Finite Group Databasebeta

← #38741282

Subgroups — #38741282

Order 25216.

288 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 288 conjugacy classes of subgroups — too many to draw. Only its 71 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
1C22⋊Q8×D197C211yesmaximal, characteristic
2C22⋊C8×D197C211yesmaximal, characteristic
3#27160938C211yesmaximal, characteristic
4#27160817C211yesmaximal, characteristic, complemented
5#27160816C211yesmaximal, characteristic, complemented
6#27160815C211yesmaximal, characteristic, complemented
7#328×C197C211yesmaximal, characteristic, complemented
8C2×Q8×D197C411yescharacteristic, complemented
9C22×C4×D197C2211yescharacteristic
10C2×C8×D197—21no—
11#20263306C411yescharacteristic, complemented
12#20263305C2211yescharacteristic
13#20263304—21no—
14#20263303—21no—
15#20263302C411yescharacteristic, complemented
16C2×#15691337—21no—
17C4⋊C4×D197—21no—
18C4⋊C4×D197C411yescharacteristic, complemented
19C2×#15691335—21no—
20C22⋊C4×D197—21no—
21C22⋊#13096802C2211yescharacteristic
22#20263238—21no—
23#20263233C2211yescharacteristic
24C22⋊Q8×C197C2211yescharacteristic
25C22⋊C8×C197C2211yescharacteristic
26#95×C197C2211yescharacteristic
27C23×D197D411yescharacteristic
28C2×Dic394—21no—
29C2×Dic394C2×C411yescharacteristic
30Q8×D197—42no—
31C22×Dic197—21no—
32C22×Dic197D411yescharacteristic
33C2×C4×D197—21no—
34C2×C4×D197—21no—
35C2×C4×D197—42no—
36C2×C4×D197C2×C411yescharacteristic
37C2×C4×D197C2×C411yescharacteristic
38C2×C4×D197D411yescharacteristic
39C2×C4×D197D411yescharacteristic
40C2×#13096802—21no—
41C8×D197—42no—
42C4⋊Dic197—21no—
43C4⋊Dic197C2×C411yescharacteristic
44#15691337—21no—
45#15691337—21no—
46#15691337—21no—
47#15691337—42no—
48C22⋊Dic197—21no—
49#15691335—21no—
50#15691335—21no—
51#15691335—42no—
52C2×Q8×C197C2×C411yescharacteristic
53C22×C4×C197—21no—
54C22×C4×C197C2311yescharacteristic
55C2×C8×C197—21no—
56C4⋊C4×C197—21no—
57C4⋊C4×C197C2×C411yescharacteristic
58C22⋊C4×C197—21no—
59C22×D197—41no—
60C22×D197—41no—
61C22×D197C22⋊C411yescharacteristic
62C22×D197C22⋊C411yescharacteristic
63C22×D197SD1612yes—
64C22×D197Q1612yes—
65Dic394—42no—
66Dic394—41no—
67C2×Dic197—22no—
68C2×Dic197—21no—
69C2×Dic197—21no—
70C2×Dic197—41no—
71C2×Dic197—41no—
72C2×Dic197—41no—
73C2×Dic197C22⋊C411yescharacteristic
74C2×Dic197C22⋊C411yescharacteristic
75C4×D197—22no—
76C4×D197—22no—
77C4×D197—41no—
78C4×D197—41no—
79C4×D197—42no—
80C4×D197—42no—
81#13096802—41no—
82C23×C197C2×D411yescharacteristic
83Q8×C197—41no—
84C2×C4×C197—21no—
85C2×C4×C197—21no—
86C2×C4×C197—41no—
87C2×C4×C197—41no—
88C2×C4×C197—41no—
89C2×C4×C197—41no—
90C2×C4×C197C22×C411yescharacteristic
91C2×C4×C197C2×D411yescharacteristic
92C8×C197—41no—
93C22×C197—21no—
94C22×C197—21no—
95C22×C197C2×C22⋊C411yescharacteristic
96C22×C197C2×SD1611yescharacteristic
97C22×C197C2×Q1611yescharacteristic
98C2×D197—21no—
99C2×D197—21no—
100C2×D197—21no—
101C2×D197—21no—
102C2×D197—42no—
103C2×D197—42no—
104C2×D197#10312yes—
105C2×D197#10312yes—
106C4×C197—21no—
107C4×C197—21no—
108C4×C197—41no—
109C4×C197—41no—
110C4×C197—81no—
111Dic197—21no—
112Dic197—41no—
113Dic197—41no—
114Dic197—81no—
115Dic197#9912yes—
116C2×C197—21no—
117C2×C197—21no—
118C2×C197C2×#9911yescharacteristic
