Finite Group Databasebeta

← #29738890

Subgroups — #29738890

Order 15616.

782 conjugacy classes of subgroups.

The subgroup lattice has 610 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840172C211yesmaximal, characteristic, complemented
2C2×#16839802C211yesmaximal, characteristic
3#21449510C211yesmaximal, characteristic
4#21447301C211yesmaximal, characteristic, complemented
5#21447300C211yesmaximal, characteristic, complemented
6#21447154C211yesmaximal, characteristic, complemented
7#21447147C211yesmaximal, characteristic, complemented
8#21446554C211yesmaximal, characteristic, complemented
9#21446549C211yesmaximal, characteristic, complemented
10#21446016C211yesmaximal, characteristic
11#21446007C211yesmaximal, characteristic
12#21445981C211yesmaximal, characteristic, complemented
13#21445850C211yesmaximal, characteristic
14#21445603C211yesmaximal, characteristic
15C61×#2442C211yesmaximal, characteristic
16C22×#11758194C2211yescharacteristic
17C2×#14519777C2211yescharacteristic
18C2×#14519736C2211yescharacteristic
19#16840172—24no—
20#16839802—24no—
21C2×#14519710C2211yescharacteristic
22C2×#14519710C2211yescharacteristic
23C2×#14519710C2211yescharacteristic
24C2×#14519710C2211yescharacteristic
25C2×#14519710C2211yescharacteristic
26C2×#14519710C2211yescharacteristic
27C2×#14519708C2211yescharacteristic
28C2×#14519708C2211yescharacteristic
29C2×#14519708C2211yescharacteristic
30C2×#14519708C2211yescharacteristic
31C2×C22⋊#92418C2211yescharacteristic
32C2×#14519688C2211yescharacteristic
33C2×C4×#92418C2211yescharacteristic
34#16839685—21no—
35#16839674—21no—
36#16839666—21no—
37#16839660C2211yescharacteristic
38#16839660C2211yescharacteristic
39#16839649—21no—
40#16839628C2211yescharacteristic
41#16839622C2211yescharacteristic
42#16839524C2211yescharacteristic
43#16839524C2211yescharacteristic
44#16839524C2211yescharacteristic
45#16839524C2211yescharacteristic
46#16839524C2211yescharacteristic
47#16839524C2211yescharacteristic
48#16839517C2211yescharacteristic
49#16839517C2211yescharacteristic
50C2×#125×C61C2211yescharacteristic
51C22×C4⋊C4×C61C2211yescharacteristic
52C61×#400—21no—
53C61×#398—21no—
54C61×#389C2211yescharacteristic
55C61×#389C2211yescharacteristic
56C61×#385—21no—
57C61×#382C2211yescharacteristic
58C61×#380—21no—
59C61×#380C2211yescharacteristic
60C4×C22⋊C4×C61C2211yescharacteristic
61C22×Dic122—21no—
62C2×#11758194—22no—
63C2×#11758194—22no—
64C2×#11758194Q814yescomplemented
65C22×#92418C2311yescharacteristic
66C22×#92418C2311yescharacteristic
67C22×#92418C2311yescharacteristic
68C22×#92418C2311yescharacteristic
69#14519777—22no—
70#14519777—41no—
71#14519736—22no—
72#14519736—41no—
73C2×C4⋊Dic61—21no—
74C2×C4⋊Dic61C2311yescharacteristic
75C2×C4⋊Dic61C2311yescharacteristic
76C2×C4⋊Dic61C2311yescharacteristic
77C2×C4⋊Dic61C2311yescharacteristic
78C2×#11758196—21no—
