Finite Group Databasebeta

← #37582769

Subgroups — #37582769

Order 24448.

394 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#27133271C211yesmaximal, characteristic, complemented
2#27133135C211yesmaximal, characteristic
3#27133007C211yesmaximal, characteristic, complemented
4#27133002C211yesmaximal, characteristic, complemented
5#27132965C211yesmaximal, characteristic, complemented
6#27132962C211yesmaximal, characteristic, complemented
7#27132867C211yesmaximal, characteristic, complemented
8#27132866C211yesmaximal, characteristic, complemented
9#27132815C211yesmaximal, characteristic
10#27132812C211yesmaximal, characteristic
11#27132763C211yesmaximal, characteristic, complemented
12#27132761C211yesmaximal, characteristic, complemented
13C4×#15690276C211yesmaximal, characteristic, complemented
14#27132722C211yesmaximal, characteristic, complemented
15C4×SD16×C191C211yesmaximal, characteristic, complemented
16C2×D4×D191C2211yescharacteristic
17C2×#15690276C2211yescharacteristic
18C2×#15690275C2211yescharacteristic
19C2×#15690274C2211yescharacteristic
20#20259320C2211yescharacteristic, complemented
21#20259317—21no—
22#20259316—21no—
23#20259314—21no—
24#20259313C2211yescharacteristic, complemented
25#20259312—21no—
26#20259302C2211yescharacteristic
27#20259298C2211yescharacteristic
28C4×Dic382C2211yescharacteristic, complemented
29C4×D764C2211yescharacteristic
30D4×Dic191C2211yescharacteristic
31#20259284C2211yescharacteristic
32C4×C22⋊D191—21no—
33(C22⋊D191)⋊C4—21no—
34#20259280C2211yescharacteristic
35#20259279C2211yescharacteristic
36C2×(C2×D191)⋊C4—21no—
37C22⋊C4×D191—21no—
38#20259255C2211yescharacteristic, complemented
39#20259255C2211yescharacteristic, complemented
40#20259254C2211yescharacteristic, complemented
41Dic191⋊C8C2211yescharacteristic, complemented
42Dic191⋊C8C2211yescharacteristic, complemented
43#20259247C2211yescharacteristic, complemented
44#20259246C2211yescharacteristic, complemented
45#20259245C2211yescharacteristic, complemented
46#20259244C2211yescharacteristic
47#20259243C2211yescharacteristic
48#20259242C2211yescharacteristic
49(C2×D191)⋊C8C2211yescharacteristic
50(C2×D191)⋊C8C2211yescharacteristic
51#20259233C2211yescharacteristic, complemented
52C2×SD16×C191C2211yescharacteristic
53C4×Q8×C191C2211yescharacteristic, complemented
54C4×D4×C191C2211yescharacteristic
55#106×C191C2211yescharacteristic, complemented
56#103×C191C2211yescharacteristic, complemented
57#102×C191C2211yescharacteristic
58C4×C8×C191C2211yescharacteristic, complemented
59C23×D191—21no—
60C2×Dic382C2311yescharacteristic
61C2×D764C2311yescharacteristic
62C2×C22⋊D191—21no—
63D4×D191—22no—
64D4×D191—22no—
65C22×Dic191—21no—
66C2×C4×D191—21no—
67C2×C4×D191—21no—
68C2×C4×D191C2311yescharacteristic
69#15690276—22no—
70#15690275—22no—
71#15690274—22no—
72C2×C191⋊C8C2311yescharacteristic
73C2×C191⋊C8C2311yescharacteristic
74C4⋊Dic191—21no—
75C4⋊Dic191—21no—
76C4⋊Dic191C2311yescharacteristic
77C4⋊Dic191C2311yescharacteristic
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)⋊C4—21no—
89(C2×D191)⋊C4—21no—
90(C2×D191)⋊C4—21no—
91(C2×D191)⋊C4D411yescharacteristic
92(C2×D191)⋊C4D411yescharacteristic
93C4×Dic191—21no—
94C4×Dic191—21no—
95C4×Dic191C2311yescharacteristic
96C2×Q8×C191C2311yescharacteristic
97C2×D4×C191C2311yescharacteristic
98C22×C4×C191—21no—
99SD16×C191—22no—
100C2×C8×C191C2311yescharacteristic
101C2×C8×C191C2311yescharacteristic
102C4⋊C4×C191—21no—
103C4⋊C4×C191C2311yescharacteristic
104C4⋊C4×C191C2311yescharacteristic
105C22⋊C4×C191—21no—
106C42×C191—21no—
107C42×C191C2311yescharacteristic
108C22×D191—21no—
109C22×D191—24no—
110C22×D191—81no—
111C22×D191—81no—
112C22×D191C2×D411yescharacteristic
113Dic382—21no—
114Dic382—21no—
115D764—21no—
116D764#4112yes—
117C22⋊D191—24no—
118C2×Dic191—21no—
119C2×Dic191—21no—
120C2×Dic191—21no—
121C2×Dic191—81no—
122C2×Dic191C2×D411yescharacteristic
123C2×Dic191C2×D411yescharacteristic
124C2×Dic191C2×D411yescharacteristic
125C4×D191—21no—
126C4×D191—21no—
127C4×D191—81no—
128C4×D191—81no—
129C191⋊C8—21no—
130C191⋊C8—21no—
131C23×C191—21no—
132Q8×C191—21no—
133Q8×C191—21no—
134D4×C191—21no—
135D4×C191#4112yes—
136C2×C4×C191—21no—
137C2×C4×C191—21no—
138C2×C4×C191—21no—
139C2×C4×C191—81no—
140C2×C4×C191C2×D411yescharacteristic
141C2×C4×C191C2×D411yescharacteristic
142C2×C4×C191C2411yescharacteristic
