Finite Group Databasebeta

← #484×D191 #37582422

Subgroups — #484×D191 #37582422

Order 24448.

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

normal subgroups only

This group has 460 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
1C22⋊Q8×D191C211yesmaximal, characteristic
2C2×C4⋊C4×D191C211yesmaximal, characteristic
3#27133000C212yesmaximal, complemented
4#27132999C211yesmaximal, characteristic, complemented
5#27132998C211yesmaximal, characteristic, complemented
6#27132997C212yesmaximal, complemented
7#27132996C211yesmaximal, characteristic, complemented
8#107×D191C212yesmaximal, complemented
9#103×D191C212yesmaximal, complemented
10C22⋊C8×D191C211yesmaximal, characteristic
11#484×C191C211yesmaximal, characteristic, complemented
12C2×Q8×D191C2211yescharacteristic
13C22×C4×D191—21no—
14C22×C4×D191C2211yescharacteristic
15C2×C8×D191C2212yes—
16#20259302C2211yescharacteristic
17#20259301C2211yescharacteristic
18#20259300—21no—
19#20259299—21no—
20#20259298C2211yescharacteristic
21C2×C4⋊Dic191C2211yescharacteristic
22C2×Dic191⋊C4—21no—
23C4⋊C4×D191—21no—
24C4⋊C4×D191—21no—
25C4⋊C4×D191—21no—
26C4⋊C4×D191C2211yescharacteristic
27C4⋊C4×D191C2212yes—
28C4⋊C4×D191C2212yes—
29C22⋊C4×D191—21no—
30#20259256C2212yescomplemented
31#20259255C2212yescomplemented
32#20259255C2212yescomplemented
33#20259247C2212yescomplemented
34#20259246C2212yescomplemented
35#20259245C2212yescomplemented
36C22⋊(C191⋊C8)C2211yescharacteristic
37(C2×D191)⋊C8C2212yes—
38C22⋊Q8×C191C2211yescharacteristic
39C2×C4⋊C4×C191C2211yescharacteristic
40#107×C191C2212yescomplemented
41#103×C191C2212yescomplemented
42C22⋊C8×C191C2211yescharacteristic
43C23×D191D411yescharacteristic
44C2×Dic382—21no—
45C2×Dic382C2311yescharacteristic
46Q8×D191—24no—
47C22×Dic191—21no—
48C22×Dic191D411yescharacteristic
49C2×C4×D191—24no—
50C2×C4×D191—22no—
51C2×C4×D191—22no—
52C2×C4×D191—21no—
53C2×C4×D191—21no—
54C2×C4×D191—44no—
55C2×C4×D191D411yescharacteristic
56C2×C4×D191D411yescharacteristic
57C2×C4×D191C2311yescharacteristic
58C2×C4×D191C2311yescharacteristic
59C2×C191⋊C8C2312yes—
60C8×D191—24no—
61C4⋊Dic191—21no—
62C4⋊Dic191—21no—
63C4⋊Dic191C2312yes—
64C4⋊Dic191C2311yescharacteristic
65Dic191⋊C4—22no—
66Dic191⋊C4—22no—
67Dic191⋊C4—21no—
68Dic191⋊C4—21no—
69Dic191⋊C4—21no—
70C22⋊Dic191—21no—
71(C2×D191)⋊C4—21no—
72(C2×D191)⋊C4—21no—
73C2×Q8×C191C2311yescharacteristic
74C22×C4×C191—21no—
75C22×C4×C191C2311yescharacteristic
76C2×C8×C191C2312yes—
77C4⋊C4×C191—21no—
78C4⋊C4×C191—21no—
79C4⋊C4×C191C2312yes—
80C4⋊C4×C191C2311yescharacteristic
81C22⋊C4×C191—21no—
82C22×D191—22no—
83C22×D191—22no—
