Finite Group Databasebeta

← #37582731

Subgroups — #37582731

Order 24448.

372 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#27133274C211yesmaximal, characteristic, complemented
2#27133142C211yesmaximal, characteristic
3#27132997C211yesmaximal, characteristic, complemented
4#27132992C211yesmaximal, characteristic, complemented
5#27132971C211yesmaximal, characteristic, complemented
6#27132963C211yesmaximal, characteristic, complemented
7#27132870C211yesmaximal, characteristic, complemented
8#27132822C211yesmaximal, characteristic
9#27132822C211yesmaximal, characteristic
10#27132767C211yesmaximal, characteristic, complemented
11#27132764C211yesmaximal, characteristic, complemented
12#27132755C211yesmaximal, characteristic, complemented
13SD16×Dic191C211yesmaximal, characteristic, complemented
14#27132723C211yesmaximal, characteristic, complemented
15#461×C191C211yesmaximal, characteristic, complemented
16C2×#15690286C2211yescharacteristic
17C2×#15690277C2211yescharacteristic
18C2×#15690277C2211yescharacteristic
19#20259317—21no—
20#20259317—21no—
21#20259314C2211yescharacteristic
22#20259312—21no—
23#20259312—21no—
24#20259298C2211yescharacteristic
25#20259298C2211yescharacteristic
26Q8×Dic191C2211yescharacteristic
27Q8×Dic191C2211yescharacteristic
28C4×Dic382C2211yescharacteristic
29#20259287C2211yescharacteristic
30D4×Dic191C2211yescharacteristic
31C4×C22⋊D191—21no—
32(C22⋊D191)⋊C4—21no—
33#20259279C2211yescharacteristic
34#20259278—21no—
35C2×Dic191⋊C4—21no—
36C4⋊C4×D191C2211yescharacteristic
37#20259256C2211yescharacteristic
38#20259255C2211yescharacteristic
39#20259254C2211yescharacteristic
40#20259253C2211yescharacteristic
41C4⋊(C191⋊C8)C2211yescharacteristic
42#20259247C2211yescharacteristic
43#20259247C2211yescharacteristic
44#20259246C2211yescharacteristic
45#20259246C2211yescharacteristic
46#20259243C2211yescharacteristic
47#20259243C2211yescharacteristic
48(C2×D191)⋊C8C2211yescharacteristic
49(C2×D191)⋊C8C2211yescharacteristic
50#20259232C2211yescharacteristic
51C8×Dic191C2211yescharacteristic
52C2×SD16×C191C2211yescharacteristic
53C2×SD16×C191C2211yescharacteristic
54C4⋊Q8×C191C2211yescharacteristic
55C4×D4×C191C2211yescharacteristic
56C4⋊C8×C191C2211yescharacteristic
57#103×C191C2211yescharacteristic
58#103×C191C2211yescharacteristic
59#15690286—22no—
60#15690286—22no—
61C2×Dic382C2311yescharacteristic
62C2×C22⋊D191—21no—
63C22×Dic191—21no—
64C22×Dic191—21no—
65C2×C4×D191—21no—
66C2×C4×D191C2311yescharacteristic
67#15690277—22no—
68#15690277—22no—
69C2×C191⋊C8C2311yescharacteristic
70C2×C191⋊C8C2311yescharacteristic
71C4⋊Dic191—21no—
72C4⋊Dic191—21no—
73C4⋊Dic191C2311yescharacteristic
74C4⋊Dic191C2311yescharacteristic
75C4⋊Dic191C2311yescharacteristic
76Dic191⋊C4—21no—
77Dic191⋊C4—21no—
78Dic191⋊C4—21no—
79Dic191⋊C4—21no—
80Dic191⋊C4—21no—
81Dic191⋊C4—21no—
82Dic191⋊C4—21no—
83Dic191⋊C4D411yescharacteristic
84Dic191⋊C4D411yescharacteristic
85C22⋊Dic191—21no—
86(C2×D191)⋊C4—21no—
87(C2×D191)⋊C4—21no—
