Finite Group Databasebeta

← #27155876

Subgroups — #27155876

Order 12416.

1058 conjugacy classes of subgroups; only the 451 normal subgroups are shown.

normal subgroups only

This group has 1058 conjugacy classes of subgroups — too many to draw. Only its 451 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#20261485C213yesmaximal, complemented
2#20261479C216yesmaximal, complemented
3#20261474C216yesmaximal, complemented
4#20261284C213yesmaximal, complemented
5#20261264C216yesmaximal, complemented
6#20261241C216yesmaximal, complemented
7H(3,𝔽4)×C97C211yesmaximal, characteristic, complemented
8C2×#13095512C2211yescharacteristic, complemented
9C2×#13095510C2213yes—
10C23×Dic97C2212yescomplemented
11#15690808C2213yescomplemented
12#15690806C2213yes—
13#15690803C22112yescomplemented
14#15690802C2216yescomplemented
15#15690798C2213yescomplemented
16#15690794C2216yescomplemented
17#15690792C22112yescomplemented
18#15690791C2216yes—
19#15690787C22112yescomplemented
20#15690786C2216yescomplemented
21#15690783C22112yescomplemented
22#15690782C22112yes—
23#15690781C2216yes—
24#15690779C2216yescomplemented
25Q8×Dic97C2212yescomplemented
26D4×Dic97C2216yes—
27#15690770C22112yescomplemented
28#15690766C2216yescomplemented
29C42×D97C2213yescomplemented
30#124×C97C2216yes—
31#124×C97C2213yes—
32#120×C97C2216yes—
33#13095512—42no—
34#13095510—46no—
35C2×Dic194C2313yes—
36C2×D388C2313yes—
37C2×C22⋊D97C23112yes—
38C2×C22⋊D97C2316yes—
39C22×Dic97—44no—
40C22×Dic97C2316yes—
41C22×Dic97C2316yes—
42C2×C4×D97C2313yes—
43C2×C4×D97C2316yes—
44C4⋊Dic97C2316yes—
45C4⋊Dic97C2316yes—
46#13095490C23112yes—
47#13095490C23112yes—
48C22⋊Dic97C2316yes—
49C22⋊Dic97C23112yes—
50#13095488C23112yes—
51C4×Dic97C2316yes—
52C4×Dic97C2313yes—
53C24×C97C2312yes—
54C2×Q8×C97C2312yes—
55C2×Q8×C97C2311yescharacteristic
56C2×D4×C97C2316yes—
57C2×D4×C97C2313yes—
58C22⋊C4×C97C2316yes—
59C22⋊C4×C97C23112yes—
60C42×C97C2313yes—
61C23×C97—44no—
62C23×C97—42no—
63C23×C97C2416yes—
64C22×D97C2413yes—
65Q8×C97—42no—
66Q8×C97—41no—
67D4×C97—46no—
68D4×C97—43no—
69C2×C4×C97C2416yes—
70C2×C4×C97C2413yes—
71C22⋊D97—412no—
72C22⋊D97—46no—
73C2×Dic97—212no—
74C2×Dic97—46no—
75C2×Dic97C2416yes—
76C2×Dic97C2416yes—
77C2×Dic97C2411yescharacteristic
78D388—43no—
79C4×D97—26no—
80C4×D97—46no—
81Dic194—43no—
82C22×C97—26no—
83C22×C97—212no—
84C22×C97—48no—
85C22×C97C2511yescharacteristic
86C2×D97—23no—
87C2×D97—26no—
88C4×C97—23no—
89C4×C97—26no—
90Dic97—26no—
91Dic97—26no—
92Dic97H(5,𝔽2)12yes—
93C2×C97—46no—
94C2×C97C2×H(5,𝔽2)11yescharacteristic
95C2×C97#58512yes—
96D97—43no—
97#3529—971nomaximal, Sylow
98C97#352911yescharacteristic, complemented, Sylow
99#563—973no—
100#562—976no—