84C22×D191Q1614yes—
85C22×D191C2×D411yescharacteristic
86C22×D191C2×D411yescharacteristic
87Dic382—22no—
88Dic382—42no—
89C2×Dic191—22no—
90C2×Dic191—21no—
91C2×Dic191—21no—
92C2×Dic191—22no—
93C2×Dic191—42no—
94C2×Dic191C2×D411yescharacteristic
95C2×Dic191C2×D411yescharacteristic
96C4×D191—22no—
97C4×D191—22no—
98C4×D191—22no—
99C4×D191—44no—
100C4×D191—42no—
101C4×D191—42no—
102C4×D191—44no—
103C4×D191#4114yes—
104C191⋊C8—22no—
105C23×C191C2×D411yescharacteristic
106Q8×C191—22no—
107C2×C4×C191—22no—
108C2×C4×C191—21no—
109C2×C4×C191—21no—
110C2×C4×C191—22no—
111C2×C4×C191—42no—
112C2×C4×C191C2×D411yescharacteristic
113C2×C4×C191C2411yescharacteristic
114C8×C191—22no—
115C22×C191—21no—
116C22×C191—21no—
117C22×C191C2×Q1612yes—
118C22×C191C22×D411yescharacteristic
119C2×D191—21no—
120C2×D191—21no—
121C2×D191—28no—
122C2×D191#12312yes—
123C2×D191C2×Q1612yes—
124C2×D191C2×Q1612yes—
125C2×D191AGL(1,ℤ/8)12yes—
126C4×C191—21no—
127C4×C191—41no—
128C4×C191—42no—
129C4×C191—41no—
130C4×C191C2×#4112yes—
131Dic191—21no—
132Dic191—41no—
133Dic191—42no—
134Dic191—41no—
135Dic191#12312yes—
136C2×C191—22no—
137C2×C191C2×#12311yescharacteristic
138C2×C191C22×Q1611yescharacteristic
139C2×C191C2×AGL(1,ℤ/8)11yescharacteristic
140D191—22no—
141D191#48414yesdirect product factor
142C191C2×#48411yescharacteristic, complemented, Sylow
143C2×#484—1911nomaximal, Sylow
144C2×C22⋊Q8—1911no—
145C22×C4⋊C4—1911no—
146#484—1911no—
147#484—1911no—
148#484—1912no—
149#484—1912no—
150#484—1911no—
151#484D19111yescharacteristic, direct product factor
152C2×#107—1912no—
153C2×#103—1912no—
154C2×C22⋊C8—1911no—
155C22×Q8—1911no—
156C23×C4—3821no—
157C23×C4—1911no—
158C22×C8—1912no—
159C22⋊Q8—3821no—
160C22⋊Q8—3821no—
161C22⋊Q8—1911no—
162C22⋊Q8—1911no—
163C22⋊Q8—1911no—
164C22⋊Q8C2×D19111yescharacteristic
165C2×C4⋊C4—3821no—
166C2×C4⋊C4—3821no—
167C2×C4⋊C4—3821no—
168C2×C4⋊C4—3821no—
169C2×C4⋊C4—1911no—
170C2×C4⋊C4—1912no—
171C2×C4⋊C4—1912no—
172C2×C4⋊C4—1911no—
173C2×C4⋊C4C2×D19111yescharacteristic
174C2×C22⋊C4—3821no—
175#107—1912no—
176#107—1912no—
177#107—1912no—
178#107C2×D19112yescomplemented
179#103—1912no—
180#103—1912no—
181#103—1912no—
182#103C2×D19112yescomplemented
183C22⋊C8—1911no—
184C22⋊C8—1912no—
185C22⋊C8C2×D19111yescharacteristic
186C24—1911no—
187C2×Q8—3821no—
188C2×Q8—3824no—
189C2×Q8—1911no—
190C2×Q8C22×D19111yescharacteristic
191C22×C4—21no—
192C22×C4—3824no—
193C22×C4—3821no—
194C22×C4—3821no—
195C22×C4—3822no—
196C22×C4—3822no—
197C22×C4—3821no—
198C22×C4—1911no—
199C22×C4—1911no—
200C22×C4—1911no—
201C22×C4—1911no—
202C22×C4—1911no—
203C22×C4—7644no—
