Finite Group Databasebeta

← #27140902

Subgroups — #27140902

Order 12416.

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

normal subgroups only

This group has 228 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#20261087C211yesmaximal, characteristic, complemented
3#20261031C211yesmaximal, characteristic, complemented
4#20261026C211yesmaximal, characteristic
5#20260465C211yesmaximal, characteristic, complemented
6#20260462C211yesmaximal, characteristic
7#332×C97C211yesmaximal, characteristic, complemented
8C2×Q8×D97—21no—
9#15690810C2211yescharacteristic
10#15690809—21no—
11#15690788—22no—
12#15690786—21no—
13C4×Dic194C2211yescharacteristic
14C4×D388C2211yescharacteristic
15#15690772—21no—
16#15690744—21no—
17#15690742—21no—
18#15690735—21no—
19#15690733—22no—
20#15690732—21no—
21C4×#11715895C2211yescharacteristic
22C4⋊Q8×C97C2211yescharacteristic
23C4⋊C8×C97C2211yescharacteristic
24C4⋊C8×C97C2211yescharacteristic
25C2×Dic194—22no—
26C2×Dic194—21no—
27C2×Dic194D411yescharacteristic
28Q8×D97—42no—
29Q8×D97—42no—
30C2×D388D411yescharacteristic
31C2×C4×D97—22no—
32C2×C4×D97—21no—
33C2×#11715895—21no—
34C4⋊Dic97—21no—
35C4⋊Dic97—21no—
36C4⋊Dic97D411yescharacteristic
37C4⋊Dic97D411yescharacteristic
38#13095490—22no—
39#13095490—21no—
40#13095488—22no—
41#13095488—21no—
42C4×Dic97—21no—
43C2×Q8×C97—21no—
44C2×C8×C97—21no—
45C2×C8×C97—21no—
46C4⋊C4×C97—21no—
47C4⋊C4×C97D412yes—
48C42×C97C2311yescharacteristic
49C22×D97—21no—
50Q8×C97—41no—
51Q8×C97—41no—
52C2×C4×C97—22no—
53C2×C4×C97C2×D411yescharacteristic
54C2×C4×C97C2×D411yescharacteristic
55C2×C4×C97C2×D411yescharacteristic
56C2×Dic97—21no—
57C2×Dic97—21no—
58C2×Dic97—21no—
59C2×Dic97—41no—
60D388—21no—
61D388Q1612yes—
62C4×D97—44no—
63C4×D97—41no—
64C4×D97—41no—
65Dic194—21no—
66Dic194—44no—
67Dic194—42no—
68Dic194SD1612yes—
69C8×C97—41no—
70C8×C97—41no—
71#11715895—41no—
72C22×C97#12011yescharacteristic
73C2×D97—22no—
74C2×D97—41no—
75C2×D97—41no—
76C4×C97—21no—
77C4×C97—42no—
78C4×C97C2×SD1611yescharacteristic
79C4×C97C2×Q1611yescharacteristic
80C4×C97AGL(1,ℤ/8)11yescharacteristic
81C4×C97#13711yescharacteristic
82Dic97—22no—
83Dic97—41no—
84Dic97—41no—
85Dic97—81no—
86C2×C97#45011yescharacteristic
87C2×C97#45111yescharacteristic
88C2×C97#45411yescharacteristic
89D97—42no—
90#1637—971nomaximal, Sylow
91C97#163711yescharacteristic, complemented, Sylow
92D4×Q8—971no—
93#487—971no—
94#477—971no—
95#476—971no—
96#332D9711yescharacteristic, complemented
97Q8⋊C8—971no—
98D4⋊C8—971no—
99C22×Q8—1941no—
100C4⋊Q8—971no—
101C4⋊Q8—1941no—
102C4⋊Q8C2×D9711yescharacteristic
103C22⋊Q8—1941no—
104C22⋊Q8—1942no—
105C4×Q8—971no—
106C4×D4—971no—
107C4×D4—1941no—
108#107—1941no—
109#106—1941no—
110C4⋊C8C2×D9711yescharacteristic
111C4⋊C8C2×D9711yescharacteristic
112#103—1941no—
113#103—1942no—
114#102—1941no—
115C4×C8—971no—
116C2×Q8—21no—
117C2×Q8—971no—
118C2×Q8—3882no—
119C2×Q8—3882no—
120C2×Q8—1942no—
121C2×Q8—1941no—
122C2×D4—971no—
123C22×C4—1942no—
124C22×C4—1941no—
125C2×C8—21no—
126C2×C8—21no—
127C2×C8—1941no—
128C4⋊C4—21no—
129C4⋊C4—971no—
130C4⋊C4—971no—
131C4⋊C4—1942no—
132C4⋊C4—1941no—
133C4⋊C4—1941no—
134C4⋊C4—1941no—
135C4⋊C4D38812yes—
136C22⋊C4—1942no—
137C22⋊C4—1941no—
138C42—1941no—
139C42C22×D9711yescharacteristic
140C23—1941no—
141Q8—41no—
142Q8—41no—
143Q8—972no—
144Q8—3882no—
145Q8—3884no—
146Q8—1941no—
147D4—972no—
148D4—1941no—
149C2×C4—22no—
150C2×C4—3884no—
151C2×C4—3881no—
152C2×C4—3881no—
153C2×C4—3881no—
154C2×C4—1941no—
155C2×C4—1941no—
156C2×C4—1941no—
157C2×C4D4×D9711yescharacteristic
158C2×C4D4×D9711yescharacteristic
159C2×C4C2×D38811yescharacteristic
160C8—41no—
161C8—41no—
162C8—3881no—
163C22—3881no—
164C22—3881no—
165C22—1942no—
166C22#1569078011yescharacteristic
167C4—21no—
168C4—42no—
169C4—7761no—
170C4—3881no—
171C4—3881no—
172C4—1942no—
173C4SD16×D9711yescharacteristic
174C4Q16×D9711yescharacteristic
175C4#1569083411yescharacteristic
176C4#1569083911yescharacteristic
177C2—3882no—
178C2#2026088711yescharacteristic
179C2#2026089211yescharacteristic
180C2#2026091011yescharacteristic
subgroupG/H
centerC22#15690780
commutator, cosocleC42×C97C23
FrattiniC42C22×D97
Fitting#332×C97C2
socleC22×C97#120
prime radicalC2×D388D4
hypercenter#332D97
hypocenterC97#1637
LCSG ▷ C42×C97 ▷ C22×C97 ▷ C97
UCSC1 ◁ C22 ◁ C42 ◁ #332

Complements

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

NH ≅ G/Ntype
2. #20261087C2semidirect
3. #20261031C2semidirect
5. #20260465C2semidirect
7. #332×C97C2semidirect
91. C97#1637semidirect
96. #332D97semidirect

Maximal subgroups

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

subgroupindex
#202614002
#202610872
#202610312
#202610262
#202604652
#202604622
#332×C972
#163797