Finite Group Databasebeta

← #29663307

Subgroups — #29663307

Order 15616.

658 conjugacy classes of subgroups.

The subgroup lattice has 400 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456764C211yesmaximal, characteristic, complemented
2#21456757C211yesmaximal, characteristic
3C2×#16840109C211yesmaximal, characteristic
4#21445387C214yesmaximal, complemented
5#21445263C214yesmaximal, complemented
6#21444567C211yesmaximal, characteristic, complemented
7#21444505C211yesmaximal, characteristic, complemented
8#21444488C211yesmaximal, characteristic
9C61×#2960C211yesmaximal, characteristic, complemented
10C2×#14519777C2211yescharacteristic
11C2×#14519774C2211yescharacteristic
12#16840141—21no—
13#16840136—21no—
14#16840114—21no—
15#16840110—21no—
16#16840109—21no—
17#16840109—21no—
18#16840109C2212yes—
19#16840109C2212yes—
20#16839972C2214yescomplemented
21#16839962C2214yes—
22#16839917—22no—
23#16839912—22no—
24#16839815C2211yescharacteristic
25#16839815C2211yescharacteristic
26C2×#14519699—21no—
27C2×#14519699—21no—
28#16839515C2212yescomplemented
29#16839513C2212yescomplemented
30#16839507C2214yescomplemented
31#16839505C2214yes—
32C61×#533C2211yescharacteristic
33C61×#446—21no—
34C61×#446C2212yescomplemented
35C8×Q8×C61—21no—
36C8×Q8×C61C2212yescomplemented
37C61×#424—21no—
38C61×#423C2211yescharacteristic
39C61×#405—21no—
40C2×C4×C8×C61C2211yescharacteristic
41C22×Dic122—21no—
42C22×D244—21no—
43C22×D244—21no—
44C22×D244—21no—
45C2×#11758202—24no—
46C2×#11758201—24no—
47#14519777—21no—
48#14519777—21no—
49#14519777—21no—
50#14519777C2312yes—
51#14519774—21no—
52#14519774—21no—
53#14519774—21no—
54#14519774C2312yes—
55C2×C4⋊Dic61—21no—
56C2×C4⋊Dic61—21no—
57C4⋊#92418C2312yes—
58C4⋊#92418C2312yes—
59#14519700—24no—
60#14519699—22no—
61#14519699—22no—
62#14519699—22no—
63#14519699—22no—
64#14519697—24no—
65C22⋊#92418—21no—
66C22⋊#92418—21no—
67C4×#92418—21no—
68C4×#92418—21no—
69C2×M16×C61—21no—
70C22×C8×C61D411yescharacteristic
71C22×C8×C61D411yescharacteristic
72C4⋊Q8×C61D411yescharacteristic
73C4⋊Q8×C61D411yescharacteristic
74#125×C61—21no—
75C22⋊Q8×C61—21no—
76C22⋊Q8×C61D412yes—
77C4×Q8×C61—21no—
78C4×Q8×C61C2312yes—
79#117×C61—21no—
80C2×C42×C61C2311yescharacteristic
81C4⋊C8×C61—22no—
82C4⋊C8×C61—22no—
83C4⋊C8×C61—21no—
84C4⋊C8×C61C2312yes—
85#97×C61—22no—
86C4×C8×C61—22no—
87C4×C8×C61D411yescharacteristic
88C4×C8×C61D411yescharacteristic
89C4×C8×C61C2311yescharacteristic
90C4×C8×C61C2311yescharacteristic
91C23×D61—41no—
92C2×Q8×C61—21no—
93C2×Q8×C61C2×D412yes—
94C22×C4×C61C2×D411yescharacteristic
95C22×C4×C61C2×D411yescharacteristic
96C22×C4×C61#4111yescharacteristic
97C22×Dic61—41no—
98C2×D244—22no—
99C2×D244—22no—
100C2×D244—22no—
101C2×D244—42no—
102C2×D244—42no—
103C2×D244—41no—
104C2×D244—41no—
105C2×D244—41no—
106C2×D244—82no—
107C2×D244—82no—
108C2×Dic122—22no—
109C2×Dic122—41no—
110C2×Dic122—82no—
111M16×C61—42no—
112C2×C8×C61—22no—
113C2×C8×C61—22no—
114C2×C8×C61—22no—
115C2×C8×C61C2×D411yescharacteristic
116C2×C8×C61C2×D411yescharacteristic
117C2×C8×C61C2×D411yescharacteristic
118C2×C8×C61C2×D411yescharacteristic
119C4⋊C4×C61—21no—
120C4⋊C4×C61—21no—
121C4⋊C4×C61—21no—
122C4⋊C4×C61—22no—
123C4⋊C4×C61—22no—
124C4⋊C4×C61C2×D412yes—
125C22⋊C4×C61—22no—
126C42×C61—22no—
