Finite Group Databasebeta

← #37583294

Subgroups — #37583294

Order 24448.

416 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 416 conjugacy classes of subgroups — too many to draw. Only its 121 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#20259302C211yesmaximal, characteristic
2C22⋊Q8×D191C211yesmaximal, characteristic
3#27133000C212yesmaximal, complemented
4#27132998C211yesmaximal, characteristic, complemented
5#27132905C212yesmaximal, complemented
6#27132903C212yesmaximal, complemented
7#27132821C211yesmaximal, characteristic, complemented
8#27132820C212yesmaximal, complemented
9#27132818C211yesmaximal, characteristic, complemented
10C22⋊C8×D191C211yesmaximal, characteristic
11#484×C191C211yesmaximal, characteristic, complemented
12C22×Dic382C2211yescharacteristic
13C2×Q8×D191C2211yescharacteristic
14C22×C4×D191C2211yescharacteristic
15C2×Dic764C2212yescomplemented
16C2×#15690277C2212yescomplemented
17C2×C8×D191C2212yes—
18#20259302—21no—
19#20259302—21no—
20#20259302C2212yes—
21#20259302C2211yescharacteristic
22#20259302C2212yes—
23#20259301C2211yescharacteristic
24#20259300—21no—
25#20259299—21no—
26#20259298C2211yescharacteristic
27C2×C4⋊Dic191—21no—
28C4⋊C4×D191—21no—
29C4⋊C4×D191C2211yescharacteristic
30C2×(C2×D191)⋊C4—21no—
31C22⋊C4×D191—21no—
32#20259255C2212yescomplemented
33#20259247C2212yescomplemented
34#20259245C2212yescomplemented
35#20259245C2212yescomplemented
36C22⋊(C191⋊C8)C2211yescharacteristic
37(C2×D191)⋊C8C2212yes—
38C22⋊Q8×C191C2211yescharacteristic
39C2×C4⋊C4×C191C2211yescharacteristic
40#107×C191C2212yescomplemented
41#103×C191C2212yescomplemented
42C22⋊C8×C191C2211yescharacteristic
43C23×D191D411yescharacteristic
44C2×Dic382—21no—
45C2×Dic382—21no—
46C2×Dic382—42no—
47C2×Dic382C2312yes—
48C2×Dic382C2311yescharacteristic
49Q8×D191—42no—
50C22×Dic191—21no—
51C22×Dic191D411yescharacteristic
52C2×C4×D191—21no—
53C2×C4×D191—21no—
54C2×C4×D191—42no—
55C2×C4×D191D411yescharacteristic
56C2×C4×D191D411yescharacteristic
57C2×C4×D191C2311yescharacteristic
58C2×C4×D191C2311yescharacteristic
59Dic764—42no—
60#15690277—42no—
61C2×C191⋊C8C2312yes—
62C8×D191—42no—
63C4⋊Dic191—21no—
64C4⋊Dic191—22no—
65C4⋊Dic191—22no—
66C4⋊Dic191C2311yescharacteristic
67Dic191⋊C4—21no—
68Dic191⋊C4—21no—
69Dic191⋊C4—21no—
70C22⋊Dic191—21no—
71(C2×D191)⋊C4—21no—
72(C2×D191)⋊C4—21no—
73(C2×D191)⋊C4—22no—
74(C2×D191)⋊C4—22no—
75C2×Q8×C191C2311yescharacteristic
76C22×C4×C191—21no—
77C22×C4×C191C2311yescharacteristic
78C2×C8×C191C2312yes—
79C4⋊C4×C191—21no—
80C4⋊C4×C191—21no—
81C4⋊C4×C191C2312yes—
82C4⋊C4×C191C2311yescharacteristic
