Finite Group Databasebeta

← #38741682

Subgroups — #38741682

Order 25216.

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

normal subgroups only

This group has 282 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#27161656C211yesmaximal, characteristic
2#27161220C211yesmaximal, characteristic, complemented
3#27161217C211yesmaximal, characteristic, complemented
4#27161000C211yesmaximal, characteristic, complemented
5#27160812C211yesmaximal, characteristic, complemented
6#27160804C211yesmaximal, characteristic, complemented
7#327×C197C211yesmaximal, characteristic, complemented
8C2×#15691357—21no—
9C22×Dic394C2211yescharacteristic
10C2×#15691347—21no—
11C2×#15691346—21no—
12#20263304—21no—
13#20263303—21no—
14#20263300C2211yescharacteristic
15#20263299—21no—
16#20263298—21no—
17#20263298—21no—
18C2×C4⋊Dic197—21no—
19C2×#15691337C2211yescharacteristic
20#20263251—21no—
21#20263249—21no—
22C22⋊#13096802C2211yescharacteristic
23#20263233—21no—
24#121×C197C2211yescharacteristic
25C22⋊C8×C197C2211yescharacteristic
26#95×C197C2211yescharacteristic
27#15691357—42no—
28#15691357—42no—
29C2×Dic394—21no—
30C2×Dic394—21no—
31C2×Dic394—21no—
32C2×Dic394—42no—
33C2×D788D411yescharacteristic
34C2×C22⋊D197—21no—
35C2×C22⋊D197—21no—
36C2×C22⋊D197—21no—
37C22×Dic197—21no—
38C22×Dic197—21no—
39C22×Dic197D411yescharacteristic
40C22×Dic197D411yescharacteristic
41C2×C4×D197—21no—
42#15691347—42no—
43#15691346—42no—
44C2×#13096802—21no—
45C4⋊Dic197—42no—
46C4⋊Dic197D411yescharacteristic
47#15691337—21no—
48#15691337—21no—
49#15691337—21no—
50C22⋊Dic197—21no—
51#15691335—21no—
52#15691335—21no—
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—
61Dic394—22no—
62Dic394—22no—
63Dic394—42no—
64Dic394—41no—
65Dic394—41no—
66D788—41no—
67C22⋊D197—42no—
68C22⋊D197—41no—
69C22⋊D197—41no—
70C22⋊D197—81no—
71C2×Dic197—21no—
72C2×Dic197—21no—
73C2×Dic197—22no—
74C2×Dic197—21no—
75C2×Dic197—21no—
76C2×Dic197—42no—
77C2×Dic197—41no—
78C2×Dic197—41no—
79C2×Dic197—41no—
80C2×Dic197SD1612yescomplemented
81C4×D197—42no—
82#13096802—41no—
83C23×C197—21no—
84C23×C197C2×D411yescharacteristic
85D4×C197—41no—
86D4×C197—41no—
87D4×C197—41no—
88C2×C4×C197—22no—
89C2×C4×C197—21no—
90C2×C4×C197—41no—
91C2×C4×C197—41no—
92C2×C4×C197—41no—
93C2×C4×C197C2×D411yescharacteristic
94C2×C4×C197C2×D411yescharacteristic
95C8×C197—41no—
96C22×C197—21no—
97C22×C197—21no—
98C22×C197—41no—
99C22×C197—41no—
100C22×C197—41no—
101C22×C197#12011yescharacteristic
102C22×C197C2×SD1611yescharacteristic
103C22×C197AGL(1,ℤ/8)11yescharacteristic
104C2×D197—41no—
105C2×D197—41no—
106C2×D197—41no—
107C4×C197—21no—
108C4×C197—21no—
109C4×C197—42no—
110C4×C197—41no—
111Dic197—22no—
112Dic197—41no—
113Dic197—42no—
114Dic197—41no—
115C2×C197—21no—
116C2×C197—21no—
117C2×C197—81no—
118C2×C197#45011yescharacteristic
119C2×C197#45611yescharacteristic
120C2×C197B(4,𝔽2)11yescharacteristic
