Finite Group Databasebeta

← #27140293

Subgroups — #27140293

Order 12416.

284 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 284 conjugacy classes of subgroups — too many to draw. Only its 73 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#20261246C211yesmaximal, characteristic
2#20260598C211yesmaximal, characteristic, complemented
3#20260473C212yesmaximal, complemented
4#20260470C212yesmaximal, complemented
5C2×C22⋊C8×C97C211yesmaximal, characteristic, complemented
6C2×#13095511C411yescharacteristic, complemented
7#15690793—21no—
8#15690789C411yescharacteristic, complemented
9#15690789C2211yescharacteristic
10#15690784C411yescharacteristic, complemented
11#15690784C2211yescharacteristic
12#15690779—21no—
13C4×C22⋊D97—21no—
14C4×C22⋊D97—21no—
15#15690769C411yescharacteristic, complemented
16C2×C22⋊Dic97—21no—
17#15690717—21no—
18#15690717C2212yes—
19C23×C4×C97C2211yescharacteristic
20C22×C8×C97—21no—
21C22⋊C8×C97—21no—
22C22⋊C8×C97C2212yes—
23#13095511—42no—
24C2×Dic194C2×C411yescharacteristic
25C2×D388C2×C411yescharacteristic
26C2×C22⋊D97—21no—
27C2×C22⋊D97—21no—
28C2×C22⋊D97—21no—
29C22×Dic97—22no—
30C2×C4×D97—21no—
31C4⋊Dic97C2×C411yescharacteristic
32C4⋊Dic97C2×C411yescharacteristic
33#13095490—21no—
34#13095490—21no—
35C22⋊Dic97—21no—
36C22⋊Dic97—21no—
37C22⋊Dic97—21no—
38C22⋊Dic97—42no—
39#13095488—21no—
40#13095488—21no—
41C4×Dic97—21no—
42C24×C97D411yescharacteristic
43C22×C4×C97—42no—
44C22×C4×C97C2×C411yescharacteristic
45C22×C4×C97C2×C411yescharacteristic
46C22×C4×C97D411yescharacteristic
47C22×C4×C97D411yescharacteristic
48C22×C4×C97D411yescharacteristic
49C22×C4×C97C2311yescharacteristic
50C2×C8×C97—22no—
51C2×C8×C97—42no—
52C23×C97—22no—
53C23×C97—22no—
54C23×C97—41no—
55C23×C97C22⋊C411yescharacteristic
56C23×C97C22⋊C411yescharacteristic
57C23×C97C2×D411yescharacteristic
58C22×D97—21no—
59C2×C4×C97—22no—
60C2×C4×C97—22no—
61C2×C4×C97—41no—
62C2×C4×C97—41no—
63C2×C4×C97—41no—
64C2×C4×C97—41no—
65C2×C4×C97C22⋊C411yescharacteristic
66C2×C4×C97C22⋊C411yescharacteristic
67C2×C4×C97C22×C411yescharacteristic
68C2×C4×C97C2×D411yescharacteristic
69C22⋊D97—42no—
70C22⋊D97—42no—
71C22⋊D97—42no—
72C2×Dic97—21no—
73C2×Dic97—21no—
74C2×Dic97—21no—
75C2×Dic97—42no—
76C2×Dic97—42no—
77C2×Dic97—42no—
78D388—41no—
79C4×D97—42no—
80Dic194—41no—
81C8×C97—42no—
82C22×C97—21no—
83C22×C97—21no—
84C22×C97—21no—
85C22×C97—21no—
86C22×C97—21no—
87C22×C97—21no—
88C22×C97—21no—
89C22×C97—44no—
90C22×C97#9912yes—
91C22×C97C2×C22⋊C411yescharacteristic
92C22×C97#13512yes—
93C2×D97—41no—
94C2×D97—41no—
95C2×D97—41no—
96C4×C97—21no—
97C4×C97—21no—
98C4×C97—21no—
99C4×C97—21no—
100Dic97—41no—
101Dic97—41no—
102Dic97—41no—
103Dic97—82no—
104C2×C97—22no—
105C2×C97—22no—
106C2×C97—41no—
107C2×C97C2×#9911yescharacteristic
108C2×C97#41611yescharacteristic
109C2×C97#42111yescharacteristic
110D97—81no—
111#1516—971nomaximal, Sylow
112C97#151611yescharacteristic, complemented, Sylow
