Finite Group Databasebeta

← #38734660

Subgroups — #38734660

Order 24896.

346 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 346 conjugacy classes of subgroups — too many to draw. Only its 103 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×D1556C211yesmaximal, characteristic, complemented
2C2×D4×D389C211yesmaximal, characteristic, complemented
3#27159369C212yesmaximal, complemented
4#27159367C211yesmaximal, characteristic, complemented
5#27159359C212yesmaximal, complemented
6#27159357C212yesmaximal, complemented
7#27159354C211yesmaximal, characteristic, complemented
8#27159354C212yesmaximal, complemented
9#27159352C211yesmaximal, characteristic, complemented
10C22⋊C4×D389C211yesmaximal, characteristic, complemented
11#123×C389C211yesmaximal, characteristic, complemented
12C23×D389C2211yescharacteristic
13C23×D389C2211yescharacteristic, complemented
14C23×D389C2211yescharacteristic, complemented
15C2×D1556—24no—
16C2×D1556C2212yes—
17C2×D1556C2212yescomplemented
18C2×D1556C2211yescharacteristic, complemented
19C2×D1556C2212yescomplemented
20C2×C22⋊D389C2211yescharacteristic
21C2×C22⋊D389C2211yescharacteristic, complemented
22C2×C22⋊D389C2212yescomplemented
23D4×D389—24no—
24C2×C4×D389C2211yescharacteristic, complemented
25C2×C4×D389C2212yescomplemented
26#20261945C2212yescomplemented
27C22⋊Dic389C2211yescharacteristic, complemented
28#20261943C2212yescomplemented
29#20261943C2212yescomplemented
30#20261943C2212yescomplemented
31#20261943C2212yescomplemented
32C2×D4×C389C2211yescharacteristic
33C22×C4×C389C2211yescharacteristic, complemented
34C4⋊C4×C389C2212yescomplemented
35C22⋊C4×C389C2211yescharacteristic, complemented
36C22⋊C4×C389C2212yescomplemented
37C22×D389—22no—
38C22×D389—22no—
39C22×D389—42no—
40C22×D389—42no—
41C22×D389—42no—
42C22×D389D414yescomplemented
43C22×D389C2311yescharacteristic
44C22×D389C2312yes—
45C22×D389C2311yescharacteristic
46C22×D389C2311yescharacteristic
47D1556—22no—
48D1556—24no—
49D1556—24no—
50D1556—42no—
51C22⋊D389—22no—
52C22⋊D389—22no—
53C22⋊D389—42no—
54C2×Dic389C2312yes—
55C2×Dic389C2311yescharacteristic
56C4×D389—22no—
57C4×D389—42no—
58C23×C389C2311yescharacteristic
59C23×C389C2311yescharacteristic
60D4×C389—22no—
61C2×C4×C389—22no—
62C2×C4×C389C2312yes—
63C2×C4×C389C2312yes—
64C2×C4×C389C2311yescharacteristic
65C22×C389—22no—
66C22×C389—21no—
67C22×C389—22no—
68C22×C389C2×D412yes—
69C22×C389C2411yescharacteristic
70C2×D389—22no—
71C2×D389—22no—
72C2×D389—21no—
73C2×D389—22no—
74C2×D389—22no—
75C2×D389—22no—
76C2×D389—22no—
77C2×D389—44no—
78C2×D389—44no—
79C2×D389—44no—
80C2×D389C2×D412yes—
81C2×D389C2×D412yes—
82C4×C389—22no—
83C4×C389—21no—
84C4×C389—22no—
85Dic389—21no—
86Dic389—22no—
87C2×C389—22no—
88C2×C389—41no—
89C2×C389C22×D411yescharacteristic
90C2×C389H(5,𝔽2)12yes—
91D389—22no—
92D389—22no—
93D389—41no—
94D389—42no—
95C389#53411yescharacteristic, complemented, Sylow
96#534—3891nomaximal, Sylow
97C22×D4—3891no—
98C22×D4—3891no—
99#123—3891no—
100#123—3892no—
101#123D38911yescharacteristic, complemented
102#121—3892no—
103#121—3892no—
104#120—3891no—
105#120—3892no—
106#120—3891no—
107C2×C22⋊C4—3891no—
108C24—3891no—
109C24—3891no—
110C24—3891no—
111C2×D4—7784no—
112C2×D4—7784no—
113C2×D4—3891no—
114C2×D4—3891no—
115C2×D4—3892no—
116C2×D4—3892no—
117C2×D4—3891no—
118C2×D4—3892no—
119C2×D4—3892no—
120C2×D4C2×D38911yescharacteristic
121C22×C4—3891no—
122C22×C4—3892no—
123C22×C4C2×D38911yescharacteristic, complemented
124C4⋊C4—3892no—
125C4⋊C4C2×D38912yescomplemented
126C22⋊C4—3891no—
127C22⋊C4—3892no—
128C22⋊C4—3892no—
129C22⋊C4—3892no—
130C22⋊C4—3892no—
131C22⋊C4C2×D38911yescharacteristic, complemented
132C22⋊C4C2×D38912yescomplemented
133C23—7782no—
134C23—7782no—
135C23—15562no—
136C23—15562no—
137C23—15562no—
138C23—3891no—
139C23—3892no—
140C23—3891no—
141C23—3891no—
142C23—3894no—
143C23C22×D38911yescharacteristic
144C23C22×D38911yescharacteristic
145D4—22no—
146D4—7784no—
147D4—7784no—
148D4—7782no—
149D4—7782no—
150D4—7782no—
151D4—15562no—
152D4—15562no—
153C2×C4—22no—
154C2×C4—7782no—
155C2×C4—15562no—
156C2×C4—3892no—
157C2×C4—3891no—
158C2×C4C22×D38912yes—
159C2×C4C22×D38911yescharacteristic
160C2×C4C22×D38912yes—
161C22—22no—
162C22—21no—
163C22—22no—
164C22—7782no—
165C22—7782no—
166C22—7782no—
167C22—7781no—
168C22—7782no—
169C22—7782no—
170C22—7782no—
171C22—15564no—
172C22—15564no—
173C22—15564no—
174C22—3892no—
175C22—3892no—
176C22D4×D38912yes—
177C22C23×D38911yescharacteristic
178C4—22no—
179C4—21no—
180C4—22no—
181C4—7781no—
182C4—7782no—
183C2—22no—
184C2—41no—
185C2—7782no—
186C2—7782no—
187C2—15562no—
188C2—15561no—
189C2C2×D4×D38911yescharacteristic
190C2#2715943012yes—
subgroupG/H
center, FrattiniC22C23×D389
commutator, socle, cosocleC22×C389C24
Fitting#123×C389C2
hypercenter#123D389
hypocenterC389#534
LCSG ▷ C22×C389 ▷ C389
UCSC1 ◁ C22 ◁ #123

