Finite Group Databasebeta

← #27140443

Subgroups — #27140443

Order 12416.

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

normal subgroups only

This group has 198 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#20261290C211yesmaximal, characteristic
2#20260791C211yesmaximal, characteristic
3#20260787C211yesmaximal, characteristic
4#20260483C211yesmaximal, characteristic, complemented
5#20260483C211yesmaximal, characteristic, complemented
6#20260481C211yesmaximal, characteristic, complemented
7#332×C97C211yesmaximal, characteristic, complemented
8#15690810C2211yescharacteristic
9#15690801—21no—
10#15690790C412yes—
11#15690786—21no—
12Q8×Dic97—21no—
13#15690775C412yes—
14#15690768—21no—
15C42×D97C2211yescharacteristic
16C4⋊#11715895C2211yescharacteristic
17C4⋊#11715895C2211yescharacteristic
18#15690722—21no—
19#15690722—21no—
20#15690720—21no—
21C8×Dic97—21no—
22C4⋊Q8×C97C2211yescharacteristic
23C4⋊C8×C97C2211yescharacteristic
24C4⋊C8×C97C2211yescharacteristic
25C2×Dic194C2×C412yes—
26C2×C4×D97C2×C411yescharacteristic
27C2×C4×D97D411yescharacteristic
28C2×C4×D97D411yescharacteristic
29C2×#11715895—21no—
30C2×#11715895—21no—
31C4⋊Dic97—21no—
32C4⋊Dic97—21no—
33C4⋊Dic97—22no—
34#13095490—22no—
35#13095488—22no—
36C4×Dic97—22no—
37C4×Dic97C2×C411yescharacteristic
38C4×Dic97D411yescharacteristic
39C4×Dic97D411yescharacteristic
40C2×Q8×C97—21no—
41C2×C8×C97—21no—
42C2×C8×C97—21no—
43C4⋊C4×C97—21no—
44C4⋊C4×C97C2×C412yes—
45C42×C97C2311yescharacteristic
46C22×D97C22⋊C411yescharacteristic
47Q8×C97—41no—
48Q8×C97—41no—
49C2×C4×C97—22no—
50C2×C4×C97C22×C411yescharacteristic
51C2×C4×C97C2×D411yescharacteristic
52C2×C4×C97C2×D411yescharacteristic
53C2×Dic97—22no—
54C2×Dic97C22⋊C411yescharacteristic
55C2×Dic97C22⋊C411yescharacteristic
56C2×Dic97C22⋊C411yescharacteristic
57C4×D97—21no—
58C4×D97—21no—
59C4×D97—21no—
60C4×D97—21no—
61C4×D97—41no—
62Dic194—42no—
63C8×C97—41no—
64C8×C97—41no—
65#11715895—41no—
66#11715895—41no—
67C22×C97C2×C22⋊C411yescharacteristic
68C2×D97—21no—
69C2×D97—21no—
70C2×D97—21no—
71C4×C97—21no—
72C4×C97—42no—
73C4×C97#13511yescharacteristic
74C4×C97#13511yescharacteristic
75C4×C97#13711yescharacteristic
76C4×C97#13711yescharacteristic
77Dic97—21no—
78Dic97—21no—
79Dic97—21no—
80Dic97—42no—
81C2×C97#41311yescharacteristic
82C2×C97#41611yescharacteristic
83C2×C97#41911yescharacteristic
84D97—41no—
85#1554—971nomaximal, Sylow
86C97#155411yescharacteristic, complemented, Sylow
87#533—971no—
88#432—971no—
89#431—971no—
90#333—971no—
91#333—971no—
92#332—971no—
93#332D9711yescharacteristic, complemented
94C4⋊Q8—971no—
95C4⋊Q8C2×D9711yescharacteristic
96#125—1941no—
97C22⋊Q8—972no—
98C22⋊Q8—1941no—
99C4×Q8—972no—
100C4×Q8—1941no—
101#117—1941no—
102C2×C42—971no—
103C4⋊C8—971no—
104C4⋊C8—971no—
105C4⋊C8C2×D9711yescharacteristic
106C4⋊C8C2×D9711yescharacteristic
107C22⋊C8—1941no—
108C22⋊C8—1941no—
109#97—1941no—
110C4×C8—1941no—
111C2×Q8—21no—
112C2×Q8—972no—
113C22×C4—971no—
114C22×C4—971no—
115C22×C4—971no—
116C2×C8—21no—
117C2×C8—21no—
118C2×C8—1941no—
119C2×C8—1941no—
120C4⋊C4—21no—
121C4⋊C4—1942no—
122C4⋊C4—1942no—
123C4⋊C4—1941no—
124C4⋊C4—1941no—
125C4⋊C4C4×D9712yes—
126C22⋊C4—1942no—
127C42—971no—
128C42—971no—
129C42—971no—
130C42—1942no—
131C42C22×D9711yescharacteristic
132C23—971no—
133Q8—41no—
134Q8—41no—
135Q8—3882no—
136C2×C4—22no—
137C2×C4—971no—
138C2×C4—971no—
139C2×C4—971no—
140C2×C4—3881no—
141C2×C4—1942no—
142C2×C4—1941no—
143C2×C4—1941no—
144C2×C4—1941no—
145C2×C4—1941no—
146C2×C4C2×C4×D9711yescharacteristic
147C2×C4D4×D9711yescharacteristic
148C2×C4D4×D9711yescharacteristic
149C8—41no—
150C8—41no—
151C8—3881no—
152C8—3881no—
153C22—1941no—
154C22—1941no—
155C22—1941no—
156C22C22⋊C4×D9711yescharacteristic
157C4—21no—
158C4—42no—
159C4—3882no—
160C4—1941no—
161C4—1941no—
162C4—1941no—
163C4#1569082911yescharacteristic
164C4#1569083111yescharacteristic
165C4#1569083911yescharacteristic
166C4#1569084211yescharacteristic
167C2—3881no—
168C2#2026071811yescharacteristic
169C2#2026072811yescharacteristic
170C2#2026074211yescharacteristic
subgroupG/H
centerC22C22⋊C4×D97
commutatorC2×C4×C97C22×C4
FrattiniC42C22×D97
Fitting#332×C97C2
socleC22×C97C2×C22⋊C4
prime radicalC22×D97C22⋊C4
cosocleC42×C97C23
hypercenter#332D97
hypocenterC97#1554
LCSG ▷ C2×C4×C97 ▷ C22×C97 ▷ C97
UCSC1 ◁ C22 ◁ C42 ◁ #332

Complements

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

NH ≅ G/Ntype
4. #20260483C2semidirect
5. #20260483C2semidirect
6. #20260481C2semidirect
7. #332×C97C2semidirect
86. C97#1554semidirect
93. #332D97semidirect

Maximal subgroups

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

subgroupindex
#202612902
#202607912
#202607872
#202604832
#202604812
#332×C972
#155497