127C42×C61C2×D411yescharacteristic
128C42×C61C2×D411yescharacteristic
129C42×C61#4111yescharacteristic
130C42×C61C2411yescharacteristic
131#11758202—84no—
132#11758201—84no—
133C4⋊Dic61—22no—
134C4⋊Dic61—22no—
135C4⋊Dic61—41no—
136C4⋊Dic61—41no—
137C2×#92418—22no—
138C2×#92418—22no—
139C23×C61#12111yescharacteristic
140C22×D61—42no—
141C22×D61—82no—
142C22×D61—82no—
143C22×D61—82no—
144Q8×C61—24no—
145Q8×C61—42no—
146C2×C4×C61—22no—
147C2×C4×C61—22no—
148C2×C4×C61—22no—
149C2×C4×C61—22no—
150C2×C4×C61#12111yescharacteristic
151C2×C4×C61#12111yescharacteristic
152C2×C4×C61#12111yescharacteristic
153C2×C4×C61#13512yes—
154C2×C4×C61#13712yes—
155C2×C4×C61C22×D411yescharacteristic
156C2×C4×C61C22×D411yescharacteristic
157C2×C4×C61C2×#4111yescharacteristic
158C2×Dic61—42no—
159C2×Dic61—82no—
160D244—42no—
161D244—42no—
162D244—44no—
163D244—82no—
164D244—82no—
165D244—82no—
166D244—82no—
167Dic122—44no—
168Dic122—82no—
169C8×C61—21no—
170C8×C61—21no—
171C8×C61—21no—
172C8×C61—21no—
173C8×C61—42no—
174#92418—42no—
175#92418—42no—
176#19214—611nomaximal, Sylow
177C22×C61—21no—
178C22×C61—21no—
179C22×C61#46412yes—
180C22×C61C2×#12111yescharacteristic
181C2×D61—82no—
182C2×D61—82no—
183C2×D61—82no—
184C2×D61—164no—
185C4×C61—21no—
186C4×C61—21no—
187C4×C61—42no—
188C4×C61—42no—
189C4×C61#46412yes—
190C4×C61#47112yes—
191C4×C61C2×#13511yescharacteristic
192C4×C61C2×#13711yescharacteristic
193C4×C61#57612yes—
194Dic61—82no—
195#3209—611no—
196#3208—611no—
197C2×#486—611no—
198#2960D6111yescharacteristic, complemented
199#1702—614no—
200#1682—614no—
201#1564—611no—
202#1549—611no—
203#1545—611no—
204C2×C61—22no—
205C2×C61C2×#46411yescharacteristic
206C2×C61#305811yescharacteristic
207C2×C61C2×#47111yescharacteristic
208D61—162no—
209#533C2×D6111yescharacteristic
210C2×C4⋊Q8—611no—
211C2×#127—611no—
212#494—1221no—
213#492—1221no—
214#487—1221no—
215#486—612no—
216#486—612no—
217#486—1221no—
218#486—1221no—
219#486—1221no—
220#462—614no—
221#460—614no—
222#451—1222no—
223#450—1222no—
224#446—21no—
225#446C2×D6112yescomplemented
226C8×Q8—21no—
227C8×Q8C2×D6112yescomplemented
228#431—611no—
229#431—611no—
230#424—21no—
231#423C2×D6111yescharacteristic
232C2×#102—1221no—
233C2×#102—1221no—
234#405—21no—
235C2×C4×C8C2×D6111yescharacteristic
236#335—612no—
237#334—612no—
238#332—614no—
239#331—614no—
240C61#1921411yescharacteristic, complemented, Sylow
241C22×Q8—1221no—
242C22×D4—1221no—
243C22×D4—1221no—
244C22×D4—1221no—
245C2×Q16—1224no—
246C2×SD16—1224no—
247C2×M16—21no—
248C22×C8C22⋊D6111yescharacteristic
249C22×C8C22⋊D6111yescharacteristic
250C4⋊Q8—612no—
251C4⋊Q8—1221no—
252C4⋊Q8—1221no—
253C4⋊Q8—1221no—
254C4⋊Q8D24411yescharacteristic
255C4⋊Q8D24411yescharacteristic
256#127—612no—
257#127—1221no—
258#127—1221no—
259#127—1221no—
260#125—21no—
261C22⋊Q8—21no—
262C22⋊Q8D24412yes—
263C4×Q8—21no—
264C4×Q8C22×D6112yes—
265#117—21no—
266C2×C4⋊C4—1221no—
267C2×C4⋊C4—1221no—
268C2×C42C22×D6111yescharacteristic
269C4⋊C8—22no—
270C4⋊C8—22no—
271C4⋊C8—21no—
272C4⋊C8—612no—
273C4⋊C8—612no—
274C4⋊C8C22×D6112yes—
275#103—1224no—
276#102—1222no—
277#102—1222no—
278#102—1222no—
279#102—1222no—
280#102—1224no—
281C22⋊C8—1221no—
282C22⋊C8—1221no—
283#97—22no—
284C4×C8—22no—
285C4×C8—1221no—
286C4×C8—1221no—
