Finite Group Databasebeta

← #86572

Subgroups — #86572

Order 384.

1160 conjugacy classes of subgroups; only the 471 normal subgroups are shown.

normal subgroups only

This group has 1160 conjugacy classes of subgroups — too many to draw. Only its 471 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
1C3×#543C211yesmaximal, characteristic, complemented
2#5645C216yesmaximal, complemented
3#5644C216yesmaximal, complemented
4#5627C216yesmaximal, complemented
5#5624C216yesmaximal, complemented
6#5493C213yesmaximal, complemented
7C2×#67⋊C4C213yesmaximal, complemented
8#3459—31nomaximal, Sylow
9C22×#67C2213yes—
10C23×Dic3C2211yescharacteristic
11C2×#239C2212yescomplemented
12C22×C4×D3C2213yescomplemented
13C3×C22⋊Q8C2216yescomplemented
14C3×C22⋊Q8C2216yescomplemented
15C2×C3×C22⋊C4C2213yes—
16#947C2216yescomplemented
17#946C2216yescomplemented
18Q8×Dic3C2212yescomplemented
19#930C2216yes—
20#930C2216yes—
21C2×(C2×D3)⋊C4C2213yescomplemented
22C2×Dic3⋊C4C2213yescomplemented
23C2×C4×Dic3C2213yescomplemented
24#898C22112yescomplemented
25#895C2216yescomplemented
26#894C22112yescomplemented
27#893C2216yescomplemented
28#892C2216yescomplemented
29#890C2216yescomplemented
30#886C22112yescomplemented
31#882C22112yescomplemented
32#67⋊C4C22112yescomplemented
33#67⋊C4C22112yescomplemented
34#543D311yescharacteristic, complemented
35#539—36no—
36#538—36no—
37#534—33no—
38#529—36no—
39#525—36no—
40C2×C4×D4—33no—
41C24×C3C2311yescharacteristic
42C23×D3C2313yes—
43C2×C3×Q8C2312yes—
44C2×C3×Q8C2312yes—
45C22×C3×C4C2313yes—
46C22×C3×C4C2313yes—
47C2×#67—212no—
48C2×#67C23112yes—
49C22×Dic3—42no—
50C22×Dic3C2313yes—
51C22×Dic3C2313yes—
52C22×Dic3C2313yes—
53#239—44no—
54C2×D12C2316yes—
55C2×D12C2316yes—
56C2×C4×D3—46no—
57C2×C4×D3C2316yes—
58C2×C4×D3C2316yes—
59C3×C4⋊C4C2316yes—
60C3×C4⋊C4C2316yes—
61C3×C22⋊C4C23112yes—
62(C2×D3)⋊C4C23112yes—
63(C2×D3)⋊C4C23112yes—
64(C2×D3)⋊C4C23112yes—
65C4⋊Dic3C2316yes—
66C4⋊Dic3C2316yes—
67Dic3⋊C4C23112yes—
68C4×Dic3C2316yes—
69C4×Dic3C2316yes—
70C2×#41—32no—
71C22×D4—33no—
72C23×C4—31no—
73C23×C4—33no—
74#126—312no—
75#125—36no—
76#124—36no—
77#123—312no—
78#123—312no—
79C22⋊Q8—36no—
80C22⋊Q8C2×D316yescomplemented
81C22⋊Q8C2×D316yescomplemented
82#121—36no—
83#121—36no—
84#121—36no—
85#120—312no—
86C4×Q8—32no—
87C4×D4—36no—
88C4×D4—312no—
89C4×D4—312no—
90#117—36no—
91C2×C4⋊C4—33no—
92C2×C22⋊C4—33no—
93C2×C22⋊C4C2×D313yes—
94C2×C42—33no—
95C23×C3—41no—
96C23×C3—41no—
97C23×C3#4114yescomplemented
98C23×C3C2413yes—
99C22×D3—26no—
100C22×D3—43no—
101C22×D3—43no—
102C22×D3C2416yes—
103C3×Q8—42no—
104C3×Q8—42no—
105C2×C3×C4—43no—
106C2×C3×C4—43no—
107C2×C3×C4C2416yes—
108C2×C3×C4C2416yes—
109#67—224no—
110C2×Dic3—26no—
111C2×Dic3—26no—
112C2×Dic3—43no—
113C2×Dic3C2413yes—
114C2×Dic3C2416yes—
115C2×Dic3C2411yescharacteristic
116D12—46no—
117D12—46no—
118C4×D3—212no—
119C4×D3—46no—
120C24—33no—
121C24C22×D311yescharacteristic
122#41—124no—
123C2×Q8C22×D312yes—
124C2×Q8C22×D312yes—
125C2×D4—312no—
126C2×D4—36no—
127C2×D4—36no—
128C2×D4—612no—
129C22×C4—33no—
130C22×C4—33no—
131C22×C4—33no—
132C22×C4—36no—
133C22×C4—36no—
134C22×C4—126no—
135C22×C4—122no—
136C22×C4C22×D313yes—
137C22×C4C22×D313yes—
138C4⋊C4—312no—
139C4⋊C4—36no—
140C4⋊C4—36no—
141C4⋊C4C22×D316yes—
142C4⋊C4C22×D316yes—
143C22⋊C4—312no—
144C22⋊C4—312no—
145C22⋊C4—312no—
146C22⋊C4C22×D3112yes—
147C42—36no—
148C42—36no—
149C22×C3—23no—
150C22×C3—23no—
151C22×C3—44no—
152C22×C3C2×#4116yes—
153C22×C3C2511yescharacteristic
154C2×D3—26no—
155C2×D3—26no—
156C2×D3—26no—
157C2×D3—412no—
158C3×C4—26no—
159C3×C4—26no—
160Dic3—26no—
161Dic3—23no—
162Dic3H(5,𝔽2)12yes—
163C23—36no—
164C23—41no—
165C23—41no—
166C23—66no—
167C23—123no—
168C23—123no—
169C23#23914yescomplemented
170C23C23×D313yes—
171Q8—42no—
172Q8—42no—
173D4—624no—
174D4—126no—
175D4—126no—
176C2×C4—31no—
177C2×C4—36no—
178C2×C4—33no—
179C2×C4—43no—
180C2×C4—43no—
181C2×C4—612no—
182C2×C4—66no—
183C2×C4—66no—
184C2×C4—126no—
185C2×C4—123no—
186C2×C4C23×D316yes—
187C2×C4C23×D316yes—
188C2×C3—26no—
189C2×C3C22×#4111yescharacteristic
190C2×C3C2×H(5,𝔽2)11yescharacteristic
191C2×C3#58511yescharacteristic
192D3—46no—
193C22—23no—
194C22—23no—
195C22—44no—
196C22—66no—
197C22—66no—
198C22—66no—
199C22—1212no—
200C22C2×#23916yes—
201C22C24×D311yescharacteristic
202C4—26no—
203C4—26no—
204C4—32no—
205C4—66no—
206C4—63no—
207C3#345911yescharacteristic, complemented, Sylow
208C2—26no—
209C2—126no—
210C2C22×#23911yescharacteristic
211C2D3×H(5,𝔽2)11yescharacteristic
212C2#596611yescharacteristic
subgroupG/H
center, FrattiniC22C24×D3
commutator, socle, cosocleC22×C3C25
FittingC3×#543C2
hypercenter#543D3
hypocenterC3#3459
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ #543

