Finite Group Databasebeta

← #29657030

Subgroups — #29657030

Order 15616.

1106 conjugacy classes of subgroups.

The subgroup lattice has 550 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21457013C211yesmaximal, characteristic, complemented
2#21456533C211yesmaximal, characteristic, complemented
3#21455858C211yesmaximal, characteristic, complemented
4#21445068C212yesmaximal, complemented
5#21445012C212yesmaximal, complemented
6#21444984C212yesmaximal, complemented
7#21444946C212yesmaximal, complemented
8#21444524C211yesmaximal, characteristic, complemented
9#21444266C211yesmaximal, characteristic, complemented
10#21444236C211yesmaximal, characteristic, complemented
11C61×#3497C211yesmaximal, characteristic
12#16840319C2211yescharacteristic
13#16840168—21no—
14#16840165C2211yescharacteristic
15#16840134C2211yescharacteristic
16#16840126—21no—
17#16840114C2211yescharacteristic
18#16840108C2211yescharacteristic
19#16840097—21no—
20#16840083—22no—
21#16840057C2212yes—
22#16840052C2212yes—
23#16840042C2212yes—
24#16840006—22no—
25#16839972C2212yes—
26#16839918—21no—
27#16839862—22no—
28C4×#11758202—22no—
29#16839819C2211yescharacteristic
30#16839815C2211yescharacteristic
31C4×#11758194C2211yescharacteristic
32#16839508C2212yes—
33#16839506C2212yes—
34Q8⋊#92418C2212yes—
35Q8⋊#92418C2212yes—
36D4⋊#92418C2212yes—
37D4⋊#92418C2212yes—
38C2×2−1+4×C61—21no—
39C61×#550—21no—
40C61×#550C2212yes—
41D4×Q8×C61—21no—
42D4×Q8×C61C2212yes—
43C61×#548—22no—
44C61×#548—22no—
45C61×#537—22no—
46C61×#537—21no—
47C61×#532—22no—
48C61×#532C2211yescharacteristic
49C61×#532C2211yescharacteristic
50C2×C4×Q8×C61C2211yescharacteristic
51C2×#11758202—22no—
52C2×#11758200—22no—
53C2×#11758194—21no—
54#14519777C2311yescharacteristic
55#14519777C2311yescharacteristic
56#14519769—21no—
57#14519756—21no—
58#14519756—21no—
59C4×Dic122—21no—
60C4×Dic122—21no—
61#14519736—21no—
62#14519710—22no—
63#14519710—22no—
64#14519710—22no—
65#14519708—22no—
66C4⋊#92418C2312yes—
67C4⋊#92418C2312yes—
68#14519701—22no—
69#14519700—22no—
70#14519700—22no—
71#14519698—22no—
72#14519698—22no—
73#14519698—22no—
74C22⋊#92418—21no—
75C22⋊#92418—21no—
76#14519688—21no—
77#14519688—21no—
78C4×#92418—21no—
79C4×#92418C2311yescharacteristic
80C4×#92418C2311yescharacteristic
812−1+4×C61—48no—
82C2×#41×C61—24no—
83C2×#41×C61—22no—
84C2×#41×C61—22no—
85C2×#41×C61—21no—
86C2×#41×C61—21no—
87C22×Q8×C61—22no—
88C22×Q8×C61—22no—
89C22×Q8×C61D411yescharacteristic
90C4⋊Q8×C61—22no—
91C4⋊Q8×C61D411yescharacteristic
92C4⋊Q8×C61C2311yescharacteristic
93#127×C61—22no—
94#127×C61D411yescharacteristic
95#127×C61C2311yescharacteristic
96#124×C61—24no—
97#124×C61—21no—
98#124×C61—21no—
99C22⋊Q8×C61—22no—
100C22⋊Q8×C61—24no—
101C22⋊Q8×C61—22no—
102C22⋊Q8×C61—24no—
103#121×C61—24no—
104#121×C61—22no—
105#121×C61—24no—
106#121×C61—24no—
107#121×C61—21no—
108#121×C61—21no—
109#121×C61—22no—
110#121×C61D412yes—
111#121×C61D412yes—
112C4×Q8×C61—21no—
113C4×Q8×C61—21no—
114C4×Q8×C61D412yes—
115C4×Q8×C61C2312yes—
