Finite Group Databasebeta

← #29663627

Subgroups — #29663627

Order 15616.

636 conjugacy classes of subgroups.

The subgroup lattice has 444 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21457164C211yesmaximal, characteristic, complemented
2#21456760C211yesmaximal, characteristic
3#21456193C211yesmaximal, characteristic
4#21454762C211yesmaximal, characteristic, complemented
5#21445386C212yesmaximal, complemented
6#21445358C212yesmaximal, complemented
7#21445262C212yesmaximal, complemented
8#21445234C212yesmaximal, complemented
9#21444569C211yesmaximal, characteristic, complemented
10#21444477C211yesmaximal, characteristic, complemented
11C2×C61×#331C211yesmaximal, characteristic
12#16840319C2211yescharacteristic
13#16840319C2211yescharacteristic
14#16840134—21no—
15#16840126—22no—
16#16840108—21no—
17#16840108C2212yes—
18#16840108C2212yes—
19#16840058—21no—
20#16840058C2212yescomplemented
21#16840003—22no—
22#16839973—21no—
23#16839973C2212yescomplemented
24#16839959C2212yes—
25#16839959C2212yes—
26#16839913—22no—
27#16839891—22no—
28#16839889C2212yescomplemented
29#16839886C2212yescomplemented
30#16839815C2211yescharacteristic
31#16839788—21no—
32#16839767—21no—
33#16839767—21no—
34#16839717—21no—
35C4×#11758194C2211yescharacteristic
36#16839514C2212yescomplemented
37#16839512C2212yescomplemented
38#16839510C2211yescharacteristic, complemented
39#16839510C2211yescharacteristic, complemented
40#16839507C2211yescharacteristic, complemented
41#16839507C2211yescharacteristic, complemented
42C2×#127×C61C2211yescharacteristic
43C2×C4⋊C8×C61C2211yescharacteristic
44C2×C4⋊C8×C61C2211yescharacteristic
45C61×#331C2212yes—
46C61×#331C2212yes—
47C2×Dic244—22no—
48C2×#11758200—22no—
49C2×#11758200—22no—
50C2×#11758194D411yescharacteristic
51C2×#11758194D411yescharacteristic
52#14519777D411yescharacteristic
53#14519777D411yescharacteristic
54#14519777C2311yescharacteristic
55#14519777C2311yescharacteristic
56#14519769—21no—
57#14519769—21no—
58#14519756—21no—
59#14519756—21no—
60#14519756—21no—
61#14519756D412yes—
62C4×Dic122—21no—
63C4×Dic122—21no—
64C4×Dic122—21no—
65C4×Dic122C2312yes—
66#14519736—21no—
67#14519736—21no—
68#14519711—22no—
69C4⋊#92418C2311yescharacteristic
70C4⋊#92418C2311yescharacteristic
71#14519706—22no—
72#14519706—22no—
73#14519702—22no—
74#14519702—22no—
75#14519698—22no—
76#14519698—22no—
77#14519698—22no—
78#14519698—22no—
79#14519698—22no—
80#14519698—22no—
81C22⋊#92418—21no—
82#14519688D412yes—
83#14519687—22no—
84C4×#92418—21no—
85C4×#92418C2312yes—
86C8×Dic61—22no—
87C22×D4×C61—21no—
88C22×D4×C61—21no—
89C22×D4×C61—21no—
90C22×C8×C61—21no—
91C22×C8×C61—21no—
