Finite Group Databasebeta

← #87498

Subgroups — #87498

Order 384.

759 conjugacy classes of subgroups; only the 100 normal subgroups are shown.

normal subgroups only

This group has 759 conjugacy classes of subgroups — too many to draw. Only its 100 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#5945C211yesmaximal, characteristic, complemented
2#5926C216yesmaximal, complemented
3#5925C212yesmaximal, complemented
4#5923C213yesmaximal, complemented
5C2×#986C213yesmaximal, complemented
6#3585—31nomaximal, Sylow
7#1009—41nomaximal
82−1+4⋊C3C2213yescomplemented
9#994C2211yescharacteristic, complemented
10C2×#41⋊C3C2213yescomplemented
11#986C2213yescomplemented
12#986C2216yescomplemented
13#985C2216yescomplemented
14#984C2211yescharacteristic, complemented
15#983C2216yescomplemented
16C2×GL(2,𝔽3)C2216yescomplemented
17#585D311yescharacteristic, complemented
18C2×H(5,𝔽2)—31no—
19#577—32no—
20#577—36no—
21#576—36no—
22#576—32no—
23#575—33no—
24#575—33no—
25C2×AGL(1,ℤ/8)—33no—
26C2×AGL(1,ℤ/8)—33no—
27#568—31no—
28C3×#41—41no—
29#239—42no—
30D3×D4—46no—
31#235—43no—
32C2×D12—43no—
33#41⋊C3C2311yescharacteristic
34#41⋊C3C2313yescomplemented
35C2×SL(2,𝔽3)C2313yescomplemented
36GL(2,𝔽3)C2316yescomplemented
37#226C2312yescomplemented
382−1+4—31no—
392−1+4C2×D313yescomplemented
40H(5,𝔽2)—33no—
41H(5,𝔽2)—32no—
42H(5,𝔽2)—36no—
43H(5,𝔽2)—63no—
44H(5,𝔽2)—61no—
45H(5,𝔽2)C2×D311yescharacteristic, complemented
46C2×#41—31no—
47C2×#41—33no—
48C2×#41—33no—
49C2×#41—61no—
50C2×#41C2×D313yescomplemented
51C22×D4—33no—
52C22×D4—63no—
53#137—33no—
54#137—36no—
55#137—31no—
56#137—36no—
57AGL(1,ℤ/8)—33no—
58AGL(1,ℤ/8)—36no—
59AGL(1,ℤ/8)—36no—
60AGL(1,ℤ/8)—312no—
61AGL(1,ℤ/8)—36no—
62AGL(1,ℤ/8)—36no—
63AGL(1,ℤ/8)—33no—
64AGL(1,ℤ/8)—36no—
65#135—36no—
66#135—312no—
67#135—36no—
68C2×SD16—36no—
69C2×SD16—36no—
70C2×D8—36no—
71C2×D8—36no—
72#131—32no—
73#131—36no—
74C2×M16—33no—
75C2×M16—33no—
76C22×D3—46no—
77C3×Q8—41no—
78C3×D4—43no—
79C2×C3×C4—43no—
80#67—46no—
81D12—43no—
82D12—46no—
83C4×D3—46no—
84Dic6—41no—
85SL(2,𝔽3)C2411yescharacteristic
86C24—63no—
87#41—31no—
88#41—33no—
89#41—33no—
90#41—33no—
91#41—33no—
92#41—36no—
93#41—36no—
94#41—36no—
95#41—63no—
96#41—63no—
97#41—66no—
98#41—62no—
99#41—63no—
100#41S411yescharacteristic
101#41C22×D313yes—
102#41C22×D311yescharacteristic
103C2×Q8—31no—
104C2×Q8—33no—
105C2×Q8—31no—
106C2×Q8C22×D313yes—
107C2×D4—33no—
108C2×D4—33no—
109C2×D4—33no—
110C2×D4—36no—
111C2×D4—36no—
112C2×D4—33no—
113C2×D4—63no—
114C2×D4—63no—
115C2×D4—63no—
116C2×D4—612no—
117C2×D4—66no—
118C2×D4—63no—
119C2×D4—66no—
120C2×D4—63no—
121C22×C4—33no—
122C22×C4—61no—
123C22×C4—63no—
124Q16—36no—
125Q16—32no—
126SD16—36no—
127SD16—312no—
128SD16—32no—
129SD16—36no—
130SD16—36no—
131D8—312no—
132D8—36no—
133D8—36no—
134M16—31no—
135M16—33no—
136M16—36no—
137M16—36no—
138C2×C8—36no—
139C2×C8—36no—
140C22×C3—43no—
141C2×D3—46no—
142C2×D3—166no—
143C3×C4—41no—
144C3×C4—43no—
145Dic3—42no—
146C23—33no—
147C23—61no—
148C23—63no—
149C23—66no—
150C23—66no—
151C23—246no—
152Q8—33no—
153Q8—33no—
154Q8—32no—
155Q8—61no—
156Q8C2×S411yescharacteristic
157Q8C23×D311yescharacteristic
158D4—31no—
159D4—33no—
160D4—36no—
161D4—36no—
162D4—33no—
163D4—63no—
164D4—66no—
165D4—66no—
166D4—66no—
167D4—63no—
168D4—66no—
169D4C2×S413yes—
170C2×C4—33no—
171C2×C4—33no—
172C2×C4—31no—
173C2×C4—33no—
174C2×C4—33no—
175C2×C4—63no—
176C2×C4—66no—
177C2×C4—61no—
178C2×C4—66no—
179C2×C4C2×S413yes—
180C8—32no—
181C8—36no—
182C2×C3—41no—
183C2×C3—83no—
184D3—86no—
185C22—31no—
186C22—33no—
187C22—66no—
188C22—123no—
189C22—126no—
190C22—246no—
191C22—241no—
192C22—246no—
193C22C22×S413yes—
194C4—31no—
195C4—33no—
196C4—62no—
197C4C22×S411yescharacteristic
198C4C22×S413yes—
199C3—41noSylow
200C2—23no—
201C2—61no—
202C2—63no—
203C2—126no—
204C2C23×S411yescharacteristic
subgroupG/H
center, Frattini, socleC2C23×S4
commutator, cosocle, hypocenterSL(2,𝔽3)C24
Fitting#585D3
hypercenter#41S4
LCSG ▷ SL(2,𝔽3)
UCSC1 ◁ C2 ◁ #41

Complements

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

NH ≅ G/Ntype
1. #5945C2semidirect
2. #5926C2semidirect
3. #5925C2semidirect
4. #5923C2semidirect
5. C2×#986C2semidirect
8. 2−1+4⋊C3C22semidirect
9. #994C22semidirect
10. C2×#41⋊C3C22semidirect
11. #986C22semidirect
12. #986C22semidirect
13. #985C22semidirect
14. #984C22semidirect
15. #983C22semidirect
16. C2×GL(2,𝔽3)C22semidirect
17. #585D3semidirect
34. #41⋊C3C23semidirect
35. C2×SL(2,𝔽3)C23semidirect
36. GL(2,𝔽3)C23semidirect
37. #226C23semidirect
39. 2−1+4C2×D3semidirect
45. H(5,𝔽2)C2×D3semidirect
50. C2×#41C2×D3semidirect

Maximal subgroups

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

subgroupindex
#59452
#59262
#59252
#59232
C2×#9862
#35853
#10094