Finite Group Databasebeta

← #27157169

Subgroups — #27157169

Order 12416.

220 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 220 conjugacy classes of subgroups — too many to draw. Only its 66 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#20261754C211yesmaximal, characteristic
2#20261679C211yesmaximal, characteristic
3#20261589C212yesmaximal
4#20261588C212yesmaximal
5#20261289C211yesmaximal, characteristic
6C2×C4⋊#87604—21no—
7C4⋊#11715897—21no—
8#15690867C2212yes—
9C4×#11715897—21no—
10(C2×#87604)⋊C4—21no—
11(C2×#87604)⋊C4C2212yes—
12#15690809C411yescharacteristic, complemented
13#15690808C411yescharacteristic, complemented
14#15690800—21no—
15#15690790C2211yescharacteristic
16#15690788—21no—
17#15690786C2211yescharacteristic
18C4×Dic194—21no—
19C4×Dic194C411yescharacteristic, complemented
20#15690767—21no—
21C42×D97C2211yescharacteristic
22C4×Q8×C97C411yescharacteristic, complemented
23C22×#87604—22no—
24C2×#11715897—22no—
25C4⋊#87604—42no—
26C2×Dic194—21no—
27C2×Dic194C2×C411yescharacteristic
28C2×C4×D97D411yescharacteristic
29C2×C4×D97D411yescharacteristic
30C2×C4×D97C2311yescharacteristic
31C4⋊Dic97—21no—
32C4⋊Dic97C2×C411yescharacteristic
33#13095490—21no—
34#13095490—21no—
35#13095490—21no—
36#13095490—21no—
37#13095490—21no—
38#13095488—21no—
39#13095488—21no—
40C4×Dic97—21no—
41C4×Dic97C2×C411yescharacteristic
42C4×Dic97D411yescharacteristic
43C4×Dic97D411yescharacteristic
44C2×Q8×C97C2×C411yescharacteristic
45C4⋊C4×C97—21no—
46C4⋊C4×C97C2×C411yescharacteristic
47C42×C97—21no—
48C42×C97C2×C411yescharacteristic
49C22×D97C2×D411yescharacteristic
50C2×#87604—42no—
51C2×#87604—42no—
52C2×#87604—42no—
53Q8×C97—41no—
54C2×C4×C97—21no—
55C2×C4×C97—21no—
56C2×C4×C97C22⋊C411yescharacteristic
57C2×C4×C97C22⋊C411yescharacteristic
58C2×C4×C97C22×C411yescharacteristic
59C2×Dic97—21no—
60C2×Dic97—21no—
61C2×Dic97C22⋊C411yescharacteristic
62C2×Dic97C22⋊C411yescharacteristic
63C2×Dic97C2×D411yescharacteristic
64C4×D97—41no—
65C4×D97—41no—
66C4×D97SD1612yes—
67C4×D97Q1612yes—
68Dic194—42no—
69Dic194—41no—
70#11715897—42no—
71C22×C97C2×C22⋊C411yescharacteristic
72C2×D97—21no—
73C2×D97—21no—
74C2×D97C2×SD1611yescharacteristic
75C2×D97C2×Q1611yescharacteristic
76#87604—82no—
77C4×C97—21no—
78C4×C97—21no—
79C4×C97—41no—
80C4×C97—41no—
81C4×C97#10312yes—
82Dic97—21no—
83Dic97—21no—
84Dic97—41no—
85Dic97—41no—
86Dic97#10312yes—
87C2×C97#41011yescharacteristic
88C2×C97C2×#10311yescharacteristic
89C2×C97#42111yescharacteristic
90D97—21no—
91D97—21no—
92#1518—971nomaximal, Sylow
93C97#151811yescharacteristic, complemented, Sylow
94#533—971no—
95#431—971no—
96#382—971no—
97#328—972no—
98#328—972no—
99C4⋊Q8—971no—
100C4⋊Q8—971no—
101#125—1941no—
102C22⋊Q8—971no—
103C22⋊Q8—971no—
104C22⋊Q8—1941no—
105C4×Q8—971no—
106C4×Q8—1941no—
107C4×Q8#8760411yescharacteristic, complemented
108#117—1941no—
109C2×C4⋊C4—1941no—
110C2×C42—971no—
111C4⋊C8—1941no—
112C22⋊C8—972no—
113C4×C8—1941no—
114#95—972no—
115#95—1941no—
116C2×Q8—971no—
117C2×Q8—1941no—
118C2×Q8C2×#8760411yescharacteristic
119C22×C4—971no—
120C22×C4—971no—
121C22×C4—971no—
122C22×C4—1942no—
123C2×C8—1942no—
124C4⋊C4—21no—
125C4⋊C4—971no—
126C4⋊C4—3882no—
127C4⋊C4—1941no—
128C4⋊C4—1941no—
129C4⋊C4—1941no—
130C4⋊C4—1941no—
131C4⋊C4—1941no—
132C4⋊C4—1941no—
133C4⋊C4C2×#8760411yescharacteristic
134C22⋊C4—1941no—
135C22⋊C4—1941no—
136C42—21no—
137C42—971no—
138C42—971no—
139C42—971no—
140C42—1941no—
141C42C2×#8760411yescharacteristic
142C23—971no—
143Q8—41no—
144Q8—3881no—
145Q8—3882no—
146C2×C4—21no—
147C2×C4—21no—
148C2×C4—971no—
149C2×C4—972no—
150C2×C4—972no—
151C2×C4—971no—
152C2×C4—971no—
153C2×C4—3881no—
154C2×C4—3881no—
155C2×C4—3882no—
156C2×C4—3882no—
157C2×C4—3882no—
158C2×C4—1941no—
159C2×C4—1941no—
160C2×C4C22⋊#8760411yescharacteristic
161C2×C4C22⋊#8760411yescharacteristic
162C2×C4C22×#8760411yescharacteristic
163C8—3882no—
164C22—971no—
165C22—971no—
166C22—1941no—
167C22—1941no—
168C22C2×C22⋊#8760411yescharacteristic
169C4—21no—
170C4—21no—
171C4—41no—
172C4—41no—
173C4—7762no—
174C4—972no—
175C4—3881no—
176C4—3881no—
177C4—1941no—
178C4—1941no—
179C4#1569087312yes—
180C2—1941no—
181C2—1941no—
182C2#2026171411yescharacteristic
183C2C2×#1569087311yescharacteristic
184C2#2026173211yescharacteristic
subgroupG/H
centerC22C2×C22⋊#87604
commutatorC2×C4×C97C22×C4
FrattiniC2×C4C22×#87604
FittingC4×Q8×C97C4
socleC22×C97C2×C22⋊C4
prime radicalC22×D97C2×D4
cosocleC2×C4×D97C23
hypercenterC4×Q8#87604
hypocenterC97#1518
LCSG ▷ C2×C4×C97 ▷ C22×C97 ▷ C97
UCSC1 ◁ C22 ◁ C42 ◁ C4×Q8

Complements

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

NH ≅ G/Ntype
12. #15690809C4semidirect
13. #15690808C4semidirect
19. C4×Dic194C4semidirect
22. C4×Q8×C97C4semidirect
93. C97#1518semidirect
107. C4×Q8#87604semidirect

Maximal subgroups

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

subgroupindex
#202617542
#202616792
#202615892
#202615882
#202612892
#151897