Finite Group Databasebeta

← #27143410

Subgroups — #27143410

Order 12416.

238 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 238 conjugacy classes of subgroups — too many to draw. Only its 65 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#20261306C211yesmaximal, characteristic
2#20260709C211yesmaximal, characteristic
3#20260529C214yesmaximal, complemented
4#410×C97C211yesmaximal, characteristic
5C22×Dic194C411yescharacteristic, complemented
6#15690808C412yescomplemented
7#15690797C2212yes—
8#15690791—21no—
9#15690791—21no—
10#15690789—21no—
11#15690789—21no—
12#15690769—21no—
13#15690769C411yescharacteristic, complemented
14#15690725—21no—
15#15690725C2212yes—
16C2×#41×C97C2211yescharacteristic
17#117×C97—21no—
18#99×C97—21no—
19#99×C97C2212yes—
20C2×Dic194—21no—
21C2×Dic194—41no—
22C2×Dic194—41no—
23C2×Dic194C2×C412yes—
24C22×Dic97—21no—
25C4⋊Dic97—21no—
26C4⋊Dic97—41no—
27#13095490—22no—
28#13095490—22no—
29C22⋊Dic97—21no—
30C22⋊Dic97—22no—
31C22⋊Dic97—22no—
32C22⋊Dic97—22no—
33C4×Dic97—41no—
34C4×Dic97C2×C412yes—
35#41×C97—41no—
36#41×C97—41no—
37C2×Q8×C97C2×C411yescharacteristic
38C2×D4×C97D411yescharacteristic
39C2×D4×C97D411yescharacteristic
40C2×D4×C97C2311yescharacteristic
41C22×C4×C97C2×C411yescharacteristic
42C22×C4×C97D411yescharacteristic
43C22×C4×C97D411yescharacteristic
44C4⋊C4×C97—41no—
45C22⋊C4×C97—22no—
46C42×C97—41no—
47C23×C97C22⋊C411yescharacteristic
48C23×C97C2×D411yescharacteristic
49C23×C97C2×D411yescharacteristic
50Q8×C97—41no—
51D4×C97—41no—
52D4×C97—41no—
53D4×C97—41no—
54C2×C4×C97—21no—
55C2×C4×C97—21no—
56C2×C4×C97—21no—
57C2×C4×C97—21no—
58C2×C4×C97—21no—
59C2×C4×C97—21no—
60C2×C4×C97—42no—
61C2×C4×C97C22⋊C411yescharacteristic
62C2×C4×C97C22⋊C411yescharacteristic
63C2×C4×C97C22⋊C411yescharacteristic
64C2×C4×C97C22×C411yescharacteristic
65C2×Dic97—22no—
66C2×Dic97—22no—
67C2×Dic97—42no—
68C2×Dic97—42no—
69Dic194—41no—
70Dic194—41no—
71Dic194—42no—
72C22×C97—21no—
73C22×C97—21no—
74C22×C97—21no—
75C22×C97—41no—
76C22×C97—41no—
77C22×C97—41no—
78C22×C97C2×C22⋊C411yescharacteristic
79C4×C97—21no—
80C4×C97—21no—
81C4×C97—21no—
82C4×C97—82no—
83C4×C97#9911yescharacteristic
84C4×C97#9911yescharacteristic
85Dic97—42no—
86Dic97—42no—
87Dic97—82no—
88C2×C97—21no—
89C2×C97—41no—
90C2×C97—41no—
91C2×C97—41no—
92C2×C97C2×#9911yescharacteristic
93#2129—971nomaximal, Sylow
94C97#212911yescharacteristic, complemented, Sylow
95#536—971no—
96#410—971no—
97#410D9711yescharacteristic
98#354—974no—
99C2×#41C2×D9711yescharacteristic
100C22×Q8—971no—
101C4⋊Q8—972no—
102#124—972no—
103#123—1941no—
104#123—1941no—
105C22⋊Q8—1941no—
106C22⋊Q8—1941no—
107#117—21no—
108#117—971no—
109#117—1941no—
110#99—21no—
111#99—972no—
112#99—1941no—
113#99C2×D9712yes—
114#41—41no—
115#41—41no—
116C2×Q8—972no—
117C2×Q8—3881no—
118C2×Q8—3881no—
119C2×Q8—1941no—
120C2×Q8C4×D9711yescharacteristic
121C2×D4D38811yescharacteristic
122C2×D4C22⋊D9711yescharacteristic
123C2×D4C22×D9711yescharacteristic
124C22×C4—1941no—
125C22×C4C4×D9711yescharacteristic
126C22×C4D38811yescharacteristic
127C22×C4C22⋊D9711yescharacteristic
128C4⋊C4—41no—
129C4⋊C4—3881no—
130C4⋊C4—1942no—
131C4⋊C4—1942no—
132C4⋊C4—1941no—
133C22⋊C4—22no—
134C22⋊C4—1942no—
135C22⋊C4—1941no—
136C22⋊C4—1942no—
137C22⋊C4—1942no—
138C42—41no—
139C42—972no—
140C42—3881no—
141C23#1309548811yescharacteristic
142C23C2×C22⋊D9711yescharacteristic
143C23C2×D38811yescharacteristic
144Q8—41no—
145Q8—3882no—
146Q8—3881no—
147Q8—3881no—
148D4—41no—
149D4—41no—
150D4—41no—
151C2×C4—21no—
152C2×C4—21no—
153C2×C4—21no—
154C2×C4—21no—
155C2×C4—21no—
156C2×C4—21no—
157C2×C4—42no—
158C2×C4—3882no—
159C2×C4—3882no—
160C2×C4—1942no—
161C2×C4—1942no—
162C2×C4#1309548811yescharacteristic
163C2×C4#1309548811yescharacteristic
164C2×C4#1309548811yescharacteristic
165C2×C4C2×C4×D9711yescharacteristic
166C22—21no—
167C22—21no—
168C22—21no—
169C22—41no—
170C22—41no—
171C22—41no—
172C22C2×#1309548811yescharacteristic
173C4—21no—
174C4—21no—
175C4—21no—
176C4—82no—
177C4—7762no—
178C4—3882no—
179C4—3882no—
180C4#1569072411yescharacteristic
181C4#1569072411yescharacteristic
182C2—21no—
183C2—41no—
184C2—41no—
185C2—41no—
186C2C2×#1569072411yescharacteristic
subgroupG/H
centerC2C2×#15690724
commutatorC2×C4×C97C22×C4
FrattiniC2×D4C22×D97
Fitting#410×C97C2
socleC2×C97C2×#99
prime radicalC2×#41×C97C22
cosocleC2×D4×C97C23
hypercenter#410D97
hypocenterC97#2129
LCSG ▷ C2×C4×C97 ▷ C22×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×#41 ◁ #410

Complements

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

NH ≅ G/Ntype
3. #20260529C2semidirect
5. C22×Dic194C4semidirect
6. #15690808C4semidirect
13. #15690769C4semidirect
94. C97#2129semidirect

Maximal subgroups

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

subgroupindex
#202613062
#202607092
#202605292
#410×C972
#212997