Finite Group Databasebeta

← #27141005

Subgroups — #27141005

Order 12416.

200 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 200 conjugacy classes of subgroups — too many to draw. Only its 53 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#20261455C211yesmaximal, characteristic
2#20261131C211yesmaximal, characteristic, complemented
3#20260942C211yesmaximal, characteristic, complemented
4#20260482C211yesmaximal, characteristic, complemented
5#20260465C211yesmaximal, characteristic, complemented
6D4⋊#11715895C211yesmaximal, characteristic
7#474×C97C211yesmaximal, characteristic
8C2×#13095501—21no—
9#15690810C2211yescharacteristic
10#15690803—22no—
11#15690797—21no—
12#15690789—21no—
13C4×Dic194C2211yescharacteristic
14#15690775—21no—
15#15690769—21no—
16#15690741—22no—
17C4⋊#11715895C2211yescharacteristic
18#15690734—21no—
19C4×#11715895C2211yescharacteristic
20C4⋊Q8×C97C2211yescharacteristic
21C4×D4×C97C2211yescharacteristic
22#107×C97—21no—
23C4⋊C8×C97C2211yescharacteristic
24#102×C97—21no—
25C2×Dic194—21no—
26C2×Dic194D411yescharacteristic
27#13095501—42no—
28C2×#11715895—21no—
29C2×#11715895—21no—
30C4⋊Dic97—21no—
31C4⋊Dic97D411yescharacteristic
32C4⋊Dic97D412yes—
33#13095490—22no—
34#13095490—21no—
35C22⋊Dic97—22no—
36C22⋊Dic97—22no—
37C4×Dic97—22no—
38C4×Dic97—21no—
39C2×Q8×C97—21no—
40C2×D4×C97D411yescharacteristic
41C22×C4×C97—21no—
42C2×C8×C97—21no—
43C4⋊C4×C97—21no—
44C4⋊C4×C97—21no—
45C4⋊C4×C97D411yescharacteristic
46C22⋊C4×C97—21no—
47C42×C97C2311yescharacteristic
48C23×C97—21no—
49Q8×C97—41no—
50Q8×C97—41no—
51D4×C97—21no—
52D4×C97—21no—
53C2×C4×C97—21no—
54C2×C4×C97—41no—
55C2×C4×C97—41no—
56C2×C4×C97—41no—
57C2×C4×C97C2×D411yescharacteristic
58C2×C4×C97C2×D411yescharacteristic
59C2×C4×C97C2×D411yescharacteristic
60C2×Dic97—21no—
61C2×Dic97—21no—
62C2×Dic97—22no—
63Dic194—21no—
64Dic194—21no—
65Dic194—41no—
66Dic194—41no—
67C8×C97—41no—
68#11715895—22no—
69#11715895—41no—
70C22×C97—41no—
71C22×C97—41no—
72C22×C97—41no—
73C22×C97#12011yescharacteristic
74C4×C97—21no—
75C4×C97—41no—
76C4×C97—81no—
77C4×C97#13511yescharacteristic
78C4×C97#13511yescharacteristic
79C4×C97AGL(1,ℤ/8)11yescharacteristic
80C4×C97#13711yescharacteristic
81Dic97—41no—
82Dic97—42no—
83Dic97—41no—
84C2×C97—81no—
85C2×C97#44911yescharacteristic
86C2×C97#45211yescharacteristic
87C2×C97#45611yescharacteristic
88#1652—971nomaximal, Sylow
89C97#165211yescharacteristic, complemented, Sylow
90#556—971no—
91#498—971no—
92#474D9711yescharacteristic
93#462—971no—
94#332—971no—
95Q8⋊C8—971no—
96D4⋊C8—971no—
97C2×Q16—1941no—
98C4⋊Q8—971no—
99C4⋊Q8C2×D9711yescharacteristic
100#126—1942no—
101#124—1941no—
102C22⋊Q8—1941no—
103C4×Q8—971no—
104C4×Q8—1941no—
105C4×D4C2×D9711yescharacteristic
106#117—1941no—
107#107—21no—
108#106—1942no—
109C4⋊C8—971no—
110C4⋊C8C2×D9711yescharacteristic
111#103—1941no—
112#102—21no—
113C4×C8—971no—
114C2×Q8—21no—
115C2×Q8—971no—
116C2×Q8—1941no—
117C2×D4C22⋊D9711yescharacteristic
118C22×C4—21no—
119Q16—3882no—
120C2×C8—21no—
121C2×C8—1941no—
122C2×C8—1941no—
123C4⋊C4—21no—
124C4⋊C4—21no—
125C4⋊C4—971no—
126C4⋊C4—972no—
127C4⋊C4—1942no—
128C4⋊C4—1941no—
129C4⋊C4—1941no—
130C4⋊C4C22⋊D9711yescharacteristic
131C22⋊C4—21no—
132C22⋊C4—1942no—
133C22⋊C4—1942no—
134C42—1942no—
135C42—1941no—
136C42C22×D9711yescharacteristic
137C23—21no—
138Q8—41no—
139Q8—41no—
140Q8—3881no—
141Q8—3881no—
142Q8—1941no—
143Q8—1941no—
144D4—21no—
145D4—21no—
146C2×C4—21no—
147C2×C4—41no—
148C2×C4—41no—
149C2×C4—41no—
150C2×C4—1941no—
151C2×C4—1942no—
152C2×C4—1941no—
153C2×C4D4×D9711yescharacteristic
154C2×C4D4×D9711yescharacteristic
155C2×C4C2×C22⋊D9711yescharacteristic
156C8—41no—
157C8—3881no—
158C8—1942no—
159C22—41no—
160C22—41no—
161C22—41no—
162C22#1569077811yescharacteristic
163C4—21no—
164C4—41no—
165C4—81no—
166C4—3881no—
167C4—3882no—
168C4—3881no—
169C4#1569082911yescharacteristic
170C4#1569083011yescharacteristic
171C4#1569083411yescharacteristic
172C4#1569084111yescharacteristic
173C2—81no—
174C2#2026087711yescharacteristic
175C2#2026089511yescharacteristic
176C2#2026091611yescharacteristic
subgroupG/H
centerC22#15690778
commutator, cosocleC42×C97C23
FrattiniC42C22×D97
Fitting#474×C97C2
socleC22×C97#120
prime radicalC2×D4×C97D4
hypercenter#474D97
hypocenterC97#1652
LCSG ▷ C42×C97 ▷ C22×C97 ▷ C97
UCSC1 ◁ C22 ◁ C42 ◁ #474

Complements

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

NH ≅ G/Ntype
2. #20261131C2semidirect
3. #20260942C2semidirect
4. #20260482C2semidirect
5. #20260465C2semidirect
89. C97#1652semidirect

Maximal subgroups

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

subgroupindex
#202614552
#202611312
#202609422
#202604822
#202604652
D4⋊#117158952
#474×C972
#165297