Finite Group Databasebeta

← #28384141

Subgroups — #28384141

Order 14272.

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×D892C211yesmaximal, characteristic, complemented
2C2×D4×D223C211yesmaximal, characteristic, complemented
3#20297289C212yesmaximal, complemented
4#20297287C211yesmaximal, characteristic, complemented
5#20297279C212yesmaximal, complemented
6#20297277C212yesmaximal, complemented
7#20297274C211yesmaximal, characteristic, complemented
8#20297274C212yesmaximal, complemented
9#20297272C211yesmaximal, characteristic, complemented
10C22⋊C4×D223C211yesmaximal, characteristic, complemented
11#123×C223C211yesmaximal, characteristic, complemented
12C23×D223C2211yescharacteristic
13C23×D223C2211yescharacteristic, complemented
14C23×D223C2211yescharacteristic, complemented
15C2×D892—24no—
16C2×D892C2212yes—
17C2×D892C2212yescomplemented
18C2×D892C2211yescharacteristic, complemented
19C2×D892C2212yescomplemented
20C2×C22⋊D223C2211yescharacteristic
21C2×C22⋊D223C2211yescharacteristic, complemented
22C2×C22⋊D223C2212yescomplemented
23D4×D223—24no—
24C2×C4×D223C2211yescharacteristic, complemented
25C2×C4×D223C2212yescomplemented
26Dic223⋊C4C2212yescomplemented
27C22⋊Dic223C2211yescharacteristic, complemented
28(C2×D223)⋊C4C2212yescomplemented
29(C2×D223)⋊C4C2212yescomplemented
30(C2×D223)⋊C4C2212yescomplemented
31(C2×D223)⋊C4C2212yescomplemented
32C2×D4×C223C2211yescharacteristic
33C22×C4×C223C2211yescharacteristic, complemented
34C4⋊C4×C223C2212yescomplemented
35C22⋊C4×C223C2211yescharacteristic, complemented
36C22⋊C4×C223C2212yescomplemented
37C22×D223—22no—
38C22×D223—22no—
39C22×D223—42no—
40C22×D223—42no—
41C22×D223—42no—
42C22×D223D414yescomplemented
43C22×D223C2311yescharacteristic
44C22×D223C2312yes—
45C22×D223C2311yescharacteristic
46C22×D223C2311yescharacteristic
47D892—22no—
48D892—24no—
49D892—24no—
50D892—42no—
51C22⋊D223—22no—
52C22⋊D223—22no—
53C22⋊D223—42no—
54C2×Dic223C2312yes—
55C2×Dic223C2311yescharacteristic
56C4×D223—22no—
57C4×D223—42no—
58C23×C223C2311yescharacteristic
59C23×C223C2311yescharacteristic
60D4×C223—22no—
61C2×C4×C223—22no—
62C2×C4×C223C2312yes—
63C2×C4×C223C2312yes—
64C2×C4×C223C2311yescharacteristic
65C22×C223—22no—
66C22×C223—21no—
67C22×C223—22no—
68C22×C223C2×D412yes—
69C22×C223C2411yescharacteristic
70C2×D223—22no—
71C2×D223—22no—
72C2×D223—21no—
73C2×D223—22no—
74C2×D223—22no—
75C2×D223—22no—
76C2×D223—22no—
77C2×D223—44no—
78C2×D223—44no—
79C2×D223—44no—
80C2×D223C2×D412yes—
81C2×D223C2×D412yes—
82C4×C223—22no—
83C4×C223—21no—
84C4×C223—22no—
85Dic223—21no—
86Dic223—22no—
87C2×C223—22no—
88C2×C223—41no—
89C2×C223C22×D411yescharacteristic
90C2×C223H(5,𝔽2)12yes—
91D223—22no—
92D223—22no—
93D223—41no—
94D223—42no—
95C223#53411yescharacteristic, complemented, Sylow
96#534—2231nomaximal, Sylow
97C22×D4—2231no—
98C22×D4—2231no—
99#123—2231no—
100#123—2232no—
101#123D22311yescharacteristic, complemented
102#121—2232no—
103#121—2232no—
104#120—2231no—
105#120—2232no—
106#120—2231no—
107C2×C22⋊C4—2231no—
108C24—2231no—
109C24—2231no—
110C24—2231no—
111C2×D4—4464no—
112C2×D4—4464no—
113C2×D4—2232no—
114C2×D4—2231no—
115C2×D4—2231no—
116C2×D4—2232no—
117C2×D4—2232no—
118C2×D4—2231no—
119C2×D4—2232no—
120C2×D4C2×D22311yescharacteristic
121C22×C4—2232no—
122C22×C4—2231no—
123C22×C4C2×D22311yescharacteristic, complemented
124C4⋊C4—2232no—
125C4⋊C4C2×D22312yescomplemented
126C22⋊C4—2231no—
127C22⋊C4—2232no—
128C22⋊C4—2232no—
129C22⋊C4—2232no—
130C22⋊C4—2232no—
131C22⋊C4C2×D22311yescharacteristic, complemented
132C22⋊C4C2×D22312yescomplemented
133C23—8922no—
134C23—8922no—
135C23—8922no—
136C23—4462no—
137C23—4462no—
138C23—2234no—
139C23—2232no—
140C23—2231no—
141C23—2231no—
142C23—2231no—
143C23C22×D22311yescharacteristic
144C23C22×D22311yescharacteristic
145D4—22no—
146D4—8922no—
147D4—8922no—
148D4—4464no—
149D4—4464no—
150D4—4462no—
151D4—4462no—
152D4—4462no—
153C2×C4—22no—
154C2×C4—8922no—
155C2×C4—4462no—
156C2×C4—2232no—
157C2×C4—2231no—
158C2×C4C22×D22312yes—
159C2×C4C22×D22311yescharacteristic
160C2×C4C22×D22312yes—
161C22—22no—
162C22—21no—
163C22—22no—
164C22—8924no—
165C22—8924no—
166C22—8924no—
167C22—4462no—
168C22—4461no—
169C22—4462no—
170C22—4462no—
171C22—4462no—
172C22—4462no—
173C22—4462no—
174C22—2232no—
175C22—2232no—
176C22D4×D22312yes—
177C22C23×D22311yescharacteristic
178C4—22no—
179C4—21no—
180C4—22no—
181C4—4461no—
182C4—4462no—
183C2—22no—
184C2—41no—
185C2—8922no—
186C2—8921no—
187C2—4462no—
188C2—4462no—
189C2C2×D4×D22311yescharacteristic
190C2#2029735012yes—
subgroupG/H
center, FrattiniC22C23×D223
commutator, socle, cosocleC22×C223C24
Fitting#123×C223C2
hypercenter#123D223
hypocenterC223#534
LCSG ▷ C22×C223 ▷ C223
UCSC1 ◁ C22 ◁ #123

