Finite Group Databasebeta

← #27140435

Subgroups — #27140435

Order 12416.

208 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 208 conjugacy classes of subgroups — too many to draw. Only its 67 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#20261290C211yesmaximal, characteristic
2#20260791C211yesmaximal, characteristic
3C4⋊C8×D97C211yesmaximal, characteristic
4#20260482C211yesmaximal, characteristic, complemented
5#20260481C211yesmaximal, characteristic, complemented
6#20260480C211yesmaximal, characteristic, complemented
7#332×C97C211yesmaximal, characteristic, complemented
8C2×C8×D97—21no—
9#15690810C2211yescharacteristic
10#15690801C412yes—
11#15690790—21no—
12#15690786—21no—
13Q8×Dic97—21no—
14#15690775—21no—
15#15690768C412yes—
16C42×D97C2211yescharacteristic
17C4⋊#11715895C2211yescharacteristic
18C4⋊#11715895C2211yescharacteristic
19#15690739—21no—
20#15690722—21no—
21#15690720—21no—
22C4⋊Q8×C97C2211yescharacteristic
23C4⋊C8×C97C2211yescharacteristic
24C4⋊C8×C97C2211yescharacteristic
25C2×Dic194—21no—
26C2×C4×D97C2×C411yescharacteristic
27C2×C4×D97D411yescharacteristic
28C2×C4×D97D411yescharacteristic
29C2×#11715895—21no—
30C2×#11715895—21no—
31C8×D97—42no—
32C4⋊Dic97—21no—
33C4⋊Dic97—22no—
34C4⋊Dic97C2×C412yes—
35#13095490—22no—
36#13095488—22no—
37C4×Dic97—22no—
38C4×Dic97C2×C411yescharacteristic
39C4×Dic97D411yescharacteristic
40C4×Dic97D411yescharacteristic
41C2×Q8×C97—21no—
42C2×C8×C97—21no—
43C2×C8×C97—21no—
44C4⋊C4×C97—21no—
45C4⋊C4×C97C2×C412yes—
46C42×C97C2311yescharacteristic
47C22×D97C22⋊C411yescharacteristic
48Q8×C97—41no—
49Q8×C97—41no—
50C2×C4×C97—22no—
51C2×C4×C97C22×C411yescharacteristic
52C2×C4×C97C2×D411yescharacteristic
53C2×C4×C97C2×D411yescharacteristic
54C2×Dic97—22no—
55C2×Dic97C22⋊C411yescharacteristic
56C2×Dic97C22⋊C411yescharacteristic
57C2×Dic97C22⋊C411yescharacteristic
58C4×D97—22no—
59C4×D97—21no—
60C4×D97—21no—
61C4×D97—21no—
62C4×D97—21no—
63Dic194—41no—
64Dic194—41no—
65C8×C97—41no—
66C8×C97—41no—
67#11715895—41no—
68#11715895—41no—
69C22×C97C2×C22⋊C411yescharacteristic
70C2×D97—21no—
71C2×D97—21no—
72C2×D97#10112yes—
73C4×C97—21no—
74C4×C97—42no—
75C4×C97#13511yescharacteristic
76C4×C97#13511yescharacteristic
77C4×C97AGL(1,ℤ/8)11yescharacteristic
78C4×C97#13711yescharacteristic
79Dic97—21no—
80Dic97—21no—
81Dic97—21no—
82Dic97—42no—
83C2×C97C2×#10111yescharacteristic
84C2×C97#41611yescharacteristic
85C2×C97#41711yescharacteristic
86D97—22no—
87#1552—971nomaximal, Sylow
88C97#155211yescharacteristic, complemented, Sylow
89#533—971no—
90#432—971no—
91C2×C4⋊C8—971no—
92#332—971no—
93#332—971no—
94#332—971no—
95#332D9711yescharacteristic, complemented
96C22×C8—1941no—
97C4⋊Q8—971no—
98C4⋊Q8C2×D9711yescharacteristic
99#125—972no—
100C22⋊Q8—1941no—
101C22⋊Q8—1941no—
102C4×Q8—1941no—
103C4×Q8—1941no—
104#117—972no—
105C2×C42—971no—
106C4⋊C8—971no—
107C4⋊C8—971no—
108C4⋊C8—1941no—
109C4⋊C8C2×D9711yescharacteristic
110C4⋊C8C2×D9711yescharacteristic
111C22⋊C8—1941no—
112#97—1941no—
113C2×Q8—21no—
114C2×Q8—1941no—
115C22×C4—971no—
116C22×C4—971no—
117C22×C4—971no—
118C2×C8—21no—
119C2×C8—21no—
120C2×C8—3882no—
121C2×C8—1941no—
122C2×C8—1941no—
123C4⋊C4—21no—
124C4⋊C4—972no—
125C4⋊C4—1942no—
126C4⋊C4—1942no—
127C4⋊C4—1941no—
128C4⋊C4C4×D9712yes—
129C22⋊C4—1942no—
130C42—971no—
131C42—971no—
132C42—971no—
133C42—1942no—
134C42C22×D9711yescharacteristic
135C23—971no—
136Q8—41no—
137Q8—41no—
138Q8—3881no—
139Q8—3881no—
140C2×C4—22no—
141C2×C4—971no—
142C2×C4—971no—
143C2×C4—971no—
144C2×C4—1942no—
145C2×C4—1942no—
146C2×C4—1941no—
147C2×C4—1941no—
148C2×C4—1941no—
149C2×C4—1941no—
150C2×C4C2×C4×D9711yescharacteristic
151C2×C4D4×D9711yescharacteristic
152C2×C4D4×D9711yescharacteristic
153C8—41no—
154C8—41no—
155C8—3881no—
156C8—3881no—
157C22—972no—
158C22—1941no—
159C22—1941no—
160C22C22⋊C4×D9711yescharacteristic
161C4—21no—
162C4—42no—
163C4—3882no—
164C4—1941no—
165C4—1941no—
166C4—1941no—
167C4#1569083111yescharacteristic
168C4#1569083311yescharacteristic
169C4#1569083411yescharacteristic
170C4#1569083911yescharacteristic
171C2—1942no—
172C2#101×D9711yescharacteristic
173C2#2026073011yescharacteristic
174C2#2026073311yescharacteristic
subgroupG/H
centerC22C22⋊C4×D97
commutatorC2×C4×C97C22×C4
FrattiniC42C22×D97
Fitting#332×C97C2
socleC22×C97C2×C22⋊C4
prime radicalC22×D97C22⋊C4
cosocleC42×C97C23
hypercenter#332D97
hypocenterC97#1552
LCSG ▷ C2×C4×C97 ▷ C22×C97 ▷ C97
UCSC1 ◁ C22 ◁ C42 ◁ #332

Complements

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

NH ≅ G/Ntype
4. #20260482C2semidirect
5. #20260481C2semidirect
6. #20260480C2semidirect
7. #332×C97C2semidirect
88. C97#1552semidirect
95. #332D97semidirect

Maximal subgroups

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

subgroupindex
#202612902
#202607912
C4⋊C8×D972
#202604822
#202604812
#202604802
#332×C972
#155297