Finite Group Databasebeta

← #37582439

Subgroups — #37582439

Order 24448.

458 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 458 conjugacy classes of subgroups — too many to draw. Only its 123 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#27133173C211yesmaximal, characteristic
2C2×C4⋊C4×D191C211yesmaximal, characteristic, complemented
3#27133000C212yesmaximal, complemented
4#27132994C211yesmaximal, characteristic, complemented
5#27132993C211yesmaximal, characteristic, complemented
6#27132980C212yesmaximal, complemented
7#27132974C211yesmaximal, characteristic, complemented
8#27132704C212yesmaximal, complemented
9#27132661C212yesmaximal, complemented
10#27132625C211yesmaximal, characteristic, complemented
11#480×C191C211yesmaximal, characteristic, complemented
12C2×#15690286C2211yescharacteristic
13C22×C4×D191—21no—
14C22×C4×D191C2211yescharacteristic
15C2×#15690270C2212yes—
16#20259304—21no—
17#20259302C2211yescharacteristic
18#20259301C2211yescharacteristic
19#20259293C2211yescharacteristic
20#20259290—21no—
21D4×Dic191—21no—
22(C22⋊D191)⋊C4—21no—
23#20259280C2211yescharacteristic
24C2×C4⋊Dic191C2211yescharacteristic
25C2×Dic191⋊C4—21no—
26C4⋊C4×D191—21no—
27C4⋊C4×D191—21no—
28C4⋊C4×D191C2212yes—
29C4⋊C4×D191C2212yes—
30#20259255C2212yescomplemented
31#20259254C2212yescomplemented
32#20259253C2212yescomplemented
33#20259247C2212yescomplemented
34#20259245C2212yescomplemented
35#20259243C2212yescomplemented
36C22⋊(C191⋊C8)C2211yescharacteristic
37(C2×D191)⋊C8C2212yes—
38#121×C191C2211yescharacteristic
39C2×C4⋊C4×C191C2211yescharacteristic
40#107×C191C2212yescomplemented
41#102×C191C2212yescomplemented
42C22⋊C8×C191C2211yescharacteristic
43C23×D191D411yescharacteristic
44#15690286—24no—
45C2×Dic382C2311yescharacteristic
46C2×C22⋊D191—21no—
47C2×C22⋊D191—21no—
48C22×Dic191—21no—
49C22×Dic191—21no—
50C22×Dic191D411yescharacteristic
51C2×C4×D191—24no—
52C2×C4×D191—22no—
53C2×C4×D191—22no—
54C2×C4×D191—44no—
55C2×C4×D191D411yescharacteristic
56C2×C4×D191D411yescharacteristic
57C2×C4×D191C2311yescharacteristic
58C2×C4×D191C2311yescharacteristic
59#15690270—24no—
60C2×C191⋊C8C2312yes—
61C4⋊Dic191—21no—
62C4⋊Dic191—21no—
63C4⋊Dic191C2312yes—
64C4⋊Dic191C2311yescharacteristic
65Dic191⋊C4—22no—
66Dic191⋊C4—22no—
67Dic191⋊C4—21no—
68C22⋊Dic191—21no—
69C22⋊Dic191—21no—
70(C2×D191)⋊C4—21no—
71(C2×D191)⋊C4—21no—
72C4×Dic191—21no—
73C2×D4×C191—21no—
74C2×D4×C191C2311yescharacteristic
75C22×C4×C191—21no—
76C22×C4×C191C2311yescharacteristic
77C2×C8×C191C2312yes—
78C4⋊C4×C191—21no—
79C4⋊C4×C191C2312yes—
80C4⋊C4×C191C2311yescharacteristic
81C22⋊C4×C191—21no—