79C2×#11758196—21no—
80C2×C22⋊Dic61—21no—
81C2×C22⋊Dic61—21no—
82C2×C4×Dic61—21no—
83#14519710—22no—
84#14519710—22no—
85#14519710—22no—
86#14519710—41no—
87#14519710—41no—
88#14519710D414yes—
89#14519708—22no—
90#14519708—22no—
91#14519708—22no—
92#14519708—22no—
93C22⋊#92418Q814yescomplemented
94#14519688—22no—
95C4×#92418—22no—
96#14519684—21no—
97#14519684—21no—
98#14519684—21no—
99#14519684—21no—
100#14519684—21no—
101C23×C4×C61—21no—
102C23×C4×C61C2311yescharacteristic
103#125×C61—22no—
104#125×C61—41no—
105C2×C4⋊C4×C61—22no—
106C2×C4⋊C4×C61—21no—
107C2×C4⋊C4×C61—21no—
108C2×C4⋊C4×C61—21no—
109C2×C4⋊C4×C61—21no—
110C2×C4⋊C4×C61—21no—
111C2×C4⋊C4×C61—41no—
112C2×C4⋊C4×C61C2311yescharacteristic
113C2×C4⋊C4×C61C2311yescharacteristic
114C2×C4⋊C4×C61C2311yescharacteristic
115C2×C4⋊C4×C61C2311yescharacteristic
116C2×C22⋊C4×C61—21no—
117C2×C22⋊C4×C61D411yescharacteristic
118C2×C22⋊C4×C61D411yescharacteristic
119C2×C42×C61C2311yescharacteristic
120C2×C42×C61C2311yescharacteristic
121#95×C61—21no—
122#95×C61—21no—
123#95×C61—21no—
124#95×C61—21no—
125#95×C61—21no—
126#95×C61D411yescharacteristic
127#95×C61D411yescharacteristic
128C24×C61C2×D411yescharacteristic
129C22×C4×C61—21no—
130C22×C4×C61—21no—
131C22×C4×C61—21no—
132C22×C4×C61—21no—
133C22×C4×C61—21no—
134C22×C4×C61—21no—
135C22×C4×C61—21no—
136C22×C4×C61—21no—
137C22×C4×C61—42no—
138C22×C4×C61—42no—
139C22×C4×C61—81no—
140C22×C4×C61C2×D411yescharacteristic
141C22×C4×C61C2×D411yescharacteristic
142C22×C4×C61C2×D411yescharacteristic
143C22×C4×C61C2×D411yescharacteristic
144C22×C4×C61C2×D411yescharacteristic
145C22×C4×C61C2×Q812yes—
146C22×C4×C61C2×Q812yes—
147C22×C4×C61C2411yescharacteristic
148C22×Dic61—21no—
149C22×Dic61—21no—
150C22×Dic61—21no—
151C22×Dic61—21no—
152C2×Dic122—42no—
153C2×Dic122—42no—
154C2×Dic122—81no—
155C4⋊C4×C61—21no—
156C4⋊C4×C61—21no—
157C4⋊C4×C61—21no—
158C4⋊C4×C61—21no—
159C4⋊C4×C61—21no—
160C4⋊C4×C61—21no—
161C4⋊C4×C61—21no—
162C4⋊C4×C61—42no—
163C4⋊C4×C61—42no—
164C4⋊C4×C61—42no—
165C4⋊C4×C61—81no—
166C4⋊C4×C61—81no—
167C4⋊C4×C61C2×D412yes—
168C22⋊C4×C61—41no—
169C22⋊C4×C61—41no—
170C22⋊C4×C61—42no—
171C42×C61—21no—
172C42×C61—21no—
173C42×C61—21no—
174C42×C61—21no—
175C22⋊Dic61—42no—
176C22⋊Dic61—81no—
177C4⋊Dic61—21no—
178C4⋊Dic61—21no—
179C4⋊Dic61—21no—
180C4⋊Dic61—21no—
181C4⋊Dic61—21no—
182C4⋊Dic61—21no—
183C4⋊Dic61—21no—
184C4⋊Dic61—42no—
185C4⋊Dic61C2×D412yes—
186#11758196—42no—
187#11758196—81no—
188C4×Dic61—42no—
189#11758194—28no—
190C2×#92418—21no—
191C2×#92418—21no—
192C2×#92418—21no—
193C2×#92418—21no—
194C2×#92418—21no—
195C2×#92418—21no—
196C2×#92418C2×D412yes—
197C2×#92418C2×Q812yes—
198C2×#92418C2×Q812yes—
199C2×#92418C2×Q812yes—
200C2×#92418C2×Q812yes—
201C2×#92418#4112yes—
202C23×C61—21no—
203C23×C61—21no—
204C23×C61—21no—
205C23×C61—21no—
206C23×C61—21no—
207C23×C61—21no—
208C23×C61C4⋊Q812yes—
