Finite Group Databasebeta

← #27140078

Subgroups — #27140078

Order 12416.

236 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 236 conjugacy classes of subgroups — too many to draw. Only its 69 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×#15690775C211yesmaximal, characteristic
2#20260789C212yesmaximal
3#20260461C211yesmaximal, characteristic
4#20260460C212yesmaximal
5#324×C97C211yesmaximal, characteristic
6C22×Dic194C411yescharacteristic
7#15690775—22no—
8#15690775—22no—
9C2×#13095490C411yescharacteristic
10C2×#13095490C2211yescharacteristic
11C2×C4×Dic97C411yescharacteristic
12C2×C4×Dic97C411yescharacteristic
13C2×C4×Dic97C2211yescharacteristic
14#15690739—22no—
15C22⋊#11715895C2212yes—
16#15690720—22no—
17C2×C4⋊C4×C97C2211yescharacteristic
18C22⋊C8×C97C2212yes—
19C2×Dic194—22no—
20C2×Dic194—22no—
21C2×Dic194—22no—
22C22×Dic97C2×C411yescharacteristic
23C22×Dic97C2×C411yescharacteristic
24C22×Dic97C2×C411yescharacteristic
25C22×Dic97C2×C411yescharacteristic
26C2×#11715895—22no—
27#13095490—22no—
28#13095490—22no—
29C4×Dic97—21no—
30C4×Dic97—21no—
31C4×Dic97—21no—
32C4×Dic97—21no—
33C4×Dic97D412yes—
34C4×Dic97D412yes—
35C22×C4×C97C2×C411yescharacteristic
36C22×C4×C97C2×C411yescharacteristic
37C22×C4×C97C2311yescharacteristic
38C2×C8×C97—22no—
39C4⋊C4×C97—22no—
40C23×C97C22×C411yescharacteristic
41C2×C4×C97—21no—
42C2×C4×C97—21no—
43C2×C4×C97—21no—
44C2×C4×C97—41no—
45C2×C4×C97—41no—
46C2×C4×C97M1612yes—
47C2×C4×C97C2×D412yes—
48C2×Dic97—21no—
49C2×Dic97—21no—
50C2×Dic97—21no—
51C2×Dic97—21no—
52C2×Dic97—22no—
53C2×Dic97—21no—
54C2×Dic97—41no—
55C2×Dic97C22⋊C411yescharacteristic
56C2×Dic97C22⋊C412yes—
57C2×Dic97C22⋊C411yescharacteristic
58C2×Dic97M1612yes—
59Dic194—28no—
60C8×C97—42no—
61#11715895—42no—
62C22×C97—21no—
63C22×C97—21no—
64C22×C97C2×C22⋊C411yescharacteristic
65C22×C97C2×M1611yescharacteristic
66C22×C97C2×M1611yescharacteristic
67C4×C97—22no—
68C4×C97—22no—
69C4×C97—41no—
70Dic97—22no—
71Dic97—22no—
72Dic97—21no—
73Dic97—41no—
74Dic97#10112yes—
75C2×C97—21no—
76C2×C97—21no—
77C2×C97#40711yescharacteristic
78C2×C97#41011yescharacteristic
79C2×C97C2×#10111yescharacteristic
80#1467—971nomaximal, Sylow
81C97#146711yescharacteristic, complemented, Sylow
82C2×C4×Q8—971no—
83#432—972no—
84#324—971no—
85#324—972no—
86#324D9711yescharacteristic
87C22×Q8—971no—
88C4×Q8—1942no—
89C4×Q8—1942no—
90C2×C4⋊C4—971no—
91C2×C4⋊C4—971no—
92C2×C4⋊C4C2×D9711yescharacteristic
93C2×C42—971no—
94C2×C42—971no—
95C2×C42—971no—
96C4⋊C8—1942no—
97C22⋊C8—972no—
98C22⋊C8C2×D9712yes—
99#97—1942no—
100C2×Q8—1942no—
101C2×Q8—1942no—
102C2×Q8—1942no—
103C22×C4—971no—
104C22×C4—971no—
105C22×C4—971no—
106C22×C4—971no—
107C22×C4C4×D9711yescharacteristic
108C22×C4C4×D9711yescharacteristic
109C22×C4C22×D9711yescharacteristic
110C2×C8—22no—
111C2×C8—1942no—
112C4⋊C4—22no—
113C4⋊C4—1942no—
114C4⋊C4—1942no—
115C42—972no—
116C42—972no—
117C42—1941no—
118C42—1941no—
119C42—1941no—
120C42—1941no—
121C23C2×C4×D9711yescharacteristic
122Q8—1948no—
123C2×C4—21no—
124C2×C4—21no—
125C2×C4—21no—
126C2×C4—41no—
127C2×C4—41no—
128C2×C4—971no—
129C2×C4—971no—
130C2×C4—972no—
131C2×C4—972no—
132C2×C4—3881no—
133C2×C4—1942no—
134C2×C4—1941no—
135C2×C4—1941no—
136C2×C4—1941no—
137C2×C4—1941no—
138C2×C4—1941no—
139C2×C4#1309549412yes—
140C2×C4D4×D9712yes—
141C8—42no—
142C8—3882no—
143C22—21no—
144C22—21no—
145C22C22⋊C4×D9711yescharacteristic
146C22M16×D9711yescharacteristic
147C22C2×#1309549411yescharacteristic
148C4—22no—
149C4—22no—
150C4—41no—
151C4—972no—
152C4—3881no—
153C4—1942no—
154C4—1942no—
155C4—1941no—
156C2—21no—
157C2—21no—
158C2#2026069711yescharacteristic
159C2#2026070711yescharacteristic
160C2#101×D9711yescharacteristic
subgroupG/H
centerC22C22⋊C4×D97
commutator, prime radicalC23×C97C22×C4
FrattiniC22×C4C22×D97
Fitting#324×C97C2
socleC22×C97C2×C22⋊C4
cosocleC22×C4×C97C23
hypercenter#324D97
hypocenterC97#1467
LCSG ▷ C23×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C22×C4 ◁ #324

Complements

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

NH ≅ G/Ntype
81. C97#1467semidirect

Maximal subgroups

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

subgroupindex
C2×#156907752
#202607892
#202604612
#202604602
#324×C972
#146797