143C8×C191—21no—
144C8×C191—21no—
145C22×C191—22no—
146C22×C191—41no—
147C22×C191—41no—
148C22×C191C22×D411yescharacteristic
149C2×D191—21no—
150C2×D191—22no—
151C2×D191—21no—
152C2×D191—21no—
153C2×D191—41no—
154C2×D191—41no—
155C2×D191—84no—
156C4×C191—21no—
157C4×C191—21no—
158C4×C191—41no—
159C4×C191—41no—
160C4×C191—41no—
161C4×C191C2×#4111yescharacteristic
162C4×C191C2×#4111yescharacteristic
163Dic191—21no—
164Dic191—21no—
165Dic191—21no—
166Dic191—41no—
167Dic191—41no—
168Dic191—41no—
169C2×C191—42no—
170C2×C191#52511yescharacteristic
171C2×C191#57711yescharacteristic
172C2×C191#57711yescharacteristic
173D191—41no—
174D191—42no—
175C191#312311yescharacteristic, complemented, Sylow
176#3123—1911nomaximal, Sylow
177#541—1911no—
178#518—1911no—
179#485—1911no—
180#485—1911no—
181#478—1911no—
182#478—1911no—
183#461—1911no—
184#461—1911no—
185#450—1911no—
186#450—1911no—
187#440—1911no—
188#440—1911no—
189C4×SD16—1911no—
190C4×SD16D19111yescharacteristic, complemented
191#433—1911no—
192C22×D4—1911no—
193C2×SD16—1911no—
194C2×SD16—1911no—
195C2×SD16—1911no—
196C2×SD16C2×D19111yescharacteristic
197C4⋊Q8—1911no—
198#126—3821no—
199#126—3821no—
200#125—3821no—
201#125—3821no—
202#125—1911no—
203C22⋊Q8—1911no—
204C22⋊Q8—1911no—
205C4×Q8—1911no—
206C4×Q8C2×D19111yescharacteristic, complemented
207C4×D4—3821no—
208C4×D4—3821no—
209C4×D4—1911no—
210C4×D4—1911no—
211C4×D4—1911no—
212C4×D4C2×D19111yescharacteristic
213#117—1911no—
214#117—1911no—
215C2×C22⋊C4—3821no—
216C2×C22⋊C4—3821no—
217#107—1911no—
218#107—1911no—
219#106—1911no—
220#106C2×D19111yescharacteristic, complemented
221C4⋊C8—1911no—
222C4⋊C8—1911no—
223#103—1911no—
224#103—1911no—
225#103—1911no—
226#103C2×D19111yescharacteristic, complemented
227#102—1911no—
228#102—1911no—
229#102—1911no—
230#102C2×D19111yescharacteristic
231C22⋊C8—1911no—
232C22⋊C8—1911no—
233#97—1911no—
234C4×C8C2×D19111yescharacteristic, complemented
235C24—3821no—
236C2×Q8—1911no—
237C2×Q8C22×D19111yescharacteristic
238C2×D4—3822no—
239C2×D4—3822no—
240C2×D4—3821no—
241C2×D4—1911no—
242C2×D4C22×D19111yescharacteristic
243C22×C4—21no—
244C22×C4—3821no—
245C22×C4—3821no—
246C22×C4—3821no—
247C22×C4—1911no—
248SD16—22no—
249SD16—3822no—
250SD16—3822no—
251SD16—3822no—
252C2×C8—1911no—
253C2×C8—1911no—
254C2×C8C22×D19111yescharacteristic
255C2×C8C22×D19111yescharacteristic
256C4⋊C4—21no—
257C4⋊C4—3821no—
258C4⋊C4—3821no—
259C4⋊C4—3821no—
260C4⋊C4—3821no—
261C4⋊C4—3821no—
262C4⋊C4—3821no—
263C4⋊C4—3821no—
264C4⋊C4—1911no—
265C4⋊C4—1911no—
266C4⋊C4—1911no—
267C4⋊C4—1911no—
268C4⋊C4C22×D19111yescharacteristic
269C4⋊C4C22×D19111yescharacteristic
270C22⋊C4—21no—
271C22⋊C4—3821no—
272C22⋊C4—3821no—
273C22⋊C4—3821no—
274C22⋊C4—3821no—
275C22⋊C4—3821no—
276C22⋊C4—3821no—
277C22⋊C4—1911no—
278C22⋊C4—1911no—
279C42—21no—
280C42—3821no—
281C42—3821no—
282C42—1911no—
283C42C22×D19111yescharacteristic
284C23—21no—
285C23—3824no—
286C23—3821no—
287C23—1911no—
288C23—15281no—
289C23—15281no—
290Q8—21no—
291Q8—21no—
292Q8—3821no—
293Q8—3821no—
294D4—21no—
295D4—3824no—
296D4—3821no—
297D4—1912no—
298D4#1569028712yes—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—81no—
303C2×C4—3821no—
304C2×C4—3821no—
305C2×C4—3821no—
306C2×C4—3821no—
307C2×C4—3821no—
308C2×C4—1911no—
309C2×C4—1911no—
310C2×C4—1911no—
311C2×C4—15281no—
312C2×C4—15281no—
313C2×C4—15281no—
314C2×C4D4×D19111yescharacteristic
315C2×C4D4×D19111yescharacteristic
316C2×C4C23×D19111yescharacteristic
317C8—21no—
318C8—21no—
319C8—3821no—
320C8—3821no—
321C22—22no—
322C22—41no—
323C22—41no—
324C22—3821no—
325C22—3821no—
326C22—3821no—
327C22—3822no—
328C22—15284no—
329C22—7641no—
330C22—7641no—
331C22C2×D4×D19111yescharacteristic
332C4—21no—
333C4—21no—
334C4—41no—
335C4—41no—
336C4—41no—
337C4—3821no—
338C4—3821no—
339C4—3821no—
340C4—7641no—
341C4—7641no—
342C4—7641no—
343C4#41×D19111yescharacteristic
344C4C2×#1569028711yescharacteristic
345C2—42no—
346C2—7641no—
347C2—7642no—
348C2#2713317211yescharacteristic
349C2#2713346911yescharacteristic
350C2#2713347011yescharacteristic
subgroupG/H
centerC22C2×D4×D191
commutator, cosocleC2×C4×C191C24
FrattiniC2×C4C23×D191
FittingC4×SD16×C191C2
socleC22×C191C22×D4
prime radicalC2×D4×D191C22
hypercenterC4×SD16D191
hypocenterC191#3123
LCSG ▷ C2×C4×C191 ▷ C2×C191 ▷ C191
UCSC1 ◁ C22 ◁ C42 ◁ C4×SD16

