Finite Group Databasebeta

← #27143420

Subgroups — #27143420

Order 12416.

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

normal subgroups only

This group has 250 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#20261286C211yesmaximal, characteristic
2#20260718C212yesmaximal, complemented
3#20260534C211yesmaximal, characteristic, complemented
4#20260531C212yesmaximal, complemented
5#356×C97C211yesmaximal, characteristic, complemented
6C2×#13095512C411yescharacteristic
7C2×#13095512C2211yescharacteristic
8M16×D97—22no—
9#15690807C411yescharacteristic
10#15690799C411yescharacteristic
11#15690799C2211yescharacteristic
12#15690785—22no—
13#15690772—22no—
14C42×D97C411yescharacteristic
15#15690729C2212yes—
16#15690726—22no—
17C4⋊Q8×C97C2211yescharacteristic
18#101×C97C2212yes—
19#13095512—41no—
20#13095512—41no—
21#13095512—41no—
22#13095512—41no—
23C2×D388—22no—
24C2×D388—21no—
25C2×D388C2×C411yescharacteristic
26C2×D388C2×C411yescharacteristic
27C2×D388D412yes—
28C2×C4×D97—21no—
29C2×C4×D97—21no—
30C2×C4×D97C2×C411yescharacteristic
31C2×C4×D97D412yes—
32#13095495—22no—
33#13095494—42no—
34C8×D97—42no—
35#13095488—22no—
36#13095488—22no—
37C4×Dic97—21no—
38C4×Dic97C2×C411yescharacteristic
39C2×Q8×C97C2×C411yescharacteristic
40C2×Q8×C97C2311yescharacteristic
41M16×C97—22no—
42C4⋊C4×C97—22no—
43C42×C97C2×C411yescharacteristic
44C22×D97—21no—
45C22×D97C22⋊C412yes—
46C22×D97C22⋊C411yescharacteristic
47Q8×C97—41no—
48Q8×C97—41no—
49C2×C4×C97—21no—
50C2×C4×C97—21no—
51C2×C4×C97C22×C411yescharacteristic
52C2×C4×C97C2×D412yes—
53C2×Dic97—21no—
54C2×Dic97C22⋊C411yescharacteristic
55D388—41no—
56D388—42no—
57D388—41no—
58D388—44no—
59D388—42no—
60C4×D97—22no—
61C4×D97—22no—
62C4×D97—21no—
63C4×D97—21no—
64C4×D97—42no—
65C4×D97—41no—
66C4×D97—41no—
67C8×C97—42no—
68#11715895—42no—
69C22×C97C2×C22⋊C411yescharacteristic
70C2×D97—21no—
71C2×D97—22no—
72C2×D97—41no—
73C2×D97—42no—
74C2×D97—42no—
75C2×D97—41no—
76C4×C97—22no—
77C4×C97—22no—
78C4×C97—21no—
79C4×C97—41no—
80Dic97—41no—
81Dic97#9911yescharacteristic
82Dic97#9911yescharacteristic
83C2×C97—21no—
84C2×C97C2×#9911yescharacteristic
85D97—41no—
86D97—42no—
87D97—81no—
88#2132—971nomaximal, Sylow
89C97#213211yescharacteristic, complemented, Sylow
90#532—971no—
91#413—972no—
92#356—971no—
93#356D9711yescharacteristic, complemented
94#355—972no—
95C2×#41—971no—
96C2×#41—971no—
97C2×M16—1942no—
98C4⋊Q8C2×D9711yescharacteristic
99#127—971no—
100#124—971no—
101#124—971no—
102#121—1942no—
103C4×D4—1942no—
104C2×C42—971no—
105#101—972no—
106#101C2×D9712yes—
107#100—1942no—
108#41—3881no—
109#41—3881no—
110#41—3881no—
111#41—3881no—
112C2×Q8C4×D9711yescharacteristic
113C2×Q8C22×D9711yescharacteristic
114C2×D4—971no—
115C2×D4—971no—
116C2×D4—972no—
117C2×D4—1942no—
118C2×D4—1941no—
119C22×C4—971no—
120C22×C4—972no—
121C22×C4—1941no—
122C22×C4—1941no—
123M16—22no—
124M16—3882no—
125M16—1942no—
126C2×C8—3882no—
127C4⋊C4—22no—
128C22⋊C4—1942no—
129C22⋊C4—1942no—
130C42—971no—
131C42—1941no—
132C42C4×D9711yescharacteristic
133C23—972no—
134C23—971no—
135C23—1941no—
136Q8—41no—
137Q8—41no—
138D4—3881no—
139D4—3884no—
140D4—3882no—
141D4—3881no—
142D4—3882no—
143C2×C4—21no—
144C2×C4—21no—
145C2×C4—971no—
146C2×C4—3882no—
147C2×C4—3881no—
148C2×C4—3881no—
149C2×C4—1942no—
150C2×C4—1942no—
151C2×C4—1941no—
152C2×C4—1941no—
153C2×C4—1941no—
154C2×C4C2×C4×D9711yescharacteristic
155C2×C4D4×D9712yes—
156C8—42no—
157C8—3882no—
158C22—3882no—
159C22—3881no—
160C22—3882no—
161C22—3881no—
162C22—1941no—
163C22—1942no—
164C22C22⋊C4×D9711yescharacteristic
165C4—22no—
166C4—22no—
167C4—21no—
168C4—41no—
169C4—971no—
170C4—971no—
171C4—3881no—
172C2—21no—
173C2—7761no—
174C2—3881no—
175C2—3882no—
176C2#99×D9711yescharacteristic
subgroupG/H
centerC2#99×D97
commutatorC2×C4×C97C22×C4
FrattiniC2×Q8C22×D97
Fitting#356×C97C2
socleC2×C97C2×#99
prime radical#20261286C2
cosocleC2×Q8×C97C23
hypercenter#356D97
hypocenterC97#2132
LCSG ▷ C2×C4×C97 ▷ C22×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C2 ◁ C22 ◁ C2×Q8 ◁ #356

Complements

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

NH ≅ G/Ntype
2. #20260718C2semidirect
3. #20260534C2semidirect
4. #20260531C2semidirect
5. #356×C97C2semidirect
89. C97#2132semidirect
93. #356D97semidirect

Maximal subgroups

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

subgroupindex
#202612862
#202607182
#202605342
#202605312
#356×C972
#213297