92#127×C61—21no—
93#127×C61—21no—
94#127×C61—21no—
95#127×C61C2312yes—
96C2×C42×C61C2311yescharacteristic
97C4⋊C8×C61—21no—
98C4⋊C8×C61—21no—
99C4⋊C8×C61C2312yes—
100C4⋊C8×C61C2311yescharacteristic
101C4⋊C8×C61C2311yescharacteristic
102C24×C61—41no—
103C2×D4×C61—22no—
104C2×D4×C61—22no—
105C2×D4×C61—22no—
106C2×D4×C61—41no—
107C2×D4×C61—41no—
108C2×D4×C61—41no—
109C2×D4×C61—42no—
110C2×D4×C61—42no—
111C2×D4×C61—81no—
112C2×D4×C61—81no—
113C2×D4×C61—82no—
114C22×C4×C61C2×D411yescharacteristic
115C22×C4×C61C2×D411yescharacteristic
116C22×C4×C61#4111yescharacteristic
117C2×Dic122—21no—
118C2×Dic122—21no—
119C2×Dic122—21no—
120C2×Dic122C2×D412yes—
121C2×C8×C61—21no—
122C2×C8×C61—21no—
123C2×C8×C61—22no—
124C2×C8×C61—42no—
125C2×C8×C61—81no—
126C42×C61C2×D411yescharacteristic
127C42×C61C2×D411yescharacteristic
128C42×C61#4111yescharacteristic
129C42×C61C2411yescharacteristic
130C22⋊Dic61—22no—
131C22⋊Dic61—41no—
132#11758200—82no—
133#11758200—82no—
134C4⋊Dic61—21no—
135C4⋊Dic61—21no—
136C4⋊Dic61—21no—
137C4⋊Dic61—21no—
138C4⋊Dic61—21no—
139C4⋊Dic61—21no—
140C4⋊Dic61—21no—
141C4⋊Dic61C2×D412yes—
142#11758196—22no—
143#11758196—22no—
144#11758196—41no—
145#11758196—41no—
146C4×Dic61—22no—
147C4×Dic61—41no—
148#11758194—22no—
149#11758194—22no—
150C2×#92418—21no—
151C2×#92418—21no—
152C2×#92418C2×D412yes—
153C2×#92418C2×D412yes—
154Dic244—82no—
155C23×C61—42no—
156C23×C61—82no—
157C23×C61—82no—
158C23×C61—82no—
159C23×C61#12111yescharacteristic
160D4×C61—42no—
161D4×C61—42no—
162D4×C61—42no—
163D4×C61—42no—
164D4×C61—82no—
165D4×C61—82no—
166D4×C61—82no—
167D4×C61—82no—
168C2×C4×C61—21no—
169C2×C4×C61—21no—
170C2×C4×C61—41no—
171C2×C4×C61#12111yescharacteristic
172C2×C4×C61#12111yescharacteristic
173C2×C4×C61#12111yescharacteristic
174C2×C4×C61#13512yes—
175C2×C4×C61#13712yes—
176C2×C4×C61C22×D411yescharacteristic
177C2×C4×C61C22×D411yescharacteristic
178C2×C4×C61C2×#4111yescharacteristic
179C2×Dic61—22no—
180C2×Dic61—22no—
181C2×Dic61—41no—
182C2×Dic61—41no—
183Dic122—22no—
184Dic122—22no—
185Dic122—41no—
186Dic122—41no—
187Dic122—81no—
188Dic122—81no—
189C8×C61—41no—
190C8×C61—41no—
191C8×C61—42no—
192#92418—22no—
193#92418—22no—
194#92418—41no—
195#92418—41no—
196#19242—611nomaximal, Sylow
197C22×C61—21no—
198C22×C61—21no—
199C22×C61—82no—
200C22×C61—82no—
201C22×C61—82no—
202C22×C61—164no—
203C22×C61#46412yes—
204C22×C61C2×#12111yescharacteristic
205C4×C61—21no—
206C4×C61—21no—
207C4×C61—21no—
208C4×C61#46412yes—
209C4×C61C2×#13511yescharacteristic
210C4×C61C2×#13711yescharacteristic
211C4×C61#57611yescharacteristic
212C4×C61#57711yescharacteristic
213Dic61—42no—
214Dic61—42no—
215Dic61—81no—
