Finite Group Databasebeta

← #37582314

Subgroups — #37582314

Order 24448.

440 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 440 conjugacy classes of subgroups — too many to draw. Only its 143 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×D4×Dic191C211yesmaximal, characteristic
3#27132965C212yesmaximal, complemented
4#27132961C212yesmaximal, complemented
5#27132951C212yesmaximal, complemented
6#27132941C212yesmaximal, complemented
7#27132706C211yesmaximal, characteristic, complemented
8#27132705C211yesmaximal, characteristic, complemented
9#27132686C211yesmaximal, characteristic, complemented
10#27132668C211yesmaximal, characteristic
11#418×C191C211yesmaximal, characteristic
12C23×Dic191—21no—
13C2×#15690271C2211yescharacteristic
14#20259321C2211yescharacteristic, complemented
15#20259321C2211yescharacteristic, complemented
16#20259313C2211yescharacteristic, complemented
17#20259313C2211yescharacteristic, complemented
18#20259299—21no—
19#20259299—21no—
20C4×Dic382—21no—
21C4×Dic382—21no—
22D4×Dic191—21no—
23D4×Dic191—21no—
24D4×Dic191C2212yes—
25D4×Dic191C2212yes—
26#20259281C2211yescharacteristic
27C2×C4⋊Dic191C2211yescharacteristic
28C2×C22⋊Dic191—21no—
29C2×C4×Dic191C2211yescharacteristic
30#20259256C2212yescomplemented
31#20259255C2212yescomplemented
32#20259254C2212yescomplemented
33#20259253C2212yescomplemented
34Dic191⋊C8C2212yescomplemented
35Dic191⋊C8C2212yescomplemented
36#20259243C2212yes—
37#20259243C2212yes—
38C22×D4×C191C2211yescharacteristic
39C2×M16×C191C2211yescharacteristic
40#117×C191C2211yescharacteristic
41#102×C191C2212yes—
42#102×C191C2212yes—
43C2×Dic382—21no—
44C2×Dic382—21no—
45C22×Dic191—22no—
46C22×Dic191—22no—
47C22×Dic191—21no—
48C22×Dic191—44no—
49C22×Dic191D411yescharacteristic
50C22×Dic191D411yescharacteristic
51#15690271—22no—
52C2×C191⋊C8C2312yes—
53C4⋊Dic191—21no—
54C4⋊Dic191—21no—
55C4⋊Dic191—21no—
56C4⋊Dic191C2311yescharacteristic
57C4⋊Dic191C2311yescharacteristic
58C4⋊Dic191C2312yes—
59Dic191⋊C4—21no—
60Dic191⋊C4—21no—
61Dic191⋊C4—21no—
62Dic191⋊C4—21no—
63C22⋊Dic191—22no—
64C22⋊Dic191—22no—
65C22⋊Dic191—21no—
66C4×Dic191—21no—
67C4×Dic191D411yescharacteristic
68C4×Dic191D411yescharacteristic
69C4×Dic191C2311yescharacteristic
70C4×Dic191C2311yescharacteristic
71C24×C191—21no—
72C2×D4×C191—21no—
73C2×D4×C191—22no—
74C2×D4×C191Q814yescomplemented
75C2×D4×C191C2312yes—
76C22×C4×C191C2311yescharacteristic
77M16×C191—22no—
78C2×C8×C191C2312yes—
79C4⋊C4×C191C2311yescharacteristic
80C4⋊C4×C191C2311yescharacteristic
81C22⋊C4×C191—21no—
82C42×C191—21no—
83Dic382—42no—
84Dic382—42no—
85C2×Dic191—22no—
86C2×Dic191—21no—
87C2×Dic191—21no—
88C2×Dic191—22no—
89C2×Dic191—41no—
90C2×Dic191—44no—
91C2×Dic191—42no—
92C2×Dic191—44no—
93C2×Dic191C2×D411yescharacteristic
94C2×Dic191C2×D411yescharacteristic
95C2×Dic191C2×D411yescharacteristic
96C2×Dic191C2×D411yescharacteristic
97C191⋊C8—22no—
98C23×C191—24no—
99C23×C191—22no—
100C23×C191—42no—
101C23×C191—42no—
102C23×C191C2×D411yescharacteristic
103D4×C191—24no—
104D4×C191—22no—
105D4×C191C2×Q814yes—
106C2×C4×C191—21no—
107C2×C4×C191—21no—
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×C191—24no—
116C22×C191—48no—
117C22×C191—42no—
118C22×C191—42no—
119C22×C191C22⋊Q812yes—
120C22×C191C22×D411yescharacteristic
121C4×C191—41no—
122C4×C191—41no—
123C4×C191C22⋊Q812yes—
124C4×C191C22×Q811yescharacteristic
125C4×C191C2×#4111yescharacteristic
