Finite Group Databasebeta

← #29742975

Subgroups — #29742975

Order 15616.

788 conjugacy classes of subgroups.

The subgroup lattice has 602 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839854C211yesmaximal, characteristic
2#21449654C211yesmaximal, characteristic
3#21448737C211yesmaximal, characteristic
4#21447368C211yesmaximal, characteristic
5#21447304C211yesmaximal, characteristic
6#21447199C211yesmaximal, characteristic
7#21447144C211yesmaximal, characteristic
8#21446545C211yesmaximal, characteristic
9#21446373C211yesmaximal, characteristic
10#21446160C211yesmaximal, characteristic
11#21446158C211yesmaximal, characteristic
12#21446089C211yesmaximal, characteristic
13#21446084C211yesmaximal, characteristic
14#21446071C211yesmaximal, characteristic
15C2×C61×#475C211yesmaximal, characteristic
16C22×#11758202C2211yescharacteristic
17C2×Q8×Dic61C2211yescharacteristic
18C2×Q8×Dic61C2211yescharacteristic
19C2×#14519742C2211yescharacteristic
20#16839854—24no—
21C2×#14519710C2211yescharacteristic
22C2×#14519710C2211yescharacteristic
23C2×#14519709C2211yescharacteristic
24C2×#14519702C2211yescharacteristic
25C2×#14519702C2211yescharacteristic
26C2×#14519701C2211yescharacteristic
27C2×#14519700C2211yescharacteristic
28C2×C8×Dic61C2211yescharacteristic
29#16839695—21no—
30#16839681—21no—
31#16839669—21no—
32#16839668C2211yescharacteristic
33#16839667C2211yescharacteristic
34#16839661C2211yescharacteristic
35#16839660—21no—
36#16839660C2211yescharacteristic
37#16839658—21no—
38#16839658C2211yescharacteristic
39#16839658C2211yescharacteristic
40#16839643—21no—
41#16839627—21no—
42#16839627C2211yescharacteristic
43C4×#11758196—21no—
44C4×#11758196C2211yescharacteristic
45#16839525C2211yescharacteristic
46#16839524C2211yescharacteristic
47#16839524C2211yescharacteristic
48#16839524C2211yescharacteristic
49#16839517C2211yescharacteristic
50#16839516C2211yescharacteristic
51#16839516C2211yescharacteristic
52#16839516C2211yescharacteristic
53C2×C4⋊Q8×C61C2211yescharacteristic
54C2×C4×Q8×C61C2211yescharacteristic
55C61×#475—24no—
56C2×#106×C61C2211yescharacteristic
57C2×#106×C61C2211yescharacteristic
58C2×C4⋊C8×C61C2211yescharacteristic
59C2×#103×C61C2211yescharacteristic
60C2×#103×C61C2211yescharacteristic
61C22×Dic122C2311yescharacteristic
62C2×#11758202—22no—
63C2×#11758202—22no—
64C2×#11758202Q814yescomplemented
65C22×#92418C2311yescharacteristic
66C22×#92418C2311yescharacteristic
67Q8×Dic61—22no—
68Q8×Dic61—22no—
69Q8×Dic61—22no—
70Q8×Dic61—41no—
71#14519742—22no—
72#14519742—41no—
73C2×C4⋊Dic61—21no—
74C2×C4⋊Dic61—21no—
75C2×C4⋊Dic61—21no—
76C2×C4⋊Dic61C2311yescharacteristic
77C2×C4⋊Dic61C2311yescharacteristic
78C2×C4⋊Dic61C2311yescharacteristic
