22×C4C22×Dic6111yescharacteristic
461C22×C4C2×C22⋊D6111yescharacteristic
462C22×C4C2×C22⋊D6111yescharacteristic
463C22×C4C2×C22⋊D6111yescharacteristic
464C22×C4C2×C22⋊D6111yescharacteristic
465C22×C4C2×C22⋊D6111yescharacteristic
466M16—1222no—
467M16—1222no—
468M16—2441no—
469M16—2441no—
470M16—2441no—
471M16—2441no—
472C2×C8—1222no—
473C2×C8—2441no—
474C2×C8—2441no—
475C2×C8—2441no—
476C4⋊C4—22no—
477C4⋊C4—22no—
478C4⋊C4—22no—
479C4⋊C4—22no—
480C4⋊C4—22no—
481C4⋊C4—22no—
482C4⋊C4—21no—
483C4⋊C4—21no—
484C4⋊C4—21no—
485C4⋊C4—21no—
486C4⋊C4—21no—
487C4⋊C4—2441no—
488C4⋊C4—2441no—
489C4⋊C4C22⋊Dic6112yes—
490C4⋊C4C22⋊Dic6111yescharacteristic
491C4⋊C4C22⋊Dic6111yescharacteristic
492C22⋊C4—22no—
493C22⋊C4—22no—
494C22⋊C4—22no—
495C22⋊C4—22no—
496C22⋊C4—22no—
497C22⋊C4—21no—
498C22⋊C4—21no—
499C22⋊C4—22no—
500C22⋊C4—22no—
501C22⋊C4—21no—
502C22⋊C4—22no—
503C22⋊C4—21no—
504C22⋊C4—22no—
505C22⋊C4—21no—
506C22⋊C4—1222no—
507C22⋊C4—2442no—
508C22⋊C4—2441no—
509C42—22no—
510C42—21no—
511C42—2441no—
512C42—2441no—
513C23—21no—
514C23—21no—
515C23—22no—
516C23—21no—
517C23—41no—
518C23—41no—
519C23—41no—
520C23—41no—
521C23—81no—
522C23—81no—
523C23C2×C22⋊Dic6111yescharacteristic
524C23#1451974611yescharacteristic
525C23#1451974611yescharacteristic
526Q8—41no—
527Q8—41no—
528Q8—41no—
529Q8—81no—
530Q8—82no—
531D4—42no—
532D4—41no—
533D4—44no—
534D4—42no—
535D4—41no—
536D4—41no—
537D4—41no—
538D4—41no—
539D4—82no—
540D4—81no—
541D4—82no—
542D4—82no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—22no—
551C2×C4—21no—
552C2×C4—22no—
553C2×C4—21no—
554C2×C4—22no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—81no—
563C2×C4—82no—
564C2×C4—82no—
565C2×C4—82no—
566C2×C4—4881no—
567C2×C4—4881no—
568C2×C4—4881no—
569C2×C4—4881no—
570C2×C4—4881no—
571C2×C4—2442no—
572C2×C4C2×C22⋊Dic6111yescharacteristic
573C2×C4C2×C22⋊Dic6111yescharacteristic
574C2×C4#1451974611yescharacteristic
575C2×C4#1451974611yescharacteristic
576C8—4881no—
577C8—2442no—
578C22—21no—
579C22—21no—
580C22—21no—
581C22—21no—
582C22—21no—
583C22—21no—
584C22—21no—
585C22—21no—
586C22—41no—
587C22—42no—
588C22—41no—
589C22—42no—
590C22—41no—
591C22—42no—
592C22—81no—
593C22—81no—
594C22—161no—
595C22—161no—
596C22#1683961811yescharacteristic
597C22#1683993511yescharacteristic
598C22#1683994511yescharacteristic
599C4—21no—
600C4—21no—
601C4—21no—
602C4—41no—
603C4—42no—
604C4—41no—
605C4—42no—
606C4—42no—
607C4—41no—
608C4—9761no—
609C4—4882no—
610C4#1683994911yescharacteristic
611C4#1683995211yescharacteristic
612C2—21no—
613C2—21no—
614C2—41no—
615C2—41no—
616C2—82no—
617C2—81no—
618C2#2144576711yescharacteristic
619C2#2144577811yescharacteristic
620C2#2144578911yescharacteristic
subgroupG/H
centerC22#16839618
commutatorC22×C4×C61C22×C4
FrattiniC23×C4C22×D61
Fitting, prime radicalC61×#3536C2
socleC22×C61#379
cosocleC23×C4×C61C23
hypercenter#3536D61
hypocenterC61#8480
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #3536

Complements

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

NH ≅ G/Ntype
1. #21446805C2semidirect
2. #21446288C2semidirect
3. #21446258C2semidirect
4. #21446122C2semidirect
5. #21444362C2semidirect
6. #21444335C2semidirect
30. C61×#539C4semidirect
33. C61×#537C4semidirect
327. #539Dic61semidirect
330. #537Dic61semidirect
347. C61#8480semidirect

Maximal subgroups

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

subgroupindex
#214468052
#214462882
#214462582
#214461222
#214443622
#214443352
C61×#35362
#848061