116C4×D4×C61—24no—
117C4×D4×C61—22no—
118C4×D4×C61—22no—
119C4×D4×C61—21no—
120C4×D4×C61—21no—
121C4×D4×C61D412yes—
122C4×D4×C61C2312yes—
123C2×C4⋊C4×C61—21no—
124C2×C4⋊C4×C61D411yescharacteristic
125C2×C42×C61—21no—
126C2×C42×C61C2311yescharacteristic
127#41×C61—416no—
128#41×C61—44no—
129#41×C61—44no—
130#41×C61—48no—
131#41×C61—44no—
132#41×C61—44no—
133C2×Q8×C61—24no—
134C2×Q8×C61—24no—
135C2×Q8×C61—21no—
136C2×Q8×C61—21no—
137C2×Q8×C61—21no—
138C2×Q8×C61—48no—
139C2×Q8×C61—44no—
140C2×Q8×C61—44no—
141C2×Q8×C61C2×D412yes—
142C2×D4×C61—24no—
143C2×D4×C61—22no—
144C2×D4×C61—22no—
145C2×D4×C61—24no—
146C2×D4×C61—21no—
147C2×D4×C61—21no—
148C2×D4×C61—21no—
149C2×D4×C61—21no—
150C2×D4×C61C2×D412yes—
151C2×D4×C61C2×D412yes—
152C22×C4×C61—21no—
153C22×C4×C61—24no—
154C22×C4×C61—24no—
155C22×C4×C61—21no—
156C22×C4×C61—22no—
157C22×C4×C61—22no—
158C22×C4×C61C2×D411yescharacteristic
159C22×C4×C61C2×D411yescharacteristic
160C22×C4×C61C2×D411yescharacteristic
161C2×Dic122—21no—
162C2×Dic122—21no—
163C4⋊C4×C61—24no—
164C4⋊C4×C61—22no—
165C4⋊C4×C61—21no—
166C4⋊C4×C61—21no—
167C4⋊C4×C61—22no—
168C4⋊C4×C61—21no—
169C4⋊C4×C61C2×D412yes—
170C4⋊C4×C61C2×D412yes—
171C4⋊C4×C61C2×D412yes—
172C22⋊C4×C61—24no—
173C22⋊C4×C61—24no—
174C22⋊C4×C61—22no—
175C22⋊C4×C61—22no—
176C22⋊C4×C61—22no—
177C22⋊C4×C61—22no—
178C42×C61—22no—
179C42×C61—22no—
180C42×C61C2×D411yescharacteristic
181C42×C61C2×D411yescharacteristic
182C42×C61C2×D411yescharacteristic
183C42×C61C2411yescharacteristic
184C22⋊Dic61—41no—
185#11758202—44no—
186#11758200—44no—
187C4⋊Dic61—21no—
188C4⋊Dic61—21no—
189C4⋊Dic61—21no—
190C4⋊Dic61—21no—
191#11758196—41no—
192#11758196—41no—
193C4×Dic61—41no—
194#11758194—42no—
195C2×#92418—21no—
196C2×#92418—22no—
197C2×#92418—21no—
198C2×#92418—22no—
199C23×C61—22no—
200C23×C61—22no—
201C23×C61#12011yescharacteristic
202Q8×C61—24no—
203Q8×C61—22no—
204Q8×C61—48no—
205Q8×C61—42no—
206Q8×C61—42no—
207Q8×C61AGL(1,ℤ/8)14yes—
208D4×C61—24no—
209D4×C61—42no—
210D4×C61—48no—
211D4×C61—42no—
212D4×C61—44no—
213D4×C61—44no—
214D4×C61—42no—
215D4×C61—42no—
216D4×C61—81no—
217D4×C61—84no—
218C2×C4×C61—24no—
219C2×C4×C61—22no—
220C2×C4×C61—22no—
221C2×C4×C61—22no—
222C2×C4×C61—22no—
223C2×C4×C61—22no—
224C2×C4×C61—22no—
225C2×C4×C61—41no—
226C2×C4×C61—48no—
227C2×C4×C61—48no—
228C2×C4×C61—44no—
229C2×C4×C61—42no—
230C2×C4×C61—42no—
231C2×C4×C61—81no—
232C2×C4×C61#12011yescharacteristic
233C2×C4×C61#12011yescharacteristic
234C2×C4×C61#12011yescharacteristic
235C2×C4×C61C22×D411yescharacteristic
236C2×C4×C61C22×D411yescharacteristic
237C2×C4×C61C22×D411yescharacteristic
238C2×Dic61—41no—
239C2×Dic61—41no—
240Dic122—42no—
241Dic122—42no—
242#92418—22no—
243#92418—41no—
244#92418—42no—
245#92418—42no—
246#18611—611nomaximal, Sylow
247C22×C61—21no—
248C22×C61—21no—
249C22×C61—21no—
250C22×C61—42no—
251C22×C61—42no—
252C22×C61—42no—
253C22×C61—42no—
254C22×C61—42no—
255C22×C61—42no—
256C22×C61C2×#12011yescharacteristic
257C4×C61—24no—
258C4×C61—21no—
259C4×C61—21no—
