Finite Group Databasebeta

← #87087

Subgroups — #87087

Order 384.

1258 conjugacy classes of subgroups; only the 455 normal subgroups are shown.

normal subgroups only

This group has 1258 conjugacy classes of subgroups — too many to draw. Only its 455 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×H(3,𝔽4)C211yesmaximal, characteristic, complemented
2#5683C213yesmaximal, complemented
3#5674C216yesmaximal, complemented
4#5670C216yesmaximal, complemented
5#5668C213yesmaximal, complemented
6#5592C216yesmaximal, complemented
7#5590C216yesmaximal, complemented
8#3549—31nomaximal, Sylow
9C22×#67C2216yescomplemented
10C2×#239C2213yescomplemented
11C2×#237C2213yescomplemented
12C2×D3×D4C2216yescomplemented
13C3×#124C2213yescomplemented
14C3×#124C2216yescomplemented
15C3×#120C2216yescomplemented
16#953C2212yescomplemented
17#947C2216yescomplemented
18#944C2213yescomplemented
19#940C2213yes—
20#940C2216yescomplemented
21#939C22112yescomplemented
22#937C2216yescomplemented
23#885C22112yescomplemented
24#884C22112yescomplemented
25#884C22112yescomplemented
26#882C2216yescomplemented
27#882C22112yescomplemented
28#67⋊C4C22112yescomplemented
29#877C2216yescomplemented
30#875C2213yescomplemented
31#874C2212yescomplemented
32#872C2216yescomplemented
33C42×D3C2211yescharacteristic, complemented
34H(3,𝔽4)—33no—
35H(3,𝔽4)D311yescharacteristic, complemented
36#555—36no—
37D42—36no—
38#535—36no—
39#535—36no—
40#532—33no—
41C24×C3C2312yes—
42C23×D3C2316yes—
43C2×C3×Q8C2313yes—
44C2×C3×D4C2313yes—
45C2×C3×D4C2316yes—
46C2×#67—412no—
47C2×#67C23112yes—
48C2×#67C2316yes—
49C2×#67C23112yes—
50C22×Dic3C2316yes—
51#239—46no—
52#237—46no—
53D3×D4—412no—
54C2×D12C2316yes—
55C2×D12C2316yes—
56C2×D12C2313yes—
57C2×C4×D3C2313yes—
58C2×C4×D3C2316yes—
59C2×Dic6C2313yes—
60C3×C22⋊C4C2316yes—
61C3×C22⋊C4C23112yes—
62C3×C42C2311yescharacteristic
63C3×C42C2312yes—
64#217C2316yes—
65(C2×D3)⋊C4C23112yes—
66(C2×D3)⋊C4C23112yes—
67Dic3⋊C4C23112yes—
68C4×Dic3C2313yes—
69C4×Dic3C2316yes—
70C2×#41—33no—
71C2×#41—33no—
72C22×D4—36no—
73C22×D4—36no—
74C4⋊Q8—33no—
75#127—33no—
76#127—36no—
77#127—32no—
78#124—36no—
79#124—312no—
80#124—33no—
81#124C2×D313yescomplemented
82#124C2×D316yescomplemented
83#121—312no—
84#121—312no—
85#121—312no—
86#120—32no—
87#120—36no—
88#120—36no—
89#120—312no—
90#120C2×D316yescomplemented
91C4×D4—312no—
92#117—36no—
93#117—36no—
94C2×C42—31no—
95C23×C3—46no—
96C23×C3C2416yes—
97C22×D3—412no—
98C22×D3—46no—
99C22×D3C2416yes—
100C22×D3C2411yescharacteristic
101C3×Q8—43no—
102C3×D4—46no—
103C3×D4—43no—
104C2×C3×C4C2416yes—
105C2×C3×C4C2413yes—
106#67—224no—
107#67—46no—
108#67—412no—
109C2×Dic3—212no—
110C2×Dic3C2416yes—
111C2×Dic3C2413yes—
112D12—46no—
113D12—43no—
114D12—46no—
115C4×D3—212no—
116C4×D3—26no—
117Dic6—43no—
118C24—36no—
119C24C22×D312yes—
120#41—126no—
121#41—126no—
122C2×Q8—33no—
123C2×Q8C22×D313yes—
124C2×D4—312no—
125C2×D4—312no—
126C2×D4—36no—
127C2×D4—33no—
128C2×D4—36no—
129C2×D4—36no—
130C2×D4—1212no—
131C2×D4—1212no—
132C2×D4C22×D313yes—
133C2×D4C22×D316yes—
134C22×C4—36no—
135C22×C4—33no—
136C22×C4—36no—
137C4⋊C4—312no—
138C22⋊C4—36no—
139C22⋊C4—312no—
140C22⋊C4—312no—
141C22⋊C4C22×D316yes—
142C22⋊C4C22×D3112yes—
143C42—33no—
144C42—36no—
145C42C22×D311yescharacteristic
146C42C22×D312yes—
147C22×C3—26no—
148C22×C3—212no—
149C22×C3—84no—
150C22×C3C2511yescharacteristic
151C2×D3—23no—
152C2×D3—26no—
153C2×D3—212no—
154C2×D3—812no—
155C3×C4—23no—
156C3×C4—26no—
157Dic3—26no—
158Dic3H(5,𝔽2)16yes—
159C23—36no—
160C23—31no—
161C23—46no—
162C23—126no—
163C23—1212no—
164C23C23×D316yes—
165Q8—43no—
166Q8—123no—
167D4—46no—
168D4—43no—
169D4—624no—
170D4—126no—
171D4—126no—
172D4—123no—
173D4—1212no—
174D4—126no—
175C2×C4—33no—
176C2×C4—36no—
177C2×C4—612no—
178C2×C4—66no—
179C2×C4—612no—
180C2×C4C23×D313yes—
181C2×C4C23×D316yes—
182C2×C3—46no—
183C2×C3C2×H(5,𝔽2)13yes—
184D3—41no—
185D3—46no—
186C22—26no—
187C22—212no—
188C22—63no—
189C22—66no—
190C22—612no—
191C22—84no—
192C22—2412no—
193C22C24×D311yescharacteristic
194C4—26no—
195C4—23no—
196C4—36no—
197C4—66no—
198C3#354911yescharacteristic, complemented, Sylow
199C2—46no—
200C2—126no—
201C2—121no—
202C2D3×H(5,𝔽2)13yes—
subgroupG/H
center, FrattiniC22C24×D3
commutator, socle, cosocleC22×C3C25
FittingC3×H(3,𝔽4)C2
hypercenterH(3,𝔽4)D3
hypocenterC3#3549
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ H(3,𝔽4)

