Finite Group Databasebeta

← #82736

Subgroups — #82736

Order 384.

468 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 468 conjugacy classes of subgroups — too many to draw. Only its 137 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×C3×C22⋊Q8C211yesmaximal, characteristic, complemented
2C2×#930C211yesmaximal, characteristic
3#5227C212yesmaximal, complemented
4C2×#830C211yesmaximal, characteristic, complemented
5#5048C214yesmaximal, complemented
6#5046C212yesmaximal, complemented
7#5043C212yesmaximal, complemented
8#4968C212yesmaximal, complemented
9#3187—31nomaximal, Sylow
10C22×C3×Q8C2211yescharacteristic, complemented
11C23×C3×C4C2211yescharacteristic
12C22×#67—21no—
13C22×D12C2211yescharacteristic
14C3×C22⋊Q8—21no—
15C3×C22⋊Q8—21no—
16C3×C22⋊Q8C2212yes—
17C3×C22⋊Q8C2212yescomplemented
18C2×C3×C4⋊C4—21no—
19C2×C3×C4⋊C4C2211yescharacteristic, complemented
20C2×C3×C22⋊C4—21no—
21C2×#215C2214yescomplemented
22#930—21no—
23#930—21no—
24#930C2212yes—
25#930C2212yes—
26C2×(C2×D3)⋊C4—21no—
27C2×C4⋊Dic3C2211yescharacteristic
28C22×C3⋊C8C2212yescomplemented
29#835C2214yescomplemented
30#830C2212yes—
31#830C2212yescomplemented
32#809C2214yescomplemented
33#808C2214yescomplemented
34C2×C22⋊Q8D311yescharacteristic, complemented
35C2×#121—31no—
36#483—32no—
37#481—32no—
38#465—34no—
39#450—32no—
40#448—32no—
41C2×C22⋊C8—31no—
42C24×C3D411yescharacteristic
43C23×D3—21no—
44C2×C3×Q8—21no—
45C2×C3×Q8—42no—
46C2×C3×Q8C2312yes—
47C22×C3×C4—21no—
48C22×C3×C4—21no—
49C22×C3×C4—42no—
50C22×C3×C4D411yescharacteristic
51C22×C3×C4D411yescharacteristic
52C22×C3×C4D411yescharacteristic
53C22×C3×C4C2311yescharacteristic
54C22×C3×C4C2311yescharacteristic
55C22×C3×C4C2311yescharacteristic
56C2×#67—22no—
57C2×#67—22no—
58C2×#67—44no—
59C22×Dic3—21no—
60C2×D12—21no—
61C2×D12—42no—
62C2×D12C2312yes—
63C3×C4⋊C4—22no—
64C3×C4⋊C4—22no—
65C3×C4⋊C4—21no—
66C3×C4⋊C4C2312yes—
67C3×C22⋊C4—22no—
68C3×C22⋊C4—22no—
69#215—44no—
70(C2×D3)⋊C4—22no—
71(C2×D3)⋊C4—22no—
72C4⋊Dic3—21no—
73C4⋊Dic3C2312yes—
74C2×C3⋊C8—42no—
75C2×C3⋊C8C2314yes—
76C22×Q8C2×D311yescharacteristic, complemented
77C22×D4—31no—
78C22×D4—61no—
79C23×C4C2×D311yescharacteristic
80C2×SD16—34no—
81C22×C8—32no—
82C22⋊Q8—21no—
83C22⋊Q8—21no—
84C22⋊Q8C2×D312yes—
85C22⋊Q8C2×D312yescomplemented
86#121—32no—
87#121—32no—
88#121—61no—
89#121—61no—
90C2×C4⋊C4—21no—
91C2×C4⋊C4—31no—
92C2×C4⋊C4C2×D311yescharacteristic, complemented
93C2×C22⋊C4—21no—
94C2×C22⋊C4—61no—
95#106—34no—
96#103—34no—
97#102—34no—
98C22⋊C8—32no—
99C22⋊C8—32no—
100C23×C3—41no—
101C23×C3—41no—
102C23×C3SD1614yescomplemented
103C23×C3C2×D411yescharacteristic
104C23×C3C2×D411yescharacteristic
105C23×C3C2×D411yescharacteristic
106C22×D3—22no—
107C22×D3—42no—
108C22×D3—42no—
109C22×D3—42no—
110C3×Q8—24no—
111C3×Q8—42no—
112C2×C3×C4—22no—
113C2×C3×C4—22no—
114C2×C3×C4—22no—
115C2×C3×C4—22no—
116C2×C3×C4—22no—
117C2×C3×C4—41no—
118C2×C3×C4—41no—
119C2×C3×C4—41no—
120C2×C3×C4—42no—
121C2×C3×C4—42no—
122C2×C3×C4C2×D411yescharacteristic
123C2×C3×C4C2×D411yescharacteristic
124C2×C3×C4C2×D411yescharacteristic
125C2×C3×C4C2411yescharacteristic
126#67—44no—
127#67—44no—
128C2×Dic3—22no—
129C2×Dic3—42no—
130D12—24no—
131D12—42no—
132C3⋊C8—24no—
133C24—61no—
134C24#6711yescharacteristic
