Finite Group Databasebeta

← #27140258

Subgroups — #27140258

Order 12416.

242 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 242 conjugacy classes of subgroups — too many to draw. Only its 63 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#20261266C211yesmaximal, characteristic
2#20260789C211yesmaximal, characteristic, complemented
3#20260586C211yesmaximal, characteristic, complemented
4#20260473C211yesmaximal, characteristic, complemented
5#20260471C211yesmaximal, characteristic, complemented
6#20260469C211yesmaximal, characteristic
7#327×C97C211yesmaximal, characteristic
8#15690803—21no—
9#15690801C411yescharacteristic, complemented
10#15690797C411yescharacteristic, complemented
11#15690791—21no—
12#15690789C2211yescharacteristic
13#15690775C411yescharacteristic, complemented
14D4×Dic97—21no—
15D4×Dic97C411yescharacteristic, complemented
16#15690766—21no—
17C2×#13095490—21no—
18C2×C4×Dic97—21no—
19C2×C4×Dic97C2211yescharacteristic
20#15690739—21no—
21C22⋊#11715895C2211yescharacteristic
22#15690720—21no—
23#15690717C2211yescharacteristic
24#121×C97C2211yescharacteristic
25C22⋊C8×C97C2211yescharacteristic
26#95×C97C2211yescharacteristic
27C2×Dic194C2×C411yescharacteristic
28C22×Dic97—21no—
29C22×Dic97—21no—
30C22×Dic97D411yescharacteristic
31C22×Dic97D411yescharacteristic
32C2×#11715895—21no—
33C4⋊Dic97—21no—
34C4⋊Dic97C2×C411yescharacteristic
35#13095490—24no—
36#13095490—21no—
37#13095490—21no—
38#13095490—21no—
39C22⋊Dic97—21no—
40C22⋊Dic97—21no—
41C22⋊Dic97—21no—
42C22⋊Dic97—21no—
43C4×Dic97—21no—
44C4×Dic97—42no—
45C4×Dic97C2×C411yescharacteristic
46C4×Dic97C2×C411yescharacteristic
47C4×Dic97D411yescharacteristic
48C4×Dic97D411yescharacteristic
49C2×D4×C97—21no—
50C2×D4×C97C2×C411yescharacteristic
51C22×C4×C97—21no—
52C22×C4×C97C2311yescharacteristic
53C2×C8×C97—21no—
54C4⋊C4×C97C2×C411yescharacteristic
55C22⋊C4×C97—21no—
56C23×C97—21no—
57C23×C97C2×D411yescharacteristic
58D4×C97—41no—
59D4×C97—41no—
60D4×C97—41no—
61C2×C4×C97—22no—
62C2×C4×C97—21no—
63C2×C4×C97—41no—
64C2×C4×C97—41no—
65C2×C4×C97—41no—
66C2×C4×C97C22×C411yescharacteristic
67C2×C4×C97C2×D411yescharacteristic
68C2×Dic97—21no—
69C2×Dic97—21no—
70C2×Dic97—22no—
71C2×Dic97—22no—
72C2×Dic97—21no—
73C2×Dic97—21no—
74C2×Dic97—41no—
75C2×Dic97—41no—
76C2×Dic97—42no—
77C2×Dic97C22⋊C411yescharacteristic
78C2×Dic97C22⋊C411yescharacteristic
79C2×Dic97C22⋊C411yescharacteristic
80C2×Dic97C22⋊C411yescharacteristic
81Dic194—41no—
82C8×C97—41no—
83#11715895—41no—
84C22×C97—21no—
85C22×C97—21no—
86C22×C97—41no—
87C22×C97—41no—
88C22×C97—41no—
89C22×C97C2×C22⋊C411yescharacteristic
90C22×C97AGL(1,ℤ/8)11yescharacteristic
91C22×C97#13711yescharacteristic
92C4×C97—21no—
93C4×C97—21no—
94C4×C97—42no—
95C4×C97—41no—
96Dic97—21no—
97Dic97—21no—
98Dic97—21no—
99Dic97—42no—
100Dic97—41no—
101Dic97—41no—
102Dic97#10412yes—
103C2×C97—21no—
104C2×C97—21no—
105C2×C97—81no—