83C22⋊C4×C191—21no—
84C22×D191—41no—
85C22×D191—41no—
86C22×D191Q1614yescomplemented
87C22×D191C2×D411yescharacteristic
88C22×D191C2×D411yescharacteristic
89Dic382—22no—
90Dic382—22no—
91Dic382—22no—
92Dic382—42no—
93Dic382—42no—
94C2×Dic191—22no—
95C2×Dic191—21no—
96C2×Dic191—21no—
97C2×Dic191—41no—
98C2×Dic191—42no—
99C2×Dic191C2×D411yescharacteristic
100C2×Dic191C2×D411yescharacteristic
101C4×D191—22no—
102C4×D191—22no—
103C4×D191—41no—
104C4×D191—41no—
105C4×D191—42no—
106C4×D191—42no—
107C191⋊C8—22no—
108C23×C191C2×D411yescharacteristic
109Q8×C191—22no—
110C2×C4×C191—22no—
111C2×C4×C191—21no—
112C2×C4×C191—21no—
113C2×C4×C191—22no—
114C2×C4×C191—42no—
115C2×C4×C191C2×D411yescharacteristic
116C2×C4×C191C2411yescharacteristic
117C8×C191—22no—
118C22×C191—21no—
119C22×C191—21no—
120C22×C191C2×Q1612yes—
121C22×C191C22×D411yescharacteristic
122C2×D191—21no—
123C2×D191—21no—
124C2×D191—21no—
125C2×D191—21no—
126C2×D191—44no—
127C2×D191C2×Q1612yes—
128C2×D191C2×Q1612yes—
129C4×C191—21no—
130C4×C191—41no—
131C4×C191—42no—
132C4×C191—41no—
133C4×C191H(5,𝔽2)12yes—
134Dic191—21no—
135Dic191—21no—
136Dic191—41no—
137Dic191—42no—
138Dic191—41no—
139C2×C191—22no—
140C2×C191#53411yescharacteristic
141C2×C191C22×Q1611yescharacteristic
142C2×C191#57511yescharacteristic
143D191—22no—
144D191—22no—
145C191#318911yescharacteristic, complemented, Sylow
146#3189—1911nomaximal, Sylow
147C2×C22⋊Q8—1911no—
148C2×C22⋊Q8—1911no—
149#484—1911no—
150#484—1912no—
151#484D19111yescharacteristic, complemented
152#467—1912no—
153#467—1912no—
154#451—1911no—
155#451—1912no—
156#451—1911no—
157C2×C22⋊C8—1911no—
158C22×Q8—1911no—
159C22×Q8—1911no—
160C23×C4—1911no—
161C2×Q16—1912no—
162C2×Q16—1912no—
163C22×C8—1912no—
164C22⋊Q8—3821no—
165C22⋊Q8—3821no—
166C22⋊Q8—3821no—
167C22⋊Q8—3821no—
168C22⋊Q8—1911no—
169C22⋊Q8—1911no—
170C22⋊Q8—1912no—
171C22⋊Q8—1911no—
172C22⋊Q8—1912no—
173C22⋊Q8C2×D19111yescharacteristic
174C2×C4⋊C4—3821no—
175C2×C4⋊C4—3821no—
176C2×C4⋊C4—1911no—
177C2×C4⋊C4C2×D19111yescharacteristic
178C2×C22⋊C4—3821no—
179C2×C22⋊C4—3821no—
180#107—1912no—
181#107C2×D19112yescomplemented
182#103—1912no—
183#103—1912no—
184#103—1912no—
185#103C2×D19112yescomplemented
186C22⋊C8—1911no—
187C22⋊C8—1912no—
188C22⋊C8C2×D19111yescharacteristic
189C24—1911no—
190C2×Q8—3821no—
191C2×Q8—3821no—
192C2×Q8—1912no—
193C2×Q8—1911no—
194C2×Q8—7642no—
195C2×Q8—7642no—
196C2×Q8C22×D19111yescharacteristic
197C22×C4—21no—
198C22×C4—3821no—
199C22×C4—3821no—
200C22×C4—3821no—
201C22×C4—1911no—
202C22×C4—1911no—
203C22×C4—1911no—
204C22×C4—1911no—
205C22×C4—1911no—
206C22×C4—7642no—