88(C2×D191)⋊C4D411yescharacteristic
89(C2×D191)⋊C4D411yescharacteristic
90C4×Dic191—21no—
91C4×Dic191—21no—
92C4×Dic191—21no—
93C4×Dic191—21no—
94C4×Dic191C2311yescharacteristic
95C2×Q8×C191C2311yescharacteristic
96C2×Q8×C191C2311yescharacteristic
97C2×D4×C191C2311yescharacteristic
98C22×C4×C191—21no—
99SD16×C191—22no—
100SD16×C191—22no—
101C2×C8×C191C2311yescharacteristic
102C2×C8×C191C2311yescharacteristic
103C4⋊C4×C191—21no—
104C4⋊C4×C191—21no—
105C4⋊C4×C191C2311yescharacteristic
106C22⋊C4×C191—21no—
107C42×C191C2311yescharacteristic
108C22×D191C2×D411yescharacteristic
109Dic382—21no—
110Dic382#4112yes—
111C22⋊D191—24no—
112C2×Dic191—24no—
113C2×Dic191—21no—
114C2×Dic191—21no—
115C2×Dic191—21no—
116C2×Dic191—21no—
117C2×Dic191—81no—
118C2×Dic191C2×D411yescharacteristic
119C2×Dic191C2×D411yescharacteristic
120C2×Dic191C2×D411yescharacteristic
121C4×D191—21no—
122C4×D191—21no—
123C4×D191—81no—
124C191⋊C8—21no—
125C191⋊C8—21no—
126C23×C191—21no—
127Q8×C191—21no—
128Q8×C191—21no—
129Q8×C191—21no—
130Q8×C191—21no—
131D4×C191—21no—
132D4×C191#4112yes—
133C2×C4×C191—21no—
134C2×C4×C191—21no—
135C2×C4×C191—21no—
136C2×C4×C191—81no—
137C2×C4×C191C2×D411yescharacteristic
138C2×C4×C191C2×D411yescharacteristic
139C2×C4×C191C2411yescharacteristic
140C8×C191—21no—
141C8×C191—21no—
142C22×C191—22no—
143C22×C191—41no—
144C22×C191—41no—
145C22×C191C22×D411yescharacteristic
146C2×D191—21no—
147C2×D191—21no—
148C2×D191—21no—
149C4×C191—21no—
150C4×C191—21no—
151C4×C191—41no—
152C4×C191—41no—
153C4×C191—41no—
154C4×C191C2×#4111yescharacteristic
155C4×C191C2×#4111yescharacteristic
156Dic191—21no—
157Dic191—22no—
158Dic191—21no—
159Dic191—21no—
160Dic191—41no—
161Dic191—41no—
162Dic191—41no—
163C2×C191—42no—
164C2×C191#52511yescharacteristic
165C2×C191#57711yescharacteristic
166C2×C191#57811yescharacteristic
167D191—41no—
168C191#312011yescharacteristic, complemented, Sylow
169#3120—1911nomaximal, Sylow
170#541—1911no—
171#520—1911no—
172#484—1911no—
173#483—1911no—
174#479—1911no—
175#478—1911no—
176#462—1911no—
177#461D19111yescharacteristic, complemented
178#452—1911no—
179#452—1911no—
180#441—1911no—
181#440—1911no—
182C4×Q16—1911no—
183C4×SD16—1911no—
184#433—1911no—
185C2×#41—1911no—
186C2×Q16—1911no—
187C2×Q16—1911no—
188C2×SD16C2×D19111yescharacteristic
189C2×SD16C2×D19111yescharacteristic
190C4⋊Q8C2×D19111yescharacteristic
191#126—3821no—
192#126—3821no—
193#125—3821no—
194#125—3821no—
195#125—1911no—
196C22⋊Q8—1911no—
197C22⋊Q8—1911no—
198C4×Q8—1911no—
199C4×Q8—1911no—
200C4×Q8—1911no—
201C4×Q8—1911no—
202C4×D4—3821no—
203C4×D4—3821no—
204C4×D4—1911no—
205C4×D4C2×D19111yescharacteristic
206#117—3821no—
207#117—1911no—
208C2×C4⋊C4—3821no—
209C2×C4⋊C4—1911no—
210#107—1911no—
211#107—1911no—
212#106—1911no—
213#106—1911no—
214C4⋊C8—1911no—
215C4⋊C8C2×D19111yescharacteristic
216#103—1911no—
217#103—1911no—
218#103—1911no—
219#103—1911no—
220#103C2×D19111yescharacteristic
221#103C2×D19111yescharacteristic
222#102—1911no—
223#102—1911no—
224C22⋊C8—1911no—
