Finite Group Databasebeta

← #38741665

Subgroups — #38741665

Order 25216.

308 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 308 conjugacy classes of subgroups — too many to draw. Only its 51 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#27161637C211yesmaximal, characteristic
2#27161220C211yesmaximal, characteristic, complemented
3#27161217C211yesmaximal, characteristic, complemented
4#27160992C211yesmaximal, characteristic, complemented
5#27160812C211yesmaximal, characteristic, complemented
6#27160802C211yesmaximal, characteristic, complemented
7#327×C197C211yesmaximal, characteristic, complemented
8C22×Dic394C2211yescharacteristic, complemented
9C2×D4×D197—21no—
10C2×#15691347—21no—
11C2×#15691346—21no—
12#20263311—21no—
13#20263308—21no—
14#20263300C2211yescharacteristic
15#20263298—21no—
16#20263296—21no—
17#20263294—21no—
18C2×#15691335—21no—
19C2×#15691335C2211yescharacteristic
20#20263251—21no—
21#20263249—21no—
22C22⋊#13096802C2211yescharacteristic, complemented
23#20263233—21no—
24#121×C197C2211yescharacteristic
25C22⋊C8×C197C2211yescharacteristic, complemented
26#95×C197C2211yescharacteristic, complemented
27C23×D197—21no—
28C23×D197D411yescharacteristic
29C2×Dic394—21no—
30C2×Dic394—21no—
31C2×Dic394—42no—
32C2×D788—21no—
33C2×D788D411yescharacteristic
34C2×C22⋊D197—21no—
35C2×C22⋊D197—21no—
36C2×C22⋊D197—21no—
37D4×D197—42no—
38D4×D197—42no—
39C22×Dic197—21no—
40C22×Dic197D411yescharacteristic
41C2×C4×D197—21no—
42#15691347—42no—
43#15691346—42no—
44C2×#13096802—21no—
45C4⋊Dic197D411yescharacteristic
46#15691337—21no—
47C22⋊Dic197—21no—
48#15691335—21no—
49#15691335—21no—
50#15691335—21no—
51#15691335—21no—
52#15691335—42no—
53C2×D4×C197—21no—
54C2×D4×C197D411yescharacteristic
55C22×C4×C197—21no—
56C22×C4×C197C2311yescharacteristic
57C2×C8×C197—21no—
58C4⋊C4×C197D411yescharacteristic
59C22⋊C4×C197—21no—
60C22×D197—21no—
61C22×D197—21no—
62C22×D197—21no—
63C22×D197—42no—
64C22×D197—41no—
65C22×D197—41no—
66C22×D197—81no—
67C22×D197—81no—
68C22×D197SD1612yescomplemented
69Dic394—22no—
70Dic394—22no—
71Dic394—41no—
72Dic394—41no—
73D788—41no—
74D788—42no—
75C22⋊D197—42no—
76C22⋊D197—41no—
77C22⋊D197—41no—
78C22⋊D197—81no—
79C2×Dic197—22no—
80C2×Dic197—21no—
81C2×Dic197—21no—
82C2×Dic197—41no—
83C2×Dic197—41no—
84C2×Dic197—41no—
85C4×D197—42no—
86#13096802—41no—
87C23×C197—21no—
88C23×C197C2×D411yescharacteristic
89D4×C197—41no—
90D4×C197—41no—
91D4×C197—41no—
92C2×C4×C197—21no—
93C2×C4×C197—41no—
94C2×C4×C197—41no—
95C2×C4×C197—41no—
96C2×C4×C197—41no—
97C2×C4×C197C2×D411yescharacteristic
98C2×C4×C197C2×D411yescharacteristic
99C8×C197—41no—
100C22×C197—21no—
101C22×C197—21no—
102C22×C197—41no—
103C22×C197—41no—
104C22×C197—41no—
105C22×C197#12011yescharacteristic
106C22×C197C2×SD1611yescharacteristic
107C22×C197AGL(1,ℤ/8)11yescharacteristic
108C2×D197—22no—
109C2×D197—41no—
110C2×D197—41no—
111C2×D197—42no—
112C2×D197—41no—
113C2×D197—41no—
114C2×D197—41no—
115C2×D197—81no—
116C2×D197—84no—
117C4×C197—21no—
118C4×C197—21no—
119C4×C197—41no—
120C4×C197—81no—
121Dic197—41no—
122Dic197—41no—
123Dic197—42no—
124C2×C197—21no—
125C2×C197—21no—
126C2×C197—81no—
127C2×C197#45011yescharacteristic
128C2×C197#45511yescharacteristic
129C2×C197B(4,𝔽2)11yescharacteristic
130D197—42no—
131D197—81no—
132C197#160111yescharacteristic, complemented, Sylow