113#525—971no—
114C2×C22⋊C8D9711yescharacteristic, complemented
115#381—971no—
116#328—972no—
117#327—972no—
118C2×#41—971no—
119C23×C4C2×D9711yescharacteristic
120C22×C8—21no—
121#123—1941no—
122C22⋊Q8—971no—
123C22⋊Q8—971no—
124#121—971no—
125#121—971no—
126#120—1941no—
127C4×D4—1941no—
128C4×D4—1941no—
129#117—971no—
130C2×C22⋊C4—1941no—
131C22⋊C8—21no—
132C22⋊C8C2×D9712yes—
133#95—972no—
134#95—1941no—
135C24D38811yescharacteristic
136#41—3882no—
137C2×Q8—971no—
138C2×D4—971no—
139C2×D4—1941no—
140C2×D4—1941no—
141C2×D4—1941no—
142C22×C4—42no—
143C22×C4—1942no—
144C22×C4—1941no—
145C22×C4C4×D9711yescharacteristic
146C22×C4C4×D9711yescharacteristic
147C22×C4D38811yescharacteristic
148C22×C4C22⋊D9711yescharacteristic
149C22×C4C22⋊D9711yescharacteristic
150C22×C4C22×D9711yescharacteristic
151C2×C8—22no—
152C2×C8—42no—
153C4⋊C4—971no—
154C4⋊C4—971no—
155C4⋊C4—1941no—
156C4⋊C4—1941no—
157C22⋊C4—3882no—
158C22⋊C4—1941no—
159C22⋊C4—1941no—
160C22⋊C4—1941no—
161C22⋊C4—1941no—
162C22⋊C4—1941no—
163C42—1941no—
164C23—22no—
165C23—22no—
166C23—41no—
167C23—1941no—
168C23#1309548811yescharacteristic
169C23#1309548811yescharacteristic
170C23C2×D38811yescharacteristic
171Q8—3881no—
172D4—3881no—
173D4—3882no—
174D4—3882no—
175D4—3882no—
176C2×C4—22no—
177C2×C4—22no—
178C2×C4—41no—
179C2×C4—41no—
180C2×C4—41no—
181C2×C4—41no—
182C2×C4—3882no—
183C2×C4—3882no—
184C2×C4—3882no—
185C2×C4—3882no—
186C2×C4—1941no—
187C2×C4—1941no—
188C2×C4—1941no—
189C2×C4#1309548811yescharacteristic
190C2×C4#1309548811yescharacteristic
191C2×C4C2×C4×D9711yescharacteristic
192C2×C4C2×C22⋊D9711yescharacteristic
193C8—42no—
194C22—21no—
195C22—21no—
196C22—21no—
197C22—21no—
198C22—21no—
199C22—21no—
200C22—21no—
201C22—44no—
202C22—3881no—
203C22—3881no—
204C22—3881no—
205C22#1569072412yes—
206C22C2×#1309548811yescharacteristic
207C22#1569083212yes—
208C4—21no—
209C4—21no—
210C4—21no—
211C4—21no—
212C4—7762no—
213C4—3881no—
214C4—3881no—
215C4—3881no—
216C2—22no—
217C2—22no—
218C2—41no—
219C2—7761no—
220C2C2×#1569072411yescharacteristic
221C2#2026073211yescharacteristic
222C2#2026075111yescharacteristic
subgroupG/H
centerC22C2×#13095488
commutatorC2×C4×C97C22×C4
FrattiniC22×C4C22×D97
FittingC2×C22⋊C8×C97C2
socleC22×C97C2×C22⋊C4
prime radical#20261246C2
cosocleC22×C4×C97C23
hypercenterC2×C22⋊C8D97
hypocenterC97#1516
LCSG ▷ C2×C4×C97 ▷ C22×C97 ▷ C97
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×C22⋊C8

Complements

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

NH ≅ G/Ntype
2. #20260598C2semidirect
3. #20260473C2semidirect
4. #20260470C2semidirect
5. C2×C22⋊C8×C97C2semidirect
6. C2×#13095511C4semidirect
8. #15690789C4semidirect
10. #15690784C4semidirect
15. #15690769C4semidirect
112. C97#1516semidirect
114. C2×C22⋊C8D97semidirect

Maximal subgroups

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

subgroupindex
#202612462
#202605982
#202604732
#202604702
C2×C22⋊C8×C972
#151697