260C4×C61—21no—
261C4×C61—21no—
262C4×C61—42no—
263C4×C61—42no—
264C4×C61—42no—
265C4×C61C2×AGL(1,ℤ/8)12yes—
266C4×C61#57812yes—
267Dic61—81no—
268Dic61—81no—
269#3497D6111yescharacteristic
270#3235—611no—
271#3182—611no—
272#3104—611no—
273#1649—612no—
274#1641—612no—
275#1637—612no—
276#1631—612no—
277#1553—611no—
278#1491—611no—
279#1484—611no—
280C2×C61—41no—
281C2×C61—82no—
282C2×C61—82no—
283C2×C61#300311yescharacteristic
284C2×C61#301111yescharacteristic
285C2×C61#301811yescharacteristic
286C2×2−1+4—21no—
287#550—21no—
288#550C2×D6112yes—
289D4×Q8—21no—
290D4×Q8C2×D6112yes—
291#548—22no—
292#548—22no—
293#537—22no—
294#537—21no—
295#533—611no—
296#532—22no—
297#532C2×D6111yescharacteristic
298#532C2×D6111yescharacteristic
299C2×C4×Q8C2×D6111yescharacteristic
300#501—1221no—
301#500—611no—
302#492—611no—
303#490—1221no—
304#487—611no—
305#486—611no—
306#484—1221no—
307#482—1222no—
308#477—612no—
309#476—612no—
310#474—612no—
311#468—1222no—
312#462—612no—
313#451—1221no—
314#440—1222no—
315C4×Q16—1222no—
316#432—611no—
317#431—611no—
318C4×M16—611no—
319#332—612no—
320#331—612no—
321Q8⋊C8—612no—
322Q8⋊C8—612no—
323D4⋊C8—612no—
324D4⋊C8—612no—
325C61#1861111yescharacteristic, complemented, Sylow
3262−1+4—48no—
327C2×#41—24no—
328C2×#41—22no—
329C2×#41—22no—
330C2×#41—21no—
331C2×#41—21no—
332C22×Q8—22no—
333C22×Q8—22no—
334C22×Q8C22⋊D6111yescharacteristic
335C2×Q16—1222no—
336C2×SD16—1222no—
337C2×M16—1221no—
338C4⋊Q8—22no—
339C4⋊Q8—611no—
340C4⋊Q8—611no—
341C4⋊Q8C22⋊D6111yescharacteristic
342C4⋊Q8C22×D6111yescharacteristic
343#127—22no—
344#127C22⋊D6111yescharacteristic
345#127C22×D6111yescharacteristic
346#125—1221no—
347#124—24no—
348#124—21no—
349#124—21no—
350C22⋊Q8—22no—
351C22⋊Q8—24no—
352C22⋊Q8—22no—
353C22⋊Q8—24no—
354C22⋊Q8—1221no—
355C22⋊Q8—1221no—
356#121—24no—
357#121—22no—
358#121—24no—
359#121—24no—
360#121—21no—
361#121—21no—
362#121—22no—
363#121C22⋊D6112yes—
364#121C22⋊D6112yes—
365C4×Q8—21no—
366C4×Q8—21no—
367C4×Q8—1221no—
368C4×Q8—1221no—
369C4×Q8C22⋊D6112yes—
370C4×Q8C22×D6112yes—
371C4×D4—24no—
372C4×D4—22no—
373C4×D4—22no—
374C4×D4—21no—
375C4×D4—21no—
376C4×D4C22⋊D6112yes—
377C4×D4C22×D6112yes—
378#117—1221no—
379C2×C4⋊C4—21no—
380C2×C4⋊C4C22⋊D6111yescharacteristic
381C2×C42—21no—
382C2×C42C22×D6111yescharacteristic
383#107—1222no—
384#107—1222no—
385#107—1222no—
386#106—1222no—
387C4⋊C8—612no—
388C4⋊C8—612no—
389#103—1222no—
390#103—1222no—
391#103—1222no—
392#102—1222no—
393#102—1222no—
394#102—1222no—
395C22⋊C8—1221no—
396C22⋊C8—1221no—
397#97—1221no—
398#97—1221no—
399C4×C8—611no—
400C4×C8—611no—
401C4×C8—1221no—
402#41—416no—
403#41—44no—
404#41—44no—
405#41—48no—
406#41—44no—
407#41—44no—
408C2×Q8—24no—
409C2×Q8—24no—
410C2×Q8—21no—
411C2×Q8—21no—
412C2×Q8—21no—
413C2×Q8—48no—
414C2×Q8—44no—
415C2×Q8—44no—
416C2×Q8—1221no—
417C2×Q8—1221no—
418C2×Q8C2×C22⋊D6112yes—
419C2×D4—24no—
420C2×D4—22no—
421C2×D4—22no—
422C2×D4—24no—
423C2×D4—21no—
424C2×D4—21no—
425C2×D4—21no—
426C2×D4—21no—
427C2×D4C2×C22⋊D6112yes—
