Finite Group Databasebeta

← #27140869

Subgroups — #27140869

Order 12416.

214 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 214 conjugacy classes of subgroups — too many to draw. Only its 55 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#20261445C211yesmaximal, characteristic
2#20261114C211yesmaximal, characteristic, complemented
3#20261022C211yesmaximal, characteristic
4#20260483C211yesmaximal, characteristic, complemented
5Q8⋊#11715895C211yesmaximal, characteristic, complemented
6#20260462C211yesmaximal, characteristic
7#477×C97C211yesmaximal, characteristic, complemented
8C2×#13095501—22no—
9#15690810C2211yescharacteristic
10#15690802—21no—
11#15690788—22no—
12#15690786—21no—
13C4×D388—21no—
14C4×D388C2211yescharacteristic
15C4⋊C4×D97—21no—
16#15690741—21no—
17C4⋊#11715895C2211yescharacteristic
18#15690730—21no—
19C4×#11715895C2211yescharacteristic
20C4⋊Q8×C97C2211yescharacteristic
21C4×Q8×C97C2211yescharacteristic
22#107×C97—21no—
23C4⋊C8×C97C2211yescharacteristic
24#103×C97—21no—
25C2×Dic194D412yes—
26C2×D388D411yescharacteristic
27C2×C4×D97—21no—
28C2×C4×D97—22no—
29#13095501—44no—
30C2×#11715895—21no—
31C2×#11715895—21no—
32C4⋊Dic97—21no—
33C4⋊Dic97—21no—
34C4⋊Dic97D411yescharacteristic
35#13095490—22no—
36#13095490—22no—
37#13095488—22no—
38#13095488—21no—
39C2×Q8×C97—21no—
40C2×Q8×C97D411yescharacteristic
41C2×C8×C97—21no—
42C4⋊C4×C97—21no—
43C4⋊C4×C97—21no—
44C4⋊C4×C97—21no—
45C4⋊C4×C97D411yescharacteristic
46C42×C97—21no—
47C42×C97C2311yescharacteristic
48C22×D97—21no—
49Q8×C97—21no—
50Q8×C97—21no—
51Q8×C97—42no—
52C2×C4×C97—21no—
53C2×C4×C97—21no—
54C2×C4×C97—41no—
55C2×C4×C97C2×D411yescharacteristic
56C2×C4×C97C2×D411yescharacteristic
57C2×C4×C97C2×D411yescharacteristic
58C2×Dic97—21no—
59C2×Dic97—22no—
60D388—21no—
61D388Q1612yescomplemented
62C4×D97—41no—
63C4×D97—41no—
64C4×D97—82no—
65Dic194—42no—
66C8×C97—41no—
67#11715895—22no—
68#11715895—41no—
69C22×C97#12011yescharacteristic
70C2×D97—22no—
71C2×D97—41no—
72C2×D97—41no—
73C4×C97—21no—
74C4×C97—41no—
75C4×C97—41no—
76C4×C97—81no—
77C4×C97C2×Q1611yescharacteristic
78C4×C97#13511yescharacteristic
79C4×C97#13711yescharacteristic
80C4×C97#13711yescharacteristic
81Dic97—41no—
82Dic97—42no—
83C2×C97#45111yescharacteristic
84C2×C97#45211yescharacteristic
85C2×C97#45311yescharacteristic
86D97—42no—
87#1632—971nomaximal, Sylow
88C97#163211yescharacteristic, complemented, Sylow
89#554—971no—
90#494—971no—
91#477D9711yescharacteristic, complemented
92#476—971no—
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×Q8C2×D9711yescharacteristic
103C4×D4—971no—
104C4×D4—1941no—
105C2×C4⋊C4—1941no—
106#107—21no—
107#106—1941no—
108C4⋊C8—971no—
109C4⋊C8C2×D9711yescharacteristic
110#103—21no—
111#102—1941no—
112C4×C8—971no—
113C2×Q8—21no—
114C2×Q8—972no—
115C2×Q8C22⋊D9711yescharacteristic
116C2×D4—971no—
117C22×C4—1942no—
118C22×C4—1941no—
119Q16—3884no—
120C2×C8—21no—
121C2×C8—1941no—
122C2×C8—1941no—
123C4⋊C4—21no—
124C4⋊C4—21no—
125C4⋊C4—21no—
126C4⋊C4—971no—
127C4⋊C4—1942no—
128C4⋊C4—1942no—
129C4⋊C4—1941no—
130C4⋊C4—1941no—
131C4⋊C4C22⋊D9711yescharacteristic
132C22⋊C4—1942no—
133C22⋊C4—1941no—
134C42—21no—
135C42C22×D9711yescharacteristic
136C23—1941no—
137Q8—21no—
138Q8—21no—
139Q8—42no—
140Q8—3882no—
141D4—972no—
142D4—1941no—
143C2×C4—21no—
144C2×C4—21no—
145C2×C4—41no—
146C2×C4—7762no—
147C2×C4—3881no—
148C2×C4—3881no—
149C2×C4—1941no—
150C2×C4—1942no—
151C2×C4D4×D9711yescharacteristic
152C2×C4D4×D9711yescharacteristic
153C2×C4C2×C22⋊D9711yescharacteristic
154C8—41no—
155C8—3881no—
156C8—1942no—
157C22—3881no—
158C22—3881no—
159C22—1942no—
160C22#1569077811yescharacteristic
161C4—21no—
162C4—41no—
163C4—41no—
164C4—81no—
165C4—3881no—
166C4—3882no—
167C4Q16×D9711yescharacteristic
168C4#1569083011yescharacteristic
169C4#1569084111yescharacteristic
170C4#1569084211yescharacteristic
171C2—3882no—
172C2#2026088911yescharacteristic
173C2#2026089511yescharacteristic
174C2#2026090111yescharacteristic
subgroupG/H
centerC22#15690778
commutator, cosocleC42×C97C23
FrattiniC42C22×D97
Fitting#477×C97C2
socleC22×C97#120
prime radicalC2×D388D4
hypercenter#477D97
hypocenterC97#1632
LCSG ▷ C42×C97 ▷ C22×C97 ▷ C97
UCSC1 ◁ C22 ◁ C42 ◁ #477

Complements

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

NH ≅ G/Ntype
2. #20261114C2semidirect
4. #20260483C2semidirect
5. Q8⋊#11715895C2semidirect
7. #477×C97C2semidirect
61. D388Q16semidirect
88. C97#1632semidirect
91. #477D97semidirect

Maximal subgroups

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

subgroupindex
#202614452
#202611142
#202610222
#202604832
Q8⋊#117158952
#202604622
#477×C972
#163297