Complements

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

NH ≅ G/Ntype
1. #27133271C2semidirect
3. #27133007C2semidirect
4. #27133002C2semidirect
5. #27132965C2semidirect
6. #27132962C2semidirect
7. #27132867C2semidirect
8. #27132866C2semidirect
11. #27132763C2semidirect
12. #27132761C2semidirect
13. C4×#15690276C2semidirect
14. #27132722C2semidirect
15. C4×SD16×C191C2semidirect
20. #20259320C22semidirect
24. #20259313C22semidirect
28. C4×Dic382C22semidirect
38. #20259255C22semidirect
39. #20259255C22semidirect
40. #20259254C22semidirect
41. Dic191⋊C8C22semidirect
42. Dic191⋊C8C22semidirect
43. #20259247C22semidirect
44. #20259246C22semidirect
45. #20259245C22semidirect
51. #20259233C22semidirect
53. C4×Q8×C191C22semidirect
55. #106×C191C22semidirect
56. #103×C191C22semidirect
58. C4×C8×C191C22semidirect
175. C191#3123semidirect
190. C4×SD16D191semidirect
206. C4×Q8C2×D191semidirect
220. #106C2×D191semidirect
226. #103C2×D191semidirect
234. C4×C8C2×D191semidirect

Maximal subgroups

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

subgroupindex
#271332712
#271331352
#271330072
#271330022
#271329652
#271329622
#271328672
#271328662
#271328152
#271328122
#271327632
#271327612
C4×#156902762
#271327222
C4×SD16×C1912
#3123191