209C23×C61C22×D411yescharacteristic
210C2×C4×C61—21no—
211C2×C4×C61—21no—
212C2×C4×C61—22no—
213C2×C4×C61—22no—
214C2×C4×C61—22no—
215C2×C4×C61—22no—
216C2×C4×C61—21no—
217C2×C4×C61—21no—
218C2×C4×C61—22no—
219C2×C4×C61—21no—
220C2×C4×C61—21no—
221C2×C4×C61—21no—
222C2×C4×C61—21no—
223C2×C4×C61—21no—
224C2×C4×C61—21no—
225C2×C4×C61—21no—
226C2×C4×C61—21no—
227C2×C4×C61—21no—
228C2×C4×C61—21no—
229C2×C4×C61—41no—
230C2×C4×C61—41no—
231C2×C4×C61—41no—
232C2×C4×C61—41no—
233C2×C4×C61—41no—
234C2×C4×C61—41no—
235C2×C4×C61—41no—
236C2×C4×C61—41no—
237C2×C4×C61—41no—
238C2×C4×C61—41no—
239C2×C4×C61—41no—
240C2×C4×C61—84no—
241C2×C4×C61C4⋊Q812yes—
242C2×C4×C61C22×D411yescharacteristic
243C2×C4×C61C22×Q811yescharacteristic
244C2×C4×C61C22×Q811yescharacteristic
245C2×C4×C61C2×#4111yescharacteristic
246C2×C4×C61H(5,𝔽2)11yescharacteristic
247C2×C4×C612−1+411yescharacteristic
248C2×Dic61—22no—
249C2×Dic61—41no—
250C2×Dic61—41no—
251C2×Dic61—41no—
252C2×Dic61—41no—
253C2×Dic61—41no—
254C2×Dic61—41no—
255C2×Dic61—41no—
256C2×Dic61—41no—
257C2×Dic61—41no—
258C2×Dic61—41no—
259C2×Dic61—41no—
260Dic122—84no—
261#92418—22no—
262#92418—22no—
263#92418—22no—
264#92418C22⋊Q814yes—
265#26079—611nomaximal, Sylow
266C22×C61—22no—
267C22×C61—22no—
268C22×C61—22no—
269C22×C61—41no—
270C22×C61—41no—
271C22×C61—41no—
272C22×C61—41no—
273C22×C61C2×C4⋊Q811yescharacteristic
274C22×C61D4211yescharacteristic
275C22×C61#54711yescharacteristic
276C22×C61#57511yescharacteristic
277C22×C61#57511yescharacteristic
278C22×C61#57611yescharacteristic
279C22×C61#57811yescharacteristic
280C4×C61—22no—
281C4×C61—41no—
282C4×C61—41no—
283C4×C61—41no—
284C4×C61—42no—
285C4×C61—41no—
286C4×C61—81no—
287C4×C61—81no—
288C4×C61—81no—
289C4×C61C2×C22⋊Q811yescharacteristic
290C4×C61#54411yescharacteristic
291C4×C61D4×Q811yescharacteristic
292C4×C61#55411yescharacteristic
293Dic61—42no—
294Dic61—81no—
295Dic61—81no—
296Dic61—81no—
297C2×#501—611no—
298C2×#428—611no—
299#2442D6111yescharacteristic
300#2439—611no—
301#2085—611no—
302#2085—611no—
303#2059—611no—
304#2057—611no—
305#1943—611no—
306#1942—611no—
307#1854—611no—
308#1852—611no—
309#1846—611no—
310#1809—611no—
311#1753—611no—
312C2×C61—42no—
313C2×C61#244011yescharacteristic
314C2×C61#316411yescharacteristic
315C2×C61#316611yescharacteristic
316C2×C61#327311yescharacteristic
317C2×C61#327711yescharacteristic
318C2×C61#334711yescharacteristic
319C2×C61#334911yescharacteristic
320C22×M16—611no—
321C2×C4⋊Q8—611no—
322C2×#125C2×D6111yescharacteristic
323C2×#117—611no—
324C22×C4⋊C4C2×D6111yescharacteristic
325#501—1224no—
326#428—1224no—
327C2×#107—611no—
328C2×#107—611no—
329C2×#107—611no—
330C2×#107—611no—
331C2×#107—611no—
332C2×#107—611no—
333C2×#106—611no—
334C2×#106—611no—
335C2×#106—611no—
336C2×#106—611no—
337C2×C22⋊C8—611no—
338C2×#97—611no—
339C2×C4×C8—611no—
