Finite Group Databasebeta

← #27140221

Subgroups — #27140221

Order 12416.

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

normal subgroups only

This group has 238 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#20261271C211yesmaximal, characteristic
2#20260785C211yesmaximal, characteristic, complemented
3#20260580C211yesmaximal, characteristic, complemented
4#20260472C211yesmaximal, characteristic, complemented
5#20260470C211yesmaximal, characteristic
6#20260468C211yesmaximal, characteristic
7#328×C97C211yesmaximal, characteristic, complemented
8#15690805—21no—
9#15690801C411yescharacteristic, complemented
10#15690799C411yescharacteristic, complemented
11#15690794—21no—
12#15690784C2211yescharacteristic
13Q8×Dic97C411yescharacteristic, complemented
14#15690772C411yescharacteristic, complemented
15#15690770—21no—
16#15690768—21no—
17C2×C4×Dic97—21no—
18C2×C4×Dic97C2211yescharacteristic
19#15690739—21no—
20C22⋊#11715895C2211yescharacteristic
21C8×Dic97—21no—
22#15690717—21no—
23#15690717C2211yescharacteristic
24C22⋊Q8×C97C2211yescharacteristic
25C22⋊C8×C97C2211yescharacteristic
26#95×C97C2211yescharacteristic
27C2×D388C2×C411yescharacteristic
28C2×C22⋊D97—21no—
29C22×Dic97—21no—
30C22×Dic97D411yescharacteristic
31C22×Dic97D411yescharacteristic
32C2×C4×D97—21no—
33C2×#11715895—21no—
34C4⋊Dic97—21no—
35C4⋊Dic97—21no—
36C4⋊Dic97C2×C411yescharacteristic
37#13095490—21no—
38#13095488—21no—
39#13095488—21no—
40#13095488—21no—
41#13095488—21no—
42C4×Dic97—24no—
43C4×Dic97—21no—
44C4×Dic97C2×C411yescharacteristic
45C4×Dic97C2×C411yescharacteristic
46C4×Dic97D411yescharacteristic
47C4×Dic97D411yescharacteristic
48C2×Q8×C97C2×C411yescharacteristic
49C22×C4×C97—21no—
50C22×C4×C97C2311yescharacteristic
51C2×C8×C97—21no—
52C4⋊C4×C97—21no—
53C4⋊C4×C97C2×C411yescharacteristic
54C22⋊C4×C97—21no—
55C23×C97C2×D411yescharacteristic
56C22×D97—21no—
57Q8×C97—41no—
58C2×C4×C97—22no—
59C2×C4×C97—21no—
60C2×C4×C97—21no—
61C2×C4×C97—41no—
62C2×C4×C97—41no—
63C2×C4×C97—41no—
64C2×C4×C97C22×C411yescharacteristic
65C2×C4×C97C2×D411yescharacteristic
66C22⋊D97—41no—
67C22⋊D97—41no—
68C2×Dic97—21no—
69C2×Dic97—21no—
70C2×Dic97—22no—
71C2×Dic97—22no—
72C2×Dic97—21no—
73C2×Dic97—41no—
74C2×Dic97—41no—
75C2×Dic97C22⋊C411yescharacteristic
76C2×Dic97C22⋊C411yescharacteristic
77C2×Dic97C22⋊C411yescharacteristic
78C2×Dic97C22⋊C411yescharacteristic
79D388—41no—
80C4×D97—42no—
81C8×C97—41no—
82#11715895—41no—
83C22×C97—21no—
84C22×C97—21no—
85C22×C97C2×C22⋊C411yescharacteristic
86C22×C97#13511yescharacteristic
87C22×C97#13511yescharacteristic
88C2×D97—41no—
89C2×D97—41no—
90C2×D97—41no—
91C4×C97—21no—
92C4×C97—21no—
93C4×C97—42no—
94C4×C97—41no—
95C4×C97—41no—
96Dic97—21no—
97Dic97—21no—
98Dic97—21no—
99Dic97—42no—
100Dic97—41no—
101Dic97#10412yes—
102C2×C97—21no—
103C2×C97—21no—
104C2×C97#41011yescharacteristic
105C2×C97#41611yescharacteristic
