Finite Group Databasebeta

← #37582020

Subgroups — #37582020

Order 24448.

526 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 526 conjugacy classes of subgroups — too many to draw. Only its 141 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×#20259348C211yesmaximal, characteristic, complemented
2#27133230C211yesmaximal, characteristic, complemented
3#27133138C211yesmaximal, characteristic, complemented
4#27132876C214yesmaximal, complemented
5#27132856C214yesmaximal, complemented
6#27132689C211yesmaximal, characteristic, complemented
7#27132668C211yesmaximal, characteristic, complemented
8#27132660C211yesmaximal, characteristic, complemented
9C22×D8×C191C211yesmaximal, characteristic, complemented
10C2×#15690287C2211yescharacteristic
11#20259348—24no—
12C2×#15690275C2212yescomplemented
13C2×#15690272C2212yescomplemented
14C2×#15690271C2211yescharacteristic
15#20259318C2212yescomplemented
16#20259309C2212yescomplemented
17#20259293—22no—
18#20259291—22no—
19D4×Dic191—22no—
20D4×Dic191C2214yescomplemented
21#20259281C2211yescharacteristic
22#20259281C2211yescharacteristic
23C2×C22⋊Dic191—22no—
24Dic191⋊C8C2214yescomplemented
25#20259247C2212yescomplemented
26#20259244C2212yescomplemented
27#20259243C2214yescomplemented
28C22×D4×C191C2211yescharacteristic
29C22×D4×C191C2211yescharacteristic
30C2×D8×C191—24no—
31C2×D8×C191C2214yes—
32C22×C8×C191C2211yescharacteristic
33#15690287—22no—
34#15690287—22no—
35C2×Dic382C2311yescharacteristic
36C2×D764C2311yescharacteristic
37C2×C22⋊D191—24no—
38C2×C22⋊D191—21no—
39C22×Dic191—24no—
40C2×C4×D191—21no—
41#15690275—24no—
42#15690272—24no—
43#15690271—22no—
44C2×C191⋊C8C2312yes—
45C4⋊Dic191D412yes—
46C4⋊Dic191C2312yes—
47C22⋊Dic191—24no—
48C22⋊Dic191—24no—
49C22⋊Dic191—21no—
50C22⋊Dic191D412yes—
51C4×Dic191—21no—
52C4×Dic191C2312yes—
53C24×C191—21no—
54C24×C191—21no—
55C2×D4×C191—21no—
56C2×D4×C191—22no—
57C2×D4×C191—21no—
58C2×D4×C191—22no—
59C2×D4×C191—22no—
60C2×D4×C191D414yescomplemented
61C2×D4×C191C2312yes—
62C2×D4×C191C2312yes—
63C22×C4×C191C2311yescharacteristic
64D8×C191—28no—
65C2×C8×C191—22no—
66C2×C8×C191C2312yes—
67C22×D191—21no—
68Dic382—21no—
69Dic382—21no—
70D764—21no—
71D764—21no—
72C22⋊D191—42no—
73C22⋊D191—84no—
74C2×Dic191—22no—
75C2×Dic191—21no—
76C2×Dic191—84no—
77C2×Dic191C2×D414yes—
78C4×D191—42no—
79C191⋊C8—22no—
80C23×C191—22no—
81C23×C191—24no—
82C23×C191—22no—
83C23×C191—42no—
84C23×C191—82no—
85C23×C191—82no—
86C23×C191C2×D411yescharacteristic
87D4×C191—24no—
88D4×C191—22no—
89D4×C191—22no—
90D4×C191—22no—
91D4×C191—42no—
92D4×C191C2×D414yes—
93C2×C4×C191C2×D411yescharacteristic
94C2×C4×C191C2×D412yes—
95C2×C4×C191#4112yes—
96C2×C4×C191C2411yescharacteristic
97C8×C191—22no—
98C22×C191—22no—
99C22×C191—24no—
100C22×C191—44no—
101C22×C191—42no—
102C22×C191—44no—
103C22×C191—84no—
104C22×C191—84no—
105C22×C191#12112yes—
106C22×C191C22×D411yescharacteristic
107C2×D191—41no—
108C2×D191—42no—
109C4×C191#12112yes—
110C4×C191C22×D411yescharacteristic
111C4×C191C2×#4111yescharacteristic
112Dic191—24no—
113Dic191—41no—
114Dic191—42no—
115C2×C191—22no—
116C2×C191—44no—
117C2×C191—82no—
118C2×C191C2×#12111yescharacteristic
119C2×C191#57612yes—
120D191—81no—
121C191#304111yescharacteristic, complemented, Sylow
122#3041—1911nomaximal, Sylow
123C2×AGL(1,ℤ/8)—1911no—
