Finite Group Databasebeta

← #37581799

Subgroups — #37581799

Order 24448.

752 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 752 conjugacy classes of subgroups — too many to draw. Only its 149 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×#20259368C211yesmaximal, characteristic
2C2×#20259355C211yesmaximal, characteristic, complemented
3C22×#15690276C211yesmaximal, characteristic, complemented
4#27133234C211yesmaximal, characteristic, complemented
5#27132827C214yesmaximal, complemented
6#27132816C214yesmaximal, complemented
7#27132669C211yesmaximal, characteristic, complemented
8#27132660C211yesmaximal, characteristic, complemented
9#408×C191C211yesmaximal, characteristic, complemented
10#20259368—24no—
11#20259368—24no—
12C2×#15690288—21no—
13C2×#15690287C2212yes—
14C2×#15690287C2211yescharacteristic
15C22×Dic382C2211yescharacteristic, complemented
16C22×D764C2212yes—
17C22×D764C2211yescharacteristic
18C2×D4×D191—22no—
19C2×D4×D191—21no—
20#20259355—24no—
21C2×Dic764C2212yescomplemented
22C2×#15690276—24no—
23C2×#15690276C2212yescomplemented
24C2×#15690276C2214yescomplemented
25#20259320—21no—
26#20259309—21no—
27#20259303—22no—
28#20259301C2214yescomplemented
29#20259281C2211yescharacteristic, complemented
30#20259247C2212yescomplemented
31#20259244C2212yescomplemented
32#20259244C2214yescomplemented
33C2×#41×C191C2211yescharacteristic
34C2×M16×C191C2211yescharacteristic, complemented
35C22×C8×C191C2211yescharacteristic, complemented
36C22⋊C8×C191C2214yescomplemented
37C23×D191—22no—
38C23×D191—21no—
39#15690288—28no—
40#15690287—24no—
41#15690287—24no—
42#15690287—22no—
43#15690287D414yes—
44C2×Dic382—21no—
45C2×Dic382—22no—
46C2×Dic382—22no—
47C2×Dic382C2312yes—
48C2×Dic382C2311yescharacteristic
49C2×D764—24no—
50C2×D764—24no—
51C2×D764—22no—
52C2×D764—21no—
53C2×D764—22no—
54C2×D764D414yes—
55C2×D764C2311yescharacteristic
56C2×D764C2312yes—
57C2×C22⋊D191—22no—
58C2×C22⋊D191—21no—
59D4×D191—216no—
60D4×D191—28no—
61C22×Dic191—21no—
62C2×C4×D191—22no—
63C2×C4×D191—21no—
64Dic764—24no—
65#15690276—28no—
66#15690276—24no—
67C4⋊Dic191C2312yes—
68Dic191⋊C4—24no—
69C22⋊Dic191—21no—
70C22⋊Dic191—24no—
71C4×Dic191—21no—
72#41×C191—22no—
73#41×C191—22no—
74C2×Q8×C191D411yescharacteristic
75C2×D4×C191D412yes—
76C2×D4×C191D411yescharacteristic
77C22×C4×C191C2312yes—
78C22×C4×C191C2311yescharacteristic
79M16×C191—22no—
80C2×C8×C191—22no—
81C2×C8×C191C2312yes—
82C2×C8×C191C2312yes—
83C22×D191—22no—
84C22×D191—21no—
85C22×D191—28no—
86C22×D191—24no—
87C22×D191—82no—
88C22×D191—84no—
89Dic382—21no—
90Dic382—22no—
91Dic382—22no—
92Dic382—42no—
93Dic382C2×D412yes—
94D764—28no—
95D764—24no—
96D764—22no—
97D764—21no—
98D764C2×D412yes—
99D764C2×D414yes—
100C22⋊D191—28no—
101C22⋊D191—24no—
102C2×Dic191—21no—
103C2×Dic191—22no—
104C2×Dic191—22no—
105C2×Dic191—42no—
106C4×D191—28no—
107C4×D191—24no—
108C23×C191C2×D412yes—
109C23×C191C2×D411yescharacteristic
110Q8×C191—22no—
111D4×C191—24no—
112D4×C191—22no—
113C2×C4×C191—22no—
114C2×C4×C191—22no—
115C2×C4×C191C2×D412yes—
116C2×C4×C191C2×D412yes—
117C2×C4×C191C2×D412yes—
118C2×C4×C191C2×D411yescharacteristic
119C2×C4×C191C2411yescharacteristic
120C8×C191—22no—
121C8×C191—22no—
122C22×C191—22no—
123C22×C191—22no—
124C22×C191—24no—
125C22×C191#12012yes—
126C22×C191C22×D411yescharacteristic
127C2×D191—24no—
128C2×D191—22no—
129C2×D191—44no—
130C2×D191—42no—
131C2×D191—84no—
132C2×D191—88no—
133C4×C191—22no—
134C4×C191#12012yes—
135C4×C191C22×D411yescharacteristic
136C4×C191C22×D411yescharacteristic
137Dic191—22no—
138Dic191—42no—
139Dic191—42no—
140C2×C191—22no—
141C2×C191—42no—
142C2×C191C2×#12011yescharacteristic
143C2×C191#57712yes—
144D191—44no—
145D191—42no—
146C191#300911yescharacteristic, complemented, Sylow
147#3009—1911nomaximal, Sylow
148C2×H(5,𝔽2)—1911no—
149C2×#137—1911no—
150C22×SD16—1911no—
151#536—1911no—