121D197—81no—
122C197#160411yescharacteristic, complemented, Sylow
123#1604—1971nomaximal, Sylow
124#537—1971no—
125#448—1971no—
126#448—1971no—
127#395—1971no—
128#327—1971no—
129#327D19711yescharacteristic, complemented
130#324—1971no—
131C2×#41—3941no—
132C22×Q8—1971no—
133C2×SD16—3941no—
134C2×SD16—3941no—
135C22⋊Q8—3941no—
136C22⋊Q8—3941no—
137#121—3941no—
138#121—3941no—
139#121—3941no—
140#121—1971no—
141#121C2×D19711yescharacteristic
142C2×C4⋊C4—3941no—
143C2×C4⋊C4—1971no—
144#103—3941no—
145#103—3941no—
146C22⋊C8—1971no—
147C22⋊C8C2×D19711yescharacteristic
148#95—3941no—
149#95C2×D19711yescharacteristic
150#41—7882no—
151#41—7882no—
152C2×Q8—7882no—
153C2×Q8—3941no—
154C2×Q8—3941no—
155C2×Q8—3941no—
156C2×D4—21no—
157C2×D4—3941no—
158C2×D4—3941no—
159C2×D4—3941no—
160C2×D4—1971no—
161C2×D4D78811yescharacteristic
162C22×C4—21no—
163C22×C4—3941no—
164C22×C4—3941no—
165C22×C4—3941no—
166C22×C4—1971no—
167C22×C4—1971no—
168C22×C4C22×D19711yescharacteristic
169SD16—7882no—
170SD16—7882no—
171C2×C8—21no—
172C2×C8—3941no—
173C4⋊C4—7882no—
174C4⋊C4—3941no—
175C4⋊C4—3941no—
176C4⋊C4—3941no—
177C4⋊C4—1971no—
178C4⋊C4D78811yescharacteristic
179C22⋊C4—21no—
180C22⋊C4—3941no—
181C22⋊C4—3941no—
182C22⋊C4—3941no—
183C23—21no—
184C23—3941no—
185C23D4×D19711yescharacteristic
186Q8—7881no—
187Q8—7881no—
188Q8—7882no—
189Q8—3942no—
190Q8—3942no—
191D4—41no—
192D4—41no—
193D4—41no—
194D4—7881no—
195D4—7881no—
196D4—7881no—
197D4—7882no—
198D4—15761no—
199C2×C4—22no—
200C2×C4—21no—
201C2×C4—41no—
202C2×C4—41no—
203C2×C4—41no—
204C2×C4—7882no—
205C2×C4—7881no—
206C2×C4—7881no—
207C2×C4—7882no—
208C2×C4—7881no—
209C2×C4—3942no—
210C2×C4—3941no—
211C2×C4—3941no—
212C2×C4—3941no—
213C2×C4—3941no—
214C2×C4—1972no—
215C2×C4D4×D19711yescharacteristic
216C2×C4C2×D78811yescharacteristic
217C8—41no—
218C8—7881no—
219C22—21no—
220C22—21no—
221C22—41no—
222C22—41no—
223C22—41no—
224C22—7881no—
225C22—7881no—
226C22—7881no—
227C22#2026329611yescharacteristic
228C22SD16×D19711yescharacteristic
229C22#2026335011yescharacteristic
230C4—21no—
231C4—21no—
232C4—42no—
233C4—41no—
234C4—7881no—
235C4—7882no—
236C4—7881no—
237C4—3942no—
238C2—21no—
239C2—21no—
240C2—81no—
241C2—15761no—
242C2#2716123111yescharacteristic
243C2#2716126311yescharacteristic
244C2#2716126711yescharacteristic
subgroupG/H
centerC22#20263296
commutator, cosocleC22×C4×C197C23
FrattiniC22×C4C22×D197
Fitting#327×C197C2
socleC22×C197#120
prime radical#27161656C2
hypercenter#327D197
hypocenterC197#1604
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. #27161000C2semidirect
5. #27160812C2semidirect
6. #27160804C2semidirect
7. #327×C197C2semidirect
80. C2×Dic197SD16semidirect
122. C197#1604semidirect
129. #327D197semidirect

Maximal subgroups

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

subgroupindex
#271616562
#271612202
#271612172
#271610002
#271608122
#271608042
#327×C1972
#1604197