133#1601—1971nomaximal, Sylow
134#534—1971no—
135#448—1971no—
136#448—1971no—
137#393—1971no—
138#327—1971no—
139#327D19711yescharacteristic, complemented
140#323—1971no—
141C22×Q8—1971no—
142C22×D4—3941no—
143C2×SD16—3941no—
144C2×SD16—3941no—
145#123—3941no—
146#123—3941no—
147#121—3941no—
148#121—1971no—
149#121C2×D19711yescharacteristic
150#120—3941no—
151#120—3941no—
152C2×C22⋊C4—3941no—
153C2×C22⋊C4—1971no—
154#103—3941no—
155#103—3941no—
156C22⋊C8—1971no—
157C22⋊C8C2×D19711yescharacteristic, complemented
158#95—3941no—
159#95C2×D19711yescharacteristic, complemented
160C24—3941no—
161C24—1971no—
162C2×Q8—7882no—
163C2×Q8—3941no—
164C2×Q8—3941no—
165C2×D4—21no—
166C2×D4—7882no—
167C2×D4—7882no—
168C2×D4—3941no—
169C2×D4—3941no—
170C2×D4—3941no—
171C2×D4—3941no—
172C2×D4—1971no—
173C2×D4D78811yescharacteristic
174C22×C4—21no—
175C22×C4—3941no—
176C22×C4—3941no—
177C22×C4—1971no—
178C22×C4C22×D19711yescharacteristic
179SD16—7882no—
180SD16—7882no—
181C2×C8—21no—
182C2×C8—3941no—
183C4⋊C4—3941no—
184C4⋊C4—1971no—
185C4⋊C4D78811yescharacteristic
186C22⋊C4—21no—
187C22⋊C4—7882no—
188C22⋊C4—3941no—
189C22⋊C4—3941no—
190C22⋊C4—3941no—
191C22⋊C4—3941no—
192C22⋊C4—3941no—
193C23—21no—
194C23—7881no—
195C23—7881no—
196C23—7882no—
197C23—15761no—
198C23—15761no—
199C23—3941no—
200C23—3941no—
201C23—3941no—
202C23—1972no—
203C23D4×D19711yescharacteristic
204Q8—7881no—
205Q8—7881no—
206Q8—3942no—
207Q8—3942no—
208D4—41no—
209D4—41no—
210D4—41no—
211D4—7881no—
212D4—7882no—
213D4—7882no—
214D4—7881no—
215D4—7881no—
216D4—15761no—
217C2×C4—21no—
218C2×C4—41no—
219C2×C4—41no—
220C2×C4—41no—
221C2×C4—41no—
222C2×C4—7882no—
223C2×C4—7881no—
224C2×C4—7881no—
225C2×C4—7881no—
226C2×C4—3942no—
227C2×C4—3941no—
228C2×C4—3941no—
229C2×C4D4×D19711yescharacteristic
230C2×C4C2×D78811yescharacteristic
231C8—41no—
232C8—7881no—
233C22—21no—
234C22—21no—
235C22—41no—
236C22—41no—
237C22—41no—
238C22—7882no—
239C22—7881no—
240C22—7881no—
241C22—7881no—
242C22—7881no—
243C22—7881no—
244C22—15761no—
245C22—15764no—
246C22—3942no—
247C22#2026329611yescharacteristic
248C22SD16×D19711yescharacteristic
249C22#2026335011yescharacteristic
250C4—21no—
251C4—21no—
252C4—41no—
253C4—81no—
254C4—7881no—
255C4—7881no—
256C4—7882no—
257C2—21no—
258C2—21no—
259C2—81no—
260C2—7882no—
261C2—15761no—
262C2#2716123111yescharacteristic
263C2#2716125811yescharacteristic
264C2#2716126711yescharacteristic
subgroupG/H
centerC22#20263296
commutator, cosocleC22×C4×C197C23
FrattiniC22×C4C22×D197
Fitting#327×C197C2
socleC22×C197#120
prime radical#27161637C2
hypercenter#327D197
hypocenterC197#1601
LCSG ▷ C22×C4×C197 ▷ C22×C197 ▷ C197
UCSC1 ◁ C22 ◁ C22×C4 ◁ #327

Complements

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

NH ≅ G/Ntype
2. #27161220C2semidirect
3. #27161217C2semidirect
4. #27160992C2semidirect
5. #27160812C2semidirect
6. #27160802C2semidirect
7. #327×C197C2semidirect
8. C22×Dic394C22semidirect
22. C22⋊#13096802C22semidirect
25. C22⋊C8×C197C22semidirect
26. #95×C197C22semidirect
68. C22×D197SD16semidirect
132. C197#1601semidirect
139. #327D197semidirect
157. C22⋊C8C2×D197semidirect
159. #95C2×D197semidirect

Maximal subgroups

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

subgroupindex
#271616372
#271612202
#271612172
#271609922
#271608122
#271608022
#327×C1972
#1601197