Finite Group Databasebeta

← #27140868

Subgroups — #27140868

Order 12416.

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

normal subgroups only

This group has 214 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#20261442C211yesmaximal, characteristic
2#20261114C211yesmaximal, characteristic, complemented
3#20261027C211yesmaximal, characteristic, complemented
4#20260483C211yesmaximal, characteristic, complemented
5#20260465C211yesmaximal, characteristic, complemented
6D4⋊#11715895C211yesmaximal, characteristic
7#476×C97C211yesmaximal, characteristic
8C2×#13095501—22no—
9#15690810C2211yescharacteristic
10#15690801—21no—
11#15690789—21no—
12#15690787—22no—
13C4×Dic194C2211yescharacteristic
14D4×Dic97—21no—
15C2×C4⋊Dic97—21no—
16#15690743—21no—
17C4⋊#11715895C2211yescharacteristic
18#15690733—21no—
19C4×#11715895C2211yescharacteristic
20C4⋊Q8×C97C2211yescharacteristic
21C4×D4×C97C2211yescharacteristic
22#106×C97—21no—
23C4⋊C8×C97C2211yescharacteristic
24#102×C97—21no—
25C2×Dic194D411yescharacteristic
26C2×Dic194D412yes—
27C22×Dic97—22no—
28#13095501—44no—
29C2×#11715895—21no—
30C2×#11715895—21no—
31C4⋊Dic97—22no—
32C4⋊Dic97—21no—
33C4⋊Dic97—21no—
34C4⋊Dic97D411yescharacteristic
35#13095490—22no—
36#13095490—21no—
37C22⋊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—42no—
50D4×C97—21no—
51D4×C97Q1612yescomplemented
52C2×C4×C97—21no—
53C2×C4×C97—41no—
54C2×C4×C97—41no—
55C2×C4×C97—41no—
56C2×C4×C97C2×D411yescharacteristic
57C2×C4×C97C2×D411yescharacteristic
58C2×C4×C97C2×D411yescharacteristic
59C2×Dic97—21no—
60C2×Dic97—21no—
61C2×Dic97—22no—
62C2×Dic97—82no—
63Dic194—21no—
64Dic194—21no—
65Dic194—42no—
66C8×C97—41no—
67#11715895—22no—
68#11715895—41no—
69C22×C97—22no—
70C22×C97—41no—
71C22×C97—41no—
72C22×C97#12011yescharacteristic
73C4×C97—21no—
74C4×C97—41no—
75C4×C97—81no—
76C4×C97C2×Q1611yescharacteristic
77C4×C97#13511yescharacteristic
78C4×C97#13711yescharacteristic
79C4×C97#13711yescharacteristic
80Dic97—41no—
81Dic97—42no—
82Dic97—41no—
83C2×C97—42no—
84C2×C97#45111yescharacteristic
85C2×C97#45211yescharacteristic
86C2×C97#45311yescharacteristic
87#1632—971nomaximal, Sylow
88C97#163211yescharacteristic, complemented, Sylow
89#554—971no—
90#494—971no—
91#477—971no—
92#476D9711yescharacteristic
93#333—971no—
94Q8⋊C8—971no—
95D4⋊C8—971no—
96C2×Q16—1942no—
97C4⋊Q8—971no—
98C4⋊Q8C2×D9711yescharacteristic
99#125—1941no—
100C22⋊Q8—1942no—
101C22⋊Q8—1941no—
102C4×Q8—971no—
103C4×D4—1941no—
104C4×D4C2×D9711yescharacteristic
105C2×C4⋊C4—1941no—
106#107—1941no—
107#106—21no—
108C4⋊C8—971no—
109C4⋊C8C2×D9711yescharacteristic
110#103—1941no—
111#102—21no—
112C4×C8—971no—
113C2×Q8—21no—
114C2×Q8—972no—
115C2×Q8—971no—
116C2×D4C22⋊D9711yescharacteristic
117C22×C4—21no—
118C22×C4—1942no—
119Q16—3884no—
120C2×C8—21no—
121C2×C8—1941no—
122C2×C8—1941no—
123C4⋊C4—21no—
124C4⋊C4—21no—
125C4⋊C4—971no—
126C4⋊C4—1942no—
127C4⋊C4—1942no—
128C4⋊C4—1941no—
129C4⋊C4—1941no—
130C4⋊C4—1941no—
131C4⋊C4C22⋊D9711yescharacteristic
132C22⋊C4—21no—
133C22⋊C4—1942no—
134C42—1941no—
135C42C22×D9711yescharacteristic
136C23—21no—
137Q8—42no—
138Q8—3882no—
139Q8—1941no—
140Q8—1941no—
141D4—21no—
142D4#1309550112yescomplemented
143C2×C4—21no—
144C2×C4—41no—
145C2×C4—41no—
146C2×C4—41no—
147C2×C4—7762no—
148C2×C4—1941no—
149C2×C4—1941no—
150C2×C4—1942no—
151C2×C4D4×D9711yescharacteristic
152C2×C4D4×D9711yescharacteristic
153C2×C4C2×C22⋊D9711yescharacteristic
154C8—41no—
155C8—3881no—
156C8—1942no—
157C22—22no—
158C22—41no—
159C22—41no—
160C22#1569077811yescharacteristic
161C4—21no—
162C4—41no—
163C4—81no—
164C4—3881no—
165C4—3881no—
166C4—3882no—
167C4C2×#1309550111yescharacteristic
168C4#1569083111yescharacteristic
169C4#1569083911yescharacteristic
170C4#1569084011yescharacteristic
171C2—42no—
172C2#2026089011yescharacteristic
173C2#2026089411yescharacteristic
174C2#2026090111yescharacteristic
subgroupG/H
centerC22#15690778
commutator, cosocleC42×C97C23
FrattiniC42C22×D97
Fitting#476×C97C2
socleC22×C97#120
prime radicalC2×D4×C97D4
hypercenter#476D97
hypocenterC97#1632
LCSG ▷ C42×C97 ▷ C22×C97 ▷ C97
UCSC1 ◁ C22 ◁ C42 ◁ #476

Complements

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

NH ≅ G/Ntype
2. #20261114C2semidirect
3. #20261027C2semidirect
4. #20260483C2semidirect
5. #20260465C2semidirect
51. D4×C97Q16semidirect
88. C97#1632semidirect
142. D4#13095501semidirect

Maximal subgroups

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

subgroupindex
#202614422
#202611142
#202610272
#202604832
#202604652
D4⋊#117158952
#476×C972
#163297