106C2×C97#41011yescharacteristic
107C2×C97#41711yescharacteristic
108C2×C97C2×#10411yescharacteristic
109#1509—971nomaximal, Sylow
110C97#150911yescharacteristic, complemented, Sylow
111#529—971no—
112#432—971no—
113C4×C4⋊C4—971no—
114#328—971no—
115#328—971no—
116#327—971no—
117#327D9711yescharacteristic
118#126—1941no—
119#125—971no—
120#124—971no—
121#123—1941no—
122C22⋊Q8—971no—
123#121C2×D9711yescharacteristic
124C4×Q8—971no—
125C4×D4—971no—
126C4×D4—1941no—
127#117—1941no—
128C2×C4⋊C4—1941no—
129C2×C42—971no—
130C2×C42—1941no—
131C4⋊C8—1941no—
132C22⋊C8—971no—
133C22⋊C8C2×D9711yescharacteristic
134#97—1941no—
135#95—971no—
136#95C2×D9711yescharacteristic
137C2×Q8—971no—
138C2×D4—21no—
139C2×D4C4×D9711yescharacteristic
140C22×C4—21no—
141C22×C4—971no—
142C22×C4—971no—
143C22×C4—1941no—
144C22×C4—1941no—
145C22×C4C22×D9711yescharacteristic
146C2×C8—21no—
147C2×C8—1941no—
148C4⋊C4—971no—
149C4⋊C4—1941no—
150C4⋊C4—1941no—
151C4⋊C4—1941no—
152C4⋊C4—1944no—
153C4⋊C4—1941no—
154C4⋊C4C4×D9711yescharacteristic
155C22⋊C4—21no—
156C22⋊C4—1941no—
157C22⋊C4—1941no—
158C22⋊C4—1941no—
159C22⋊C4—1941no—
160C42—971no—
161C42—971no—
162C42—971no—
163C42—971no—
164C42—3882no—
165C42—1941no—
166C23—21no—
167C23D4×D9711yescharacteristic
168Q8—3881no—
169D4—41no—
170D4—41no—
171D4—41no—
172C2×C4—22no—
173C2×C4—21no—
174C2×C4—41no—
175C2×C4—41no—
176C2×C4—41no—
177C2×C4—971no—
178C2×C4—971no—
179C2×C4—971no—
180C2×C4—971no—
181C2×C4—3882no—
182C2×C4—3881no—
183C2×C4—3881no—
184C2×C4—1942no—
185C2×C4—1942no—
186C2×C4—1941no—
187C2×C4—1941no—
188C2×C4—1941no—
189C2×C4—1941no—
190C2×C4C2×C4×D9711yescharacteristic
191C2×C4D4×D9711yescharacteristic
192C8—41no—
193C8—3881no—
194C22—21no—
195C22—21no—
196C22—41no—
197C22—41no—
198C22—41no—
199C22C22⋊C4×D9711yescharacteristic
200C22#1569083411yescharacteristic
201C22#1569083911yescharacteristic
202C4—21no—
203C4—21no—
204C4—42no—
205C4—41no—
206C4—972no—
207C4—3882no—
208C4—3881no—
209C4—3881no—
210C4—1941no—
211C4—1941no—
212C4—1941no—
213C2—21no—
214C2—21no—
215C2—81no—
216C2#2026070711yescharacteristic
217C2#2026073311yescharacteristic
218C2#104×D9711yescharacteristic
subgroupG/H
centerC22C22⋊C4×D97
commutatorC2×C4×C97C22×C4
FrattiniC22×C4C22×D97
Fitting#327×C97C2
socleC22×C97C2×C22⋊C4
prime radical#121×C97C22
cosocleC22×C4×C97C23
hypercenter#327D97
hypocenterC97#1509
LCSG ▷ C2×C4×C97 ▷ C22×C97 ▷ C97
UCSC1 ◁ C22 ◁ C22×C4 ◁ #327

Complements

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

NH ≅ G/Ntype
2. #20260789C2semidirect
3. #20260586C2semidirect
4. #20260473C2semidirect
5. #20260471C2semidirect
9. #15690801C4semidirect
10. #15690797C4semidirect
13. #15690775C4semidirect
15. D4×Dic97C4semidirect
110. C97#1509semidirect

Maximal subgroups

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

subgroupindex
#202612662
#202607892
#202605862
#202604732
#202604712
#202604692
#327×C972
#150997