Complements

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

NH ≅ G/Ntype
1. C22×D1556C2semidirect
2. C2×D4×D389C2semidirect
3. #27159369C2semidirect
4. #27159367C2semidirect
5. #27159359C2semidirect
6. #27159357C2semidirect
7. #27159354C2semidirect
8. #27159354C2semidirect
9. #27159352C2semidirect
10. C22⋊C4×D389C2semidirect
11. #123×C389C2semidirect
13. C23×D389C22semidirect
14. C23×D389C22semidirect
17. C2×D1556C22semidirect
18. C2×D1556C22semidirect
19. C2×D1556C22semidirect
21. C2×C22⋊D389C22semidirect
22. C2×C22⋊D389C22semidirect
24. C2×C4×D389C22semidirect
25. C2×C4×D389C22semidirect
26. #20261945C22semidirect
27. C22⋊Dic389C22semidirect
28. #20261943C22semidirect
29. #20261943C22semidirect
30. #20261943C22semidirect
31. #20261943C22semidirect
33. C22×C4×C389C22semidirect
34. C4⋊C4×C389C22semidirect
35. C22⋊C4×C389C22semidirect
36. C22⋊C4×C389C22semidirect
42. C22×D389D4semidirect
95. C389#534semidirect
101. #123D389semidirect
123. C22×C4C2×D389semidirect
125. C4⋊C4C2×D389semidirect
131. C22⋊C4C2×D389semidirect
132. C22⋊C4C2×D389semidirect

Maximal subgroups

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

subgroupindex
C22×D15562
C2×D4×D3892
#271593692
#271593672
#271593592
#271593572
#271593542
#271593522
C22⋊C4×D3892
#123×C3892
#534389