Finite Group Databasebeta

← #27161756

Subgroups — #27161756

Order 12608.

248 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 248 conjugacy classes of subgroups — too many to draw. Only its 97 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#20263321C211yesmaximal, characteristic, complemented
2#20263321C211yesmaximal, characteristic, complemented
3#20263315C211yesmaximal, characteristic, complemented
4#20263311C211yesmaximal, characteristic, complemented
5#20263311C211yesmaximal, characteristic, complemented
6#20263310C211yesmaximal, characteristic, complemented
7#20263302C211yesmaximal, characteristic, complemented
8#20263296C211yesmaximal, characteristic
9#20263288C211yesmaximal, characteristic, complemented
10C4×C22⋊D197C211yesmaximal, characteristic, complemented
11#20263284C211yesmaximal, characteristic, complemented
12#20263282C211yesmaximal, characteristic, complemented
13C2×#15691335C211yesmaximal, characteristic, complemented
14C22⋊C4×D197C211yesmaximal, characteristic, complemented
15C22⋊Q8×C197C211yesmaximal, characteristic, complemented
16C23×D197C2211yescharacteristic
17C2×D788C2211yescharacteristic
18C2×D788C2211yescharacteristic
19C2×C22⋊D197C2211yescharacteristic
20C22×Dic197C2211yescharacteristic
21C2×C4×D197C2211yescharacteristic
22C2×C4×D197C2211yescharacteristic
23C2×C4×D197C2211yescharacteristic, complemented
24C4⋊Dic197C2211yescharacteristic, complemented
25C4⋊Dic197C2211yescharacteristic, complemented
26#15691337C2211yescharacteristic, complemented
27#15691337C2211yescharacteristic, complemented
28#15691337C2211yescharacteristic, complemented
29C22⋊Dic197C2211yescharacteristic
30#15691335C2211yescharacteristic
31#15691335C2211yescharacteristic
32#15691335C2211yescharacteristic
33#15691335C2211yescharacteristic
34#15691335C2211yescharacteristic
35#15691335C2211yescharacteristic
36#15691335C2211yescharacteristic
37#15691335C2211yescharacteristic
38#15691335C2211yescharacteristic, complemented
39#15691335C2211yescharacteristic, complemented
40#15691335C2211yescharacteristic, complemented
41C4×Dic197C2211yescharacteristic, complemented
42C4×Dic197C2211yescharacteristic, complemented
43C4×Dic197C2211yescharacteristic, complemented
44C2×Q8×C197C2211yescharacteristic, complemented
45C22×C4×C197C2211yescharacteristic
46C4⋊C4×C197C2211yescharacteristic, complemented
47C4⋊C4×C197C2211yescharacteristic, complemented
48C4⋊C4×C197C2211yescharacteristic, complemented
49C22⋊C4×C197C2211yescharacteristic
50C22⋊C4×C197C2211yescharacteristic
51C22×D197—22no—
52C22×D197—22no—
53C22×D197—41no—
54C22×D197C2311yescharacteristic
55C22×D197C2311yescharacteristic
56C22×D197C2311yescharacteristic
57D788—22no—
58D788—41no—
59C22⋊D197—22no—
60C2×Dic197—22no—
61C2×Dic197C2311yescharacteristic
62C2×Dic197C2311yescharacteristic
63C2×Dic197C2311yescharacteristic
64C2×Dic197C2311yescharacteristic
65C2×Dic197C2311yescharacteristic
66C4×D197—22no—
67C4×D197—41no—
68C4×D197—41no—
69C23×C197C2311yescharacteristic
70Q8×C197—41no—
71C2×C4×C197—41no—
72C2×C4×C197C2311yescharacteristic
73C2×C4×C197C2311yescharacteristic
74C2×C4×C197C2311yescharacteristic
75C2×C4×C197C2311yescharacteristic
76C2×C4×C197C2311yescharacteristic
77C2×C4×C197C2311yescharacteristic
78C22×C197—21no—
79C22×C197—21no—
80C22×C197#4112yes—
81C22×C197C2411yescharacteristic
82C2×D197—21no—
83C2×D197—21no—
84C2×D197—21no—
85C2×D197—21no—
86C2×D197—21no—
87C2×D197—21no—
88C2×D197—21no—
89C2×D197—21no—
90C2×D197—44no—
91C2×D197#4112yes—
92C4×C197—21no—
93C4×C197—21no—
94C4×C197—21no—
95C4×C197—21no—
96C4×C197—21no—
97C4×C197—21no—
98Dic197—21no—
99Dic197—21no—
100Dic197—21no—
101Dic197—21no—
102Dic197—21no—
103C2×C197—22no—
104C2×C197C2×#4111yescharacteristic
105C2×C197C2×#4111yescharacteristic
106C2×C197H(5,𝔽2)11yescharacteristic