126Dic191—21no—
127Dic191—21no—
128Dic191—21no—
129Dic191—42no—
130Dic191—41no—
131Dic191—41no—
132Dic191AGL(1,ℤ/8)12yes—
133C2×C191—22no—
134C2×C191—44no—
135C2×C191C2×C22⋊Q811yescharacteristic
136C2×C191C2×AGL(1,ℤ/8)11yescharacteristic
137C2×C191#57511yescharacteristic
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#428—1911no—
148#423—1911no—
149#418—1911no—
150#418D19111yescharacteristic
151C22×D4C2×D19111yescharacteristic
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⋊C4—1911no—
170C2×C22⋊C4—3821no—
171C2×C42—1911no—
172#107—1912no—
173#107—1912no—
174#106—1912no—
175#106—1912no—
176C4⋊C8—1912no—
177C4⋊C8—1912no—
178#102—1912no—
179#102—1912no—
180#102C2×D19112yes—
181#102C2×D19112yes—
182C24—21no—
183C2×Q8—3821no—
184C2×Q8—3821no—
185C2×D4—21no—
186C2×D4—22no—
187C2×D4Dic38214yescomplemented
188C2×D4C22×D19112yes—
189C22×C4—3821no—
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—3821no—
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⋊C4—1912no—
210C4⋊C4C22×D19111yescharacteristic
211C4⋊C4C22×D19111yescharacteristic
212C22⋊C4—21no—
213C22⋊C4—3821no—
214C22⋊C4—3822no—
215C22⋊C4—3822no—
216C42—21no—
217C42—3821no—
218C42—1911no—
219C42—1911no—
220C42—1911no—
221C42—1911no—
222C23—22no—
223C23—24no—
224C23—42no—
225C23—42no—
226C23D4×D19111yescharacteristic
227Q8—7642no—
228Q8—7642no—
229D4—22no—
230D4—24no—
231D4C2×Dic38214yes—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—3822no—
235C2×C4—3821no—
236C2×C4—3821no—
237C2×C4—3822no—
238C2×C4—1911no—
239C2×C4—1911no—
240C2×C4—1911no—
241C2×C4—1911no—
242C2×C4—7641no—
243C2×C4—7644no—
244C2×C4—7644no—
245C2×C4—7642no—
246C2×C4D4×D19111yescharacteristic
247C2×C4C2×Dic38212yes—
248C2×C4#1569028612yes—
249C2×C4C23×D19111yescharacteristic
250C8—22no—
251C8—3822no—
252C22—21no—
253C22—21no—
254C22—24no—
255C22—48no—
256C22—42no—
257C22—42no—
258C22#2025929912yes—
259C22C2×D4×D19111yescharacteristic
260C4—41no—
261C4—41no—
262C4—3821no—
263C4—3821no—
264C4—3821no—
265C4—1912no—
266C4—7642no—
267C4—7641no—
268C4—7641no—
269C4#2025929912yes—
270C4C22×Dic38211yescharacteristic
271C4C2×#1569028611yescharacteristic
272C2—22no—
273C2—44no—
274C2C2×#2025929911yescharacteristic
275C2AGL(1,ℤ/8)×D19111yescharacteristic
276C2#2713345911yescharacteristic
subgroupG/H
centerC22C2×D4×D191
commutator, cosocleC2×C4×C191C24
FrattiniC2×C4C23×D191
Fitting#418×C191C2
socleC22×C191C22×D4
prime radicalC22×D4×C191C22
hypercenter#418D191
hypocenterC191#3077
LCSG ▷ C2×C4×C191 ▷ C2×C191 ▷ C191
UCSC1 ◁ C22 ◁ C22×C4 ◁ #418

Complements

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

NH ≅ G/Ntype
1. #27133216C2semidirect
3. #27132965C2semidirect
4. #27132961C2semidirect
5. #27132951C2semidirect
6. #27132941C2semidirect
7. #27132706C2semidirect
8. #27132705C2semidirect
9. #27132686C2semidirect
14. #20259321C22semidirect
15. #20259321C22semidirect
16. #20259313C22semidirect
17. #20259313C22semidirect
30. #20259256C22semidirect
31. #20259255C22semidirect
32. #20259254C22semidirect
33. #20259253C22semidirect
34. Dic191⋊C8C22semidirect
35. Dic191⋊C8C22semidirect
74. C2×D4×C191Q8semidirect
138. C191#3077semidirect
187. C2×D4Dic382semidirect

Maximal subgroups

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

subgroupindex
#271332162
C2×D4×Dic1912
#271329652
#271329612
#271329512
#271329412
#271327062
#271327052
#271326862
#271326682
#418×C1912
#3077191