207C22×C4C22×D19111yescharacteristic
208Q16—7642no—
209Q16—7642no—
210C2×C8—1912no—
211C2×C8—7642no—
212C2×C8C22×D19112yes—
213C4⋊C4—21no—
214C4⋊C4—21no—
215C4⋊C4—3821no—
216C4⋊C4—3821no—
217C4⋊C4—3821no—
218C4⋊C4—3822no—
219C4⋊C4—3822no—
220C4⋊C4—3821no—
221C4⋊C4—1911no—
222C4⋊C4C22×D19111yescharacteristic
223C4⋊C4C22×D19112yes—
224C22⋊C4—21no—
225C22⋊C4—3821no—
226C22⋊C4—3821no—
227C22⋊C4—3822no—
228C22⋊C4—3822no—
229C22⋊C4—3821no—
230C23—1911no—
231C23—1911no—
232C23—1914no—
233C23—7641no—
234C23—7641no—
235C23D4×D19111yescharacteristic
236Q8—22no—
237Q8—3822no—
238Q8—3822no—
239Q8—3822no—
240Q8—7642no—
241Q8—7642no—
242C2×C4—22no—
243C2×C4—21no—
244C2×C4—21no—
245C2×C4—22no—
246C2×C4—42no—
247C2×C4—3822no—
248C2×C4—3822no—
249C2×C4—3821no—
250C2×C4—3822no—
251C2×C4—3821no—
252C2×C4—1911no—
253C2×C4—1911no—
254C2×C4—7641no—
255C2×C4—7641no—
256C2×C4—7642no—
257C2×C4—7642no—
258C2×C4—7641no—
259C2×C4—7642no—
260C2×C4D4×D19111yescharacteristic
261C2×C4C23×D19111yescharacteristic
262C8—22no—
263C8—3822no—
264C22—21no—
265C22—21no—
266C22—3821no—
267C22—3821no—
268C22—3821no—
269C22—3821no—
270C22—1912no—
271C22—1912no—
272C22—7644no—
273C22Q16×D19112yes—
274C22C2×D4×D19111yescharacteristic
275C4—21no—
276C4—42no—
277C4—41no—
278C4—41no—
279C4—3821no—
280C4—3821no—
281C4—7641no—
282C4—7641no—
283C4—7642no—
284C4#2025936812yes—
285C2—22no—
286C2—3822no—
287C2—3822no—
288C2#2713322411yescharacteristic
289C2C2×Q16×D19111yescharacteristic
290C2#2713345911yescharacteristic
subgroupG/H
centerC22C2×D4×D191
commutator, cosocleC2×C4×C191C24
FrattiniC2×C4C23×D191
Fitting#484×C191C2
socleC22×C191C22×D4
prime radicalC23×D191D4
hypercenter#484D191
hypocenterC191#3189
LCSG ▷ C2×C4×C191 ▷ C2×C191 ▷ C191
UCSC1 ◁ C22 ◁ C22×C4 ◁ #484

Complements

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

NH ≅ G/Ntype
3. #27133000C2semidirect
4. #27132998C2semidirect
5. #27132905C2semidirect
6. #27132903C2semidirect
7. #27132821C2semidirect
8. #27132820C2semidirect
9. #27132818C2semidirect
11. #484×C191C2semidirect
15. C2×Dic764C22semidirect
16. C2×#15690277C22semidirect
32. #20259255C22semidirect
33. #20259247C22semidirect
34. #20259245C22semidirect
35. #20259245C22semidirect
40. #107×C191C22semidirect
41. #103×C191C22semidirect
86. C22×D191Q16semidirect
145. C191#3189semidirect
151. #484D191semidirect
181. #107C2×D191semidirect
185. #103C2×D191semidirect

Maximal subgroups

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

subgroupindex
C2×#202593022
C22⋊Q8×D1912
#271330002
#271329982
#271329052
#271329032
#271328212
#271328202
#271328182
C22⋊C8×D1912
#484×C1912
#3189191