Finite Group Databasebeta

← #37582316

Subgroups — #37582316

Order 24448.

440 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 440 conjugacy classes of subgroups — too many to draw. Only its 135 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
1#27133216C211yesmaximal, characteristic, complemented
2C2×#20259284C211yesmaximal, characteristic
3#27132966C212yesmaximal, complemented
4#27132962C212yesmaximal, complemented
5#27132952C212yesmaximal, complemented
6#27132942C212yesmaximal, complemented
7#27132703C211yesmaximal, characteristic, complemented
8#27132686C211yesmaximal, characteristic, complemented
9#27132669C211yesmaximal, characteristic
10#27132667C211yesmaximal, characteristic
11#428×C191C211yesmaximal, characteristic, complemented
12C22×D764C2211yescharacteristic
13C22×C4×D191—21no—
14C2×#15690271C2211yescharacteristic
15#20259321C2211yescharacteristic, complemented
16#20259321C2211yescharacteristic, complemented
17#20259313C2211yescharacteristic, complemented
18#20259313C2211yescharacteristic, complemented
19#20259299—21no—
20#20259299—21no—
21C4×Dic382—21no—
22C4×Dic382—21no—
23#20259284—21no—
24#20259284—21no—
25#20259284C2212yes—
26#20259284C2212yes—
27#20259281C2211yescharacteristic
28C2×(C2×D191)⋊C4—21no—
29C2×C4×Dic191C2211yescharacteristic
30#20259255C2212yescomplemented
31#20259253C2212yescomplemented
32Dic191⋊C8C2212yescomplemented
33Dic191⋊C8C2212yescomplemented
34#20259244C2212yes—
35#20259244C2212yes—
36#20259242C2212yes—
37#20259242C2212yes—
38C2×M16×C191C2211yescharacteristic
39#117×C191C2211yescharacteristic
40C2×C4⋊C4×C191C2211yescharacteristic
41#107×C191C2212yescomplemented
42#106×C191C2212yescomplemented
43C23×D191—21no—
44C2×Dic382—21no—
45C2×Dic382—21no—
46C2×D764—21no—
47C2×D764—22no—
48C2×D764Q814yescomplemented
49C2×D764C2312yes—
50C22×Dic191D411yescharacteristic
51C22×Dic191D411yescharacteristic
52C2×C4×D191—22no—
53C2×C4×D191—22no—
54C2×C4×D191—44no—
55#15690271—22no—
56C2×C191⋊C8C2312yes—
57C4⋊Dic191—21no—
58C4⋊Dic191—21no—
59C4⋊Dic191C2311yescharacteristic
60C4⋊Dic191C2311yescharacteristic
61Dic191⋊C4—21no—
62Dic191⋊C4—21no—
63Dic191⋊C4—21no—
64Dic191⋊C4—21no—
65C22⋊Dic191—21no—
66(C2×D191)⋊C4—22no—
67(C2×D191)⋊C4—22no—
68C4×Dic191—21no—
69C4×Dic191D411yescharacteristic
70C4×Dic191D411yescharacteristic
71C4×Dic191C2311yescharacteristic
72C4×Dic191C2311yescharacteristic
73C22×C4×C191—21no—
74C22×C4×C191C2311yescharacteristic
75M16×C191—22no—
76C2×C8×C191C2312yes—
77C4⋊C4×C191—21no—
78C4⋊C4×C191C2311yescharacteristic
79C4⋊C4×C191C2312yes—
80C4⋊C4×C191C2311yescharacteristic
81C22⋊C4×C191—21no—
82C42×C191—21no—
83C22×D191—24no—
84C22×D191—22no—
85C22×D191—42no—
86C22×D191—42no—
87Dic382—42no—
88Dic382—42no—
89D764—24no—
90D764—22no—
91D764C2×Q814yes—
92C2×Dic191—22no—
93C2×Dic191—21no—
94C2×Dic191—21no—
95C2×Dic191—41no—
96C2×Dic191C2×D411yescharacteristic
97C2×Dic191C2×D411yescharacteristic
98C2×Dic191C2×D411yescharacteristic
99C2×Dic191C2×D411yescharacteristic
100C4×D191—44no—
101C4×D191—44no—
102C191⋊C8—22no—
103C23×C191C2×D411yescharacteristic
104C2×C4×C191—21no—
105C2×C4×C191—21no—
106C2×C4×C191—22no—
107C2×C4×C191—42no—
108C2×C4×C191C2×D411yescharacteristic
109C2×C4×C191C2×Q812yes—
110C2×C4×C191#4112yes—
111C2×C4×C191C2411yescharacteristic
112C8×C191—22no—
113C22×C191—21no—
114C22×C191—21no—
115C22×C191C22⋊Q812yes—
116C22×C191C22×D411yescharacteristic
117C2×D191—24no—
118C2×D191—48no—
119C2×D191—42no—
120C2×D191—42no—
121C4×C191—42no—
122C4×C191—41no—
123C4×C191—41no—
124C4×C191C22⋊Q812yes—
125C4×C191C22×Q811yescharacteristic
126C4×C191C2×#4111yescharacteristic
127Dic191—21no—
128Dic191—21no—
129Dic191—21no—
130Dic191—41no—
131Dic191—41no—
132Dic191AGL(1,ℤ/8)12yes—
133C2×C191—22no—
134C2×C191C2×C22⋊Q811yescharacteristic
135C2×C191C2×AGL(1,ℤ/8)11yescharacteristic
136C2×C191#57511yescharacteristic
137D191—44no—