428C2×D4C2×C22⋊D6112yes—
429C22×C4—21no—
430C22×C4—24no—
431C22×C4—24no—
432C22×C4—21no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4C2×C22⋊D6111yescharacteristic
436C22×C4C2×C22⋊D6111yescharacteristic
437C22×C4C2×C22⋊D6111yescharacteristic
438Q16—2444no—
439SD16—2444no—
440M16—2442no—
441C2×C8—1221no—
442C2×C8—1221no—
443C2×C8—1222no—
444C2×C8—1222no—
445C4⋊C4—24no—
446C4⋊C4—22no—
447C4⋊C4—21no—
448C4⋊C4—22no—
449C4⋊C4—21no—
450C4⋊C4—21no—
451C4⋊C4—1221no—
452C4⋊C4—1221no—
453C4⋊C4—1221no—
454C4⋊C4—1221no—
455C4⋊C4—2441no—
456C4⋊C4—2441no—
457C4⋊C4C2×C22⋊D6112yes—
458C4⋊C4C2×C22⋊D6112yes—
459C4⋊C4C2×C22⋊D6112yes—
460C22⋊C4—24no—
461C22⋊C4—24no—
462C22⋊C4—22no—
463C22⋊C4—22no—
464C22⋊C4—22no—
465C22⋊C4—22no—
466C22⋊C4—2441no—
467C42—22no—
468C42—22no—
469C42—2441no—
470C42C2×C22⋊D6111yescharacteristic
471C42C2×C22⋊D6111yescharacteristic
472C42C2×C22⋊D6111yescharacteristic
473C42C23×D6111yescharacteristic
474C23—22no—
475C23—22no—
476C23#1451974611yescharacteristic
477Q8—24no—
478Q8—22no—
479Q8—48no—
480Q8—42no—
481Q8—42no—
482Q8—2442no—
483Q8—2442no—
484Q8#1451980314yes—
485D4—24no—
486D4—42no—
487D4—48no—
488D4—42no—
489D4—44no—
490D4—44no—
491D4—42no—
492D4—42no—
493D4—81no—
494D4—84no—
495C2×C4—24no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—41no—
503C2×C4—48no—
504C2×C4—48no—
505C2×C4—44no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—81no—
509C2×C4—2441no—
510C2×C4—2441no—
511C2×C4#1451974611yescharacteristic
512C2×C4#1451974611yescharacteristic
513C2×C4#1451974611yescharacteristic
514C2×C4C22×C22⋊D6111yescharacteristic
515C2×C4C22×C22⋊D6111yescharacteristic
516C2×C4C22×C22⋊D6111yescharacteristic
517C8—1222no—
518C8—2441no—
519C8—2442no—
520C8—2442no—
521C22—21no—
522C22—21no—
523C22—21no—
524C22—42no—
525C22—42no—
526C22—42no—
527C22—42no—
528C22—42no—
529C22—42no—
530C22C2×#1451974611yescharacteristic
531C4—21no—
532C4—21no—
533C4—21no—
534C4—24no—
535C4—21no—
536C4—42no—
537C4—42no—
538C4—42no—
539C4—4881no—
540C4—4881no—
541C4C2×#1451980312yes—
542C4#1684057712yes—
543C2—41no—
544C2—82no—
545C2—82no—
546C2#2145504611yescharacteristic
547C2#2145511411yescharacteristic
548C2#2145516811yescharacteristic
subgroupG/H
centerC22C2×#14519746
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
Fitting, prime radicalC61×#3497C2
socleC22×C61C2×#120
hypercenter#3497D61
hypocenterC61#18611
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3497

Complements

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

NH ≅ G/Ntype
1. #21457013C2semidirect
2. #21456533C2semidirect
3. #21455858C2semidirect
4. #21445068C2semidirect
5. #21445012C2semidirect
6. #21444984C2semidirect
7. #21444946C2semidirect
8. #21444524C2semidirect
9. #21444266C2semidirect
10. #21444236C2semidirect
325. C61#18611semidirect

Maximal subgroups

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

subgroupindex
#214570132
#214565332
#214558582
#214450682
#214450122
#214449842
#214449462
#214445242
#214442662
#214442362
C61×#34972
#1861161