Finite Group Databasebeta

← #28384090

Subgroups — #28384090

Order 14272.

330 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 330 conjugacy classes of subgroups — too many to draw. Only its 109 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×#16813169C211yesmaximal, characteristic, complemented
2C22×C4×D223C211yesmaximal, characteristic, complemented
3#20297286C212yesmaximal, complemented
4#20297284C211yesmaximal, characteristic, complemented
5#20297283C211yesmaximal, characteristic, complemented
6#20297275C211yesmaximal, characteristic, complemented
7#20297272C212yesmaximal, complemented
8D4×Dic223C212yesmaximal, complemented
9(C22⋊D223)⋊C4C212yesmaximal, complemented
10#20297262C211yesmaximal, characteristic, complemented
11#121×C223C211yesmaximal, characteristic, complemented
12C23×D223C2211yescharacteristic
13#16813169—24no—
14C2×Dic446C2211yescharacteristic, complemented
15C2×C22⋊D223C2212yes—
16C2×C22⋊D223C2212yes—
17C22×Dic223C2211yescharacteristic
18C22×Dic223C2212yescomplemented
19C2×C4×D223—24no—
20C2×C4×D223C2211yescharacteristic
21C2×C4×D223C2211yescharacteristic
22C2×C4×D223C2211yescharacteristic
23C2×C4×D223C2211yescharacteristic
24C4⋊Dic223C2211yescharacteristic, complemented
25C4⋊Dic223C2212yescomplemented
26Dic223⋊C4C2212yescomplemented
27C22⋊Dic223C2212yes—
28C22⋊Dic223C2212yescomplemented
29(C2×D223)⋊C4C2212yes—
30(C2×D223)⋊C4C2212yes—
31C4×Dic223C2212yescomplemented
32C2×D4×C223C2212yes—
33C2×D4×C223C2211yescharacteristic, complemented
34C22×C4×C223C2211yescharacteristic
35C4⋊C4×C223C2211yescharacteristic, complemented
36C22⋊C4×C223C2212yescomplemented
37C22×D223—22no—
38C22×D223—22no—
39C22×D223—22no—
40C22×D223C2311yescharacteristic
41C22×D223C2311yescharacteristic
42Dic446—22no—
43C22⋊D223—24no—
44C22⋊D223—24no—
45C2×Dic223—22no—
46C2×Dic223—24no—
47C2×Dic223C2312yes—
48C2×Dic223C2312yes—
49C2×Dic223C2311yescharacteristic
50C2×Dic223C2311yescharacteristic
51C4×D223—22no—
52C4×D223—22no—
53C4×D223—22no—
54C4×D223D414yescomplemented
55C23×C223C2312yes—
56C23×C223C2311yescharacteristic
57D4×C223—24no—
58D4×C223—22no—
59C2×C4×C223—22no—
60C2×C4×C223C2311yescharacteristic
61C2×C4×C223C2312yes—
62C2×C4×C223C2311yescharacteristic
63C22×C223—21no—
64C22×C223—21no—
65C22×C223—22no—
66C22×C223—22no—
67C22×C223—22no—
68C22×C223#4112yescomplemented
69C22×C223C2411yescharacteristic
70C2×D223—28no—
71C2×D223—21no—
72C2×D223—21no—
73C2×D223—21no—
74C2×D223—21no—
75C2×D223C2×D412yes—
76C2×D223#4112yescomplemented
77C4×C223—21no—
78C4×C223—22no—
79C4×C223C2×D412yes—
80Dic223—21no—
81Dic223—22no—
82Dic223—22no—
83Dic223C2×D412yes—
84C2×C223—22no—
85C2×C223—42no—
86C2×C223C22×D411yescharacteristic
87C2×C223C2×#4111yescharacteristic
88C2×C223C2×#4111yescharacteristic
89D223—22no—
90D223—22no—
91C223#52511yescharacteristic, complemented, Sylow
92#525—2231nomaximal, Sylow
93C2×#41—2231no—
94C23×C4—2231no—
95#123—2232no—
96C22⋊Q8—2231no—
97C22⋊Q8—2231no—
98#121—2231no—
99#121D22311yescharacteristic, complemented
100#120—2232no—
101C4×D4—2232no—
102C4×D4—2232no—
103#117—2231no—
104C24—2231no—
105#41—4464no—
106C2×Q8—2231no—
107C2×D4—2232no—
108C2×D4—2232no—
109C2×D4C2×D22312yes—
110C2×D4C2×D22311yescharacteristic, complemented
111C22×C4—4464no—
112C22×C4—2231no—
113C22×C4—2232no—
114C22×C4—2231no—
115C22×C4—2231no—
116C22×C4—2231no—
117C22×C4—2231no—
118C22×C4C2×D22311yescharacteristic
119C4⋊C4—2232no—
120C4⋊C4—2231no—
121C4⋊C4—2232no—
122C4⋊C4C2×D22311yescharacteristic, complemented
123C22⋊C4—2232no—
124C22⋊C4—2232no—
125C22⋊C4—2232no—
126C22⋊C4—2232no—
127C22⋊C4C2×D22312yescomplemented
128C42—2232no—
129C23—4462no—
130C23—4462no—
131C23—4462no—
132C23—2231no—
133C23—2231no—
134C23C22×D22312yes—
135C23C22×D22311yescharacteristic
136Q8—4462no—
137D4—24no—
138D4—22no—
139D4—4464no—
140D4—4464no—
141C2×C4—22no—
142C2×C4—4462no—
143C2×C4—4462no—
144C2×C4—4462no—
145C2×C4—4464no—
146C2×C4—4462no—
147C2×C4—2232no—
148C2×C4—2231no—
149C2×C4—2232no—
150C2×C4—2231no—
151C2×C4—2234no—
152C2×C4C22×D22312yes—
153C2×C4C22×D22311yescharacteristic
154C2×C4C22×D22311yescharacteristic
155C22—21no—
156C22—21no—
157C22—22no—
158C22—22no—
159C22—22no—
160C22—4468no—
161C22—4461no—
162C22—4461no—
163C22—4461no—
164C22—4461no—
165C22—2232no—
166C22—2232no—
167C22#1681316912yescomplemented
168C22C23×D22311yescharacteristic
169C4—21no—
170C4—22no—
171C4—4461no—
172C4—4462no—
173C4—4462no—
174C4—2232no—
175C4D4×D22312yes—
176C2—22no—
177C2—42no—
178C2—4462no—
179C2—4462no—
180C2C2×D4×D22311yescharacteristic
181C2#41×D22311yescharacteristic
182C2C2×#1681316911yescharacteristic
subgroupG/H
center, FrattiniC22C23×D223
commutator, socle, cosocleC22×C223C24
Fitting#121×C223C2
hypercenter#121D223
hypocenterC223#525
LCSG ▷ C22×C223 ▷ C223
UCSC1 ◁ C22 ◁ #121