Complements

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

NH ≅ G/Ntype
1. C22×D892C2semidirect
2. C2×D4×D223C2semidirect
3. #20297289C2semidirect
4. #20297287C2semidirect
5. #20297279C2semidirect
6. #20297277C2semidirect
7. #20297274C2semidirect
8. #20297274C2semidirect
9. #20297272C2semidirect
10. C22⋊C4×D223C2semidirect
11. #123×C223C2semidirect
13. C23×D223C22semidirect
14. C23×D223C22semidirect
17. C2×D892C22semidirect
18. C2×D892C22semidirect
19. C2×D892C22semidirect
21. C2×C22⋊D223C22semidirect
22. C2×C22⋊D223C22semidirect
24. C2×C4×D223C22semidirect
25. C2×C4×D223C22semidirect
26. Dic223⋊C4C22semidirect
27. C22⋊Dic223C22semidirect
28. (C2×D223)⋊C4C22semidirect
29. (C2×D223)⋊C4C22semidirect
30. (C2×D223)⋊C4C22semidirect
31. (C2×D223)⋊C4C22semidirect
33. C22×C4×C223C22semidirect
34. C4⋊C4×C223C22semidirect
35. C22⋊C4×C223C22semidirect
36. C22⋊C4×C223C22semidirect
42. C22×D223D4semidirect
95. C223#534semidirect
101. #123D223semidirect
123. C22×C4C2×D223semidirect
125. C4⋊C4C2×D223semidirect
131. C22⋊C4C2×D223semidirect
132. C22⋊C4C2×D223semidirect

Maximal subgroups

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

subgroupindex
C22×D8922
C2×D4×D2232
#202972892
#202972872
#202972792
#202972772
#202972742
#202972722
C22⋊C4×D2232
#123×C2232
#534223