Complements

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

NH ≅ G/Ntype
1. C3×#543C2semidirect
2. #5645C2semidirect
3. #5644C2semidirect
4. #5627C2semidirect
5. #5624C2semidirect
6. #5493C2semidirect
7. C2×#67⋊C4C2semidirect
11. C2×#239C22semidirect
12. C22×C4×D3C22semidirect
13. C3×C22⋊Q8C22semidirect
14. C3×C22⋊Q8C22semidirect
16. #947C22semidirect
17. #946C22semidirect
18. Q8×Dic3C22semidirect
21. C2×(C2×D3)⋊C4C22semidirect
22. C2×Dic3⋊C4C22semidirect
23. C2×C4×Dic3C22semidirect
24. #898C22semidirect
25. #895C22semidirect
26. #894C22semidirect
27. #893C22semidirect
28. #892C22semidirect
29. #890C22semidirect
30. #886C22semidirect
31. #882C22semidirect
32. #67⋊C4C22semidirect
33. #67⋊C4C22semidirect
34. #543D3semidirect
80. C22⋊Q8C2×D3semidirect
81. C22⋊Q8C2×D3semidirect
97. C23×C3#41semidirect
169. C23#239semidirect
207. C3#3459semidirect

Maximal subgroups

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

subgroupindex
C3×#5432
#56452
#56442
#56272
#56242
#54932
C2×#67⋊C42
#34593