Finite Group Databasebeta

← #37583293

Subgroups — #37583293

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×#20259298C211yesmaximal, characteristic
2C22⋊Q8×D191C211yesmaximal, characteristic
3#27132999C211yesmaximal, characteristic, complemented
4#27132997C212yesmaximal, complemented
5#27132996C211yesmaximal, characteristic, complemented
6#27132903C212yesmaximal, complemented
7#27132901C212yesmaximal, complemented
8#27132819C211yesmaximal, characteristic, complemented
9#27132817C212yesmaximal, complemented
10C22⋊C8×D191C211yesmaximal, characteristic
11#451×C191C211yesmaximal, characteristic, complemented
12C2×Q8×D191C2211yescharacteristic
13C22×C4×D191C2211yescharacteristic
14C2×#15690277C2212yescomplemented
15C2×C8×D191C2212yes—
16#20259302C2211yescharacteristic
17#20259301C2211yescharacteristic
18#20259300—21no—
19#20259299—21no—
20#20259298—21no—
21#20259298—21no—
22#20259298C2212yes—
23#20259298C2211yescharacteristic
24#20259298C2212yes—
25C2×C4⋊Dic191C2211yescharacteristic
26C2×Dic191⋊C4—21no—
27C4⋊C4×D191—21no—
28C4⋊C4×D191C2211yescharacteristic
29C2×(C2×D191)⋊C4—21no—
30C22⋊C4×D191—21no—
31#20259256C2212yescomplemented
32#20259255C2212yescomplemented
33#20259247C2212yescomplemented
34#20259246C2212yescomplemented
35#20259246C2212yescomplemented
36C22⋊(C191⋊C8)C2211yescharacteristic
37(C2×D191)⋊C8C2212yes—
38C22×Q8×C191C2211yescharacteristic
39C2×Q16×C191C2212yescomplemented
40C22⋊Q8×C191C2211yescharacteristic
41#103×C191C2212yescomplemented
42C22⋊C8×C191C2211yescharacteristic
43C23×D191D411yescharacteristic
44C2×Dic382—21no—
45C2×Dic382C2311yescharacteristic
46Q8×D191—42no—
47C22×Dic191—21no—
48C22×Dic191D411yescharacteristic
49C2×C4×D191—21no—
50C2×C4×D191—21no—
51C2×C4×D191—42no—
52C2×C4×D191D411yescharacteristic
53C2×C4×D191D411yescharacteristic
54C2×C4×D191C2311yescharacteristic
55C2×C4×D191C2311yescharacteristic
56#15690277—42no—
57C2×C191⋊C8C2312yes—
58C8×D191—42no—
59C4⋊Dic191—21no—
60C4⋊Dic191—21no—
61C4⋊Dic191C2311yescharacteristic
62C4⋊Dic191C2312yes—
63Dic191⋊C4—21no—
64Dic191⋊C4—21no—
65Dic191⋊C4—22no—
66Dic191⋊C4—22no—
67Dic191⋊C4—21no—
68C22⋊Dic191—21no—
69(C2×D191)⋊C4—21no—
70(C2×D191)⋊C4—21no—
71(C2×D191)⋊C4—22no—
72(C2×D191)⋊C4—22no—
73C2×Q8×C191—21no—
74C2×Q8×C191—42no—
75C2×Q8×C191C2312yes—
76C2×Q8×C191C2311yescharacteristic
77C22×C4×C191—21no—
78C22×C4×C191C2311yescharacteristic
79Q16×C191—42no—
80C2×C8×C191C2312yes—
81C4⋊C4×C191—21no—
82C4⋊C4×C191C2311yescharacteristic
83C22⋊C4×C191—21no—
84C22×D191—41no—
85C22×D191—41no—
86C22×D191Q1614yescomplemented
87C22×D191C2×D411yescharacteristic
88C22×D191C2×D411yescharacteristic
89Dic382—22no—
90Dic382—42no—
91C2×Dic191—22no—
92C2×Dic191—21no—
93C2×Dic191—21no—
94C2×Dic191—41no—
95C2×Dic191—42no—
96C2×Dic191C2×D411yescharacteristic
97C2×Dic191C2×D411yescharacteristic
98C4×D191—22no—
99C4×D191—22no—
100C4×D191—41no—
101C4×D191—41no—
102C4×D191—42no—
103C4×D191—42no—
104C191⋊C8—22no—
105C23×C191C2×D411yescharacteristic
106Q8×C191—22no—
107Q8×C191—22no—
