Finite Group Databasebeta

← #27143398

Subgroups — #27143398

Order 12416.

258 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 258 conjugacy classes of subgroups — too many to draw. Only its 55 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#20261283C211yesmaximal, characteristic
2#20260707C212yesmaximal, complemented
3#20260529C212yesmaximal, complemented
4#20260528C211yesmaximal, characteristic, complemented
5#353×C97C211yesmaximal, characteristic, complemented
6C2×#13095510C411yescharacteristic, complemented
7C2×#13095510C2211yescharacteristic
8#15690810C411yescharacteristic, complemented
9#15690797C411yescharacteristic, complemented
10#15690797C2211yescharacteristic
11#15690781—22no—
12D4×Dic97—22no—
13#15690766—22no—
14C42×D97C411yescharacteristic, complemented
15#15690725C2212yes—
16#15690724—22no—
17#127×C97C2211yescharacteristic
18#99×C97C2212yes—
19#13095510—41no—
20#13095510—41no—
21#13095510—41no—
22#13095510—41no—
23C2×Dic194C2×C411yescharacteristic
24C2×Dic194C2×C411yescharacteristic
25C2×C22⋊D97—21no—
26C2×C22⋊D97D412yes—
27C22×Dic97—21no—
28C22×Dic97D412yes—
29C2×C4×D97—21no—
30C2×C4×D97C2×C411yescharacteristic
31C4⋊Dic97—22no—
32#13095490—42no—
33C22⋊Dic97—22no—
34C22⋊Dic97—22no—
35C22⋊Dic97—22no—
36C4×Dic97—21no—
37C4×Dic97—42no—
38C4×Dic97C2×C411yescharacteristic
39C2×D4×C97—22no—
40C2×D4×C97C2×C411yescharacteristic
41C2×D4×C97C2311yescharacteristic
42C22⋊C4×C97—22no—
43C42×C97C2×C411yescharacteristic
44C23×C97—21no—
45C23×C97C2×D412yes—
46C22×D97C22⋊C411yescharacteristic
47D4×C97—41no—
48D4×C97—41no—
49D4×C97—44no—
50C2×C4×C97—21no—
51C2×C4×C97—42no—
52C2×C4×C97C22×C411yescharacteristic
53C22⋊D97—42no—
54C22⋊D97—42no—
55C2×Dic97—22no—
56C2×Dic97—22no—
57C2×Dic97—21no—
58C2×Dic97—21no—
59C2×Dic97—41no—
60C2×Dic97—41no—
61C2×Dic97—42no—
62C2×Dic97C22⋊C412yes—
63C2×Dic97C22⋊C411yescharacteristic
64C4×D97—21no—
65C4×D97—21no—
66C4×D97—42no—
67Dic194—41no—
68Dic194—41no—
69C22×C97—22no—
70C22×C97—42no—
71C22×C97—42no—
72C22×C97—41no—
73C22×C97C2×C22⋊C411yescharacteristic
74C2×D97—21no—
75C2×D97—41no—
76C4×C97—22no—
77C4×C97—21no—
78C4×C97—82no—
79Dic97—22no—
80Dic97—41no—
81Dic97—41no—
82Dic97—82no—
83Dic97#9911yescharacteristic
84Dic97#9911yescharacteristic
85C2×C97—21no—
86C2×C97—42no—
87C2×C97—81no—
88C2×C97C2×#9911yescharacteristic
89D97—41no—
90#2126—971nomaximal, Sylow
91C97#212611yescharacteristic, complemented, Sylow
92#532—971no—
93#410—972no—
94#354—972no—
95#353—971no—
96#353D9711yescharacteristic, complemented
97C2×#41—971no—
98C2×#41—971no—
99C4⋊Q8—971no—
100#127C2×D9711yescharacteristic
101#124—971no—
102#124—971no—
103#121—1942no—
104C4×D4—1942no—
105#117—1942no—
106C2×C42—971no—
107#99—972no—
108#99—1942no—
109#99C2×D9712yes—
110#41—3881no—
111#41—3881no—
112#41—3881no—
113#41—3881no—
114C2×Q8—971no—
115C2×Q8—971no—
116C2×D4—22no—
117C2×D4—972no—
118C2×D4—1941no—
119C2×D4C4×D9711yescharacteristic
120C2×D4C22×D9711yescharacteristic
121C22×C4—972no—
122C22×C4—971no—
123C22×C4—1941no—
124C22×C4—1941no—
125C4⋊C4—3882no—
126C4⋊C4—1942no—
127C22⋊C4—22no—
128C22⋊C4—1942no—
129C22⋊C4—1942no—
130C22⋊C4—1942no—
131C42—971no—
132C42—3882no—
133C42—1941no—
134C42C4×D9711yescharacteristic
135C23—21no—
136C23—971no—
137C23D4×D9712yes—
138Q8—3881no—
139Q8—3881no—
140D4—41no—
141D4—41no—
142D4—44no—
143D4—3882no—
144D4—3882no—
145C2×C4—21no—
146C2×C4—42no—
147C2×C4—971no—
148C2×C4—972no—
149C2×C4—3882no—
150C2×C4—3882no—
151C2×C4—3881no—
152C2×C4—3881no—
153C2×C4—1941no—
154C2×C4—1941no—
155C2×C4—1942no—
156C2×C4—1942no—
157C2×C4—1941no—
158C2×C4—1941no—
159C2×C4C2×C4×D9711yescharacteristic
160C22—22no—
161C22—42no—
162C22—42no—
163C22—41no—
164C22—3881no—
165C22—1941no—
166C22C22⋊C4×D9711yescharacteristic
167C4—22no—
168C4—21no—
169C4—82no—
170C4—7762no—
171C4—971no—
172C4—971no—
173C4—3881no—
174C4—3881no—
175C4—1942no—
176C2—21no—
177C2—42no—
178C2—81no—
179C2—3881no—
180C2#99×D9711yescharacteristic
subgroupG/H
centerC2#99×D97
commutatorC2×C4×C97C22×C4
FrattiniC2×D4C22×D97
Fitting#353×C97C2
socleC2×C97C2×#99
prime radical#20261283C2
cosocleC2×D4×C97C23
hypercenter#353D97
hypocenterC97#2126
LCSG ▷ C2×C4×C97 ▷ C22×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ #353

Complements

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

NH ≅ G/Ntype
2. #20260707C2semidirect
3. #20260529C2semidirect
4. #20260528C2semidirect
5. #353×C97C2semidirect
6. C2×#13095510C4semidirect
8. #15690810C4semidirect
9. #15690797C4semidirect
14. C42×D97C4semidirect
91. C97#2126semidirect
96. #353D97semidirect

Maximal subgroups

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

subgroupindex
#202612832
#202607072
#202605292
#202605282
#353×C972
#212697