Finite Group Databasebeta

← #27140665

Subgroups — #27140665

Order 12416.

272 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 272 conjugacy classes of subgroups — too many to draw. Only its 53 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#20260885C211yesmaximal, characteristic, complemented
2#20260868C211yesmaximal, characteristic
3#20260652C211yesmaximal, characteristic, complemented
4#20260472C211yesmaximal, characteristic, complemented
5#20260469C211yesmaximal, characteristic, complemented
6#20260461C211yesmaximal, characteristic, complemented
7#539×C97C211yesmaximal, characteristic, complemented
8C22×D388C2211yescharacteristic
9C2×#13095498—21no—
10C2×#13095496—21no—
11C2×#13095488—21no—
12C2×#13095488—21no—
13#15690730—21no—
14#15690730—21no—
15C22⋊#11715895C2211yescharacteristic
16C22⋊#11715895C2211yescharacteristic
17#15690717C2211yescharacteristic
18#126×C97—21no—
19#125×C97—21no—
20#123×C97—21no—
21#123×C97—21no—
22C22⋊Q8×C97—21no—
23C22⋊Q8×C97C2211yescharacteristic
24#121×C97C2211yescharacteristic
25C4×D4×C97—21no—
26C2×C4⋊C4×C97C2211yescharacteristic
27C23×D97—21no—
28C2×D388—21no—
29C2×D388—21no—
30C2×D388—42no—
31C22×Dic97—21no—
32#13095498—42no—
33#13095496—42no—
34C2×#11715895—21no—
35C2×#11715895—21no—
36#13095488—42no—
37#13095488—42no—
38C2×Q8×C97D411yescharacteristic
39C2×D4×C97—21no—
40C2×D4×C97D411yescharacteristic
41C22×C4×C97—21no—
42C22×C4×C97D411yescharacteristic
43C22×C4×C97D411yescharacteristic
44C22×C4×C97C2311yescharacteristic
45C4⋊C4×C97—21no—
46C4⋊C4×C97—21no—
47C4⋊C4×C97—21no—
48C4⋊C4×C97—21no—
49C4⋊C4×C97—21no—
50C4⋊C4×C97—21no—
51C4⋊C4×C97D411yescharacteristic
52C4⋊C4×C97D411yescharacteristic
53C22⋊C4×C97—21no—
54C22⋊C4×C97—21no—
55C22⋊C4×C97—21no—
56C22⋊C4×C97—21no—
57C22⋊C4×C97—21no—
58C42×C97—21no—
59C23×C97—21no—
60C23×C97C2×D411yescharacteristic
61C22×D97—22no—
62C22×D97—41no—
63C22×D97—41no—
64C22×D97—41no—
65C22×D97—41no—
66C22×D97—41no—
67C22×D97—41no—
68Q8×C97—41no—
69D4×C97—41no—
70D4×C97—41no—
71D4×C97—41no—
72C2×C4×C97—21no—
73C2×C4×C97—21no—
74C2×C4×C97—21no—
75C2×C4×C97—21no—
76C2×C4×C97—21no—
77C2×C4×C97—21no—
78C2×C4×C97—21no—
79C2×C4×C97—41no—
80C2×C4×C97—41no—
81C2×C4×C97—81no—
82C2×C4×C97C2×D411yescharacteristic
83C2×C4×C97C2×D411yescharacteristic
84C2×Dic97—41no—
85C2×Dic97—41no—
86C2×Dic97—41no—
87D388—22no—
88D388—22no—
89D388—41no—
90D388—41no—
91#11715895—41no—
92#11715895—41no—
93C22×C97—21no—
94C22×C97—21no—
95C22×C97—41no—
96C22×C97—41no—
97C22×C97—41no—
98C22×C97#12011yescharacteristic
99C22×C97#13511yescharacteristic
100C22×C97AGL(1,ℤ/8)11yescharacteristic
101C2×D97—42no—
102C2×D97—42no—
103C2×D97—42no—
104C2×D97—81no—
105C2×D97—81no—
106C2×D97—81no—
107C2×D97—81no—
108C4×C97—21no—
109C4×C97—21no—
110C4×C97—41no—
111C4×C97—41no—
112C4×C97—41no—
113C4×C97—41no—
114C4×C97—41no—
115Dic97—81no—
116C2×C97—21no—
117C2×C97—21no—
118C2×C97—81no—
119C2×C97#44911yescharacteristic
120C2×C97#45411yescharacteristic
121C2×C97B(4,𝔽2)11yescharacteristic
122D97—82no—