Complements

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

NH ≅ G/Ntype
1. C2×#16813169C2semidirect
2. C22×C4×D223C2semidirect
3. #20297286C2semidirect
4. #20297284C2semidirect
5. #20297283C2semidirect
6. #20297275C2semidirect
7. #20297272C2semidirect
8. D4×Dic223C2semidirect
9. (C22⋊D223)⋊C4C2semidirect
10. #20297262C2semidirect
11. #121×C223C2semidirect
14. C2×Dic446C22semidirect
18. C22×Dic223C22semidirect
24. C4⋊Dic223C22semidirect
25. C4⋊Dic223C22semidirect
26. Dic223⋊C4C22semidirect
28. C22⋊Dic223C22semidirect
31. C4×Dic223C22semidirect
33. C2×D4×C223C22semidirect
35. C4⋊C4×C223C22semidirect
36. C22⋊C4×C223C22semidirect
54. C4×D223D4semidirect
68. C22×C223#41semidirect
76. C2×D223#41semidirect
91. C223#525semidirect
99. #121D223semidirect
110. C2×D4C2×D223semidirect
122. C4⋊C4C2×D223semidirect
127. C22⋊C4C2×D223semidirect
167. C22#16813169semidirect

Maximal subgroups

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

subgroupindex
C2×#168131692
C22×C4×D2232
#202972862
#202972842
#202972832
#202972752
#202972722
D4×Dic2232
(C22⋊D223)⋊C42
#202972622
#121×C2232
#525223