Finite Group Databasebeta

← #27140472

Subgroups — #27140472

Order 12416.

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

normal subgroups only

This group has 210 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#20260757C211yesmaximal, characteristic
3#20260757C211yesmaximal, characteristic
4#20260482C211yesmaximal, characteristic, complemented
5#20260481C211yesmaximal, characteristic, complemented
6#20260480C211yesmaximal, characteristic, complemented
7#332×C97C211yesmaximal, characteristic, complemented
8C2×#13095494—21no—
9C2×#13095494—21no—
10#15690810C2211yescharacteristic
11#15690801C412yes—
12#15690790—21no—
13#15690786—21no—
14Q8×Dic97—21no—
15#15690775—21no—
16#15690768C412yes—
17C42×D97C2211yescharacteristic
18C4⋊#11715895C2211yescharacteristic
19C4⋊#11715895C2211yescharacteristic
20#15690739—21no—
21#15690739—21no—
22C4⋊Q8×C97C2211yescharacteristic
23C4⋊C8×C97C2211yescharacteristic
24C4⋊C8×C97C2211yescharacteristic
25C2×Dic194—21no—
26C2×C4×D97C2×C411yescharacteristic
27C2×C4×D97D411yescharacteristic
28C2×C4×D97D411yescharacteristic
29#13095494—42no—
30#13095494—42no—
31C2×#11715895—21no—
32C2×#11715895—21no—
33C4⋊Dic97—21no—
34C4⋊Dic97—22no—
35C4⋊Dic97C2×C412yes—
36#13095490—22no—
37#13095488—22no—
38C4×Dic97—22no—
39C4×Dic97C2×C411yescharacteristic
40C4×Dic97D411yescharacteristic
41C4×Dic97D411yescharacteristic
42C2×Q8×C97—21no—
43C2×C8×C97—21no—
44C2×C8×C97—21no—
45C4⋊C4×C97—21no—
46C4⋊C4×C97C2×C412yes—
47C42×C97C2311yescharacteristic
48C22×D97C22⋊C411yescharacteristic
49Q8×C97—41no—
50Q8×C97—41no—
51C2×C4×C97—22no—
52C2×C4×C97C22×C411yescharacteristic
53C2×C4×C97C2×D411yescharacteristic
54C2×C4×C97C2×D411yescharacteristic
55C2×Dic97—22no—
56C2×Dic97C22⋊C411yescharacteristic
57C2×Dic97C22⋊C411yescharacteristic
58C2×Dic97C22⋊C411yescharacteristic
59C4×D97—21no—
60C4×D97—21no—
61C4×D97—21no—
62C4×D97—21no—
63C4×D97—22no—
64Dic194—41no—
65Dic194—41no—
66C8×C97—41no—
67C8×C97—41no—
68#11715895—41no—
69#11715895—41no—
70C22×C97C2×C22⋊C411yescharacteristic
71C2×D97—21no—
72C2×D97—21no—
73C2×D97—21no—
74C4×C97—21no—
75C4×C97—42no—
76C4×C97AGL(1,ℤ/8)11yescharacteristic
77C4×C97AGL(1,ℤ/8)11yescharacteristic
78C4×C97#13711yescharacteristic
79C4×C97#13711yescharacteristic
80Dic97—21no—
81Dic97—21no—
82Dic97—42no—
83Dic97#10112yes—
84C2×C97C2×#10111yescharacteristic
85C2×C97#41711yescharacteristic
86C2×C97#41711yescharacteristic
87D97—41no—
88#1561—971nomaximal, Sylow
89C97#156111yescharacteristic, complemented, Sylow
90#533—971no—
91#423—971no—
92#423—971no—
93#332—971no—
94#332—971no—
95#332—971no—
96#332D9711yescharacteristic, complemented
97C2×M16—1941no—
98C2×M16—1941no—
99C4⋊Q8—971no—
100C4⋊Q8C2×D9711yescharacteristic
101#125—972no—
102C22⋊Q8—1941no—
103C22⋊Q8—1941no—
104C4×Q8—1941no—
105C4×Q8—1941no—
106#117—972no—
107C2×C42—971no—
108C4⋊C8—971no—
109C4⋊C8—971no—
110C4⋊C8—1941no—
111C4⋊C8—1941no—
112C4⋊C8C2×D9711yescharacteristic
113C4⋊C8C2×D9711yescharacteristic
114C2×Q8—21no—
115C2×Q8—1941no—
116C22×C4—971no—
117C22×C4—971no—
118C22×C4—971no—
119M16—3882no—
120M16—3882no—
121C2×C8—21no—
122C2×C8—21no—
123C2×C8—1941no—
124C2×C8—1941no—
125C4⋊C4—21no—
126C4⋊C4—972no—
127C4⋊C4—1942no—
128C4⋊C4—1942no—
129C4⋊C4—1941no—
130C4⋊C4C4×D9712yes—
131C22⋊C4—1942no—
132C42—971no—
133C42—971no—
134C42—971no—
135C42—1942no—
136C42C22×D9711yescharacteristic
137C23—971no—
138Q8—41no—
139Q8—41no—
140Q8—3881no—
141Q8—3881no—
142C2×C4—22no—
143C2×C4—971no—
144C2×C4—971no—
145C2×C4—971no—
146C2×C4—1942no—
147C2×C4—1942no—
148C2×C4—1941no—
149C2×C4—1941no—
150C2×C4—1941no—
151C2×C4—1941no—
152C2×C4C2×C4×D9711yescharacteristic
153C2×C4D4×D9711yescharacteristic
154C2×C4D4×D9711yescharacteristic
155C8—41no—
156C8—41no—
157C8—3881no—
158C8—3881no—
159C22—1941no—
160C22—1941no—
161C22—1941no—
162C22C22⋊C4×D9711yescharacteristic
163C4—21no—
164C4—42no—
165C4—972no—
166C4—3882no—
167C4—1941no—
168C4—1941no—
169C4#1569083411yescharacteristic
170C4#1569083711yescharacteristic
171C4#1569083911yescharacteristic
172C4#1569084211yescharacteristic
173C2—3881no—
174C2#101×D9711yescharacteristic
175C2#2026073311yescharacteristic
176C2#2026073511yescharacteristic
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#1561
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
89. C97#1561semidirect
96. #332D97semidirect

Maximal subgroups

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

subgroupindex
#202612902
#202607572
#202604822
#202604812
#202604802
#332×C972
#156197