216Dic61—81no—
217#3249—611no—
218#3208—611no—
219#3138—611no—
220#2962—611no—
221#1702—612no—
222#1698—612no—
223#1682—612no—
224#1678—612no—
225#1565—611no—
226#1542—611no—
227C2×#331D6111yescharacteristic
228C2×C61—22no—
229C2×C61—162no—
230C2×C61C2×#46411yescharacteristic
231C2×C61#306211yescharacteristic
232C2×C61#306811yescharacteristic
233#533—611no—
234#533—611no—
235C2×#127C2×D6111yescharacteristic
236#492—1221no—
237#490—1222no—
238#486—612no—
239#486—612no—
240#486—1221no—
241#477—612no—
242#477—1221no—
243#467—1222no—
244#462—612no—
245#462—1221no—
246#460—612no—
247#460—612no—
248#450—1222no—
249#446—612no—
250#446—1222no—
251C8×Q8—612no—
252#431—611no—
253#424—1221no—
254C2×C4⋊C8C2×D6111yescharacteristic
255C2×C4⋊C8C2×D6111yescharacteristic
256#418—1221no—
257#418—1221no—
258#405—1221no—
259C4×M16—611no—
260#335—612no—
261#334—612no—
262#333—611no—
263#333—611no—
264#332—611no—
265#332—611no—
266#331C2×D6112yes—
267#331C2×D6112yes—
268C61#1924211yescharacteristic, complemented, Sylow
269C22×D4—21no—
270C22×D4—21no—
271C22×D4—21no—
272C2×Q16—1222no—
273C2×SD16—1222no—
274C2×SD16—1222no—
275C2×M16—611no—
276C2×M16—611no—
277C22×C8—21no—
278C22×C8—21no—
279C4⋊Q8—611no—
280C4⋊Q8—611no—
281C4⋊Q8—611no—
282C4⋊Q8—611no—
283#127—21no—
284#127—21no—
285#127—21no—
286#127C22×D6112yes—
287#125—1221no—
288#125—1221no—
289C22⋊Q8—612no—
290C22⋊Q8—1221no—
291C22⋊Q8—1221no—
292C22⋊Q8—1221no—
293C4×Q8—612no—
294C4×Q8—1221no—
295C4×Q8—1221no—
296C4×Q8—1221no—
297#117—1221no—
298#117—1221no—
299C2×C42C22×D6111yescharacteristic
300#107—1222no—
301C4⋊C8—21no—
302C4⋊C8—21no—
303C4⋊C8—611no—
304C4⋊C8—611no—
305C4⋊C8—1222no—
306C4⋊C8—1222no—
307C4⋊C8C22×D6111yescharacteristic
308C4⋊C8C22×D6111yescharacteristic
309C4⋊C8C22×D6112yes—
310#103—1222no—
311#103—1222no—
312#102—1222no—
313#102—1222no—
314#102—1222no—
315#102—1222no—
316#102—1222no—
317#102—1222no—
318C22⋊C8—1221no—
319#97—612no—
320#97—1222no—
321C4×C8—612no—
322C4×C8—1222no—
323C4×C8—1221no—
324C24—41no—
325C2×Q8—612no—
326C2×Q8—1221no—
327C2×Q8—1221no—
328C2×Q8—1221no—
329C2×D4—22no—
330C2×D4—22no—
331C2×D4—22no—
332C2×D4—41no—
333C2×D4—41no—
334C2×D4—41no—
335C2×D4—42no—
336C2×D4—42no—
337C2×D4—81no—
338C2×D4—81no—
339C2×D4—82no—
340C22×C4#1175822211yescharacteristic
341C22×C4D4×D6111yescharacteristic
342C22×C4D4×D6111yescharacteristic
343Q16—4882no—
344SD16—4882no—
345SD16—4882no—
346M16—1222no—
347M16—1222no—
348C2×C8—21no—
349C2×C8—21no—
350C2×C8—22no—
351C2×C8—42no—
352C2×C8—81no—
353C2×C8—612no—
354C2×C8—612no—
355C2×C8—1221no—
356C2×C8—1221no—
357C4⋊C4—612no—
358C4⋊C4—1222no—
359C4⋊C4—1222no—
360C4⋊C4—1221no—
361C4⋊C4—1221no—