204C22×C4C22×D19111yescharacteristic
205C2×C8—3824no—
206C2×C8—1912no—
207C2×C8C22×D19112yes—
208C4⋊C4—21no—
209C4⋊C4—21no—
210C4⋊C4—3822no—
211C4⋊C4—3822no—
212C4⋊C4—3821no—
213C4⋊C4—3821no—
214C4⋊C4—3821no—
215C4⋊C4—3821no—
216C4⋊C4—3821no—
217C4⋊C4—1912no—
218C4⋊C4—1911no—
219C4⋊C4C22×D19112yes—
220C4⋊C4C22×D19111yescharacteristic
221C22⋊C4—21no—
222C22⋊C4—3821no—
223C22⋊C4—3821no—
224C22⋊C4—3821no—
225C23—3822no—
226C23—3822no—
227C23—1914no—
228C23—1911no—
229C23—1911no—
230C23D4×D19111yescharacteristic
231Q8—22no—
232Q8—3822no—
233Q8—7642no—
234C2×C4—22no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—22no—
238C2×C4—42no—
239C2×C4—3822no—
240C2×C4—3822no—
241C2×C4—3822no—
242C2×C4—3822no—
243C2×C4—3821no—
244C2×C4—3821no—
245C2×C4—3822no—
246C2×C4—1911no—
247C2×C4—1911no—
248C2×C4—1914no—
249C2×C4—7644no—
250C2×C4—7644no—
251C2×C4—7642no—
252C2×C4—7642no—
253C2×C4—7642no—
254C2×C4D4×D19111yescharacteristic
255C2×C4C23×D19111yescharacteristic
256C8—22no—
257C8—3822no—
258C22—21no—
259C22—21no—
260C22—3828no—
261C22—3821no—
262C22—3821no—
263C22—1912no—
264C22—1912no—
265C22—1912no—
266C22—1912no—
267C22Q16×D19112yes—
268C22C2×D4×D19111yescharacteristic
269C4—21no—
270C4—42no—
271C4—41no—
272C4—41no—
273C4—3821no—
274C4—1912no—
275C4—7642no—
276C4—7641no—
277C4—7641no—
278C4#41×D19112yes—
279C2—22no—
280C2—3822no—
281C2—1914no—
282C2#123×D19111yescharacteristic
283C2C2×Q16×D19111yescharacteristic
284C2AGL(1,ℤ/8)×D19111yescharacteristic
subgroupG/H
centerC22C2×D4×D191
commutator, cosocleC2×C4×C191C24
FrattiniC2×C4C23×D191
Fitting#484×C191C2
socleC22×C191C22×D4
prime radicalC23×D191D4
hypercenter#484D191
hypocenterC191C2×#484
LCSG ▷ C2×C4×C191 ▷ C2×C191 ▷ C191
UCSC1 ◁ C22 ◁ C22×C4 ◁ #484

Complements

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

NH ≅ G/Ntype
3. #27133000C2semidirect
4. #27132999C2semidirect
5. #27132998C2semidirect
6. #27132997C2semidirect
7. #27132996C2semidirect
8. #107×D191C2semidirect
9. #103×D191C2semidirect
11. #484×C191C2semidirect
30. #20259256C22semidirect
31. #20259255C22semidirect
32. #20259255C22semidirect
33. #20259247C22semidirect
34. #20259246C22semidirect
35. #20259245C22semidirect
40. #107×C191C22semidirect
41. #103×C191C22semidirect
141. D191#484direct (N × H)
142. C191C2×#484semidirect
151. #484D191direct (N × H)
178. #107C2×D191semidirect
182. #103C2×D191semidirect

Maximal subgroups

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

subgroupindex
C22⋊Q8×D1912
C2×C4⋊C4×D1912
#271330002
#271329992
#271329982
#271329972
#271329962
#107×D1912
#103×D1912
C22⋊C8×D1912
#484×C1912
C2×#484191