Finite Group Databasebeta

← #37581800

Subgroups — #37581800

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×#20259352C211yesmaximal, characteristic, complemented
3C22×#15690276C211yesmaximal, characteristic, complemented
4#27133239C211yesmaximal, characteristic, complemented
5#27132828C214yesmaximal, complemented
6#27132815C214yesmaximal, complemented
7#27132667C211yesmaximal, characteristic, complemented
8#27132629C211yesmaximal, characteristic, complemented
9#416×C191C211yesmaximal, characteristic, complemented
10#20259368—24no—
11#20259368—24no—
12C2×#15690288C2212yes—
13C2×#15690287—21no—
14C2×#15690287C2211yescharacteristic
15C22×Dic382C2211yescharacteristic, complemented
16C22×D764—21no—
17C22×D764C2211yescharacteristic
18C2×D4×D191—22no—
19C2×D4×D191C2212yes—
20#20259352—24no—
21C2×#15690277C2212yescomplemented
22C2×#15690276—24no—
23C2×#15690276C2214yescomplemented
24C2×#15690275C2212yescomplemented
25C2×#15690271C2211yescharacteristic, complemented
26#20259322—21no—
27#20259310—21no—
28#20259306—22no—
29#20259302C2214yescomplemented
30#20259242C2214yescomplemented
31(C2×D191)⋊C8C2214yescomplemented
32C2×#41×C191C2211yescharacteristic
33C22×C8×C191C2211yescharacteristic, complemented
34#117×C191C2211yescharacteristic, complemented
35#103×C191C2212yescomplemented
36#102×C191C2212yescomplemented
37C23×D191—22no—
38C23×D191—21no—
39#15690288—24no—
40#15690288—24no—
41#15690287—28no—
42#15690287—22no—
43#15690287—22no—
44C2×Dic382—21no—
45C2×Dic382—22no—
46C2×Dic382—22no—
47C2×Dic382D411yescharacteristic
48C2×Dic382C2312yes—
49C2×D764—28no—
50C2×D764—22no—
51C2×D764—21no—
52C2×D764—22no—
53C2×D764D411yescharacteristic
54C2×D764D414yes—
55C2×D764C2312yes—
56C2×C22⋊D191—22no—
57C2×C22⋊D191D412yes—
58D4×D191—216no—
59D4×D191—24no—
60D4×D191—24no—
61C22×Dic191—21no—
62C2×C4×D191—22no—
63C2×C4×D191C2312yes—
64#15690277—24no—
65#15690276—28no—
66#15690275—24no—
67#15690271—22no—
68C2×C191⋊C8C2312yes—
69C4⋊Dic191—24no—
70(C2×D191)⋊C4—24no—
71#41×C191—22no—
72#41×C191D414yes—
73C2×Q8×C191C2311yescharacteristic
74C2×D4×C191—21no—
75C2×D4×C191C2311yescharacteristic
76C22×C4×C191—21no—
77C22×C4×C191C2311yescharacteristic
78C2×C8×C191—22no—
79C2×C8×C191C2312yes—
80C4⋊C4×C191C2312yes—
81C22⋊C4×C191—21no—
82C42×C191—21no—
83C22×D191—22no—
84C22×D191—28no—
85C22×D191—24no—
86C22×D191—84no—
87C22×D191—82no—
88C22×D191C2×D412yes—
89Dic382—22no—
90Dic382—22no—
91Dic382—22no—
92Dic382—42no—
93D764—28no—
94D764—22no—
95D764—24no—
96D764—22no—
97D764C2×D414yes—
98C22⋊D191—28no—
99C22⋊D191—24no—
100C2×Dic191—22no—
101C2×Dic191—42no—
102C2×Dic191C2×D412yes—
103C4×D191—28no—
104C4×D191—22no—
105C4×D191—22no—
106C191⋊C8—22no—
107C23×C191—21no—
108C23×C191C2×D411yescharacteristic
109Q8×C191—21no—
110Q8×C191C2×D412yes—
111D4×C191—24no—
112D4×C191—21no—
113D4×C191C2×D412yes—
114C2×C4×C191—21no—
115C2×C4×C191—24no—
116C2×C4×C191—22no—
117C2×C4×C191C2×D411yescharacteristic
118C2×C4×C191C2×D412yes—
119C2×C4×C191C2×D412yes—
120C2×C4×C191C2411yescharacteristic
121C8×C191—22no—
122C22×C191—22no—
123C22×C191—22no—
124C22×C191—42no—
125C22×C191#12012yes—
126C22×C191C22×D411yescharacteristic
127C2×D191—22no—
128C2×D191—24no—
129C2×D191—24no—
130C2×D191—44no—
131C2×D191—84no—
132C2×D191—88no—
133C4×C191—22no—
134C4×C191—42no—
135C4×C191#12012yes—
136C4×C191C22×D411yescharacteristic
137C4×C191C22×D411yescharacteristic
138Dic191—22no—
139Dic191—42no—
140C2×C191—22no—
141C2×C191—42no—
142C2×C191C2×#12011yescharacteristic
143C2×C191#57712yes—
144D191—42no—
145D191—44no—
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#416D19111yescharacteristic, complemented
156#408—1911no—