138C191#307711yescharacteristic, complemented, Sylow
139#3077—1911nomaximal, Sylow
140#533—1911no—
141C2×C4×D4—1911no—
142#478—1912no—
143#478—1912no—
144#476—1912no—
145#474—1912no—
146#428—1911no—
147#428D19111yescharacteristic, complemented
148#423—1911no—
149#418—1911no—
150#418—1911no—
151C22×D4—1911no—
152C23×C4—3821no—
153C2×M16—1911no—
154C2×M16C2×D19111yescharacteristic
155C4⋊Q8—1911no—
156C4⋊Q8—1911no—
157#125—1911no—
158#125—1911no—
159C22⋊Q8—3821no—
160C22⋊Q8—3821no—
161C4×Q8—3821no—
162C4×Q8—3821no—
163C4×D4—3821no—
164C4×D4—3821no—
165C4×D4—1912no—
166C4×D4—1912no—
167#117—1911no—
168#117C2×D19111yescharacteristic
169C2×C4⋊C4C2×D19111yescharacteristic
170C2×C22⋊C4—3821no—
171C2×C42—1911no—
172#107—1912no—
173#107C2×D19112yescomplemented
174#106—1912no—
175#106C2×D19112yescomplemented
176C4⋊C8—1912no—
177C4⋊C8—1912no—
178#102—1912no—
179#102—1912no—
180#102—1912no—
181#102—1912no—
182C24—3821no—
183C2×Q8—3821no—
184C2×Q8—3821no—
185C2×D4—3821no—
186C2×D4—3822no—
187C2×D4—1914no—
188C2×D4—1912no—
189C22×C4—21no—
190C22×C4—3822no—
191C22×C4—3822no—
192C22×C4—1911no—
193C22×C4—1911no—
194C22×C4—7644no—
195C22×C4C22×D19111yescharacteristic
196M16—22no—
197M16—3822no—
198C2×C8—1912no—
199C2×C8C22×D19112yes—
200C4⋊C4—21no—
201C4⋊C4—3821no—
202C4⋊C4—3821no—
203C4⋊C4—3821no—
204C4⋊C4—3821no—
205C4⋊C4—3821no—
206C4⋊C4—3821no—
207C4⋊C4—1911no—
208C4⋊C4—1911no—
209C4⋊C4C22×D19111yescharacteristic
210C4⋊C4C22×D19111yescharacteristic
211C4⋊C4C22×D19112yes—
212C22⋊C4—21no—
213C22⋊C4—3822no—
214C22⋊C4—3822no—
215C22⋊C4—3821no—
216C42—21no—
217C42—3821no—
218C42—1911no—
219C42—1911no—
220C42—1911no—
221C42—1911no—
222C23—3822no—
223C23—3824no—
224C23—7642no—
225C23—7642no—
226C23D4×D19111yescharacteristic
227Q8—7642no—
228Q8—7642no—
229D4—3822no—
230D4—3824no—
231D4—1914no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—22no—
235C2×C4—42no—
236C2×C4—3822no—
237C2×C4—3821no—
238C2×C4—3821no—
239C2×C4—1911no—
240C2×C4—1911no—
241C2×C4—1911no—
242C2×C4—1911no—
243C2×C4—7644no—
244C2×C4—7644no—
245C2×C4—7641no—
246C2×C4D4×D19111yescharacteristic
247C2×C4Q8×D19112yes—
248C2×C4#1569028712yes—
249C2×C4C23×D19111yescharacteristic
250C8—22no—
251C8—3822no—
252C22—21no—
253C22—21no—
254C22—3824no—
255C22—7648no—
256C22—7642no—
257C22—7642no—
258C22#2025930012yes—
259C22C2×D4×D19111yescharacteristic
260C4—42no—
261C4—41no—
262C4—41no—
263C4—3821no—
264C4—3821no—
265C4—3821no—
266C4—1912no—
267C4—7641no—
268C4—7641no—
269C4#2025930012yes—
270C4C2×Q8×D19111yescharacteristic
271C4C2×#1569028711yescharacteristic
272C2—22no—
273C2—7644no—
274C2C2×#2025930011yescharacteristic
275C2AGL(1,ℤ/8)×D19111yescharacteristic
276C2#2713346211yescharacteristic
subgroupG/H
centerC22C2×D4×D191
commutator, cosocleC2×C4×C191C24
FrattiniC2×C4C23×D191
Fitting#428×C191C2
socleC22×C191C22×D4
prime radicalC22×D764C22
hypercenter#428D191
hypocenterC191#3077
LCSG ▷ C2×C4×C191 ▷ C2×C191 ▷ C191
UCSC1 ◁ C22 ◁ C22×C4 ◁ #428

Complements

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

NH ≅ G/Ntype
1. #27133216C2semidirect
3. #27132966C2semidirect
4. #27132962C2semidirect
5. #27132952C2semidirect
6. #27132942C2semidirect
7. #27132703C2semidirect
8. #27132686C2semidirect
11. #428×C191C2semidirect
15. #20259321C22semidirect
16. #20259321C22semidirect
17. #20259313C22semidirect
18. #20259313C22semidirect
30. #20259255C22semidirect
31. #20259253C22semidirect
32. Dic191⋊C8C22semidirect
33. Dic191⋊C8C22semidirect
41. #107×C191C22semidirect
42. #106×C191C22semidirect
48. C2×D764Q8semidirect
138. C191#3077semidirect
147. #428D191semidirect
173. #107C2×D191semidirect
175. #106C2×D191semidirect

Maximal subgroups

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

subgroupindex
#271332162
C2×#202592842
#271329662
#271329622
#271329522
#271329422
#271327032
#271326862
#271326692
#271326672
#428×C1912
#3077191