340#400—21no—
341#398—21no—
342#396—1221no—
343#393—1221no—
344#391—1221no—
345#389—611no—
346#389—611no—
347#389C2×D6111yescharacteristic
348#389C2×D6111yescharacteristic
349#387—1221no—
350#385—21no—
351#382—611no—
352#382C2×D6111yescharacteristic
353#380—21no—
354#380—611no—
355#380C2×D6111yescharacteristic
356C4×C22⋊C4C2×D6111yescharacteristic
357#340—611no—
358#340—611no—
359#340—611no—
360#340—611no—
361#340—611no—
362#340—611no—
363#336—611no—
364#336—611no—
365C61#2607911yescharacteristic, complemented, Sylow
366C22×Q8—1221no—
367C23×C4—21no—
368C23×C4C22×D6111yescharacteristic
369C2×M16—614no—
370C2×M16—1222no—
371C2×M16—1222no—
372C22×C8—611no—
373C22×C8—611no—
374C22×C8—611no—
375C22×C8—611no—
376C4⋊Q8—1222no—
377C4⋊Q8—2441no—
378#125—22no—
379#125—41no—
380#117—1222no—
381#117—2441no—
382C2×C4⋊C4—22no—
383C2×C4⋊C4—21no—
384C2×C4⋊C4—21no—
385C2×C4⋊C4—21no—
386C2×C4⋊C4—21no—
387C2×C4⋊C4—21no—
388C2×C4⋊C4—41no—
389C2×C4⋊C4—611no—
390C2×C4⋊C4—611no—
391C2×C4⋊C4—611no—
392C2×C4⋊C4—611no—
393C2×C4⋊C4—1221no—
394C2×C4⋊C4—1221no—
395C2×C4⋊C4—1221no—
396C2×C4⋊C4C22×D6111yescharacteristic
397C2×C4⋊C4C22×D6111yescharacteristic
398C2×C4⋊C4C22×D6111yescharacteristic
399C2×C4⋊C4C22×D6111yescharacteristic
400C2×C22⋊C4—21no—
401C2×C22⋊C4—1221no—
402C2×C22⋊C4—1221no—
403C2×C22⋊C4C22⋊D6111yescharacteristic
404C2×C22⋊C4C22⋊D6111yescharacteristic
405C2×C42—1221no—
406C2×C42C22×D6111yescharacteristic
407C2×C42C22×D6111yescharacteristic
408#107—614no—
409#107—1222no—
410#107—1222no—
411#107—1222no—
412#107—2441no—
413#107—2441no—
414#106—1222no—
415#106—1222no—
416#106—1222no—
417#106—1222no—
418C22⋊C8—614no—
419#97—1222no—
420C4×C8—1222no—
421#95—21no—
422#95—21no—
423#95—21no—
424#95—21no—
425#95—21no—
426#95—1221no—
427#95—1221no—
428#95—1221no—
429#95—1221no—
430#95—1221no—
431#95C22⋊D6111yescharacteristic
432#95C22⋊D6111yescharacteristic
433C24C2×C22⋊D6111yescharacteristic
434C2×Q8—4881no—
435C2×Q8—2442no—
436C2×Q8—2442no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—42no—
446C22×C4—42no—
447C22×C4—81no—
448C22×C4—1221no—
449C22×C4—1221no—
450C22×C4—1221no—
451C22×C4—1221no—
452C22×C4Q8×D6112yes—
453C22×C4Q8×D6112yes—
454C22×C4C2×C22⋊D6111yescharacteristic
455C22×C4C2×C22⋊D6111yescharacteristic
456C22×C4C2×C22⋊D6111yescharacteristic
457C22×C4C2×C22⋊D6111yescharacteristic
458C22×C4C2×C22⋊D6111yescharacteristic
459C22×C4C23×D6111yescharacteristic
460M16—1228no—
461C2×C8—612no—
462C2×C8—612no—
463C2×C8—612no—
464C2×C8—612no—
465C2×C8—612no—
466C2×C8—612no—
467C2×C8—1221no—
468C2×C8—1221no—
469C2×C8—1221no—
470C2×C8—1221no—
471C2×C8—1221no—
472C2×C8—1221no—
473C4⋊C4—21no—
474C4⋊C4—21no—
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—21no—
478C4⋊C4—21no—
479C4⋊C4—21no—
480C4⋊C4—42no—
481C4⋊C4—42no—
482C4⋊C4—42no—
483C4⋊C4—81no—
484C4⋊C4—81no—
485C4⋊C4—612no—
486C4⋊C4—1221no—
487C4⋊C4—1221no—
