Finite Group Databasebeta

← #27140898

Subgroups — #27140898

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#20261027C211yesmaximal, characteristic, complemented
3#20261022C211yesmaximal, characteristic
4#20260481C211yesmaximal, characteristic, complemented
5#20260467C211yesmaximal, characteristic, complemented
6#20260464C211yesmaximal, characteristic
7#487×C97C211yesmaximal, characteristic, complemented
8C2×Q8×D97—21no—
9#15690809—21no—
10#15690788—22no—
11#15690786—21no—
12C4×Dic194C2211yescharacteristic
13C4×D388C2211yescharacteristic
14#15690772—21no—
15#15690743—21no—
16#15690741—21no—
17C4⋊#11715895C2211yescharacteristic
18C4⋊#11715895C2211yescharacteristic
19#15690733—21no—
20#15690730—21no—
21C4⋊Q8×C97C2211yescharacteristic
22C4⋊Q8×C97C2211yescharacteristic
23#103×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—
32C2×#11715895—21no—
33C2×#11715895—21no—
34C4⋊Dic97D411yescharacteristic
35C4⋊Dic97D411yescharacteristic
36#13095490—22no—
37#13095490—21no—
38#13095488—22no—
39#13095488—21no—
40C4×Dic97—21no—
41C2×Q8×C97—21no—
42C2×Q8×C97—21no—
43C2×C8×C97—21no—
44C4⋊C4×C97—21no—
45C4⋊C4×C97—21no—
46C4⋊C4×C97—21no—
47C4⋊C4×C97D412yes—
48C42×C97C2311yescharacteristic
49C22×D97—21no—
50Q8×C97—42no—
51Q8×C97—41no—
52Q8×C97—41no—
53C2×C4×C97—22no—
54C2×C4×C97—41no—
55C2×C4×C97C2×D411yescharacteristic
56C2×C4×C97C2×D411yescharacteristic
57C2×C4×C97C2×D411yescharacteristic
58C2×Dic97—21no—
59C2×Dic97—21no—
60C2×Dic97—21no—
61D388—21no—
62D388Q1612yes—
63C4×D97—44no—
64C4×D97—41no—
65C4×D97—41no—
66Dic194—21no—
67Dic194—44no—
68Dic194SD1612yes—
69C8×C97—41no—
70#11715895—41no—
71#11715895—41no—
72C22×C97#12011yescharacteristic
73C2×D97—22no—
74C2×D97—41no—
75C2×D97—41no—
76C4×C97—21no—
77C4×C97—42no—
78C4×C97—81no—
79C4×C97C2×SD1611yescharacteristic
80C4×C97C2×Q1611yescharacteristic
81C4×C97AGL(1,ℤ/8)11yescharacteristic
82C4×C97#13711yescharacteristic
83Dic97—22no—
84Dic97—41no—
85Dic97—41no—
86C2×C97#45011yescharacteristic
87C2×C97#45111yescharacteristic
88C2×C97#45411yescharacteristic
89D97—42no—
90#1637—971nomaximal, Sylow
91C97#163711yescharacteristic, complemented, Sylow
92D4×Q8—971no—
93#487D9711yescharacteristic, complemented
94#477—971no—
95#476—971no—
96#332—971no—
97Q8⋊C8—971no—
98D4⋊C8—971no—
99C22×Q8—1941no—
100C4⋊Q8—1941no—
101C4⋊Q8C2×D9711yescharacteristic
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⋊C8—971no—
111C4⋊C8—971no—
112#103—22no—
113#103—1941no—
114#102—1941no—
115C4×C8C2×D9711yescharacteristic
116C2×Q8—21no—
117C2×Q8—21no—
118C2×Q8—971no—
119C2×Q8—3882no—
120C2×Q8—3882no—
121C2×Q8—1942no—
122C2×D4—971no—
123C22×C4—1942no—
124C22×C4—1941no—
125C2×C8—21no—
126C2×C8—1941no—
127C2×C8—1941no—
128C4⋊C4—21no—
129C4⋊C4—21no—
130C4⋊C4—21no—
131C4⋊C4—971no—
132C4⋊C4—971no—
133C4⋊C4—1942no—
134C4⋊C4—1941no—
135C4⋊C4C22⋊D9712yes—
136C22⋊C4—1942no—
137C22⋊C4—1941no—
138C42—1941no—
139C42C22×D9711yescharacteristic
140C23—1941no—
141Q8—42no—
142Q8—41no—
143Q8—41no—
144Q8—972no—
145Q8—3884no—
146Q8—1941no—
147D4—972no—
148D4—1941no—
149C2×C4—22no—
150C2×C4—41no—
151C2×C4—3884no—
152C2×C4—3881no—
153C2×C4—3881no—
154C2×C4—1941no—
155C2×C4—1941no—
156C2×C4—1941no—
157C2×C4D4×D9711yescharacteristic
158C2×C4D4×D9711yescharacteristic
159C2×C4C2×C22⋊D9711yescharacteristic
160C8—41no—
161C8—3881no—
162C8—3881no—
163C22—3881no—
164C22—3881no—
165C22—1942no—
166C22#1569077811yescharacteristic
167C4—21no—
168C4—42no—
169C4—81no—
170C4—3881no—
171C4—3881no—
172C4—1942no—
173C4SD16×D9711yescharacteristic
174C4Q16×D9711yescharacteristic
175C4#1569083711yescharacteristic
176C4#1569084211yescharacteristic
177C2—3882no—
178C2#2026088411yescharacteristic
179C2#2026088911yescharacteristic
180C2#2026090711yescharacteristic
subgroupG/H
centerC22#15690778
commutator, cosocleC42×C97C23
FrattiniC42C22×D97
Fitting#487×C97C2
socleC22×C97#120
prime radicalC2×D388D4
hypercenter#487D97
hypocenterC97#1637
LCSG ▷ C42×C97 ▷ C22×C97 ▷ C97
UCSC1 ◁ C22 ◁ C42 ◁ #487

Complements

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

NH ≅ G/Ntype
2. #20261027C2semidirect
4. #20260481C2semidirect
5. #20260467C2semidirect
7. #487×C97C2semidirect
91. C97#1637semidirect
93. #487D97semidirect

Maximal subgroups

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

subgroupindex
#202614002
#202610272
#202610222
#202604812
#202604672
#202604642
#487×C972
#163797