82C22×D191—22no—
83C22×D191—22no—
84C22×D191—22no—
85C22×D191C2×D411yescharacteristic
86C22×D191C2×D411yescharacteristic
87Dic382—22no—
88C22⋊D191—42no—
89C22⋊D191—42no—
90C2×Dic191—22no—
91C2×Dic191—21no—
92C2×Dic191—21no—
93C2×Dic191—22no—
94C2×Dic191—42no—
95C2×Dic191—42no—
96C2×Dic191C2×D411yescharacteristic
97C2×Dic191C2×D411yescharacteristic
98C4×D191—22no—
99C4×D191—22no—
100C4×D191—22no—
101C4×D191—44no—
102C4×D191—44no—
103C4×D191#4114yes—
104C191⋊C8—22no—
105C23×C191—21no—
106C23×C191C2×D411yescharacteristic
107D4×C191—22no—
108D4×C191—42no—
109C2×C4×C191—22no—
110C2×C4×C191—21no—
111C2×C4×C191—22no—
112C2×C4×C191—42no—
113C2×C4×C191C2×D411yescharacteristic
114C2×C4×C191C2411yescharacteristic
115C8×C191—22no—
116C22×C191—21no—
117C22×C191—21no—
118C22×C191—41no—
119C22×C191—41no—
120C22×C191—41no—
121C22×C191AGL(1,ℤ/8)12yes—
122C22×C191C22×D411yescharacteristic
123C2×D191—28no—
124C2×D191—21no—
125C2×D191—21no—
126C2×D191—21no—
127C2×D191—21no—
128C2×D191#12312yes—
129C2×D191#13712yes—
130C4×C191—21no—
131C4×C191—42no—
132C4×C191—41no—
133C4×C191C2×#4112yes—
134Dic191—21no—
135Dic191—41no—
136Dic191—42no—
137Dic191—41no—
138Dic191#12312yes—
139C2×C191—22no—
140C2×C191—81no—
141C2×C191C2×#12311yescharacteristic
142C2×C191C2×AGL(1,ℤ/8)11yescharacteristic
143C2×C191C2×#13711yescharacteristic
144D191—22no—
145D191—22no—
146C191#309211yescharacteristic, complemented, Sylow
147#3092—1911nomaximal, Sylow
148#525—1911no—
149C22×C4⋊C4—1911no—
150#484—1912no—
151#483—1911no—
152#483—1911no—
153#481—1912no—
154#480—1911no—
155#480D19111yescharacteristic, complemented
156#428—1912no—
157#417—1912no—
158#407—1911no—
159C2×#41—1911no—
160C23×C4—3821no—
161C23×C4—1911no—
162C2×M16—1912no—
163#123—3821no—
164C22⋊Q8—1911no—
165C22⋊Q8—1911no—
166#121—1911no—
167#121C2×D19111yescharacteristic
168#120—3821no—
169C4×D4—3821no—
170C4×D4—3821no—
171#117—1911no—
172C2×C4⋊C4—3821no—
173C2×C4⋊C4—3821no—
174C2×C4⋊C4—3821no—
175C2×C4⋊C4—1911no—
176C2×C4⋊C4—1912no—
177C2×C4⋊C4—1912no—
178C2×C4⋊C4C2×D19111yescharacteristic
179#107—1912no—
180#107C2×D19112yescomplemented
181#106—1912no—
182#106—1912no—
183#103—1912no—
184#103—1912no—
185#102—1912no—
186#102C2×D19112yescomplemented
187C22⋊C8—1911no—
188C22⋊C8—1912no—
189C22⋊C8C2×D19111yescharacteristic
190C24—1911no—
191#41—3824no—
192C2×Q8—1911no—
193C2×D4—21no—
194C2×D4—3821no—
195C2×D4—3821no—
196C2×D4C22×D19111yescharacteristic
197C22×C4—21no—
198C22×C4—3824no—
199C22×C4—3822no—
200C22×C4—3822no—
201C22×C4—3821no—
202C22×C4—3821no—
203C22×C4—1911no—
204C22×C4—1911no—
205C22×C4—1911no—
206C22×C4—1911no—
207C22×C4—1911no—
208C22×C4—7644no—
