Finite Group Databasebeta

← #27142715

Subgroups — #27142715

Order 12416.

244 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 244 conjugacy classes of subgroups — too many to draw. Only its 83 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
1#20261288C211yesmaximal, characteristic
2#20260758C211yesmaximal, characteristic
3#20260581C211yesmaximal, characteristic
4#20260496C211yesmaximal, characteristic
5#20260495C211yesmaximal, characteristic
6#20260494C211yesmaximal, characteristic
7#339×C97C211yesmaximal, characteristic
8C2×#13095495C2211yescharacteristic
9#15690809C412yescomplemented
10#15690801C412yescomplemented
11#15690787—22no—
12C4×Dic194—22no—
13#15690769C2211yescharacteristic
14C2×C4×Dic97C2211yescharacteristic
15C2×C4×Dic97C2211yescharacteristic
16#15690739—22no—
17#15690717—22no—
18C2×M16×C97C2211yescharacteristic
19#117×C97C2211yescharacteristic
20C2×C42×C97C2211yescharacteristic
21C2×Dic194—22no—
22C22×Dic97—21no—
23C22×Dic97D412yes—
24#13095495—21no—
25#13095495—21no—
26C2×#11715895—21no—
27C4⋊Dic97—22no—
28C4⋊Dic97C2×C412yes—
29#13095490—22no—
30#13095490—22no—
31C22⋊Dic97—21no—
32C4×Dic97—21no—
33C4×Dic97—21no—
34C4×Dic97C2×C411yescharacteristic
35C4×Dic97C2×C411yescharacteristic
36C4×Dic97D412yes—
37C4×Dic97Q812yescomplemented
38C22×C4×C97—21no—
39C22×C4×C97C2311yescharacteristic
40M16×C97—21no—
41M16×C97—21no—
42C2×C8×C97—21no—
43C4⋊C4×C97C2×C412yes—
44C22⋊C4×C97—21no—
45C42×C97—21no—
46C42×C97—21no—
47C42×C97Q812yescomplemented
48C23×C97C2×D411yescharacteristic
49C2×C4×C97—22no—
50C2×C4×C97—22no—
51C2×C4×C97—42no—
52C2×C4×C97C22×C411yescharacteristic
53C2×C4×C97C2×D411yescharacteristic
54C2×C4×C97C2×Q811yescharacteristic
55C2×C4×C97C2×Q811yescharacteristic
56C2×C4×C97#4111yescharacteristic
57C2×C4×C97#4111yescharacteristic
58C2×Dic97—22no—
59C2×Dic97—22no—
60C2×Dic97—42no—
61C2×Dic97—42no—
62C2×Dic97C22⋊C412yes—
63C2×Dic97C22⋊C412yes—
64Dic194—44no—
65C8×C97—41no—
66#11715895—41no—
67C22×C97—22no—
68C22×C97C2×C22⋊C411yescharacteristic
69C22×C97C22⋊Q811yescharacteristic
70C22×C97C22⋊Q811yescharacteristic
71C4×C97—42no—
72C4×C97—42no—
73C4×C97C4×Q811yescharacteristic
74C4×C97C4×Q811yescharacteristic
75C4×C97#12411yescharacteristic
76C4×C97C4⋊Q811yescharacteristic
77Dic97—22no—
78Dic97—22no—
79Dic97—42no—
80Dic97—42no—
81C2×C97—21no—
82C2×C97—21no—
83C2×C97#39111yescharacteristic
84C2×C97#42112yes—
85#1995—971nomaximal, Sylow
86C97#199511yescharacteristic, complemented, Sylow
87#533—971no—
88#423—971no—
89#376—971no—
90#339—971no—
91#339—971no—
92#339—971no—
93#339D9711yescharacteristic
94C2×M16—971no—
95C2×M16C2×D9711yescharacteristic
96C4⋊Q8—972no—
97#125—972no—
98C22⋊Q8—1942no—
99C4×Q8—1942no—
100#117—971no—
101#117C2×D9711yescharacteristic
102C2×C42—971no—
103C2×C42—971no—
104C2×C42C2×D9711yescharacteristic
105C4⋊C8—1942no—
106#95—1942no—
107C2×Q8—1942no—
108C22×C4—21no—
109C22×C4—972no—
110C22×C4—1941no—
111C22×C4C22×D9711yescharacteristic
112M16—21no—
113M16—21no—
114M16—1941no—
115M16—1941no—
116C2×C8—21no—
117C2×C8—1941no—
118C4⋊C4—972no—
119C4⋊C4—1942no—
120C4⋊C4—1942no—
121C4⋊C4—1942no—
122C4⋊C4C4×D9712yes—
123C22⋊C4—21no—
124C22⋊C4—1941no—
125C42—21no—
126C42—21no—
127C42—971no—
128C42—971no—
129C42—972no—
130C42—972no—
131C42—1941no—
132C42—1941no—
133C42Dic19412yescomplemented
134C23D4×D9711yescharacteristic
135Q8—3884no—
136C2×C4—22no—
137C2×C4—22no—
138C2×C4—42no—
139C2×C4—972no—
140C2×C4—972no—
141C2×C4—3882no—
142C2×C4—3882no—
143C2×C4—1942no—
144C2×C4—1942no—
145C2×C4C2×C4×D9711yescharacteristic
146C2×C4D4×D9711yescharacteristic
147C2×C4Q8×D9711yescharacteristic
148C2×C4C2×Dic19411yescharacteristic
149C2×C4#1309551011yescharacteristic
150C2×C4#1309551111yescharacteristic
151C8—41no—
152C8—3881no—
153C22—22no—
154C22C22⋊C4×D9711yescharacteristic
155C22#1569078711yescharacteristic
156C22#1569078811yescharacteristic
157C4—42no—
158C4—42no—
159C4—3882no—
160C4—3882no—
161C4—1942no—
162C4—1942no—
163C4#1569077511yescharacteristic
164C4C4×Dic19411yescharacteristic
165C4#1569079611yescharacteristic
166C4#1569080911yescharacteristic
167C2—21no—
168C2—21no—
169C2#2026063911yescharacteristic
170C2#2026075012yes—
subgroupG/H
centerC2×C4D4×D97
commutatorC2×C4×C97C22×C4
FrattiniC22×C4C22×D97
Fitting#339×C97C2
socleC22×C97C2×C22⋊C4
prime radicalC23×C97C2×D4
cosocleC22×C4×C97C23
hypercenter#339D97
hypocenterC97#1995
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ #339

Complements

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

NH ≅ G/Ntype
9. #15690809C4semidirect
10. #15690801C4semidirect
37. C4×Dic97Q8semidirect
47. C42×C97Q8semidirect
86. C97#1995semidirect
133. C42Dic194semidirect

Maximal subgroups

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

subgroupindex
#202612882
#202607582
#202605812
#202604962
#202604952
#202604942
#339×C972
#199597