135C2×Q8—21no—
136C2×Q8—42no—
137C2×Q8C22×D312yes—
138C2×D4—32no—
139C2×D4—62no—
140C2×D4—62no—
141C2×D4—61no—
142C2×D4—122no—
143C2×D4—124no—
144C22×C4—21no—
145C22×C4—21no—
146C22×C4—42no—
147C22×C4—61no—
148C22×C4#6711yescharacteristic
149C22×C4#6711yescharacteristic
150C22×C4#6711yescharacteristic
151C22×C4C22×D311yescharacteristic
152C22×C4C22×D311yescharacteristic
153C22×C4C22×D311yescharacteristic
154SD16—124no—
155C2×C8—34no—
156C2×C8—122no—
157C4⋊C4—22no—
158C4⋊C4—22no—
159C4⋊C4—21no—
160C4⋊C4—32no—
161C4⋊C4—61no—
162C4⋊C4C22×D312yes—
163C22⋊C4—22no—
164C22⋊C4—22no—
165C22⋊C4—62no—
166C22⋊C4—62no—
167C22×C3—21no—
168C22×C3—21no—
169C22×C3—21no—
170C22×C3—21no—
171C22×C3—21no—
172C22×C3—21no—
173C22×C3—44no—
174C22×C3C2×SD1612yes—
175C22×C3C2×SD1612yes—
176C22×C3C2×SD1612yes—
177C22×C3C22×D411yescharacteristic
178C2×D3—42no—
179C2×D3—42no—
180C2×D3—42no—
181C2×D3—84no—
182C3×C4—21no—
183C3×C4—21no—
184C3×C4—21no—
185C3×C4—42no—
186C3×C4—42no—
187C3×C4H(5,𝔽2)12yes—
188Dic3—42no—
189C23—41no—
190C23—41no—
191C23—62no—
192C23—122no—
193C23—122no—
194C23—122no—
195C23#21514yescomplemented
196C23C2×#6711yescharacteristic
197C23C2×#6711yescharacteristic
198C23C2×#6711yescharacteristic
199Q8—24no—
200Q8—42no—
201D4—64no—
202D4—122no—
203D4—124no—
204D4—124no—
205C2×C4—22no—
206C2×C4—22no—
207C2×C4—22no—
208C2×C4—22no—
209C2×C4—22no—
210C2×C4—41no—
211C2×C4—41no—
212C2×C4—41no—
213C2×C4—42no—
214C2×C4—42no—
215C2×C4—62no—
216C2×C4—122no—
217C2×C4C2×#6711yescharacteristic
218C2×C4C2×#6711yescharacteristic
219C2×C4C2×#6711yescharacteristic
220C2×C4C23×D311yescharacteristic
221C8—64no—
222C2×C3—22no—
223C2×C3—22no—
224C2×C3—22no—
225C2×C3#53411yescharacteristic
226C2×C3C22×SD1611yescharacteristic
227C2×C3#57511yescharacteristic
228D3—82no—
229C22—21no—
230C22—21no—
231C22—21no—
232C22—21no—
233C22—21no—
234C22—21no—
235C22—44no—
236C22—122no—
237C22—122no—
238C22—122no—
239C22—244no—
240C22C2×#21512yes—
241C22C2×#21512yes—
242C22C2×#21512yes—
243C22C22×#6711yescharacteristic
244C4—21no—
245C4—21no—
246C4—21no—
247C4—42no—
248C4—42no—
249C4—122no—
250C4#100412yes—
251C3#318711yescharacteristic, complemented, Sylow
252C2—22no—
253C2—22no—
254C2—22no—
255C2—242no—
256C2#580411yescharacteristic
257C2C22×#21511yescharacteristic
258C2#582211yescharacteristic
subgroupG/H
centerC22C22×#67
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC2×C3×C22⋊Q8C2
socleC22×C3C22×D4
prime radicalC2×#930C2
hypercenterC2×C22⋊Q8D3
hypocenterC3#3187
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. C2×C3×C22⋊Q8C2semidirect
3. #5227C2semidirect
4. C2×#830C2semidirect
5. #5048C2semidirect
6. #5046C2semidirect
7. #5043C2semidirect
8. #4968C2semidirect
10. C22×C3×Q8C22semidirect
17. C3×C22⋊Q8C22semidirect
19. C2×C3×C4⋊C4C22semidirect
21. C2×#215C22semidirect
28. C22×C3⋊C8C22semidirect
29. #835C22semidirect
31. #830C22semidirect
32. #809C22semidirect
33. #808C22semidirect
34. C2×C22⋊Q8D3semidirect
76. C22×Q8C2×D3semidirect
85. C22⋊Q8C2×D3semidirect
92. C2×C4⋊C4C2×D3semidirect
102. C23×C3SD16semidirect
195. C23#215semidirect
251. C3#3187semidirect

Maximal subgroups

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

subgroupindex
C2×C3×C22⋊Q82
C2×#9302
#52272
C2×#8302
#50482
#50462
#50432
#49682
#31873