Finite Group Databasebeta

← #28384086

Subgroups — #28384086

Order 14272.

330 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 330 conjugacy classes of subgroups — too many to draw. Only its 111 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
2C23×Dic223C211yesmaximal, characteristic, complemented
3#20297288C211yesmaximal, characteristic, complemented
4#20297285C211yesmaximal, characteristic, complemented
5#20297281C212yesmaximal, complemented
6#20297276C212yesmaximal, complemented
7#20297273C211yesmaximal, characteristic, complemented
8D4×Dic223C212yesmaximal, complemented
9(C22⋊D223)⋊C4C212yesmaximal, complemented
10#20297260C211yesmaximal, characteristic, complemented
11#120×C223C211yesmaximal, characteristic, complemented
12#16813169—24no—
13C2×Dic446C2211yescharacteristic, complemented
14C2×C22⋊D223C2212yes—
15C2×C22⋊D223C2211yescharacteristic
16C2×C22⋊D223C2211yescharacteristic, complemented
17C22×Dic223—24no—
18C22×Dic223C2212yes—
19C22×Dic223C2212yes—
20C22×Dic223C2211yescharacteristic
21C22×Dic223C2211yescharacteristic
22C22×Dic223C2211yescharacteristic, complemented
23C2×C4×D223C2211yescharacteristic, complemented
24C4⋊Dic223C2212yescomplemented
25Dic223⋊C4C2212yescomplemented
26Dic223⋊C4C2212yescomplemented
27C22⋊Dic223C2211yescharacteristic
28C22⋊Dic223C2212yes—
29C22⋊Dic223C2212yescomplemented
30(C2×D223)⋊C4C2212yescomplemented
31C4×Dic223C2212yescomplemented
32C24×C223C2211yescharacteristic
33C2×D4×C223C2212yes—
34C2×D4×C223C2211yescharacteristic
35C22⋊C4×C223C2211yescharacteristic
36C22⋊C4×C223C2212yes—
37C22×D223C2311yescharacteristic
38Dic446—22no—
39C22⋊D223—22no—
40C22⋊D223—24no—
41C22⋊D223—22no—
42C2×Dic223—22no—
43C2×Dic223—24no—
44C2×Dic223—24no—
45C2×Dic223—22no—
46C2×Dic223D414yescomplemented
47C2×Dic223C2312yes—
48C2×Dic223C2312yes—
49C2×Dic223C2311yescharacteristic
50C2×Dic223C2311yescharacteristic
51C2×Dic223C2311yescharacteristic
52C4×D223—22no—
53C23×C223—22no—
54C23×C223—24no—
55C23×C223C2311yescharacteristic
56C23×C223C2311yescharacteristic
57C23×C223C2312yes—
58D4×C223—22no—
59D4×C223—24no—
60C2×C4×C223C2312yes—
61C2×C4×C223C2311yescharacteristic
62C22×C223—28no—
63C22×C223—22no—
64C22×C223—22no—
65C22×C223—22no—
66C22×C223—21no—
67C22×C223—22no—
68C22×C223C2×D412yes—
69C22×C223#4114yescomplemented
70C22×C223C2411yescharacteristic
71C2×D223—21no—
72C2×D223—22no—
73C4×C223—21no—
74C4×C223—22no—
75Dic223—22no—
76Dic223—21no—
77Dic223—22no—
78Dic223C2×D412yes—
79Dic223C2×D412yes—
80C2×C223—22no—
81C2×C223—24no—
82C2×C223—41no—
83C2×C223C22×D411yescharacteristic
84C2×C223C2×#4112yes—
85D223—41no—
86C223#52511yescharacteristic, complemented, Sylow
87#525—2231nomaximal, Sylow
88C2×#41—2231no—
89C23×C4—2231no—
90#123—2231no—
91#123—2231no—
92C22⋊Q8—2232no—
93#121—2232no—
94#120—2231no—
95#120D22311yescharacteristic, complemented
96C4×D4—2232no—
97C4×D4—2232no—
98#117—2231no—
99C24C2×D22311yescharacteristic
100#41—4464no—
101C2×Q8—2231no—
102C2×D4—2232no—
103C2×D4—2231no—
104C2×D4—2231no—
105C2×D4C2×D22312yes—
106C2×D4C2×D22311yescharacteristic
107C22×C4—4464no—
108C22×C4—2232no—
109C22×C4—2232no—
110C22×C4—2231no—
111C22×C4—2231no—
112C22×C4—2231no—
113C22×C4—2231no—
114C4⋊C4—2232no—
115C4⋊C4—2232no—
116C4⋊C4—2232no—
117C22⋊C4—2231no—
118C22⋊C4—2232no—
119C22⋊C4—2232no—
120C22⋊C4—2232no—
121C22⋊C4C2×D22311yescharacteristic
122C22⋊C4C2×D22312yes—
123C42—2232no—
124C23—22no—
125C23—24no—
126C23—2231no—
127C23C22×D22311yescharacteristic
128C23C22×D22311yescharacteristic
129C23C22×D22312yes—
130Q8—4462no—
131D4—22no—
132D4—24no—
133D4—4462no—
134D4—4464no—
135D4—4462no—
136C2×C4—4462no—
137C2×C4—4462no—
138C2×C4—4464no—
139C2×C4—4464no—
140C2×C4—4462no—
141C2×C4—2232no—
142C2×C4—2232no—
143C2×C4—2231no—
144C2×C4—2234no—
145C2×C4—2231no—
146C2×C4—2231no—
147C2×C4C22×D22312yes—
148C2×C4C22×D22311yescharacteristic
149C22—28no—
150C22—22no—
151C22—22no—
152C22—22no—
153C22—21no—
154C22—22no—
155C22—4461no—
156C22—4462no—
157C22D4×D22312yes—
158C22#1681316914yescomplemented
159C22C23×D22311yescharacteristic
160C4—21no—
161C4—22no—
162C4—4462no—
163C4—4461no—
164C4—4462no—
165C4—2232no—
166C4—2232no—
167C2—22no—
168C2—24no—
169C2—41no—
170C2—8921no—
171C2C2×D4×D22311yescharacteristic
172C2C2×#1681316912yes—
subgroupG/H
center, FrattiniC22C23×D223
commutator, socle, cosocleC22×C223C24
Fitting#120×C223C2
hypercenter#120D223
hypocenterC223#525
LCSG ▷ C22×C223 ▷ C223
UCSC1 ◁ C22 ◁ #120