79C2×#11758196—21no—
80C2×#11758196—21no—
81C2×#11758196—21no—
82C2×#11758196—21no—
83C2×#11758196D411yescharacteristic
84C2×#11758196D411yescharacteristic
85C2×C4×Dic61—21no—
86C2×C4×Dic61—21no—
87C2×C4×Dic61—21no—
88C2×C4×Dic61C2311yescharacteristic
89C2×C4×Dic61C2311yescharacteristic
90#14519710—22no—
91#14519710—41no—
92#14519709—22no—
93#14519702—22no—
94#14519702—22no—
95#14519701—22no—
96#14519700—41no—
97C8×Dic61—22no—
98#14519684—21no—
99#14519684—21no—
100#14519684—21no—
101#14519684—21no—
102#14519684—21no—
103#14519684—21no—
104#14519684—21no—
105#14519684—21no—
106#14519684—21no—
107#14519684—21no—
108#14519684D411yescharacteristic
109#14519684D411yescharacteristic
110C22×Q8×C61—21no—
111C22×Q8×C61C2311yescharacteristic
112C22×C8×C61C2311yescharacteristic
113C22×C8×C61C2311yescharacteristic
114C4⋊Q8×C61—22no—
115C4⋊Q8×C61—41no—
116C4×Q8×C61—22no—
117C4×Q8×C61—41no—
118C2×C4⋊C4×C61—21no—
119C2×C4⋊C4×C61—21no—
120C2×C4⋊C4×C61C2311yescharacteristic
121C2×C4⋊C4×C61C2311yescharacteristic
122C2×C4⋊C4×C61C2311yescharacteristic
123C2×C42×C61—21no—
124C2×C42×C61C2311yescharacteristic
125#106×C61—22no—
126#106×C61—22no—
127C4⋊C8×C61—22no—
128#103×C61—22no—
129#103×C61D414yes—
130C2×Q8×C61—21no—
131C2×Q8×C61—21no—
132C2×Q8×C61—21no—
133C2×Q8×C61—42no—
134C2×Q8×C61—42no—
135C2×Q8×C61—81no—
136C2×Q8×C61C2×D412yes—
137C2×Q8×C61C2×Q812yes—
138C2×Q8×C61#4112yes—
139C22×C4×C61—21no—
140C22×C4×C61—21no—
141C22×C4×C61—21no—
142C22×C4×C61—21no—
143C22×C4×C61C2×D411yescharacteristic
144C22×C4×C61C2×D411yescharacteristic
145C22×C4×C61C2411yescharacteristic
146C22×Dic61—21no—
147C22×Dic61—21no—
148C22×Dic61—21no—
149C22×Dic61—21no—
150C22×Dic61C2×D411yescharacteristic
151C22×Dic61C2×D411yescharacteristic
152C22×Dic61C2×D411yescharacteristic
153C22×Dic61C2×D411yescharacteristic
154C2×Dic122—21no—
155C2×Dic122—21no—
156C2×Dic122—21no—
157C2×Dic122—21no—
158C2×Dic122—21no—
159C2×Dic122C2×Q812yes—
160C2×C8×C61—21no—
161C2×C8×C61—21no—
162C2×C8×C61—21no—
163C2×C8×C61—21no—
164C2×C8×C61—21no—
165C2×C8×C61C2×D412yes—
166C4⋊C4×C61—21no—
167C4⋊C4×C61—21no—
168C4⋊C4×C61—21no—
169C4⋊C4×C61—21no—
170C4⋊C4×C61—21no—
171C4⋊C4×C61—42no—
172C4⋊C4×C61—42no—
173C4⋊C4×C61C2×D412yes—
174C42×C61—21no—
175C42×C61—21no—
176C42×C61—42no—
177#11758202—28no—
178C4⋊Dic61—24no—
179C4⋊Dic61—21no—
180C4⋊Dic61—21no—
181C4⋊Dic61—21no—
182C4⋊Dic61—21no—
183C4⋊Dic61—21no—
184C4⋊Dic61—42no—
185C4⋊Dic61—81no—
186C4⋊Dic61#4112yes—
187#11758196—22no—
188#11758196—22no—
189#11758196—42no—
190#11758196—42no—
191#11758196—42no—
192#11758196—81no—
193C4×Dic61—24no—
194C4×Dic61—21no—
195C4×Dic61—21no—
196C4×Dic61—21no—
197C4×Dic61—21no—
198C4×Dic61—42no—
199C4×Dic61—42no—
