Finite Group Databasebeta

← #27157253

Subgroups — #27157253

Order 12416.

216 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 68 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#20261740C211yesmaximal, characteristic
2#20261736C211yesmaximal, characteristic
3#20261595C212yesmaximal, complemented
4#20261594C212yesmaximal, complemented
5#20261289C211yesmaximal, characteristic
6C2×#13095521—21no—
7C2×#13095519—21no—
8#15690877—21no—
9#15690877C2212yes—
10#15690877C2212yes—
11C4⋊#11715897—21no—
12#15690809C2211yescharacteristic
13#15690808C2211yescharacteristic
14#15690800—21no—
15#15690790C411yescharacteristic
16#15690788—21no—
17#15690786C411yescharacteristic
18C4×Dic194—21no—
19C4×Dic194C411yescharacteristic
20#15690767—21no—
21C42×D97C2211yescharacteristic
22C4×Q8×C97C411yescharacteristic
23#13095521—42no—
24#13095519—42no—
25C2×#11715897—22no—
26C2×#11715897—22no—
27C2×Dic194—21no—
28C2×Dic194C2×C411yescharacteristic
29C2×C4×D97C2×C411yescharacteristic
30C2×C4×D97D411yescharacteristic
31C2×C4×D97D411yescharacteristic
32C4⋊Dic97—21no—
33C4⋊Dic97C2×C411yescharacteristic
34#13095490—21no—
35#13095490—21no—
36#13095490—21no—
37#13095490—21no—
38#13095490—21no—
39#13095488—21no—
40#13095488—21no—
41C4×Dic97—21no—
42C4×Dic97D411yescharacteristic
43C4×Dic97D411yescharacteristic
44C4×Dic97C2311yescharacteristic
45C2×Q8×C97C2×C411yescharacteristic
46C4⋊C4×C97—21no—
47C4⋊C4×C97C2×C411yescharacteristic
48C42×C97—21no—
49C42×C97C2×C411yescharacteristic
50C22×D97C22⋊C411yescharacteristic
51Q8×C97—41no—
52C2×C4×C97—21no—
53C2×C4×C97—21no—
54C2×C4×C97C22⋊C411yescharacteristic
55C2×C4×C97C22⋊C411yescharacteristic
56C2×C4×C97C22×C411yescharacteristic
57C2×Dic97—21no—
58C2×Dic97—21no—
59C2×Dic97C22⋊C411yescharacteristic
60C2×Dic97C2×D411yescharacteristic
61C2×Dic97C2×D411yescharacteristic
62C4×D97—21no—
63C4×D97—21no—
64C4×D97—41no—
65C4×D97SD1612yes—
66C4×D97Q1612yes—
67Dic194—41no—
68Dic194—41no—
69Dic194—41no—
70#11715897—42no—
71#11715897—42no—
72C22×C97C2×C22⋊C411yescharacteristic
73C2×D97—21no—
74C2×D97—21no—
75C2×D97#10312yes—
76C4×C97—21no—
77C4×C97—21no—
78C4×C97—41no—
79C4×C97—41no—
80C4×C97#10312yes—
81Dic97—21no—
82Dic97—41no—
83Dic97—41no—
84Dic97C2×SD1611yescharacteristic
85Dic97C2×Q1611yescharacteristic
86Dic97AGL(1,ℤ/8)11yescharacteristic
87Dic97#13711yescharacteristic
88C2×C97#41311yescharacteristic
89C2×C97C2×#10311yescharacteristic
90C2×C97#41711yescharacteristic
91D97—22no—
92#1550—971nomaximal, Sylow
93C97#155011yescharacteristic, complemented, Sylow
94#533—971no—
95#423—971no—
96C2×C4⋊C8—971no—
97#333—972no—
98#332—972no—
99C2×M16—1941no—
100C22×C8—1941no—
101C4⋊Q8—971no—
102C4⋊Q8—971no—
103#125—1941no—
104C22⋊Q8—971no—
105C22⋊Q8—971no—
106C22⋊Q8—1941no—
107C4×Q8—971no—
108C4×Q8—1941no—
109C4×Q8#8760411yescharacteristic
110#117—1941no—
111C2×C42—971no—
112C4⋊C8—972no—
113C4⋊C8—972no—
114C4⋊C8—1941no—
115C4⋊C8—1941no—
116C2×Q8—971no—
117C2×Q8—1941no—
118C2×Q8C2×#8760411yescharacteristic
119C22×C4—971no—
120C22×C4—971no—
121C22×C4—971no—
122M16—3882no—
123C2×C8—3882no—
124C2×C8—1942no—
125C2×C8—1942no—
126C4⋊C4—21no—
127C4⋊C4—971no—
128C4⋊C4—1941no—
129C4⋊C4—1941no—
130C4⋊C4—1941no—
131C4⋊C4—1941no—
132C4⋊C4—1941no—
133C4⋊C4—1941no—
134C4⋊C4C2×#8760411yescharacteristic
135C22⋊C4—1941no—
136C22⋊C4—1941no—
137C42—21no—
138C42—971no—
139C42—971no—
140C42—971no—
141C42—1941no—
142C42C2×#8760411yescharacteristic
143C23—971no—
144Q8—41no—
145Q8—3881no—
146Q8—3881no—
147Q8—3881no—
148C2×C4—21no—
149C2×C4—21no—
150C2×C4—971no—
151C2×C4—971no—
152C2×C4—972no—
153C2×C4—972no—
154C2×C4—971no—
155C2×C4—3881no—
156C2×C4—1941no—
157C2×C4—1941no—
158C2×C4—1941no—
159C2×C4—1941no—
160C2×C4C22⋊#8760411yescharacteristic
161C2×C4C22⋊#8760411yescharacteristic
162C2×C4C22×#8760411yescharacteristic
163C8—3882no—
164C8—3882no—
165C22—972no—
166C22—1941no—
167C22—1941no—
168C22C2×C22⋊#8760411yescharacteristic
169C4—21no—
170C4—21no—
171C4—41no—
172C4—41no—
173C4—971no—
174C4—971no—
175C4—971no—
176C4—971no—
177C4—3881no—
178C4—3881no—
179C4—1941no—
180C4#1569087312yes—
181C2—1942no—
182C2#2026172111yescharacteristic
183C2C2×#1569087311yescharacteristic
184C2#2026172611yescharacteristic
subgroupG/H
centerC22C2×C22⋊#87604
commutatorC2×C4×C97C22×C4
FrattiniC2×C4C22×#87604
FittingC4×Q8×C97C4
socleC22×C97C2×C22⋊C4
prime radicalC22×D97C22⋊C4
cosocleC4×Dic97C23
hypercenterC4×Q8#87604
hypocenterC97#1550
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
3. #20261595C2semidirect
4. #20261594C2semidirect
93. C97#1550semidirect

Maximal subgroups

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

subgroupindex
#202617402
#202617362
#202615952
#202615942
#202612892
#155097