Complements

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

NH ≅ G/Ntype
1. C2×#16813169C2semidirect
2. C23×Dic223C2semidirect
3. #20297288C2semidirect
4. #20297285C2semidirect
5. #20297281C2semidirect
6. #20297276C2semidirect
7. #20297273C2semidirect
8. D4×Dic223C2semidirect
9. (C22⋊D223)⋊C4C2semidirect
10. #20297260C2semidirect
11. #120×C223C2semidirect
13. C2×Dic446C22semidirect
16. C2×C22⋊D223C22semidirect
22. C22×Dic223C22semidirect
23. C2×C4×D223C22semidirect
24. C4⋊Dic223C22semidirect
25. Dic223⋊C4C22semidirect
26. Dic223⋊C4C22semidirect
29. C22⋊Dic223C22semidirect
30. (C2×D223)⋊C4C22semidirect
31. C4×Dic223C22semidirect
46. C2×Dic223D4semidirect
69. C22×C223#41semidirect
86. C223#525semidirect
95. #120D223semidirect
158. C22#16813169semidirect

Maximal subgroups

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

subgroupindex
C2×#168131692
C23×Dic2232
#202972882
#202972852
#202972812
#202972762
#202972732
D4×Dic2232
(C22⋊D223)⋊C42
#202972602
#120×C2232
#525223