Finite Group Databasebeta

← #27140849

Subgroups — #27140849

Order 12416.

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

normal subgroups only

This group has 216 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#20261458C211yesmaximal, characteristic, complemented
2#20261082C211yesmaximal, characteristic, complemented
3#20260935C211yesmaximal, characteristic, complemented
4#20260931C211yesmaximal, characteristic
5#20260482C211yesmaximal, characteristic, complemented
6#20260464C211yesmaximal, characteristic, complemented
7Q8⋊C8×C97C211yesmaximal, characteristic, complemented
8C2×#13095498—21no—
9C2×D776—21no—
10#15690810C2211yescharacteristic
11#15690807C2211yescharacteristic
12#15690805—22no—
13#15690798—21no—
14#15690786—21no—
15C4×Dic194—21no—
16C4×D388C2211yescharacteristic
17#15690768—21no—
18C4⋊#11715895C2211yescharacteristic
19#15690732—21no—
20#15690732—22no—
21#15690730—21no—
22C4×Q8×C97C2211yescharacteristic
23C4⋊C8×C97C2211yescharacteristic
24C4×C8×C97C2211yescharacteristic
25C2×Dic194—21no—
26C2×D388—21no—
27C2×D388—21no—
28C2×D388—21no—
29C2×D388D411yescharacteristic
30C2×C4×D97—21no—
31#13095498—42no—
32D776—42no—
33C2×#11715895—21no—
34C4⋊Dic97—21no—
35C4⋊Dic97D411yescharacteristic
36C4⋊Dic97D412yes—
37#13095490—22no—
38#13095488—22no—
39#13095488—22no—
40#13095488—21no—
41C4×Dic97—22no—
42C2×Q8×C97D411yescharacteristic
43C2×C8×C97—21no—
44C2×C8×C97—21no—
45C4⋊C4×C97—21no—
46C4⋊C4×C97D411yescharacteristic
47C42×C97—21no—
48C42×C97C2311yescharacteristic
49C22×D97—21no—
50C22×D97—41no—
51Q8×C97—21no—
52Q8×C97—21no—
53C2×C4×C97—21no—
54C2×C4×C97—21no—
55C2×C4×C97C2×D411yescharacteristic
56C2×C4×C97C2×D411yescharacteristic
57C2×C4×C97C2×D411yescharacteristic
58C2×Dic97—21no—
59C2×Dic97—22no—
60D388—21no—
61D388—21no—
62D388—42no—
63D388—41no—
64D388—41no—
65D388—41no—
66D388—41no—
67C4×D97—41no—
68C4×D97—41no—
69Dic194—41no—
70Dic194—41no—
71C8×C97—41no—
72C8×C97—41no—
73#11715895—41no—
74C22×C97#12011yescharacteristic
75C2×D97—41no—
76C2×D97—41no—
77C2×D97—41no—
78C2×D97—81no—
79C2×D97—81no—
80C2×D97—81no—
81C4×C97—21no—
82C4×C97—41no—
83C4×C97—41no—
84C4×C97#13511yescharacteristic
85C4×C97#13511yescharacteristic
86C4×C97AGL(1,ℤ/8)11yescharacteristic
87C4×C97AGL(1,ℤ/8)11yescharacteristic
88Dic97—42no—
89Dic97—41no—
90C2×C97#44911yescharacteristic
91C2×C97#44911yescharacteristic
92C2×C97#45411yescharacteristic
93D97—81no—
94D97—161no—
95#1628—971nomaximal, Sylow
96C97#162811yescharacteristic, complemented, Sylow
97#556—971no—
98#486—971no—
99#460—971no—
100#459—971no—
101#332—971no—
102Q8⋊C8D9711yescharacteristic, complemented
103D4⋊C8—971no—
104C2×SD16—1941no—
105C2×D8—1941no—
106C4⋊Q8—971no—
107#127—971no—
108#126—1942no—
109#124—1941no—
110C22⋊Q8—1941no—
111C4×Q8—1941no—
112C4×Q8C2×D9711yescharacteristic
113C4×D4—971no—
114#117—1941no—
115C4⋊C8—971no—
116C4⋊C8C2×D9711yescharacteristic
117#102—1941no—
118#102—1942no—
119#102—1941no—
120C4×C8C2×D9711yescharacteristic
121C2×Q8—1941no—
122C2×Q8D38811yescharacteristic
123C2×D4—971no—
124C2×D4—1941no—
125C2×D4—1941no—
126C2×D4—1941no—
127C22×C4—1941no—
128SD16—3882no—
129D8—3882no—
130C2×C8—21no—
131C2×C8—21no—
132C2×C8—1941no—
133C4⋊C4—21no—
134C4⋊C4—971no—
135C4⋊C4—972no—
136C4⋊C4—1942no—
137C4⋊C4—1941no—
138C4⋊C4D38811yescharacteristic
139C22⋊C4—1942no—
140C22⋊C4—1942no—
141C22⋊C4—1941no—
142C42—21no—
143C42—1942no—
144C42C22×D9711yescharacteristic
145C23—3881no—
146C23—1941no—
147Q8—21no—
148Q8—21no—
149Q8—3881no—
150Q8—3881no—
151D4—3882no—
152D4—3881no—
153D4—3881no—
154D4—3881no—
155D4—3881no—
156D4—1941no—
157D4—1941no—
158C2×C4—21no—
159C2×C4—21no—
160C2×C4—3881no—
161C2×C4—3881no—
162C2×C4—1941no—
163C2×C4—1942no—
164C2×C4D4×D9711yescharacteristic
165C2×C4D4×D9711yescharacteristic
166C2×C4C2×D38811yescharacteristic
167C8—41no—
168C8—41no—
169C8—3881no—
170C22—7761no—
171C22—7761no—
172C22—7761no—
173C22—3881no—
174C22—3881no—
175C22—3881no—
176C22#1569078011yescharacteristic
177C4—21no—
178C4—41no—
179C4—41no—
180C4—3881no—
181C4—3882no—
182C4#1569083211yescharacteristic
183C4#1569083311yescharacteristic
184C4#1569083711yescharacteristic
185C4#1569083811yescharacteristic
186C2—7761no—
187C2—15521no—
188C2#2026088211yescharacteristic
189C2#2026088311yescharacteristic
190C2#2026090911yescharacteristic
subgroupG/H
centerC22#15690780
commutator, cosocleC42×C97C23
FrattiniC42C22×D97
FittingQ8⋊C8×C97C2
socleC22×C97#120
prime radical#20260931C2
hypercenterQ8⋊C8D97
hypocenterC97#1628
LCSG ▷ C42×C97 ▷ C22×C97 ▷ C97
UCSC1 ◁ C22 ◁ C42 ◁ Q8⋊C8

Complements

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

NH ≅ G/Ntype
1. #20261458C2semidirect
2. #20261082C2semidirect
3. #20260935C2semidirect
5. #20260482C2semidirect
6. #20260464C2semidirect
7. Q8⋊C8×C97C2semidirect
96. C97#1628semidirect
102. Q8⋊C8D97semidirect

Maximal subgroups

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

subgroupindex
#202614582
#202610822
#202609352
#202609312
#202604822
#202604642
Q8⋊C8×C972
#162897