Complements

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

NH ≅ G/Ntype
1. C3×H(3,𝔽4)C2semidirect
2. #5683C2semidirect
3. #5674C2semidirect
4. #5670C2semidirect
5. #5668C2semidirect
6. #5592C2semidirect
7. #5590C2semidirect
9. C22×#67C22semidirect
10. C2×#239C22semidirect
11. C2×#237C22semidirect
12. C2×D3×D4C22semidirect
13. C3×#124C22semidirect
14. C3×#124C22semidirect
15. C3×#120C22semidirect
16. #953C22semidirect
17. #947C22semidirect
18. #944C22semidirect
20. #940C22semidirect
21. #939C22semidirect
22. #937C22semidirect
23. #885C22semidirect
24. #884C22semidirect
25. #884C22semidirect
26. #882C22semidirect
27. #882C22semidirect
28. #67⋊C4C22semidirect
29. #877C22semidirect
30. #875C22semidirect
31. #874C22semidirect
32. #872C22semidirect
33. C42×D3C22semidirect
35. H(3,𝔽4)D3semidirect
81. #124C2×D3semidirect
82. #124C2×D3semidirect
90. #120C2×D3semidirect
198. C3#3549semidirect

Maximal subgroups

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

subgroupindex
C3×H(3,𝔽4)2
#56832
#56742
#56702
#56682
#55922
#55902
#35493