Finite Group Databasebeta

← #27140996

Subgroups — #27140996

Order 12416.

234 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 234 conjugacy classes of subgroups — too many to draw. Only its 57 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#20261400C211yesmaximal, characteristic
2#20261017C211yesmaximal, characteristic, complemented
3#20260937C211yesmaximal, characteristic
4#20260481C211yesmaximal, characteristic, complemented
5#20260467C211yesmaximal, characteristic, complemented
6#20260464C211yesmaximal, characteristic
7#498×C97C211yesmaximal, characteristic, complemented
8C2×Q8×D97—21no—
9C2×#13095498—21no—
10#15690809—21no—
11#15690788—22no—
12#15690786—21no—
13C4×Dic194C2211yescharacteristic
14C4×D388C2211yescharacteristic
15#15690772—21no—
16#15690741—21no—
17C4⋊#11715895C2211yescharacteristic
18C4⋊#11715895C2211yescharacteristic
19#15690734—21no—
20#15690733—21no—
21C4⋊Q8×C97C2211yescharacteristic
22C4⋊Q8×C97C2211yescharacteristic
23#106×C97—22no—
24C4×C8×C97C2211yescharacteristic
25C2×Dic194—22no—
26C2×Dic194D411yescharacteristic
27Q8×D97—42no—
28Q8×D97—42no—
29C2×D388D411yescharacteristic
30C2×C4×D97—22no—
31C2×C4×D97—21no—
32#13095498—42no—
33C2×#11715895—21no—
34C2×#11715895—21no—
35C4⋊Dic97D411yescharacteristic
36C4⋊Dic97D411yescharacteristic
37#13095490—22no—
38#13095490—21no—
39#13095488—22no—
40#13095488—21no—
41C4×Dic97—21no—
42C2×Q8×C97—21no—
43C2×Q8×C97—21no—
44C2×C8×C97—21no—
45C4⋊C4×C97—21no—
46C4⋊C4×C97—21no—
47C4⋊C4×C97—21no—
48C4⋊C4×C97D412yes—
49C42×C97C2311yescharacteristic
50C22×D97—21no—
51Q8×C97—41no—
52Q8×C97—41no—
53Q8×C97—41no—
54Q8×C97—41no—
55C2×C4×C97—22no—
56C2×C4×C97—41no—
57C2×C4×C97C2×D411yescharacteristic
58C2×C4×C97C2×D411yescharacteristic
59C2×C4×C97C2×D411yescharacteristic
60C2×Dic97—21no—
61C2×Dic97—21no—
62C2×Dic97—21no—
63D388—21no—
64D388SD1612yes—
65C4×D97—44no—
66C4×D97—41no—
67C4×D97—41no—
68Dic194—21no—
69Dic194—44no—
70Dic194SD1612yescomplemented
71C8×C97—22no—
72#11715895—41no—
73#11715895—41no—
74C22×C97#12011yescharacteristic
75C2×D97—22no—
76C2×D97—41no—
77C2×D97—41no—
78C4×C97—21no—
79C4×C97—42no—
80C4×C97—81no—
81C4×C97C2×SD1611yescharacteristic
82C4×C97C2×SD1611yescharacteristic
83C4×C97AGL(1,ℤ/8)11yescharacteristic
84C4×C97#13711yescharacteristic
85Dic97—22no—
86Dic97—41no—
87Dic97—41no—
88C2×C97#44811yescharacteristic
89C2×C97#45011yescharacteristic
90C2×C97#45611yescharacteristic
91D97—42no—
92#1651—971nomaximal, Sylow
93C97#165111yescharacteristic, complemented, Sylow
94D4×Q8—971no—
95#498D9711yescharacteristic, complemented
96#475—971no—
97#461—971no—
98#332—971no—
99Q8⋊C8—971no—
100D4⋊C8—971no—
101C22×Q8—1941no—
102C2×SD16—1941no—
103C4⋊Q8—1941no—
104C4⋊Q8C2×D9711yescharacteristic
105C4⋊Q8C2×D9711yescharacteristic
106C22⋊Q8—1941no—
107C22⋊Q8—1942no—
108C4×Q8—971no—
109C4×D4—971no—
110C4×D4—1941no—
111#106—22no—
112#106—1941no—
113C4⋊C8—971no—
114C4⋊C8—971no—
115#103—1941no—
116#103—1941no—
117C4×C8C2×D9711yescharacteristic
118C2×Q8—21no—
119C2×Q8—21no—
120C2×Q8—971no—
121C2×Q8—3882no—
122C2×Q8—3882no—
123C2×Q8—1942no—
124C2×D4—971no—
125C22×C4—1942no—
126C22×C4—1941no—
127SD16—3882no—
128C2×C8—21no—
129C2×C8—1941no—
130C2×C8—1941no—
131C4⋊C4—21no—
132C4⋊C4—21no—
133C4⋊C4—21no—
134C4⋊C4—971no—
135C4⋊C4—971no—
136C4⋊C4—1942no—
137C4⋊C4—1941no—
138C4⋊C4C22⋊D9712yes—
139C22⋊C4—1942no—
140C22⋊C4—1941no—
141C42—1941no—
142C42C22×D9711yescharacteristic
143C23—1941no—
144Q8—41no—
145Q8—41no—
146Q8—41no—
147Q8—41no—
148Q8—972no—
149Q8—3884no—
150Q8—1941no—
151D4—972no—
152D4—1941no—
153C2×C4—22no—
154C2×C4—41no—
155C2×C4—3884no—
156C2×C4—3881no—
157C2×C4—3881no—
158C2×C4—1941no—
159C2×C4—1941no—
160C2×C4—1941no—
161C2×C4D4×D9711yescharacteristic
162C2×C4D4×D9711yescharacteristic
163C2×C4C2×C22⋊D9711yescharacteristic
164C8—22no—
165C8—3881no—
166C8—3881no—
167C22—3881no—
168C22—3881no—
169C22—1942no—
170C22#1569077811yescharacteristic
171C4—21no—
172C4—42no—
173C4—81no—
174C4—3881no—
175C4—3881no—
176C4—1942no—
177C4SD16×D9711yescharacteristic
178C4SD16×D9711yescharacteristic
179C4#1569083711yescharacteristic
180C4#1569083911yescharacteristic
181C2—3882no—
182C2#2026087211yescharacteristic
183C2#2026088411yescharacteristic
184C2#2026091711yescharacteristic
subgroupG/H
centerC22#15690778
commutator, cosocleC42×C97C23
FrattiniC42C22×D97
Fitting#498×C97C2
socleC22×C97#120
prime radicalC2×D388D4
hypercenter#498D97
hypocenterC97#1651
LCSG ▷ C42×C97 ▷ C22×C97 ▷ C97
UCSC1 ◁ C22 ◁ C42 ◁ #498

Complements

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

NH ≅ G/Ntype
2. #20261017C2semidirect
4. #20260481C2semidirect
5. #20260467C2semidirect
7. #498×C97C2semidirect
70. Dic194SD16semidirect
93. C97#1651semidirect
95. #498D97semidirect

Maximal subgroups

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

subgroupindex
#202614002
#202610172
#202609372
#202604812
#202604672
#202604642
#498×C972
#165197