Finite Group Databasebeta

← #27140837

Subgroups — #27140837

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#20261081C211yesmaximal, characteristic, complemented
3#20260927C211yesmaximal, characteristic
4#20260482C211yesmaximal, characteristic, complemented
5Q8⋊#11715895C211yesmaximal, characteristic, complemented
6#20260462C211yesmaximal, characteristic, complemented
7#460×C97C211yesmaximal, characteristic, complemented
8C2×Q8×D97—21no—
9C2×#13095496—21no—
10#15690810—21no—
11#15690810C2211yescharacteristic
12#15690790—22no—
13#15690790—21no—
14C4×D388—21no—
15C4×D388C2211yescharacteristic
16C4⋊#11715895C2211yescharacteristic
17#15690731—21no—
18#15690731—22no—
19C4×#11715895C2211yescharacteristic
20C2×SD16×C97—21no—
21#127×C97C2211yescharacteristic
22C4×Q8×C97C2211yescharacteristic
23C4⋊C8×C97C2211yescharacteristic
24#102×C97—21no—
25C2×Dic194—22no—
26C2×Dic194—21no—
27Q8×D97—42no—
28Q8×D97—42no—
29C2×D388D411yescharacteristic
30C2×C4×D97—22no—
31C2×C4×D97—21no—
32#13095496—42no—
33C2×#11715895—21no—
34C2×#11715895—21no—
35C4⋊Dic97—22no—
36C4⋊Dic97—21no—
37C4⋊Dic97D411yescharacteristic
38C4⋊Dic97D412yes—
39#13095488—22no—
40#13095488—21no—
41C2×Q8×C97D411yescharacteristic
42C2×D4×C97—21no—
43C2×D4×C97—21no—
44C2×D4×C97—21no—
45SD16×C97—42no—
46C2×C8×C97—21no—
47C4⋊C4×C97—21no—
48C4⋊C4×C97D411yescharacteristic
49C42×C97—21no—
50C42×C97C2311yescharacteristic
51C23×C97—41no—
52C22×D97—21no—
53Q8×C97—21no—
54Q8×C97D812yescomplemented
55D4×C97—42no—
56D4×C97—41no—
57D4×C97—41no—
58D4×C97—41no—
59D4×C97—41no—
60C2×C4×C97—21no—
61C2×C4×C97—21no—
62C2×C4×C97C2×D411yescharacteristic
63C2×C4×C97C2×D411yescharacteristic
64C2×C4×C97C2×D411yescharacteristic
65C2×Dic97—22no—
66C2×Dic97—21no—
67D388—21no—
68D388SD1612yescomplemented
69C4×D97—44no—
70C4×D97—41no—
71C4×D97—41no—
72Dic194—44no—
73Dic194—41no—
74Dic194—41no—
75C8×C97—41no—
76#11715895—41no—
77#11715895—41no—
78C22×C97—81no—
79C22×C97—81no—
80C22×C97—81no—
81C22×C97#12011yescharacteristic
82C2×D97—22no—
83C2×D97—41no—
84C2×D97—41no—
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—161no—
95C2×C97#44711yescharacteristic
96C2×C97#44811yescharacteristic
97C2×C97#45411yescharacteristic
98D97—42no—
99#1627—971nomaximal, Sylow
100C97#162711yescharacteristic, complemented, Sylow
101D4×Q8—971no—
102#486—971no—
103#460D9711yescharacteristic, complemented
104#459—971no—
105#332—971no—
106Q8⋊C8—971no—
107D4⋊C8—971no—
108C22×Q8—1941no—
109C2×SD16—21no—
110C2×D8—1941no—
111C4⋊Q8—971no—
112C4⋊Q8—1941no—
113#127C2×D9711yescharacteristic
114C22⋊Q8—1942no—
115C22⋊Q8—1941no—
116C4×Q8C2×D9711yescharacteristic
117C4×D4—971no—
118C4×D4—1941no—
119C4⋊C8—971no—
120C4⋊C8C2×D9711yescharacteristic
121#102—21no—
122#102—1941no—
123#102—1942no—
124C4×C8—971no—
125C2×Q8—3882no—
126C2×Q8—3882no—
127C2×Q8—1942no—
128C2×Q8—1941no—
129C2×Q8C22⋊D9711yescharacteristic
130C2×D4—21no—
131C2×D4—21no—
132C2×D4—21no—
133C2×D4—971no—
134C22×C4—1942no—
135C22×C4—1941no—
136SD16—42no—
137D8—3882no—
138C2×C8—21no—
139C2×C8—1941no—
140C2×C8—1941no—
141C4⋊C4—21no—
142C4⋊C4—971no—
143C4⋊C4—972no—
144C4⋊C4—1942no—
145C4⋊C4—1941no—
146C4⋊C4C22⋊D9711yescharacteristic
147C22⋊C4—1942no—
148C22⋊C4—1941no—
149C42—21no—
150C42C22×D9711yescharacteristic
151C23—41no—
152C23—1941no—
153Q8—21no—
154Q8—3884no—
155Q8—3881no—
156Q8—3881no—
157Q8#1309549612yescomplemented
158D4—42no—
159D4—41no—
160D4—41no—
161D4—41no—
162D4—41no—
163D4—972no—
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×C22⋊D9711yescharacteristic
175C8—41no—
176C8—3881no—
177C8—3881no—
178C22—81no—
179C22—81no—
180C22—81no—
181C22—3881no—
182C22—3881no—
183C22—1942no—
184C22#1569077811yescharacteristic
185C4—22no—
186C4—21no—
187C4—41no—
188C4—3882no—
189C4—3881no—
190C4C2×#1309549611yescharacteristic
191C4SD16×D9711yescharacteristic
192C4#1569083511yescharacteristic
193C4#1569083911yescharacteristic
194C2—161no—
195C2—3882no—
196C2#2026086811yescharacteristic
197C2#2026087211yescharacteristic
198C2#2026090611yescharacteristic
subgroupG/H
centerC22#15690778
commutator, cosocleC42×C97C23
FrattiniC42C22×D97
Fitting#460×C97C2
socleC22×C97#120
prime radical#20260927C2
hypercenter#460D97
hypocenterC97#1627
LCSG ▷ C42×C97 ▷ C22×C97 ▷ C97
UCSC1 ◁ C22 ◁ C42 ◁ #460

Complements

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

NH ≅ G/Ntype
1. Q8×D388C2semidirect
2. #20261081C2semidirect
4. #20260482C2semidirect
5. Q8⋊#11715895C2semidirect
6. #20260462C2semidirect
7. #460×C97C2semidirect
54. Q8×C97D8semidirect
68. D388SD16semidirect
100. C97#1627semidirect
103. #460D97semidirect
157. Q8#13095496semidirect

Maximal subgroups

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

subgroupindex
Q8×D3882
#202610812
#202609272
#202604822
Q8⋊#117158952
#202604622
#460×C972
#162797