209C22×C4C22×D19111yescharacteristic
210M16—3824no—
211C2×C8—1912no—
212C2×C8C22×D19112yes—
213C4⋊C4—21no—
214C4⋊C4—3822no—
215C4⋊C4—3822no—
216C4⋊C4—3821no—
217C4⋊C4—3821no—
218C4⋊C4—3821no—
219C4⋊C4—1912no—
220C4⋊C4—1911no—
221C4⋊C4C22×D19112yes—
222C4⋊C4C22×D19111yescharacteristic
223C22⋊C4—21no—
224C22⋊C4—3821no—
225C22⋊C4—3821no—
226C22⋊C4—3821no—
227C22⋊C4—3821no—
228C42—3821no—
229C23—21no—
230C23—3822no—
231C23—3822no—
232C23—3822no—
233C23—1911no—
234C23—1911no—
235C23D4×D19111yescharacteristic
236Q8—3822no—
237D4—22no—
238D4—42no—
239D4—7642no—
240D4—7642no—
241C2×C4—22no—
242C2×C4—21no—
243C2×C4—22no—
244C2×C4—42no—
245C2×C4—3822no—
246C2×C4—3822no—
247C2×C4—3822no—
248C2×C4—3822no—
249C2×C4—3821no—
250C2×C4—3821no—
251C2×C4—3822no—
252C2×C4—1911no—
253C2×C4—1911no—
254C2×C4—1914no—
255C2×C4—7644no—
256C2×C4—7644no—
257C2×C4—7642no—
258C2×C4—7642no—
259C2×C4D4×D19111yescharacteristic
260C2×C4C23×D19111yescharacteristic
261C8—22no—
262C8—3822no—
263C22—21no—
264C22—21no—
265C22—41no—
266C22—41no—
267C22—41no—
268C22—3828no—
269C22—3821no—
270C22—3821no—
271C22—3821no—
272C22—3821no—
273C22—1912no—
274C22—1912no—
275C22#2025934612yes—
276C22C2×D4×D19111yescharacteristic
277C4—21no—
278C4—42no—
279C4—41no—
280C4—3821no—
281C4—1912no—
282C4—7642no—
283C4—7641no—
284C4—7641no—
285C4#41×D19112yes—
286C2—22no—
287C2—81no—
288C2—3822no—
289C2—3822no—
290C2#123×D19111yescharacteristic
291C2C2×#2025934611yescharacteristic
292C2#137×D19111yescharacteristic
subgroupG/H
centerC22C2×D4×D191
commutator, cosocleC2×C4×C191C24
FrattiniC2×C4C23×D191
Fitting#480×C191C2
socleC22×C191C22×D4
prime radical#27133173C2
hypercenter#480D191
hypocenterC191#3092
LCSG ▷ C2×C4×C191 ▷ C2×C191 ▷ C191
UCSC1 ◁ C22 ◁ C22×C4 ◁ #480

Complements

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

NH ≅ G/Ntype
2. C2×C4⋊C4×D191C2semidirect
3. #27133000C2semidirect
4. #27132994C2semidirect
5. #27132993C2semidirect
6. #27132980C2semidirect
7. #27132974C2semidirect
8. #27132704C2semidirect
9. #27132661C2semidirect
10. #27132625C2semidirect
11. #480×C191C2semidirect
30. #20259255C22semidirect
31. #20259254C22semidirect
32. #20259253C22semidirect
33. #20259247C22semidirect
34. #20259245C22semidirect
35. #20259243C22semidirect
40. #107×C191C22semidirect
41. #102×C191C22semidirect
146. C191#3092semidirect
155. #480D191semidirect
180. #107C2×D191semidirect
186. #102C2×D191semidirect

Maximal subgroups

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

subgroupindex
#271331732
C2×C4⋊C4×D1912
#271330002
#271329942
#271329932
#271329802
#271329742
#271327042
#271326612
#271326252
#480×C1912
#3092191