119C2×C197C2×#10311yescharacteristic
120C2×C197#42111yescharacteristic
121D197—22no—
122D197—22no—
123C197#152011yescharacteristic, complemented, Sylow
124#1520—1971nomaximal, Sylow
125C2×C22⋊Q8—1971no—
126C2×C22⋊C8—1971no—
127#380—1971no—
128#328—1971no—
129#328—1971no—
130#328—1971no—
131#328D19711yescharacteristic, complemented
132C22×Q8—1971no—
133C23×C4—1971no—
134C22×C8—3941no—
135C22⋊Q8—3941no—
136C22⋊Q8—3941no—
137C22⋊Q8—1971no—
138C22⋊Q8—1971no—
139C22⋊Q8—1971no—
140C22⋊Q8C2×D19711yescharacteristic
141C2×C4⋊C4—3941no—
142C2×C4⋊C4—3941no—
143C2×C4⋊C4—1971no—
144C2×C22⋊C4—3941no—
145C2×C22⋊C4—3941no—
146C22⋊C8—3941no—
147C22⋊C8—1971no—
148C22⋊C8C2×D19711yescharacteristic
149#95—1971no—
150#95C2×D19711yescharacteristic
151C24—1971no—
152C2×Q8—7882no—
153C2×Q8—3941no—
154C2×Q8—1971no—
155C2×Q8C4×D19711yescharacteristic
156C22×C4—21no—
157C22×C4—7882no—
158C22×C4—3941no—
159C22×C4—3941no—
160C22×C4—3941no—
161C22×C4—1971no—
162C22×C4—1971no—
163C22×C4—1971no—
164C22×C4—1971no—
165C22×C4—1971no—
166C22×C4C22×D19711yescharacteristic
167C2×C8—21no—
168C2×C8—7882no—
169C2×C8—3941no—
170C4⋊C4—21no—
171C4⋊C4—7882no—
172C4⋊C4—3941no—
173C4⋊C4—3941no—
174C4⋊C4—3941no—
175C4⋊C4—3941no—
176C4⋊C4—1971no—
177C4⋊C4C4×D19711yescharacteristic
178C22⋊C4—21no—
179C22⋊C4—7882no—
180C22⋊C4—3941no—
181C22⋊C4—3941no—
182C22⋊C4—3941no—
183C23—7881no—
184C23—7881no—
185C23—1972no—
186C23—1972no—
187C23—1971no—
188C23—1971no—
189C23D4×D19711yescharacteristic
190Q8—41no—
191Q8—7881no—
192Q8—7882no—
193C2×C4—21no—
194C2×C4—21no—
195C2×C4—41no—
196C2×C4—41no—
197C2×C4—41no—
198C2×C4—41no—
199C2×C4—7881no—
200C2×C4—7881no—
201C2×C4—7882no—
202C2×C4—7882no—
203C2×C4—7881no—
204C2×C4—7881no—
205C2×C4—7881no—
206C2×C4—3942no—
207C2×C4—3942no—
208C2×C4—3941no—
209C2×C4—3941no—
210C2×C4—3942no—
211C2×C4—1971no—
212C2×C4—1971no—
213C2×C4C2×C4×D19711yescharacteristic
214C2×C4D4×D19711yescharacteristic
215C8—41no—
216C8—7881no—
217C22—21no—
218C22—21no—
219C22—7882no—
220C22—7882no—
221C22—3941no—
222C22—3941no—
223C22—3941no—
224C22—3941no—
225C22—1972no—
226C22—1972no—
227C22C22⋊C4×D19711yescharacteristic
228C22SD16×D19711yescharacteristic
229C22Q16×D19711yescharacteristic
230C4—21no—
231C4—21no—
232C4—41no—
233C4—41no—
234C4—81no—
235C4—7881no—
236C4—7881no—
237C4—15761no—
238C4—3941no—
239C4—1972no—
240C2—21no—
241C2—21no—
242C2—3942no—
243C2—3942no—
244C2#99×D19711yescharacteristic
245C2#103×D19711yescharacteristic
246C2#2716109411yescharacteristic
subgroupG/H
centerC22C22⋊C4×D197
commutatorC2×C4×C197C22×C4
FrattiniC22×C4C22×D197
Fitting#328×C197C2
socleC22×C197C2×C22⋊C4
prime radicalC23×D197D4
cosocleC22×C4×C197C23
hypercenter#328D197
hypocenterC197#1520
LCSG ▷ C2×C4×C197 ▷ C22×C197 ▷ C197
UCSC1 ◁ C22 ◁ C22×C4 ◁ #328

Complements

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

NH ≅ G/Ntype
4. #27160817C2semidirect
5. #27160816C2semidirect
6. #27160815C2semidirect
7. #328×C197C2semidirect
8. C2×Q8×D197C4semidirect
11. #20263306C4semidirect
15. #20263302C4semidirect
18. C4⋊C4×D197C4semidirect
123. C197#1520semidirect
131. #328D197semidirect

Maximal subgroups

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

subgroupindex
C22⋊Q8×D1972
C22⋊C8×D1972
#271609382
#271608172
#271608162
#271608152
#328×C1972
#1520197