Finite Group Databasebeta

← #38752221

Subgroups — #38752221

Order 25216.

494 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 494 conjugacy classes of subgroups — too many to draw. Only its 141 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
1C2×#20263352C211yesmaximal, characteristic
2#27161653C211yesmaximal, characteristic, complemented
3#27161554C211yesmaximal, characteristic, complemented
4#27161296C214yesmaximal, complemented
5#27161282C214yesmaximal, complemented
6#27161105C211yesmaximal, characteristic, complemented
7#27161088C211yesmaximal, characteristic, complemented
8#27161076C211yesmaximal, characteristic, complemented
9C22×SD16×C197C211yesmaximal, characteristic, complemented
10C2×#15691358C2211yescharacteristic
11#20263352—24no—
12C2×#15691346C2212yes—
13C2×#15691343C2212yes—
14C2×#15691342C2211yescharacteristic
15#20263324C2212yescomplemented
16#20263315C2212yescomplemented
17#20263309—22no—
18#20263302—22no—
19D4×Dic197—22no—
20D4×Dic197C2214yes—
21#20263285C2211yescharacteristic
22#20263285C2211yescharacteristic
23C2×C22⋊Dic197—22no—
24#20263255C2214yescomplemented
25#20263251C2212yescomplemented
26#20263250C2214yescomplemented
27#20263248C2212yescomplemented
28C22×Q8×C197C2211yescharacteristic
29C22×D4×C197C2211yescharacteristic
30C2×SD16×C197—24no—
31C2×SD16×C197C2214yes—
32C22×C8×C197C2211yescharacteristic
33#15691358—22no—
34#15691358—22no—
35C2×Dic394C2311yescharacteristic
36C2×D788C2311yescharacteristic
37C2×C22⋊D197—21no—
38C22×Dic197—24no—
39C2×C4×D197—21no—
40#15691346—24no—
41#15691343—24no—
42#15691342—22no—
43C2×#13096802C2312yes—
44C4⋊Dic197D412yes—
45C4⋊Dic197C2312yes—
46#15691337—24no—
47C22⋊Dic197—24no—
48C22⋊Dic197—21no—
49C22⋊Dic197D412yes—
50#15691335—24no—
51C4×Dic197—21no—
52C4×Dic197C2312yes—
53C24×C197—21no—
54C2×Q8×C197—21no—
55C2×Q8×C197—22no—
56C2×Q8×C197—22no—
57C2×Q8×C197C2312yes—
58C2×D4×C197—21no—
59C2×D4×C197—22no—
60C2×D4×C197D414yes—
61C2×D4×C197C2312yes—
62C22×C4×C197—21no—
63C22×C4×C197C2311yescharacteristic
64SD16×C197—28no—
65C2×C8×C197—22no—
66C2×C8×C197C2312yes—
67C22×D197—21no—
68Dic394—21no—
69Dic394—21no—
70D788—21no—
71D788—21no—
72C22⋊D197—42no—
73C2×Dic197—22no—
74C2×Dic197—21no—
75C2×Dic197—84no—
76C2×Dic197C2×D414yes—
77C4×D197—42no—
78#13096802—22no—
79C23×C197—22no—
80C23×C197—24no—
81C23×C197—82no—
82C23×C197C2×D411yescharacteristic
83Q8×C197—22no—
84Q8×C197—22no—
85Q8×C197—42no—
86D4×C197—24no—
87D4×C197—22no—
88D4×C197C2×D414yes—
89C2×C4×C197—22no—
90C2×C4×C197—42no—
91C2×C4×C197C2×D411yescharacteristic
92C2×C4×C197C2×D412yes—
93C2×C4×C197#4112yes—
94C2×C4×C197C2411yescharacteristic
95C8×C197—22no—
96C22×C197—22no—
97C22×C197—24no—
98C22×C197—44no—
99C22×C197—84no—
100C22×C197#12112yes—
101C22×C197C22×D411yescharacteristic
102C2×D197—41no—
103C2×D197—42no—
104C4×C197—42no—
105C4×C197#12112yes—
106C4×C197C22×D411yescharacteristic
107C4×C197C2×#4111yescharacteristic
108Dic197—24no—
109Dic197—41no—
110Dic197—42no—
111C2×C197—22no—
112C2×C197—44no—
113C2×C197C2×#12111yescharacteristic
114C2×C197#57712yes—
115D197—81no—
116C197#304211yescharacteristic, complemented, Sylow
117#3042—1971nomaximal, Sylow