101H(3,𝔽4)D9711yescharacteristic, complemented
102#560—976no—
103#532—973no—
104#529—976no—
105#525—976no—
106C2×#41—971no—
107C2×#41—973no—
108C23×C4—972no—
109C4⋊Q8—973no—
110#127—973no—
111#126—9712no—
112#125—976no—
113#124—973no—
114#124C2×D9716yes—
115#124C2×D9713yes—
116#123—976no—
117#123—976no—
118#123—9712no—
119C22⋊Q8—976no—
120C22⋊Q8—9712no—
121#121—9712no—
122#121—976no—
123#121—9712no—
124#120—976no—
125#120C2×D9716yes—
126C4×Q8—972no—
127C4×D4—976no—
128C4×D4—9712no—
129#117—976no—
130C2×C42—973no—
131C24C22×D9712yes—
132#41—3886no—
133#41—3882no—
134C2×Q8—973no—
135C2×Q8C22×D9712yes—
136C2×Q8C22×D9711yescharacteristic
137C2×D4—9712no—
138C2×D4—976no—
139C2×D4—973no—
140C2×D4C22×D9716yes—
141C2×D4C22×D9713yes—
142C22×C4—976no—
143C22×C4—976no—
144C22×C4—976no—
145C22×C4—973no—
146C22×C4—3884no—
147C4⋊C4—9712no—
148C4⋊C4—9712no—
149C4⋊C4—976no—
150C4⋊C4—976no—
151C22⋊C4—976no—
152C22⋊C4—9712no—
153C22⋊C4—9712no—
154C22⋊C4C22×D9716yes—
155C22⋊C4C22×D97112yes—
156C42—976no—
157C42—973no—
158C42C22×D9713yes—
159C23—42no—
160C23—44no—
161C23—973no—
162C23C23×D9716yes—
163Q8—41no—
164Q8—42no—
165Q8—3883no—
166D4—43no—
167D4—46no—
168D4—3883no—
169D4—3886no—
170D4—38812no—
171C2×C4—971no—
172C2×C4—976no—
173C2×C4—976no—
174C2×C4—3886no—
175C2×C4—3886no—
176C2×C4—1946no—
177C2×C4—19412no—
178C2×C4C23×D9716yes—
179C2×C4C23×D9713yes—
180C22—26no—
181C22—212no—
182C22—48no—
183C22—1943no—
184C22—1946no—
185C22C24×D9711yescharacteristic
186C4—23no—
187C4—26no—
188C4—972no—
189C4—1946no—
190C4—1946no—
191C2—46no—
192C2—3883no—
193C2H(5,𝔽2)×D9711yescharacteristic
194C2#2026157012yes—
subgroupG/H
center, FrattiniC22C24×D97
commutator, socle, cosocleC22×C97C25
FittingH(3,𝔽4)×C97C2
hypercenterH(3,𝔽4)D97
hypocenterC97#3529
LCSG ▷ C22×C97 ▷ C97
UCSC1 ◁ C22 ◁ H(3,𝔽4)

Complements

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

NH ≅ G/Ntype
1. #20261485C2semidirect
2. #20261479C2semidirect
3. #20261474C2semidirect
4. #20261284C2semidirect
5. #20261264C2semidirect
6. #20261241C2semidirect
7. H(3,𝔽4)×C97C2semidirect
8. C2×#13095512C22semidirect
10. C23×Dic97C22semidirect
11. #15690808C22semidirect
13. #15690803C22semidirect
14. #15690802C22semidirect
15. #15690798C22semidirect
16. #15690794C22semidirect
17. #15690792C22semidirect
19. #15690787C22semidirect
20. #15690786C22semidirect
21. #15690783C22semidirect
24. #15690779C22semidirect
25. Q8×Dic97C22semidirect
27. #15690770C22semidirect
28. #15690766C22semidirect
29. C42×D97C22semidirect
98. C97#3529semidirect
101. H(3,𝔽4)D97semidirect

Maximal subgroups

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

subgroupindex
#202614852
#202614792
#202614742
#202612842
#202612642
#202612412
H(3,𝔽4)×C972
#352997