200C2×#92418—21no—
201C2×#92418—21no—
202C2×#92418—21no—
203C2×#92418—21no—
204C2×#92418—21no—
205C2×#92418C2×Q812yes—
206C23×C61C22×D411yescharacteristic
207Q8×C61—22no—
208Q8×C61—22no—
209Q8×C61—22no—
210Q8×C61—84no—
211Q8×C61C22⋊Q814yes—
212C2×C4×C61—22no—
213C2×C4×C61—22no—
214C2×C4×C61—21no—
215C2×C4×C61—22no—
216C2×C4×C61—21no—
217C2×C4×C61—22no—
218C2×C4×C61—21no—
219C2×C4×C61—21no—
220C2×C4×C61—21no—
221C2×C4×C61—41no—
222C2×C4×C61—41no—
223C2×C4×C61—41no—
224C2×C4×C61—41no—
225C2×C4×C61—41no—
226C2×C4×C61—41no—
227C2×C4×C61—41no—
228C2×C4×C61—41no—
229C2×C4×C61#13512yes—
230C2×C4×C61C22×D411yescharacteristic
231C2×C4×C61C22×Q811yescharacteristic
232C2×C4×C61C2×#4111yescharacteristic
233C2×C4×C61C2×#4111yescharacteristic
234C2×C4×C61H(5,𝔽2)11yescharacteristic
235C2×C4×C612−1+411yescharacteristic
236C2×Dic61—21no—
237C2×Dic61—21no—
238C2×Dic61—21no—
239C2×Dic61—21no—
240C2×Dic61—22no—
241C2×Dic61—21no—
242C2×Dic61—22no—
243C2×Dic61—21no—
244C2×Dic61—21no—
245C2×Dic61—21no—
246C2×Dic61—21no—
247C2×Dic61—21no—
248C2×Dic61—21no—
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#13712yes—
260Dic122—22no—
261Dic122—22no—
262Dic122—41no—
263Dic122—41no—
264C8×C61—22no—
265C8×C61—41no—
266#92418—22no—
267#92418—41no—
268#26369—611nomaximal, Sylow
269C22×C61#52511yescharacteristic
270C22×C61D4211yescharacteristic
271C22×C61D4×Q811yescharacteristic
272C22×C61C2×#13511yescharacteristic
273C22×C61C2×#13711yescharacteristic
274C22×C61#57711yescharacteristic
275C22×C61#57811yescharacteristic
276C4×C61—24no—
277C4×C61—22no—
278C4×C61—41no—
279C4×C61—42no—
280C4×C61—81no—
281C4×C61—81no—
282C4×C61C2×C22⋊Q811yescharacteristic
283C4×C61#53911yescharacteristic
284C4×C61#54711yescharacteristic
285C4×C61#55411yescharacteristic
286Dic61—22no—
287Dic61—41no—
288Dic61—42no—
289Dic61—41no—
290Dic61—42no—
291Dic61—41no—
292Dic61—81no—
293Dic61—81no—
294C2×#475D6111yescharacteristic
295C2×C4×Q16—611no—
296#2451—611no—
297#2351—611no—
298#2097—611no—
299#2085—611no—
300#2066—611no—
301#2057—611no—
302#1941—611no—
303#1916—611no—
304#1878—611no—
305#1878—611no—
306#1865—611no—
307#1865—611no—
308#1863—611no—
309C2×C61#244911yescharacteristic
310C2×C61#310711yescharacteristic
311C2×C61#312111yescharacteristic
312C2×C61#327011yescharacteristic
313C2×C61#328211yescharacteristic
314C2×C61#338311yescharacteristic
315C2×C61#338711yescharacteristic
316C22×Q16—611no—
317C2×C4⋊Q8C2×D6111yescharacteristic
318C2×C4×Q8—611no—
319C2×C4×Q8—611no—
320C2×C4×Q8—611no—
321C2×C4×Q8C2×D6111yescharacteristic
322#475—24no—
323C4×Q16—1224no—
324C2×#107—611no—
325C2×#107—611no—
326C2×#106—611no—
327C2×#106C2×D6111yescharacteristic
328C2×#106C2×D6111yescharacteristic