108Q8×C191—22no—
109Q8×C191—42no—
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—42no—
131C4×C191—41no—
132C4×C191—41no—
133C4×C191H(5,𝔽2)12yes—
134Dic191—21no—
135Dic191—21no—
136Dic191—41no—
137Dic191—41no—
138Dic191—42no—
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#484—1911no—
152#467—1912no—
153#467—1912no—
154#451—1911no—
155#451—1912no—
156#451D19111yescharacteristic, complemented
157C2×C22⋊C8—1911no—
158C22×Q8—1911no—
159C22×Q8C2×D19111yescharacteristic
160C23×C4—1911no—
161C2×Q16—1912no—
162C2×Q16C2×D19112yescomplemented
163C22×C8—1912no—
164C22⋊Q8—3821no—
165C22⋊Q8—3821no—
166C22⋊Q8—3821no—
167C22⋊Q8—3821no—
168C22⋊Q8—1912no—
169C22⋊Q8—1911no—
170C22⋊Q8—1912no—
171C22⋊Q8—1911no—
172C22⋊Q8—1911no—
173C22⋊Q8C2×D19111yescharacteristic
174C2×C4⋊C4—3821no—
175C2×C4⋊C4—3821no—
176C2×C4⋊C4—1911no—
177C2×C4⋊C4—1911no—
178C2×C22⋊C4—3821no—
179C2×C22⋊C4—3821no—
180#107—1912no—
181#107—1912no—
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—21no—
191C2×Q8—42no—
192C2×Q8—3821no—
193C2×Q8—1911no—
194C2×Q8—7642no—
195C2×Q8C22×D19111yescharacteristic
196C2×Q8C22×D19112yes—
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—42no—
209Q16—7642no—
210C2×C8—1912no—
211C2×C8—7642no—
212C2×C8C22×D19112yes—
213C4⋊C4—21no—
214C4⋊C4—3821no—
215C4⋊C4—3822no—
216C4⋊C4—3822no—
217C4⋊C4—3821no—
218C4⋊C4—3821no—
219C4⋊C4—3821no—
220C4⋊C4—3821no—
221C4⋊C4—1911no—
222C4⋊C4—1912no—
223C4⋊C4C22×D19111yescharacteristic
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—22no—
238Q8—22no—
239Q8—42no—
240Q8—3822no—
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—3822no—
250C2×C4—3821no—
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—41no—
277C4—42no—
278C4—41no—
279C4—3821no—
280C4—3821no—
281C4—7641no—
282C4—7642no—
283C4—7641no—
284C4#2025936712yes—
285C2—22no—
286C2—3822no—
287C2—3822no—
288C2#2713322311yescharacteristic
289C2C2×Q16×D19111yescharacteristic
290C2#2713346211yescharacteristic
subgroupG/H
centerC22C2×D4×D191
commutator, cosocleC2×C4×C191C24
FrattiniC2×C4C23×D191
Fitting#451×C191C2
socleC22×C191C22×D4
prime radicalC23×D191D4
hypercenter#451D191
hypocenterC191#3189
LCSG ▷ C2×C4×C191 ▷ C2×C191 ▷ C191
UCSC1 ◁ C22 ◁ C22×C4 ◁ #451

Complements

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

NH ≅ G/Ntype
3. #27132999C2semidirect
4. #27132997C2semidirect
5. #27132996C2semidirect
6. #27132903C2semidirect
7. #27132901C2semidirect
8. #27132819C2semidirect
9. #27132817C2semidirect
11. #451×C191C2semidirect
14. C2×#15690277C22semidirect
31. #20259256C22semidirect
32. #20259255C22semidirect
33. #20259247C22semidirect
34. #20259246C22semidirect
35. #20259246C22semidirect
39. C2×Q16×C191C22semidirect
41. #103×C191C22semidirect
86. C22×D191Q16semidirect
145. C191#3189semidirect
156. #451D191semidirect
162. C2×Q16C2×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×#202592982
C22⋊Q8×D1912
#271329992
#271329972
#271329962
#271329032
#271329012
#271328192
#271328172
C22⋊C8×D1912
#451×C1912
#3189191