Finite Group Databasebeta

← #27140841

Subgroups — #27140841

Order 12416.

250 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 250 conjugacy classes of subgroups — too many to draw. Only its 59 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
1Q8×D388C211yesmaximal, characteristic, complemented
2#20261082C211yesmaximal, characteristic, complemented
3#20260935C211yesmaximal, characteristic, complemented
4#20260931C211yesmaximal, characteristic
5#20260480C211yesmaximal, characteristic, complemented
6#20260464C211yesmaximal, characteristic, complemented
7Q8⋊C8×C97C211yesmaximal, characteristic, complemented
8C2×Q8×D97—21no—
9C2×#13095498—21no—
10C2×D776—21no—
11#15690810—21no—
12#15690810C2211yescharacteristic
13#15690807C2211yescharacteristic
14#15690790—22no—
15#15690790—21no—
16C4×D388—21no—
17C4×D388C2211yescharacteristic
18C4⋊#11715895C2211yescharacteristic
19#15690732—21no—
20#15690732—22no—
21#15690730—21no—
22C4×Q8×C97C2211yescharacteristic
23C4⋊C8×C97C2211yescharacteristic
24C4×C8×C97C2211yescharacteristic
25C2×Dic194—22no—
26C2×Dic194—21no—
27Q8×D97—42no—
28Q8×D97—42no—
29C2×D388—21no—
30C2×D388—21no—
31C2×D388—21no—
32C2×D388D411yescharacteristic
33C2×C4×D97—22no—
34C2×C4×D97—21no—
35#13095498—42no—
36D776—42no—
37C2×#11715895—21no—
38C4⋊Dic97—22no—
39C4⋊Dic97—21no—
40C4⋊Dic97D411yescharacteristic
41C4⋊Dic97D412yes—
42#13095488—22no—
43#13095488—21no—
44C2×Q8×C97D411yescharacteristic
45C2×C8×C97—21no—
46C2×C8×C97—21no—
47C4⋊C4×C97—21no—
48C4⋊C4×C97D411yescharacteristic
49C42×C97—21no—
50C42×C97C2311yescharacteristic
51C22×D97—21no—
52C22×D97—41no—
53Q8×C97—21no—
54Q8×C97D812yescomplemented
55C2×C4×C97—21no—
56C2×C4×C97—21no—
57C2×C4×C97C2×D411yescharacteristic
58C2×C4×C97C2×D411yescharacteristic
59C2×C4×C97C2×D411yescharacteristic
60C2×Dic97—22no—
61C2×Dic97—21no—
62D388—21no—
63D388—42no—
64D388—41no—
65D388—41no—
66D388—41no—
67D388—41no—
68D388SD1612yescomplemented
69C4×D97—44no—
70C4×D97—41no—
71C4×D97—41no—
72Dic194—44no—
73Dic194—41no—
74Dic194—41no—
75C8×C97—41no—
76C8×C97—41no—
77#11715895—41no—
78C22×C97#12011yescharacteristic
79C2×D97—22no—
80C2×D97—41no—
81C2×D97—41no—
82C2×D97—81no—
83C2×D97—81no—
84C2×D97—81no—
85C4×C97—22no—
86C4×C97—21no—
87C4×C97—41no—
88C4×C97C2×D811yescharacteristic
89C4×C97C2×SD1611yescharacteristic
90C4×C97AGL(1,ℤ/8)11yescharacteristic
91C4×C97#13711yescharacteristic
92Dic97—41no—
93Dic97—42no—
94C2×C97#44711yescharacteristic
95C2×C97#44811yescharacteristic
96C2×C97#45411yescharacteristic
97D97—42no—
98D97—161no—
99#1627—971nomaximal, Sylow
100C97#162711yescharacteristic, complemented, Sylow
101D4×Q8—971no—
102#486—971no—
103#460—971no—
104#459—971no—
105#332—971no—
106Q8⋊C8D9711yescharacteristic, complemented
107D4⋊C8—971no—
108C22×Q8—1941no—
109C2×SD16—1941no—
110C2×D8—1941no—
111C4⋊Q8—971no—
112C4⋊Q8—1941no—
113#127—971no—
114C22⋊Q8—1942no—
115C22⋊Q8—1941no—
116C4×Q8C2×D9711yescharacteristic
117C4×D4—971no—
118C4×D4—1941no—
119C4⋊C8—971no—
120C4⋊C8C2×D9711yescharacteristic
121#102—1941no—
122#102—1942no—
123#102—1941no—
124C4×C8C2×D9711yescharacteristic
125C2×Q8—3882no—
126C2×Q8—3882no—
127C2×Q8—1942no—
128C2×Q8—1941no—
129C2×Q8D38811yescharacteristic
130C2×D4—971no—
131C2×D4—1941no—
132C2×D4—1941no—
133C2×D4—1941no—
134C22×C4—1942no—
135C22×C4—1941no—
136SD16—3882no—
137D8—3882no—
138C2×C8—21no—
139C2×C8—21no—
140C2×C8—1941no—
141C4⋊C4—21no—
142C4⋊C4—971no—
143C4⋊C4—972no—
144C4⋊C4—1942no—
145C4⋊C4—1941no—
146C4⋊C4D38811yescharacteristic
147C22⋊C4—1942no—
148C22⋊C4—1941no—
149C42—21no—
150C42C22×D9711yescharacteristic
151C23—3881no—
152C23—1941no—
153Q8—21no—
154Q8—3884no—
155Q8—3881no—
156Q8—3881no—
157Q8D77612yescomplemented
158D4—972no—
159D4—3882no—
160D4—3881no—
161D4—3881no—
162D4—3881no—
163D4—3881no—
164D4—1941no—
165C2×C4—21no—
166C2×C4—21no—
167C2×C4—3884no—
168C2×C4—3881no—
169C2×C4—3881no—
170C2×C4—1942no—
171C2×C4—1941no—
172C2×C4D4×D9711yescharacteristic
173C2×C4D4×D9711yescharacteristic
174C2×C4C2×D38811yescharacteristic
175C8—41no—
176C8—41no—
177C8—3881no—
178C22—7761no—
179C22—7761no—
180C22—7761no—
181C22—3881no—
182C22—3881no—
183C22—1942no—
184C22#1569078011yescharacteristic
185C4—22no—
186C4—21no—
187C4—41no—
188C4—3882no—
189C4—3881no—
190C4C2×D77611yescharacteristic
191C4SD16×D9711yescharacteristic
192C4#1569083811yescharacteristic
193C4#1569084211yescharacteristic
194C2—15521no—
195C2—3882no—
196C2#2026087111yescharacteristic
197C2#2026087511yescharacteristic
198C2#2026090911yescharacteristic
subgroupG/H
centerC22#15690780
commutator, cosocleC42×C97C23
FrattiniC42C22×D97
FittingQ8⋊C8×C97C2
socleC22×C97#120
prime radical#20260931C2
hypercenterQ8⋊C8D97
hypocenterC97#1627
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. Q8×D388C2semidirect
2. #20261082C2semidirect
3. #20260935C2semidirect
5. #20260480C2semidirect
6. #20260464C2semidirect
7. Q8⋊C8×C97C2semidirect
54. Q8×C97D8semidirect
68. D388SD16semidirect
100. C97#1627semidirect
106. Q8⋊C8D97semidirect
157. Q8D776semidirect

Maximal subgroups

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

subgroupindex
Q8×D3882
#202610822
#202609352
#202609312
#202604802
#202604642
Q8⋊C8×C972
#162797