Finite Group Databasebeta

← #27140415

Subgroups — #27140415

Order 12416.

240 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 240 conjugacy classes of subgroups — too many to draw. Only its 75 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
1C4⋊Q8×D97C211yesmaximal, characteristic
2#20260787C211yesmaximal, characteristic
3#20260757C211yesmaximal, characteristic
4#20260484C211yesmaximal, characteristic, complemented
5#20260483C211yesmaximal, characteristic, complemented
6#20260483C211yesmaximal, characteristic, complemented
7#333×C97C211yesmaximal, characteristic, complemented
8C2×Q8×D97C412yes—
9C2×#13095494—21no—
10#15690810C2211yescharacteristic
11#15690809—21no—
12#15690809—21no—
13#15690808C412yes—
14C4⋊C4×D97—21no—
15C4⋊C4×D97—21no—
16C42×D97C2211yescharacteristic
17C4⋊#11715895C2211yescharacteristic
18C4⋊#11715895C2211yescharacteristic
19#15690739—21no—
20#15690722—21no—
21C8×Dic97—21no—
22C4⋊Q8×C97C2211yescharacteristic
23C4⋊C8×C97C2211yescharacteristic
24C4⋊C8×C97C2211yescharacteristic
25C2×Dic194—22no—
26C2×Dic194C2×C412yes—
27Q8×D97—44no—
28C2×C4×D97—22no—
29C2×C4×D97C2×C411yescharacteristic
30C2×C4×D97D411yescharacteristic
31C2×C4×D97D411yescharacteristic
32#13095494—42no—
33C2×#11715895—21no—
34C2×#11715895—21no—
35C4⋊Dic97—21no—
36C4⋊Dic97—21no—
37#13095490—22no—
38#13095490—22no—
39C4×Dic97C2×C411yescharacteristic
40C4×Dic97D411yescharacteristic
41C4×Dic97D411yescharacteristic
42C2×Q8×C97C2×C412yes—
43C2×C8×C97—21no—
44C2×C8×C97—21no—
45C4⋊C4×C97—21no—
46C4⋊C4×C97—21no—
47C42×C97C2311yescharacteristic
48C22×D97C22⋊C411yescharacteristic
49Q8×C97—42no—
50C2×C4×C97—22no—
51C2×C4×C97C22×C411yescharacteristic
52C2×C4×C97C2×D411yescharacteristic
53C2×C4×C97C2×D411yescharacteristic
54C2×Dic97—22no—
55C2×Dic97C22⋊C411yescharacteristic
56C2×Dic97C22⋊C411yescharacteristic
57C2×Dic97C22⋊C411yescharacteristic
58C4×D97—21no—
59C4×D97—21no—
60C4×D97—22no—
61C4×D97—44no—
62C4×D97SD1612yes—
63C4×D97Q1612yes—
64Dic194—44no—
65Dic194—42no—
66C8×C97—41no—
67C8×C97—41no—
68#11715895—41no—
69#11715895—41no—
70C22×C97C2×C22⋊C411yescharacteristic
71C2×D97—21no—
72C2×D97—21no—
73C2×D97#10312yes—
74C4×C97—21no—
75C4×C97—42no—
76C4×C97C2×SD1611yescharacteristic
77C4×C97C2×Q1611yescharacteristic
78C4×C97#13711yescharacteristic
79C4×C97#13711yescharacteristic
80Dic97—21no—
81Dic97—42no—
82Dic97#10012yes—
83Dic97#10312yes—
84C2×C97C2×#10011yescharacteristic
85C2×C97C2×#10311yescharacteristic
86C2×C97#41911yescharacteristic
87D97—22no—
88#1548—971nomaximal, Sylow
89C97#154811yescharacteristic, complemented, Sylow
90C2×C4⋊Q8—971no—
91#431—971no—
92#423—971no—
93#333—971no—
94#333—971no—
95#333—971no—
96#333D9711yescharacteristic, complemented
97C22×Q8—972no—
98C2×M16—1941no—
99C4⋊Q8—972no—
100C4⋊Q8—971no—
101C4⋊Q8—1941no—
102C4⋊Q8—1941no—
103C4⋊Q8C2×D9711yescharacteristic
104C2×C4⋊C4—1941no—
105C2×C4⋊C4—1941no—
106C2×C42—971no—
107C4⋊C8—971no—
108C4⋊C8—971no—
109C4⋊C8—1941no—
110C4⋊C8C2×D9711yescharacteristic
111C4⋊C8C2×D9711yescharacteristic
112C22⋊C8—1941no—
113C4×C8—1941no—
114C2×Q8—972no—
115C2×Q8—3884no—
116C2×Q8—1942no—
117C2×Q8C4×D9712yes—
118C22×C4—971no—
119C22×C4—971no—
120C22×C4—971no—
121C22×C4—1942no—
122M16—3882no—
123C2×C8—21no—
124C2×C8—21no—
125C2×C8—1941no—
126C2×C8—1941no—
127C4⋊C4—21no—
128C4⋊C4—21no—
129C4⋊C4—1942no—
130C4⋊C4—1942no—
131C4⋊C4—1941no—
132C4⋊C4—1941no—
133C42—971no—
134C42—971no—
135C42—971no—
136C42C22×D9711yescharacteristic
137C23—971no—
138Q8—42no—
139Q8—3882no—
140Q8—3884no—
141C2×C4—22no—
142C2×C4—971no—
143C2×C4—971no—
144C2×C4—972no—
145C2×C4—972no—
146C2×C4—971no—
147C2×C4—3884no—
148C2×C4—1942no—
149C2×C4—1942no—
150C2×C4—1941no—
151C2×C4—1941no—
152C2×C4C2×C4×D9711yescharacteristic
153C2×C4D4×D9711yescharacteristic
154C2×C4D4×D9711yescharacteristic
155C8—41no—
156C8—41no—
157C8—3881no—
158C8—3881no—
159C22—972no—
160C22—1941no—
161C22—1941no—
162C22C22⋊C4×D9711yescharacteristic
163C4—21no—
164C4—42no—
165C4—972no—
166C4—972no—
167C4—3882no—
168C4—1941no—
169C4SD16×D9711yescharacteristic
170C4Q16×D9711yescharacteristic
171C4#1569083911yescharacteristic
172C4#1569084211yescharacteristic
173C2—1942no—
174C2#100×D9711yescharacteristic
175C2#103×D9711yescharacteristic
176C2#2026074211yescharacteristic
subgroupG/H
centerC22C22⋊C4×D97
commutatorC2×C4×C97C22×C4
FrattiniC42C22×D97
Fitting#333×C97C2
socleC22×C97C2×C22⋊C4
prime radicalC22×D97C22⋊C4
cosocleC42×C97C23
hypercenter#333D97
hypocenterC97#1548
LCSG ▷ C2×C4×C97 ▷ C22×C97 ▷ C97
UCSC1 ◁ C22 ◁ C42 ◁ #333

Complements

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

NH ≅ G/Ntype
4. #20260484C2semidirect
5. #20260483C2semidirect
6. #20260483C2semidirect
7. #333×C97C2semidirect
89. C97#1548semidirect
96. #333D97semidirect

Maximal subgroups

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

subgroupindex
C4⋊Q8×D972
#202607872
#202607572
#202604842
#202604832
#333×C972
#154897