362C4⋊C4—1221no—
363C4⋊C4—1221no—
364C4⋊C4—1221no—
365C4⋊C4—1221no—
366C4⋊C4—1221no—
367C4⋊C4—2441no—
368C4⋊C4—2441no—
369C22⋊C4—1222no—
370C22⋊C4—2441no—
371C42—1222no—
372C42—2441no—
373C42#1175822211yescharacteristic
374C42D4×D6111yescharacteristic
375C42D4×D6111yescharacteristic
376C42C23×D6111yescharacteristic
377C23—42no—
378C23—82no—
379C23—82no—
380C23—82no—
381C23#1451975011yescharacteristic
382Q8—1222no—
383Q8—1222no—
384Q8—4881no—
385Q8—4881no—
386Q8—2441no—
387Q8—2441no—
388D4—42no—
389D4—42no—
390D4—42no—
391D4—42no—
392D4—82no—
393D4—82no—
394D4—82no—
395D4—82no—
396C2×C4—21no—
397C2×C4—21no—
398C2×C4—41no—
399C2×C4—1222no—
400C2×C4—1222no—
401C2×C4—2441no—
402C2×C4—2441no—
403C2×C4#1451975011yescharacteristic
404C2×C4#1451975011yescharacteristic
405C2×C4#1451975011yescharacteristic
406C2×C4#1451979612yes—
407C2×C4#1451980612yes—
408C2×C4C2×D4×D6111yescharacteristic
409C2×C4C2×D4×D6111yescharacteristic
410C2×C4C2×#1175822211yescharacteristic
411C8—41no—
412C8—41no—
413C8—42no—
414C8—1222no—
415C8—1222no—
416C8—2441no—
417C8—2441no—
418C22—21no—
419C22—21no—
420C22—82no—
421C22—82no—
422C22—82no—
423C22—164no—
424C22#1683998312yes—
425C22C2×#1451975011yescharacteristic
426C4—21no—
427C4—21no—
428C4—21no—
429C4—4881no—
430C4—4881no—
431C4—2442no—
432C4—2442no—
433C4#1683998312yes—
434C4C2×#1451979611yescharacteristic
435C4C2×#1451980611yescharacteristic
436C4#1684056311yescharacteristic
437C4#1684056811yescharacteristic
438C2—22no—
439C2—162no—
440C2C2×#1683998311yescharacteristic
441C2#2145549711yescharacteristic
442C2#2145554611yescharacteristic
subgroupG/H
centerC22C2×#14519750
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC2×C61×#331C2
socleC22×C61C2×#121
prime radicalC2×#127×C61C22
hypercenterC2×#331D61
hypocenterC61#19242
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#331

Complements

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

NH ≅ G/Ntype
1. #21457164C2semidirect
4. #21454762C2semidirect
5. #21445386C2semidirect
6. #21445358C2semidirect
7. #21445262C2semidirect
8. #21445234C2semidirect
9. #21444569C2semidirect
10. #21444477C2semidirect
20. #16840058C22semidirect
23. #16839973C22semidirect
28. #16839889C22semidirect
29. #16839886C22semidirect
36. #16839514C22semidirect
37. #16839512C22semidirect
38. #16839510C22semidirect
39. #16839510C22semidirect
40. #16839507C22semidirect
41. #16839507C22semidirect
268. C61#19242semidirect

Maximal subgroups

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

subgroupindex
#214571642
#214567602
#214561932
#214547622
#214453862
#214453582
#214452622
#214452342
#214445692
#214444772
C2×C61×#3312
#1924261