123#1597—971nomaximal, Sylow
124C97#159711yescharacteristic, complemented, Sylow
125#539D9711yescharacteristic, complemented
126#450—971no—
127#447—971no—
128#394—971no—
129#328—971no—
130#327—971no—
131#324—971no—
132C22×D4—971no—
133C2×SD16—1941no—
134C2×D8—1941no—
135#126—21no—
136#125—21no—
137#123—21no—
138#123—21no—
139C22⋊Q8—21no—
140C22⋊Q8C2×D9711yescharacteristic
141#121C2×D9711yescharacteristic
142C4×D4—21no—
143C2×C4⋊C4C2×D9711yescharacteristic
144C2×C22⋊C4—1941no—
145C2×C22⋊C4—1941no—
146#102—1941no—
147#102—1941no—
148C22⋊C8—971no—
149C22⋊C8—971no—
150#95—971no—
151C24—1941no—
152C2×Q8C22⋊D9711yescharacteristic
153C2×D4—21no—
154C2×D4—3882no—
155C2×D4—1941no—
156C2×D4—1941no—
157C2×D4C22⋊D9711yescharacteristic
158C22×C4—21no—
159C22×C4—1941no—
160C22×C4C22⋊D9711yescharacteristic
161C22×C4C22⋊D9711yescharacteristic
162C22×C4C22×D9711yescharacteristic
163SD16—3882no—
164D8—3882no—
165C2×C8—1941no—
166C2×C8—1941no—
167C4⋊C4—21no—
168C4⋊C4—21no—
169C4⋊C4—21no—
170C4⋊C4—21no—
171C4⋊C4—21no—
172C4⋊C4—21no—
173C4⋊C4C22⋊D9711yescharacteristic
174C4⋊C4C22⋊D9711yescharacteristic
175C22⋊C4—21no—
176C22⋊C4—21no—
177C22⋊C4—21no—
178C22⋊C4—21no—
179C22⋊C4—21no—
180C22⋊C4—3882no—
181C22⋊C4—3882no—
182C42—21no—
183C23—21no—
184C23—3881no—
185C23—3881no—
186C23—3881no—
187C23—3881no—
188C23—3881no—
189C23—3881no—
190C23—1942no—
191C23C2×C22⋊D9711yescharacteristic
192Q8—41no—
193D4—41no—
194D4—41no—
195D4—41no—
196D4—3881no—
197D4—3881no—
198D4—1942no—
199D4—1942no—
200C2×C4—21no—
201C2×C4—21no—
202C2×C4—21no—
203C2×C4—21no—
204C2×C4—21no—
205C2×C4—21no—
206C2×C4—21no—
207C2×C4—41no—
208C2×C4—41no—
209C2×C4—81no—
210C2×C4—3881no—
211C2×C4—3881no—
212C2×C4—3881no—
213C2×C4C2×C22⋊D9711yescharacteristic
214C2×C4C2×C22⋊D9711yescharacteristic
215C8—3881no—
216C8—3881no—
217C22—21no—
218C22—21no—
219C22—41no—
220C22—41no—
221C22—41no—
222C22—7761no—
223C22—7761no—
224C22—7761no—
225C22—7761no—
226C22—3882no—
227C22—3882no—
228C22—3882no—
229C22#1569077911yescharacteristic
230C22#1569083011yescharacteristic
231C22#1569083511yescharacteristic
232C4—21no—
233C4—21no—
234C4—41no—
235C4—41no—
236C4—41no—
237C4—41no—
238C4—41no—
239C4—7761no—
240C2—21no—
241C2—21no—
242C2—81no—
243C2—7762no—
244C2#2026088011yescharacteristic
245C2#2026090811yescharacteristic
246C2#2026092211yescharacteristic
subgroupG/H
centerC22#15690779
commutator, cosocleC22×C4×C97C23
FrattiniC22×C4C22×D97
Fitting#539×C97C2
socleC22×C97#120
prime radical#20260868C2
hypercenter#539D97
hypocenterC97#1597
LCSG ▷ C22×C4×C97 ▷ C22×C97 ▷ C97
UCSC1 ◁ C22 ◁ C22×C4 ◁ #539

Complements

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

NH ≅ G/Ntype
1. #20260885C2semidirect
3. #20260652C2semidirect
4. #20260472C2semidirect
5. #20260469C2semidirect
6. #20260461C2semidirect
7. #539×C97C2semidirect
124. C97#1597semidirect
125. #539D97semidirect

Maximal subgroups

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

subgroupindex
#202608852
#202608682
#202606522
#202604722
#202604692
#202604612
#539×C972
#159797