287C4×C8C22⋊D6111yescharacteristic
288C4×C8C22⋊D6111yescharacteristic
289C4×C8C22×D6111yescharacteristic
290C4×C8C22×D6111yescharacteristic
291C24—2441no—
292C2×Q8—21no—
293C2×Q8—1222no—
294C2×Q8—4882no—
295C2×Q8—2441no—
296C2×Q8C2×D24412yes—
297C2×D4—1222no—
298C2×D4—1222no—
299C2×D4—1222no—
300C2×D4—4882no—
301C2×D4—4882no—
302C2×D4—2442no—
303C2×D4—2441no—
304C2×D4—2441no—
305C2×D4—2441no—
306C2×D4—2442no—
307C22×C4—2441no—
308C22×C4C2×D24411yescharacteristic
309C22×C4#1175822211yescharacteristic
310C22×C4C2×C22⋊D6111yescharacteristic
311Q16—4884no—
312SD16—4884no—
313M16—42no—
314C2×C8—22no—
315C2×C8—22no—
316C2×C8—22no—
317C2×C8—1222no—
318C2×C8—1222no—
319C2×C8C2×C22⋊D6111yescharacteristic
320C2×C8C2×C22⋊D6111yescharacteristic
321C2×C8C2×C22⋊D6111yescharacteristic
322C2×C8C2×C22⋊D6111yescharacteristic
323C4⋊C4—22no—
324C4⋊C4—21no—
325C4⋊C4—21no—
326C4⋊C4—21no—
327C4⋊C4—22no—
328C4⋊C4—1222no—
329C4⋊C4—1222no—
330C4⋊C4—2441no—
331C4⋊C4—2441no—
332C4⋊C4C2×D24412yes—
333C22⋊C4—22no—
334C42—22no—
335C42C2×D24411yescharacteristic
336C42#1175822211yescharacteristic
337C42C2×C22⋊D6111yescharacteristic
338C42C23×D6111yescharacteristic
339C23—4882no—
340C23—4882no—
341C23—4882no—
342C23—2442no—
343C23#1451975111yescharacteristic
344Q8—24no—
345Q8—42no—
346Q8—4882no—
347Q8—2444no—
348D4—4882no—
349D4—4882no—
350D4—4882no—
351D4—4882no—
352D4—2442no—
353D4—2442no—
354D4—2444no—
355C2×C4—22no—
356C2×C4—22no—
357C2×C4—22no—
358C2×C4—22no—
359C2×C4—4882no—
360C2×C4—2442no—
361C2×C4#1451975111yescharacteristic
362C2×C4#1451975111yescharacteristic
363C2×C4#1451975111yescharacteristic
364C2×C4#1451979712yes—
365C2×C4#1451980812yes—
366C2×C4C22×C22⋊D6111yescharacteristic
367C2×C4C22×D24411yescharacteristic
368C2×C4C2×#1175822211yescharacteristic
369C8—21no—
370C8—21no—
371C8—21no—
372C8—21no—
373C8—42no—
374C8—2442no—
375C8—2442no—
376C22—21no—
377C22—21no—
378C22—9764no—
379C22—4882no—
380C22—4882no—
381C22—4882no—
382C22#1683998412yes—
383C22C2×#1451975111yescharacteristic
384C4—21no—
385C4—21no—
386C4—42no—
387C4—42no—
388C4—4882no—
389C4#1683998412yes—
390C4#1684002512yes—
391C4C2×#1451979711yescharacteristic
392C4C2×#1451980811yescharacteristic
393C4#1684056712yes—
394C2—22no—
395C2—9762no—
396C2C2×#1683998411yescharacteristic
397C2#2145546711yescharacteristic
398C2C2×#1684002511yescharacteristic
subgroupG/H
centerC22C2×#14519751
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#2960C2
socleC22×C61C2×#121
prime radicalC2×#14519774C22
hypercenter#2960D61
hypocenterC61#19214
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2960

Complements

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

NH ≅ G/Ntype
1. #21456764C2semidirect
4. #21445387C2semidirect
5. #21445263C2semidirect
6. #21444567C2semidirect
7. #21444505C2semidirect
9. C61×#2960C2semidirect
20. #16839972C22semidirect
28. #16839515C22semidirect
29. #16839513C22semidirect
30. #16839507C22semidirect
34. C61×#446C22semidirect
36. C8×Q8×C61C22semidirect
198. #2960D61semidirect
225. #446C2×D61semidirect
227. C8×Q8C2×D61semidirect
240. C61#19214semidirect

Maximal subgroups

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

subgroupindex
#214567642
#214567572
C2×#168401092
#214453872
#214452632
#214445672
#214445052
#214444882
C61×#29602
#1921461