329C2×C4⋊C8C2×D6111yescharacteristic
330C2×#103—611no—
331C2×#103—611no—
332C2×#103—611no—
333C2×#103—611no—
334C2×#103C2×D6111yescharacteristic
335C2×#103C2×D6111yescharacteristic
336C2×C4×C8—611no—
337#398—1221no—
338#395—1221no—
339#391—611no—
340#391—611no—
341#391—1221no—
342#389—611no—
343#389—611no—
344#389—611no—
345#389—611no—
346#389—1221no—
347#389—1221no—
348#387—1221no—
349#382—611no—
350#382—1221no—
351C4×C4⋊C4—611no—
352C4×C4⋊C4—1221no—
353#340—611no—
354#340—611no—
355#340—611no—
356#340—611no—
357#336—611no—
358#336—611no—
359#336—611no—
360#336—611no—
361C61#2636911yescharacteristic, complemented, Sylow
362C22×Q8—21no—
363C22×Q8—611no—
364C22×Q8C22×D6111yescharacteristic
365C2×Q16—614no—
366C2×Q16—1222no—
367C2×Q16—1222no—
368C22×C8—611no—
369C22×C8—611no—
370C22×C8C22×D6111yescharacteristic
371C22×C8C22×D6111yescharacteristic
372C4⋊Q8—22no—
373C4⋊Q8—41no—
374C4×Q8—22no—
375C4×Q8—41no—
376C4×Q8—1222no—
377C4×Q8—1222no—
378C4×Q8—1222no—
379C4×Q8—1222no—
380C4×Q8—2441no—
381C4×Q8—2441no—
382C2×C4⋊C4—21no—
383C2×C4⋊C4—21no—
384C2×C4⋊C4—611no—
385C2×C4⋊C4—611no—
386C2×C4⋊C4—611no—
387C2×C4⋊C4—611no—
388C2×C4⋊C4—611no—
389C2×C4⋊C4—1221no—
390C2×C4⋊C4—1221no—
391C2×C4⋊C4—1221no—
392C2×C4⋊C4—1221no—
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×C42—21no—
400C2×C42—611no—
401C2×C42—611no—
402C2×C42—1221no—
403C2×C42—1221no—
404C2×C42—1221no—
405C2×C42C22×D6111yescharacteristic
406#107—1222no—
407#107—2441no—
408#106—22no—
409#106—22no—
410#106—1222no—
411C4⋊C8—22no—
412#103—22no—
413#103—1222no—
414#103—1222no—
415#103—1222no—
416#103—2441no—
417#103C22⋊D6114yes—
418C4×C8—1222no—
419#95—611no—
420#95—611no—
421#95—1221no—
422#95—1221no—
423#95—1221no—
424#95—1221no—
425#95—1221no—
426#95—1221no—
427#95—1221no—
428#95—1221no—
429#95—1221no—
430#95—1221no—
431C2×Q8—21no—
432C2×Q8—21no—
433C2×Q8—21no—
434C2×Q8—42no—
435C2×Q8—42no—
436C2×Q8—81no—
437C2×Q8—612no—
438C2×Q8—1221no—
439C2×Q8—1221no—
440C2×Q8—1221no—
441C2×Q8—1221no—
442C2×Q8—1221no—
443C2×Q8Q8×D6112yes—
444C2×Q8#1175822612yes—
445C2×Q8C2×C22⋊D6112yes—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—21no—
450C22×C4—611no—
451C22×C4—611no—
452C22×C4—611no—
453C22×C4—611no—
454C22×C4—1221no—
455C22×C4—1221no—
456C22×C4—1221no—
457C22×C4—1221no—
458C22×C4D4×D6111yescharacteristic
459C22×C4D4×D6111yescharacteristic
460C22×C4C23×D6111yescharacteristic
461Q16—1228no—
462C2×C8—21no—
463C2×C8—21no—
464C2×C8—21no—
465C2×C8—21no—
466C2×C8—21no—
467C2×C8—612no—
468C2×C8—1221no—
469C2×C8—1221no—
470C2×C8—1221no—
471C2×C8—1221no—
472C2×C8—1221no—
473C2×C8C2×C22⋊D6112yes—
474C4⋊C4—21no—
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—21no—
478C4⋊C4—21no—
479C4⋊C4—42no—