107D197—22no—
108D197—41no—
109D197—41no—
110C197#55111yescharacteristic, complemented, Sylow
111#551—1971nomaximal, Sylow
112#126—1971no—
113#126—1971no—
114#124—1971no—
115#123—1971no—
116#123—1971no—
117#123—1971no—
118C22⋊Q8—1971no—
119C22⋊Q8D19711yescharacteristic, complemented
120#120—1971no—
121C4×D4—1971no—
122C4×D4—1971no—
123#117—1971no—
124#117—1971no—
125C2×C22⋊C4—1971no—
126C2×C22⋊C4—1971no—
127C24—1971no—
128C2×Q8C2×D19711yescharacteristic, complemented
129C2×D4—1971no—
130C2×D4—1971no—
131C2×D4—1971no—
132C22×C4—1971no—
133C22×C4—1971no—
134C22×C4—1971no—
135C22×C4—1971no—
136C22×C4C2×D19711yescharacteristic
137C4⋊C4—1971no—
138C4⋊C4—1971no—
139C4⋊C4—1971no—
140C4⋊C4—1971no—
141C4⋊C4—1971no—
142C4⋊C4C2×D19711yescharacteristic, complemented
143C4⋊C4C2×D19711yescharacteristic, complemented
144C4⋊C4C2×D19711yescharacteristic, complemented
145C22⋊C4—1971no—
146C22⋊C4—1971no—
147C22⋊C4—1971no—
148C22⋊C4—1971no—
149C22⋊C4—1971no—
150C22⋊C4—1971no—
151C22⋊C4—1971no—
152C22⋊C4—1971no—
153C22⋊C4—1971no—
154C22⋊C4—1971no—
155C22⋊C4—1971no—
156C22⋊C4—1971no—
157C22⋊C4C2×D19711yescharacteristic
158C22⋊C4C2×D19711yescharacteristic
159C42—1971no—
160C42—1971no—
161C42—1971no—
162C23—7881no—
163C23—3942no—
164C23—3942no—
165C23—1971no—
166C23—1971no—
167C23—1971no—
168C23C22×D19711yescharacteristic
169Q8—41no—
170D4—7881no—
171D4—3942no—
172D4—3942no—
173C2×C4—41no—
174C2×C4—7881no—
175C2×C4—7881no—
176C2×C4—3942no—
177C2×C4—3942no—
178C2×C4—1971no—
179C2×C4—1971no—
180C2×C4—1971no—
181C2×C4—1971no—
182C2×C4—1971no—
183C2×C4C22×D19711yescharacteristic
184C2×C4C22×D19711yescharacteristic
185C2×C4C22×D19711yescharacteristic
186C2×C4C22×D19711yescharacteristic
187C2×C4C22×D19711yescharacteristic
188C2×C4C22×D19711yescharacteristic
189C22—21no—
190C22—21no—
191C22—7884no—
192C22—3941no—
193C22—3941no—
194C22—3941no—
195C22—3941no—
196C22—3941no—
197C22—3941no—
198C22—3941no—
199C22—3941no—
200C22—1972no—
201C22#1569135912yes—
202C22C23×D19711yescharacteristic
203C4—21no—
204C4—21no—
205C4—21no—
206C4—21no—
207C4—21no—
208C4—21no—
209C4—3941no—
210C4—3941no—
211C4—3941no—
212C4—3941no—
213C4—3941no—
214C2—22no—
215C2—7881no—
216C2—7881no—
217C2—3942no—
218C2#41×D19711yescharacteristic
219C2C2×#1569135911yescharacteristic
220C2#2026337111yescharacteristic
subgroupG/H
center, FrattiniC22C23×D197
commutator, socle, cosocleC22×C197C24
FittingC22⋊Q8×C197C2
prime radical#20263296C2
hypercenterC22⋊Q8D197
hypocenterC197#551
LCSG ▷ C22×C197 ▷ C197
UCSC1 ◁ C22 ◁ C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. #20263321C2semidirect
2. #20263321C2semidirect
3. #20263315C2semidirect
4. #20263311C2semidirect
5. #20263311C2semidirect
6. #20263310C2semidirect
7. #20263302C2semidirect
9. #20263288C2semidirect
10. C4×C22⋊D197C2semidirect
11. #20263284C2semidirect
12. #20263282C2semidirect
13. C2×#15691335C2semidirect
14. C22⋊C4×D197C2semidirect
15. C22⋊Q8×C197C2semidirect
23. C2×C4×D197C22semidirect
24. C4⋊Dic197C22semidirect
25. C4⋊Dic197C22semidirect
26. #15691337C22semidirect
27. #15691337C22semidirect
28. #15691337C22semidirect
38. #15691335C22semidirect
39. #15691335C22semidirect
40. #15691335C22semidirect
41. C4×Dic197C22semidirect
42. C4×Dic197C22semidirect
43. C4×Dic197C22semidirect
44. C2×Q8×C197C22semidirect
46. C4⋊C4×C197C22semidirect
47. C4⋊C4×C197C22semidirect
48. C4⋊C4×C197C22semidirect
110. C197#551semidirect
119. C22⋊Q8D197semidirect
128. C2×Q8C2×D197semidirect
142. C4⋊C4C2×D197semidirect
143. C4⋊C4C2×D197semidirect
144. C4⋊C4C2×D197semidirect

Maximal subgroups

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

subgroupindex
#202633212
#202633152
#202633112
#202633102
#202633022
#202632962
#202632882
C4×C22⋊D1972
#202632842
#202632822
C2×#156913352
C22⋊C4×D1972
C22⋊Q8×C1972
#551197