124C22×D8D19111yescharacteristic, complemented
125#535—1911no—
126#518—1911no—
127#463—1914no—
128#459—1914no—
129#424—1911no—
130#418—1911no—
131#416—1911no—
132C2×#41—1911no—
133C22×D4C2×D19111yescharacteristic
134C22×D4C2×D19111yescharacteristic
135AGL(1,ℤ/8)—3824no—
136C2×SD16—1912no—
137C2×D8—24no—
138C2×D8—1912no—
139C2×D8C2×D19114yes—
140C2×M16—1911no—
141C22×C8C2×D19111yescharacteristic
142#127—1912no—
143#124—1912no—
144#121—3822no—
145#120—3822no—
146C4×D4—3822no—
147C4×D4—1914no—
148#117—1911no—
149#117—1911no—
150C2×C22⋊C4—3822no—
151C4⋊C8—1914no—
152#103—1912no—
153#102—1914no—
154#102—1912no—
155C24—21no—
156C24—21no—
157#41—3822no—
158#41—3822no—
159C2×Q8—1911no—
160C2×D4—21no—
161C2×D4—21no—
162C2×D4—22no—
163C2×D4—22no—
164C2×D4—22no—
165C2×D4—3821no—
166C2×D4—3824no—
167C2×D4—1911no—
168C2×D4C22⋊D19114yescomplemented
169C2×D4C22×D19112yes—
170C2×D4C22×D19112yes—
171C22×C4—3824no—
172C22×C4—3821no—
173C22×C4C22×D19111yescharacteristic
174SD16—3824no—
175D8—28no—
176D8—3824no—
177M16—3822no—
178C2×C8—22no—
179C2×C8—1912no—
180C2×C8C22×D19112yes—
181C4⋊C4—1912no—
182C4⋊C4—1912no—
183C22⋊C4—3821no—
184C22⋊C4—3824no—
185C22⋊C4—3824no—
186C22⋊C4—1912no—
187C42—3821no—
188C42—1912no—
189C23—22no—
190C23—24no—
191C23—22no—
192C23—42no—
193C23—82no—
194C23—82no—
195C23—3821no—
196C23D4×D19111yescharacteristic
197Q8—3821no—
198Q8—3821no—
199D4—22no—
200D4—22no—
201D4—22no—
202D4—24no—
203D4—42no—
204D4—3821no—
205D4—3821no—
206D4—15284no—
207D4—7642no—
208D4C2×C22⋊D19114yes—
209C2×C4—3822no—
210C2×C4—3821no—
211C2×C4—1914no—
212C2×C4—15284no—
213C2×C4—7642no—
214C2×C4D4×D19111yescharacteristic
215C2×C4C2×C22⋊D19112yes—
216C2×C4#1569028612yes—
217C2×C4C23×D19111yescharacteristic
218C8—22no—
219C8—3822no—
220C22—22no—
221C22—24no—
222C22—44no—
223C22—44no—
224C22—42no—
225C22—84no—
226C22—84no—
227C22—7642no—
228C22—7641no—
229C22#2025929412yes—
230C22C2×D4×D19111yescharacteristic
231C4—3824no—
232C4—7642no—
233C4—7641no—
234C4#2025929412yes—
235C4C22×C22⋊D19111yescharacteristic
236C4C2×#1569028611yescharacteristic
237C2—22no—
238C2—44no—
239C2—82no—
240C2—15281no—
241C2C2×#2025929411yescharacteristic
242C2#2713346412yes—
subgroupG/H
centerC22C2×D4×D191
commutator, cosocleC2×C4×C191C24
FrattiniC2×C4C23×D191
FittingC22×D8×C191C2
socleC22×C191C22×D4
hypercenterC22×D8D191
hypocenterC191#3041
LCSG ▷ C2×C4×C191 ▷ C2×C191 ▷ C191
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×D8

Complements

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

NH ≅ G/Ntype
1. C2×#20259348C2semidirect
2. #27133230C2semidirect
3. #27133138C2semidirect
4. #27132876C2semidirect
5. #27132856C2semidirect
6. #27132689C2semidirect
7. #27132668C2semidirect
8. #27132660C2semidirect
9. C22×D8×C191C2semidirect
12. C2×#15690275C22semidirect
13. C2×#15690272C22semidirect
15. #20259318C22semidirect
16. #20259309C22semidirect
20. D4×Dic191C22semidirect
24. Dic191⋊C8C22semidirect
25. #20259247C22semidirect
26. #20259244C22semidirect
27. #20259243C22semidirect
60. C2×D4×C191D4semidirect
121. C191#3041semidirect
124. C22×D8D191semidirect
168. C2×D4C22⋊D191semidirect

Maximal subgroups

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

subgroupindex
C2×#202593482
#271332302
#271331382
#271328762
#271328562
#271326892
#271326682
#271326602
C22×D8×C1912
#3041191