118C2×AGL(1,ℤ/8)—1971no—
119C22×SD16D19711yescharacteristic, complemented
120#536—1971no—
121#518—1971no—
122#463—1974no—
123#461—1974no—
124#424—1971no—
125#419—1971no—
126#416—1971no—
127C2×#41—1971no—
128C22×Q8C2×D19711yescharacteristic
129C22×D4C2×D19711yescharacteristic
130AGL(1,ℤ/8)—3944no—
131C2×SD16—24no—
132C2×SD16—1972no—
133C2×SD16C2×D19714yes—
134C2×D8—1972no—
135C2×M16—1971no—
136C22×C8C2×D19711yescharacteristic
137C4⋊Q8—1972no—
138#124—1972no—
139#123—3942no—
140C22⋊Q8—3942no—
141C4×D4—3942no—
142C4×D4—1974no—
143#117—1971no—
144#117—1971no—
145C2×C22⋊C4—3942no—
146C4⋊C8—1974no—
147#103—1974no—
148#103—1972no—
149#102—1972no—
150C24—21no—
151#41—3942no—
152#41—3942no—
153C2×Q8—21no—
154C2×Q8—22no—
155C2×Q8—22no—
156C2×Q8—1971no—
157C2×Q8C22×D19712yes—
158C2×D4—21no—
159C2×D4—22no—
160C2×D4—3941no—
161C2×D4—1971no—
162C2×D4C22⋊D19714yes—
163C2×D4C22×D19712yes—
164C22×C4—21no—
165C22×C4—3941no—
166C22×C4—3944no—
167C22×C4C22×D19711yescharacteristic
168SD16—28no—
169SD16—3944no—
170D8—3944no—
171M16—3942no—
172C2×C8—22no—
173C2×C8—1972no—
174C2×C8C22×D19712yes—
175C4⋊C4—3944no—
176C4⋊C4—1972no—
177C4⋊C4—1972no—
178C22⋊C4—3944no—
179C22⋊C4—3941no—
180C22⋊C4—3944no—
181C22⋊C4—1972no—
182C42—3941no—
183C42—1972no—
184C23—22no—
185C23—24no—
186C23—82no—
187C23—3941no—
188C23D4×D19711yescharacteristic
189Q8—22no—
190Q8—22no—
191Q8—42no—
192Q8—3941no—
193Q8—3941no—
194D4—24no—
195D4—22no—
196D4—7882no—
197D4—3941no—
198D4—3941no—
199D4C2×C22⋊D19714yes—
200C2×C4—22no—
201C2×C4—42no—
202C2×C4—7882no—
203C2×C4—15764no—
204C2×C4—3942no—
205C2×C4—3941no—
206C2×C4—1974no—
207C2×C4D4×D19711yescharacteristic
208C2×C4C2×C22⋊D19712yes—
209C2×C4#1569135712yes—
210C2×C4C23×D19711yescharacteristic
211C8—22no—
212C8—3942no—
213C22—22no—
214C22—24no—
215C22—44no—
216C22—84no—
217C22—7882no—
218C22—7881no—
219C22#2026329812yes—
220C22C2×D4×D19711yescharacteristic
221C4—42no—
222C4—7882no—
223C4—7881no—
224C4—3944no—
225C4#2026329812yes—
226C4C22×C22⋊D19711yescharacteristic
227C4C2×#1569135711yescharacteristic
228C2—22no—
229C2—44no—
230C2—15761no—
231C2C2×#2026329811yescharacteristic
232C2#2716188512yes—
subgroupG/H
centerC22C2×D4×D197
commutator, cosocleC2×C4×C197C24
FrattiniC2×C4C23×D197
FittingC22×SD16×C197C2
socleC22×C197C22×D4
prime radicalC2×#20263352C2
hypercenterC22×SD16D197
hypocenterC197#3042
LCSG ▷ C2×C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×SD16

Complements

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

NH ≅ G/Ntype
2. #27161653C2semidirect
3. #27161554C2semidirect
4. #27161296C2semidirect
5. #27161282C2semidirect
6. #27161105C2semidirect
7. #27161088C2semidirect
8. #27161076C2semidirect
9. C22×SD16×C197C2semidirect
15. #20263324C22semidirect
16. #20263315C22semidirect
24. #20263255C22semidirect
25. #20263251C22semidirect
26. #20263250C22semidirect
27. #20263248C22semidirect
116. C197#3042semidirect
119. C22×SD16D197semidirect

Maximal subgroups

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

subgroupindex
C2×#202633522
#271616532
#271615542
#271612962
#271612822
#271611052
#271610882
#271610762
C22×SD16×C1972
#3042197