488C4⋊C4—1221no—
489C4⋊C4—1221no—
490C4⋊C4—1221no—
491C4⋊C4—1221no—
492C4⋊C4—1221no—
493C4⋊C4—4881no—
494C4⋊C4—2442no—
495C4⋊C4—2442no—
496C4⋊C4D4×D6112yes—
497C22⋊C4—41no—
498C22⋊C4—41no—
499C22⋊C4—42no—
500C22⋊C4—4881no—
501C22⋊C4—2442no—
502C42—21no—
503C42—21no—
504C42—21no—
505C42—21no—
506C42—2442no—
507C23—21no—
508C23—21no—
509C23—21no—
510C23—21no—
511C23—21no—
512C23—21no—
513C23#1451977512yes—
514C23C22×C22⋊D6111yescharacteristic
515Q8—4884no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—22no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—84no—
547C2×C4—1222no—
548C2×C4—2441no—
549C2×C4—2441no—
550C2×C4—2441no—
551C2×C4—2441no—
552C2×C4—2441no—
553C2×C4—2441no—
554C2×C4—2441no—
555C2×C4—2441no—
556C2×C4—2441no—
557C2×C4—2441no—
558C2×C4—2441no—
559C2×C4#1451977512yes—
560C2×C4C2×D4×D6111yescharacteristic
561C2×C4C2×Q8×D6111yescharacteristic
562C2×C4C2×Q8×D6111yescharacteristic
563C2×C4#41×D6111yescharacteristic
564C2×C4#1451982211yescharacteristic
565C2×C4#1451982411yescharacteristic
566C8—614no—
567C8—1222no—
568C8—1222no—
569C8—1222no—
570C22—22no—
571C22—22no—
572C22—22no—
573C22—41no—
574C22—41no—
575C22—41no—
576C22—41no—
577C22C2×#1451977511yescharacteristic
578C22D4×C22⋊D6111yescharacteristic
579C22#1684041111yescharacteristic
580C22#1684056011yescharacteristic
581C22#1684056011yescharacteristic
582C22#1684056511yescharacteristic
583C22#1684057711yescharacteristic
584C4—22no—
585C4—41no—
586C4—42no—
587C4—41no—
588C4—41no—
589C4—41no—
590C4—81no—
591C4—81no—
592C4—81no—
593C4—4881no—
594C4—4881no—
595C4—4881no—
596C4—2442no—
597C4C22⋊Q8×D6111yescharacteristic
598C4#1684039111yescharacteristic
599C4#1684042511yescharacteristic
600C4#1684046611yescharacteristic
601C2—42no—
602C2#2144951811yescharacteristic
603C2#2145635311yescharacteristic
604C2#2145636811yescharacteristic
605C2#2145742611yescharacteristic
606C2#2145747211yescharacteristic
607C2#2145831311yescharacteristic
608C2#2145834911yescharacteristic
subgroupG/H
centerC23C22×C22⋊D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2442C2
socleC23×C61C22×D4
prime radicalC24×C61C2×D4
hypercenter#2442D61
hypocenterC61#26079
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2442

Complements

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

NH ≅ G/Ntype
1. C2×#16840172C2semidirect
4. #21447301C2semidirect
5. #21447300C2semidirect
6. #21447154C2semidirect
7. #21447147C2semidirect
8. #21446554C2semidirect
9. #21446549C2semidirect
12. #21445981C2semidirect
64. C2×#11758194Q8semidirect
93. C22⋊#92418Q8semidirect
365. C61#26079semidirect

Maximal subgroups

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

subgroupindex
C2×#168401722
C2×#168398022
#214495102
#214473012
#214473002
#214471542
#214471472
#214465542
#214465492
#214460162
#214460072
#214459812
#214458502
#214456032
C61×#24422
#2607961