480C4⋊C4—42no—
481C4⋊C4—612no—
482C4⋊C4—1222no—
483C4⋊C4—1224no—
484C4⋊C4—1221no—
485C4⋊C4—1222no—
486C4⋊C4—1221no—
487C4⋊C4—1221no—
488C4⋊C4—1221no—
489C4⋊C4—1221no—
490C4⋊C4—4881no—
491C4⋊C4—4881no—
492C4⋊C4—2442no—
493C4⋊C4—2442no—
494C4⋊C4—2442no—
495C4⋊C4—2442no—
496C4⋊C4C2×C22⋊D6112yes—
497C42—21no—
498C42—21no—
499C42—42no—
500C42—1224no—
501C42—1221no—
502C42—1221no—
503C42—1221no—
504C42—1221no—
505C42—2442no—
506C42—2442no—
507C23C2×D4×D6111yescharacteristic
508Q8—22no—
509Q8—22no—
510Q8—22no—
511Q8—84no—
512Q8—1222no—
513Q8—1222no—
514Q8—2441no—
515Q8—2441no—
516Q8#1451975314yes—
517C2×C4—22no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—22no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—612no—
535C2×C4—1221no—
536C2×C4—1221no—
537C2×C4—1221no—
538C2×C4—1222no—
539C2×C4—1221no—
540C2×C4—1221no—
541C2×C4—1221no—
542C2×C4—1221no—
543C2×C4—1221no—
544C2×C4—1221no—
545C2×C4—1221no—
546C2×C4—1221no—
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#1451979812yes—
559C2×C4C22×C22⋊D6111yescharacteristic
560C2×C4C2×Q8×D6111yescharacteristic
561C2×C4#41×D6111yescharacteristic
562C2×C4C2×#1175822611yescharacteristic
563C2×C4#1451982211yescharacteristic
564C2×C4#1451982411yescharacteristic
565C8—22no—
566C8—41no—
567C8—1222no—
568C8—2441no—
569C22#1684027511yescharacteristic
570C22D4×C22⋊D6111yescharacteristic
571C22#1684042511yescharacteristic
572C22C2×#1451979811yescharacteristic
573C22#137×D6111yescharacteristic
574C22#1684056811yescharacteristic
575C22#1684057811yescharacteristic
576C4—22no—
577C4—24no—
578C4—42no—
579C4—41no—
580C4—81no—
581C4—81no—
582C4—1222no—
583C4—4881no—
584C4—4881no—
585C4—2441no—
586C4—2442no—
587C4—2441no—
588C4—2442no—
589C4—2441no—
590C4C2×#1451975311yescharacteristic
591C4#1684036111yescharacteristic
592C4#1684041111yescharacteristic
593C4#1684046611yescharacteristic
594C2#2144962411yescharacteristic
595C2#2145589311yescharacteristic
596C2#2145605511yescharacteristic
597C2#2145738311yescharacteristic
598C2#2145752611yescharacteristic
599C2#2145870911yescharacteristic
600C2#2145874611yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C61×#475C2
socle, prime radicalC23×C61C22×D4
hypercenterC2×#475D61
hypocenterC61#26369
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#475

Complements

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

NH ≅ G/Ntype
64. C2×#11758202Q8semidirect
361. C61#26369semidirect

Maximal subgroups

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

subgroupindex
C2×#168398542
#214496542
#214487372
#214473682
#214473042
#214471992
#214471442
#214465452
#214463732
#214461602
#214461582
#214460892
#214460842
#214460712
C2×C61×#4752
#2636961