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—3821no—
165C22×D4—3822no—
166C22×D4—1912no—
167C22×D4—1911no—
168#137—3824no—
169C2×Q16—1912no—
170C2×SD16—3824no—
171C2×SD16—1912no—
172C2×SD16—1914no—
173C2×M16—1911no—
174C22×C8C2×D19111yescharacteristic, complemented
175C4⋊Q8—3821no—
176#124—3821no—
177#123—3822no—
178C22⋊Q8—1914no—
179#117C2×D19111yescharacteristic, complemented
180#103C2×D19112yescomplemented
181#102—1914no—
182#102C2×D19112yescomplemented
183C22⋊C8—1914no—
184C24—3822no—
185C24—3821no—
186#41—22no—
187#41—3828no—
188#41—3824no—
189#41—3824no—
190#41—3822no—
191#41—3822no—
192#41D76414yes—
193C2×Q8—3821no—
194C2×Q8—3822no—
195C2×Q8—3822no—
196C2×Q8—1912no—
197C2×Q8—1911no—
198C2×Q8C22×D19111yescharacteristic
199C2×D4—21no—
200C2×D4—3828no—
201C2×D4—3822no—
202C2×D4—38216no—
203C2×D4—3824no—
204C2×D4—3824no—
205C2×D4—3822no—
206C2×D4—3821no—
207C2×D4—3822no—
208C2×D4—1912no—
209C2×D4—1911no—
210C2×D4—1914no—
211C2×D4—1912no—
212C2×D4C22×D19111yescharacteristic
213C22×C4—21no—
214C22×C4—3822no—
215C22×C4—3821no—
216C22×C4—1912no—
217C22×C4C22×D19111yescharacteristic
218Q16—3824no—
219SD16—3828no—
220SD16—3824no—
221M16—3822no—
222C2×C8—22no—
223C2×C8—1912no—
224C2×C8C22×D19112yes—
225C4⋊C4—3824no—
226C4⋊C4C22×D19112yes—
227C22⋊C4—21no—
228C22⋊C4—3824no—
229C42—21no—
230C23—21no—
231C23—3828no—
232C23—3824no—
233C23—3822no—
234C23—1912no—
235C23—15282no—
236C23—15284no—
237C23D4×D19111yescharacteristic
238Q8—21no—
239Q8—3822no—
240Q8—3822no—
241Q8—3822no—
242Q8—7642no—
243Q8C2×D76412yes—
244D4—24no—
245D4—21no—
246D4—3828no—
247D4—3828no—
248D4—3824no—
249D4—3822no—
250D4—3822no—
251D4—3824no—
252D4—1914no—
253D4C2×D76412yes—
254C2×C4—21no—
255C2×C4—24no—
256C2×C4—22no—
257C2×C4—3828no—
258C2×C4—3822no—
259C2×C4—3822no—
260C2×C4—3822no—
261C2×C4—1912no—
262C2×C4—7642no—
263C2×C4D4×D19112yes—
264C2×C4D4×D19111yescharacteristic
265C2×C4C2×D76412yes—
266C2×C4C23×D19111yescharacteristic
267C8—22no—
268C8—3822no—
269C22—22no—
270C22—22no—
271C22—42no—
272C22—3824no—
273C22—3822no—
274C22—3824no—
275C22—15284no—
276C22—15288no—
277C22—7644no—
278C22#2025929212yes—
279C22C2×D4×D19111yescharacteristic
280C4—22no—
281C4—42no—
282C4—3822no—
283C4—7642no—
284C4#2025929212yes—
285C4C2×D4×D19111yescharacteristic
286C4C22×D76411yescharacteristic
287C2—22no—
288C2—42no—
289C2—7642no—
290C2—7644no—
291C2C2×#2025929211yescharacteristic
292C2#2713347012yes—
subgroupG/H
centerC22C2×D4×D191
commutator, cosocleC2×C4×C191C24
FrattiniC2×C4C23×D191
Fitting#416×C191C2
socleC22×C191C22×D4
prime radicalC2×#20259368C2
hypercenter#416D191
hypocenterC191#3009
LCSG ▷ C2×C4×C191 ▷ C2×C191 ▷ C191
UCSC1 ◁ C22 ◁ C22×C4 ◁ #416

Complements

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

NH ≅ G/Ntype
2. C2×#20259352C2semidirect
3. C22×#15690276C2semidirect
4. #27133239C2semidirect
5. #27132828C2semidirect
6. #27132815C2semidirect
7. #27132667C2semidirect
8. #27132629C2semidirect
9. #416×C191C2semidirect
15. C22×Dic382C22semidirect
21. C2×#15690277C22semidirect
23. C2×#15690276C22semidirect
24. C2×#15690275C22semidirect
25. C2×#15690271C22semidirect
29. #20259302C22semidirect
30. #20259242C22semidirect
31. (C2×D191)⋊C8C22semidirect
33. C22×C8×C191C22semidirect
34. #117×C191C22semidirect
35. #103×C191C22semidirect
36. #102×C191C22semidirect
146. C191#3009semidirect
155. #416D191semidirect
174. C22×C8C2×D191semidirect
179. #117C2×D191semidirect
180. #103C2×D191semidirect
182. #102C2×D191semidirect

Maximal subgroups

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

subgroupindex
C2×#202593682
C2×#202593522
C22×#156902762
#271332392
#271328282
#271328152
#271326672
#271326292
#416×C1912
#3009191