Finite Group Databasebeta

← #87496

Subgroups — #87496

Order 384.

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

normal subgroups only

This group has 738 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
1C2×2−1+4⋊C3C211yesmaximal, characteristic, complemented
2#5927C214yesmaximal, complemented
3#5926C214yesmaximal, complemented
4#5923C212yesmaximal, complemented
5C2×#983C214yesmaximal, complemented
6#3586—31nomaximal, Sylow
7#1004—41nomaximal
82−1+4⋊C3C2214yescomplemented
9C2×#41⋊C3C2211yescharacteristic, complemented
10C22×SL(2,𝔽3)C2212yescomplemented
11#986C2212yescomplemented
12#985C2214yescomplemented
13#984C2212yescomplemented
14#983C2214yescomplemented
15#983C2218yescomplemented
16C2×GL(2,𝔽3)C2214yescomplemented
17C2×#226C2214yescomplemented
18#585—31no—
19C2×2−1+4D311yescharacteristic, complemented
20#578—34no—
21#578—34no—
22#577—34no—
23#577—34no—
24#575—34no—
25#575—32no—
26C2×#137—32no—
27C2×#137—34no—
28#568—31no—
29C2×C3×D4—41no—
30C2×#67—44no—
31#237—44no—
32D3×D4—44no—
33#235—42no—
34#41⋊C3C2312yes—
35C2×SL(2,𝔽3)C2314yes—
36C2×SL(2,𝔽3)C2311yescharacteristic
37GL(2,𝔽3)C2314yes—
38#226C2314yes—
392−1+4—34no—
402−1+4—34no—
412−1+4—61no—
422−1+4C2×D314yescomplemented
43H(5,𝔽2)—34no—
44H(5,𝔽2)—62no—
45H(5,𝔽2)—61no—
46C2×#41—31no—
47C2×#41—32no—
48C2×#41—34no—
49C2×#41—32no—
50C2×#41—64no—
51C2×#41C2×D311yescharacteristic, complemented
52C22×Q8—31no—
53C22×Q8C2×D312yescomplemented
54#137—38no—
55#137—38no—
56#137—34no—
57#137—34no—
58#137—32no—
59#137—38no—
60#137—32no—
61#137—34no—
62#137—38no—
63AGL(1,ℤ/8)—34no—
64AGL(1,ℤ/8)—32no—
65AGL(1,ℤ/8)—38no—
66AGL(1,ℤ/8)—32no—
67#135—34no—
68#135—38no—
69#135—38no—
70#135—34no—
71C2×Q16—34no—
72C2×Q16—34no—
73C2×Q16—34no—
74C2×SD16—34no—
75C2×SD16—34no—
76C2×SD16—34no—
77#131—34no—
78#131—34no—
79C2×M16—34no—
80C2×M16—32no—
81C23×C3—42no—
82C22×D3—44no—
83C3×D4—44no—
84C2×C3×C4—41no—
85#67—44no—
86#67—48no—
87C2×Dic3—44no—
88D12—42no—
89C4×D3—44no—
90Dic6—42no—
91SL(2,𝔽3)C2411yescharacteristic
92#41—34no—
93#41—34no—
94#41—32no—
95#41—38no—
96#41—38no—
97#41—34no—
98#41—34no—
99#41—64no—
100#41—64no—
101#41—62no—
102#41—64no—
103#41—68no—
104#41—64no—
105#41—64no—
106#41—62no—
107#41C22×D312yes—
108C2×Q8—31no—
109C2×Q8—34no—
110C2×Q8—32no—
111C2×Q8—34no—
112C2×Q8—34no—
113C2×Q8—32no—
114C2×Q8—62no—
115C2×Q8—64no—
116C2×Q8C22×D314yes—
117C2×Q8C22×D311yescharacteristic
118C2×D4—32no—
119C2×D4—31no—
120C2×D4—34no—
121C2×D4—32no—
122C2×D4—68no—
123C2×D4—62no—
124C2×D4—64no—
125C2×D4—64no—
126C2×D4S411yescharacteristic
127C22×C4—32no—
128C22×C4—31no—
129C22×C4—61no—
130C22×C4—64no—
131C22×C4—62no—
132Q16—38no—
133Q16—34no—
134Q16—38no—
135Q16—34no—
136SD16—38no—
137SD16—34no—
138SD16—38no—
139SD16—34no—
140SD16—34no—
141SD16—34no—
142D8—34no—
143D8—34no—
144M16—34no—
145M16—38no—
146M16—32no—
147M16—32no—
148C2×C8—34no—
149C2×C8—34no—
150C2×C8—34no—
151C22×C3—44no—
152C22×C3—41no—
153C22×C3—162no—
154C2×D3—44no—
155C2×D3—164no—
156C3×C4—42no—
157Dic3—44no—
158C23—31no—
159C23—62no—
160C23—64no—
161C23C2×S412yes—
162Q8—31no—
163Q8—34no—
164Q8—34no—
165Q8—34no—
166Q8—62no—
167Q8—64no—
168Q8—62no—
169Q8C23×D311yescharacteristic
170D4—32no—
171D4—38no—
172D4—34no—
173D4—61no—
174D4—64no—
175D4—64no—
176D4—64no—
177D4—68no—
178D4—62no—
179D4C2×S414yes—
180C2×C4—34no—
181C2×C4—31no—
182C2×C4—32no—
183C2×C4—34no—
184C2×C4—32no—
185C2×C4—32no—
186C2×C4—64no—
187C2×C4—64no—
188C2×C4—68no—
189C2×C4—62no—
190C2×C4—64no—
191C2×C4—64no—
192C2×C4C2×S411yescharacteristic
193C8—34no—
194C8—34no—
195C2×C3—41no—
196C2×C3—84no—
197C2×C3—81no—
198D3—84no—
199C22—32no—
200C22—42no—
201C22—64no—
202C22—121no—
203C22—244no—
204C22—242no—
205C22C22×S411yescharacteristic
206C22C22×S414yes—
207C4—31no—
208C4—31no—
209C4—34no—
210C4—64no—
211C4C22×S412yes—
212C3—41noSylow
213C2—21no—
214C2—24no—
215C2—62no—
216C2—124no—
217C2C23×S411yescharacteristic
subgroupG/H
center, Frattini, socleC2C23×S4
commutator, cosocle, hypocenterSL(2,𝔽3)C24
FittingC2×2−1+4D3
hypercenterC2×D4S4
LCSG ▷ SL(2,𝔽3)
UCSC1 ◁ C2 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
1. C2×2−1+4⋊C3C2semidirect
2. #5927C2semidirect
3. #5926C2semidirect
4. #5923C2semidirect
5. C2×#983C2semidirect
8. 2−1+4⋊C3C22semidirect
9. C2×#41⋊C3C22semidirect
10. C22×SL(2,𝔽3)C22semidirect
11. #986C22semidirect
12. #985C22semidirect
13. #984C22semidirect
14. #983C22semidirect
15. #983C22semidirect
16. C2×GL(2,𝔽3)C22semidirect
17. C2×#226C22semidirect
19. C2×2−1+4D3semidirect
42. 2−1+4C2×D3semidirect
51. C2×#41C2×D3semidirect
53. C22×Q8C2×D3semidirect

Maximal subgroups

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

subgroupindex
C2×2−1+4⋊C32
#59272
#59262
#59232
C2×#9832
#35863
#10044