152#452—1914no—
153#450—1914no—
154#418—1911no—
155#416—1911no—
156#408D19111yescharacteristic, complemented
157H(5,𝔽2)—3824no—
158H(5,𝔽2)—3824no—
159C2×#41—3821no—
160C2×#41—1912no—
161C2×#41—1911no—
162C2×#41C2×D19111yescharacteristic
163C22×Q8—1911no—
164C22×D4—3822no—
165C22×D4—3821no—
166C22×D4—1912no—
167C22×D4—1911no—
168#137—3824no—
169C2×Q16—1912no—
170C2×SD16—3824no—
171C2×SD16—1914no—
172C2×SD16—1912no—
173C2×M16C2×D19111yescharacteristic, complemented
174C22×C8C2×D19111yescharacteristic, complemented
175C4⋊Q8—3821no—
176#124—3821no—
177#123—3822no—
178C22⋊Q8—1914no—
179#117—1911no—
180#103—1912no—
181#102—1912no—
182#102—1914no—
183C22⋊C8C2×D19114yescomplemented
184C24—3822no—
185C24—3821no—
186#41—22no—
187#41—22no—
188#41—3824no—
189#41—3824no—
190#41—3828no—
191#41—3822no—
192#41—1914no—
193C2×Q8—3821no—
194C2×Q8—3822no—
195C2×Q8—3822no—
196C2×Q8—1912no—
197C2×Q8—1911no—
198C2×Q8D76411yescharacteristic
199C2×D4—3824no—
200C2×D4—3824no—
201C2×D4—38216no—
202C2×D4—3828no—
203C2×D4—3822no—
204C2×D4—3821no—
205C2×D4—3822no—
206C2×D4—3822no—
207C2×D4—3821no—
208C2×D4—1914no—
209C2×D4—1911no—
210C2×D4—1912no—
211C2×D4D76412yes—
212C2×D4D76411yescharacteristic
213C22×C4—3822no—
214C22×C4—3821no—
215C22×C4—3821no—
216C22×C4C22×D19112yes—
217C22×C4C22×D19111yescharacteristic
218Q16—3824no—
219SD16—3828no—
220SD16—3824no—
221M16—22no—
222C2×C8—22no—
223C2×C8C22×D19112yes—
224C2×C8C22×D19112yes—
225C4⋊C4—3824no—
226C4⋊C4—1912no—
227C22⋊C4—3821no—
228C22⋊C4—3824no—
229C42—3821no—
230C23—3828no—
231C23—3824no—
232C23—3822no—
233C23—3821no—
234C23—15282no—
235C23—15284no—
236C23C2×D76412yes—
237C23C2×D76411yescharacteristic
238Q8—22no—
239Q8—3821no—
240Q8—3822no—
241Q8—3822no—
242Q8—1912no—
243Q8—7642no—
244D4—24no—
245D4—22no—
246D4—3828no—
247D4—3828no—
248D4—3824no—
249D4—3822no—
250D4—3821no—
251D4—3824no—
252D4—1912no—
253D4—1914no—
254C2×C4—22no—
255C2×C4—22no—
256C2×C4—3828no—
257C2×C4—3821no—
258C2×C4—3824no—
259C2×C4—3822no—
260C2×C4—3822no—
261C2×C4—7642no—
262C2×C4D4×D19112yes—
263C2×C4D4×D19112yes—
264C2×C4C2×D76412yes—
265C2×C4C2×D76411yescharacteristic
266C2×C4C23×D19111yescharacteristic
267C8—22no—
268C8—22no—
269C22—22no—
270C22—24no—
271C22—22no—
272C22—3824no—
273C22—3822no—
274C22—15284no—
275C22—15288no—
276C22—7644no—
277C22—7642no—
278C22#2025929212yes—
279C22C22×D76411yescharacteristic
280C4—22no—
281C4—3822no—
282C4—7642no—
283C4—7642no—
284C4#2025929212yes—
285C4C2×D4×D19111yescharacteristic
286C4C2×D4×D19111yescharacteristic
287C2—22no—
288C2—42no—
289C2—7644no—
290C2—7642no—
291C2C2×#2025929211yescharacteristic
292C2#2713347512yes—
subgroupG/H
centerC22C22×D764
commutator, cosocleC2×C4×C191C24
FrattiniC2×C4C23×D191
Fitting#408×C191C2
socleC22×C191C22×D4
prime radicalC2×#20259368C2
hypercenter#408D191
hypocenterC191#3009
LCSG ▷ C2×C4×C191 ▷ C2×C191 ▷ C191
UCSC1 ◁ C22 ◁ C2×#41 ◁ #408

Complements

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

NH ≅ G/Ntype
2. C2×#20259355C2semidirect
3. C22×#15690276C2semidirect
4. #27133234C2semidirect
5. #27132827C2semidirect
6. #27132816C2semidirect
7. #27132669C2semidirect
8. #27132660C2semidirect
9. #408×C191C2semidirect
15. C22×Dic382C22semidirect
21. C2×Dic764C22semidirect
23. C2×#15690276C22semidirect
24. C2×#15690276C22semidirect
28. #20259301C22semidirect
29. #20259281C22semidirect
30. #20259247C22semidirect
31. #20259244C22semidirect
32. #20259244C22semidirect
34. C2×M16×C191C22semidirect
35. C22×C8×C191C22semidirect
36. C22⋊C8×C191C22semidirect
146. C191#3009semidirect
156. #408D191semidirect
173. C2×M16C2×D191semidirect
174. C22×C8C2×D191semidirect
183. C22⋊C8C2×D191semidirect

Maximal subgroups

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

subgroupindex
C2×#202593682
C2×#202593552
C22×#156902762
#271332342
#271328272
#271328162
#271326692
#271326602
#408×C1912
#3009191