106C2×C97C2×#10411yescharacteristic
107D97—81no—
108#1500—971nomaximal, Sylow
109C97#150011yescharacteristic, complemented, Sylow
110#529—971no—
111#431—971no—
112#376—971no—
113#328—971no—
114#328D9711yescharacteristic, complemented
115#327—971no—
116#327—971no—
117#126—1941no—
118#125—971no—
119#124—971no—
120#123—1941no—
121C22⋊Q8C2×D9711yescharacteristic
122#121—971no—
123C4×Q8—971no—
124C4×D4—971no—
125C4×D4—1941no—
126#117—1941no—
127C2×C42—971no—
128C2×C42—1941no—
129C4⋊C8—1941no—
130C22⋊C8—971no—
131C22⋊C8C2×D9711yescharacteristic
132C4×C8—1941no—
133#95—971no—
134#95—1941no—
135#95C2×D9711yescharacteristic
136C2×Q8C4×D9711yescharacteristic
137C2×D4—971no—
138C2×D4—1941no—
139C22×C4—21no—
140C22×C4—971no—
141C22×C4—971no—
142C22×C4—1941no—
143C22×C4—1941no—
144C22×C4C22×D9711yescharacteristic
145C2×C8—21no—
146C2×C8—1941no—
147C4⋊C4—21no—
148C4⋊C4—971no—
149C4⋊C4—1941no—
150C4⋊C4—1941no—
151C4⋊C4—1941no—
152C4⋊C4C4×D9711yescharacteristic
153C22⋊C4—21no—
154C22⋊C4—1941no—
155C22⋊C4—1941no—
156C22⋊C4—1941no—
157C22⋊C4—1941no—
158C42—971no—
159C42—971no—
160C42—971no—
161C42—971no—
162C42—1941no—
163C42—1944no—
164C23—1941no—
165C23D4×D9711yescharacteristic
166Q8—41no—
167D4—3881no—
168D4—3881no—
169D4—3881no—
170C2×C4—22no—
171C2×C4—21no—
172C2×C4—21no—
173C2×C4—41no—
174C2×C4—41no—
175C2×C4—41no—
176C2×C4—971no—
177C2×C4—971no—
178C2×C4—971no—
179C2×C4—971no—
180C2×C4—3882no—
181C2×C4—3881no—
182C2×C4—3881no—
183C2×C4—1942no—
184C2×C4—1942no—
185C2×C4—1941no—
186C2×C4—1941no—
187C2×C4—1941no—
188C2×C4C2×C4×D9711yescharacteristic
189C2×C4D4×D9711yescharacteristic
190C8—41no—
191C8—3881no—
192C22—21no—
193C22—21no—
194C22—3881no—
195C22—3881no—
196C22—3881no—
197C22C22⋊C4×D9711yescharacteristic
198C22#1569083111yescharacteristic
199C22#1569083311yescharacteristic
200C4—21no—
201C4—21no—
202C4—42no—
203C4—41no—
204C4—41no—
205C4—972no—
206C4—3882no—
207C4—3881no—
208C4—1941no—
209C4—1941no—
210C4—1941no—
211C2—21no—
212C2—21no—
213C2—7761no—
214C2#2026070711yescharacteristic
215C2#2026073011yescharacteristic
216C2#104×D9711yescharacteristic
subgroupG/H
centerC22C22⋊C4×D97
commutatorC2×C4×C97C22×C4
FrattiniC22×C4C22×D97
Fitting#328×C97C2
socleC22×C97C2×C22⋊C4
prime radical#15690784C22
cosocleC22×C4×C97C23
hypercenter#328D97
hypocenterC97#1500
LCSG ▷ C2×C4×C97 ▷ C22×C97 ▷ C97
UCSC1 ◁ C22 ◁ C22×C4 ◁ #328

Complements

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

NH ≅ G/Ntype
2. #20260785C2semidirect
3. #20260580C2semidirect
4. #20260472C2semidirect
7. #328×C97C2semidirect
9. #15690801C4semidirect
10. #15690799C4semidirect
13. Q8×Dic97C4semidirect
14. #15690772C4semidirect
109. C97#1500semidirect
114. #328D97semidirect

Maximal subgroups

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

subgroupindex
#202612712
#202607852
#202605802
#202604722
#202604702
#202604682
#328×C972
#150097