225C22⋊C8—1911no—
226#97—1911no—
227C4×C8—1911no—
228#41—3822no—
229#41—3822no—
230C2×Q8—1911no—
231C2×Q8C22×D19111yescharacteristic
232C2×Q8C22×D19111yescharacteristic
233C2×D4—3821no—
234C2×D4C22×D19111yescharacteristic
235C22×C4—21no—
236C22×C4—3821no—
237C22×C4—3821no—
238C22×C4—3821no—
239C22×C4—1911no—
240Q16—3822no—
241Q16—3822no—
242SD16—22no—
243SD16—22no—
244C2×C8—1911no—
245C2×C8—1911no—
246C2×C8C22×D19111yescharacteristic
247C2×C8C22×D19111yescharacteristic
248C4⋊C4—21no—
249C4⋊C4—21no—
250C4⋊C4—3821no—
251C4⋊C4—3821no—
252C4⋊C4—3821no—
253C4⋊C4—3821no—
254C4⋊C4—3821no—
255C4⋊C4—3821no—
256C4⋊C4—3821no—
257C4⋊C4—3821no—
258C4⋊C4—3821no—
259C4⋊C4—1911no—
260C4⋊C4—1911no—
261C4⋊C4—1911no—
262C4⋊C4—1911no—
263C4⋊C4—1911no—
264C4⋊C4C22×D19111yescharacteristic
265C22⋊C4—21no—
266C22⋊C4—3821no—
267C22⋊C4—3821no—
268C22⋊C4—3821no—
269C22⋊C4—1911no—
270C22⋊C4—1911no—
271C42—3821no—
272C42—3821no—
273C42—3821no—
274C42—3821no—
275C42—1911no—
276C42C22×D19111yescharacteristic
277C23—21no—
278C23—1911no—
279Q8—21no—
280Q8—21no—
281Q8—21no—
282Q8—21no—
283Q8—3821no—
284Q8—1912no—
285D4—21no—
286D4—3824no—
287D4#1569028612yes—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—81no—
292C2×C4—3824no—
293C2×C4—3821no—
294C2×C4—3821no—
295C2×C4—3821no—
296C2×C4—3821no—
297C2×C4—3821no—
298C2×C4—3821no—
299C2×C4—1911no—
300C2×C4—1911no—
301C2×C4—1911no—
302C2×C4—15281no—
303C2×C4—15281no—
304C2×C4D4×D19111yescharacteristic
305C2×C4D4×D19111yescharacteristic
306C2×C4C23×D19111yescharacteristic
307C8—21no—
308C8—21no—
309C8—3821no—
310C8—3821no—
311C22—22no—
312C22—41no—
313C22—41no—
314C22—3821no—
315C22—3821no—
316C22—3821no—
317C22C2×D4×D19111yescharacteristic
318C4—21no—
319C4—21no—
320C4—41no—
321C4—41no—
322C4—41no—
323C4—3821no—
324C4—3821no—
325C4—3821no—
326C4—3822no—
327C4—7641no—
328C4—7641no—
329C4—7641no—
330C4#41×D19111yescharacteristic
331C4C2×#1569028611yescharacteristic
332C2—42no—
333C2—7641no—
334C2#2713317311yescharacteristic
335C2#2713346911yescharacteristic
336C2#2713347911yescharacteristic
subgroupG/H
centerC22C2×D4×D191
commutator, cosocleC2×C4×C191C24
FrattiniC2×C4C23×D191
Fitting#461×C191C2
socleC22×C191C22×D4
prime radicalC2×#15690286C22
hypercenter#461D191
hypocenterC191#3120
LCSG ▷ C2×C4×C191 ▷ C2×C191 ▷ C191
UCSC1 ◁ C22 ◁ C42 ◁ #461

Complements

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

NH ≅ G/Ntype
1. #27133274C2semidirect
3. #27132997C2semidirect
4. #27132992C2semidirect
5. #27132971C2semidirect
6. #27132963C2semidirect
7. #27132870C2semidirect
10. #27132767C2semidirect
11. #27132764C2semidirect
12. #27132755C2semidirect
13. SD16×Dic191C2semidirect
14. #27132723C2semidirect
15. #461×C191C2semidirect
168. C191#3120semidirect
177. #461D191semidirect

Maximal subgroups

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

subgroupindex
#271332742
#271331422
#271329972
#271329922
#271329712
#271329632
#271328702
#271328222
#271327672
#271327642
#271327552
SD16×Dic1912
#271327232
#461×C1912
#3120191