Finite Group Databasebeta

← #81615

Subgroups — #81615

Order 384.

398 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 398 conjugacy classes of subgroups — too many to draw. Only its 115 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#5734C211yesmaximal, characteristic, complemented
2#5726C211yesmaximal, characteristic, complemented
3C3×#487C211yesmaximal, characteristic, complemented
4#5084C211yesmaximal, characteristic, complemented
5#5083C211yesmaximal, characteristic, complemented
6#4814C212yesmaximal, complemented
7#4810C212yesmaximal, complemented
8#4807C212yesmaximal, complemented
9#4804C212yesmaximal, complemented
10C4×D24C211yesmaximal, characteristic, complemented
11C8×D12C211yesmaximal, characteristic, complemented
12#3303—31nomaximal, Sylow
13C2×#239C2212yes—
14C3×C4⋊Q8C2211yescharacteristic
15C3×C4⋊Q8C2211yescharacteristic
16#947—22no—
17#946C2212yes—
18C2×#215C2212yes—
19C2×D24C2211yescharacteristic
20C2×D3×C8C2212yes—
21#895—21no—
22#895—21no—
23#895C2212yes—
24#894—22no—
25#892—21no—
26#892C2212yes—
27D3×C4⋊C4—21no—
28C4×D12C2211yescharacteristic
29C4×D12C2211yescharacteristic
30C4×D12C2211yescharacteristic
31C3×#103C2212yes—
32C3×#103C2212yes—
33C3×C4×C8C2211yescharacteristic
34#835C2212yes—
35#821C2211yescharacteristic
36#821C2211yescharacteristic
37(C2×D3)⋊C8C2212yes—
38#818C2211yescharacteristic
39#809C2212yes—
40#808C2212yes—
41C4⋊(C3⋊C8)C2211yescharacteristic
42#555—31no—
43#547—31no—
44#487D311yescharacteristic, complemented
45#485—32no—
46#476—31no—
47#465—32no—
48#461—31no—
49#449—32no—
50C4×D8—31no—
51C8×D4—31no—
52#416—32no—
53C2×C3×Q8—21no—
54C2×C3×Q8C2312yes—
55#239—24no—
56#239—42no—
57C2×D12—22no—
58C2×D12—22no—
59C2×D12D411yescharacteristic
60C2×D12C2311yescharacteristic
61C2×D12C2311yescharacteristic
62C2×C4×D3—21no—
63C2×C4×D3—21no—
64C2×C4×D3—22no—
65C2×C4×D3C2312yes—
66C2×C3×C8C2311yescharacteristic
67C2×C3×C8C2311yescharacteristic
68C3×C4⋊C4—21no—
69C3×C4⋊C4—21no—
70C3×C4⋊C4—21no—
71C3×C4⋊C4C2312yes—
72C3×C42C2311yescharacteristic
73#215—42no—
74(C2×D3)⋊C4—22no—
75(C2×D3)⋊C4—22no—
76(C2×D3)⋊C4—21no—
77(C2×D3)⋊C4—21no—
78(C2×D3)⋊C4—22no—
79(C2×D3)⋊C4D412yes—
80C4⋊Dic3D411yescharacteristic
81C4⋊Dic3C2311yescharacteristic
82C4⋊Dic3C2311yescharacteristic
83Dic3⋊C4—22no—
84C4×Dic3—22no—
85C2×C3⋊C8C2312yes—
86D24—22no—
87D3×C8—42no—
88C2×#41—32no—
89C2×SD16—32no—
90C2×D8—31no—
91C22×C8—32no—
92C4⋊Q8C2×D311yescharacteristic
93C4⋊Q8C2×D311yescharacteristic
94#124—62no—
95#123—62no—
96C22⋊Q8—32no—
97#121—32no—
98#121—61no—
99#121—61no—
100C4×D4—31no—
101C4×D4—31no—
102C4×D4—31no—
103#117—32no—
104#117—61no—
105C2×C4⋊C4—61no—
106#107—31no—
107#106—32no—
108C4⋊C8—31no—
109#103—32no—
110#103C2×D312yes—
111#103C2×D312yes—
112#102—31no—
113#102—31no—
114#102—32no—
115C22⋊C8—32no—
116C4×C8C2×D311yescharacteristic
117C22×D3—21no—
118C22×D3—21no—
119C22×D3C2×D412yes—
120C3×Q8—22no—
121C3×Q8—22no—
122C3×Q8—81no—
123C2×C3×C4—22no—
124C2×C3×C4—22no—
125C2×C3×C4C2×D411yescharacteristic
126C2×C3×C4C2×D411yescharacteristic
127C2×C3×C4C2411yescharacteristic
128C2×Dic3—21no—
129C2×Dic3—21no—
130C2×Dic3C2×D412yes—
131D12—22no—
132D12—21no—
133D12—21no—
134D12—21no—
135D12—44no—
136D12—82no—
137D12#4112yes—
138C4×D3—22no—
139C4×D3—22no—
140C4×D3—44no—
141C4×D3—42no—
142C4×D3—42no—
143C3×C8—21no—
144C3×C8—21no—
145C3⋊C8—22no—
146#41—64no—
147#41—122no—
148C2×Q8—21no—
149C2×Q8C22×D312yes—
150C2×D4—31no—
151C2×D4—31no—
152C2×D4—31no—
153C2×D4—62no—
154C2×D4—62no—
155C22×C4—32no—
156C22×C4—62no—
157C22×C4—61no—
158C22×C4—61no—
159SD16—122no—
160D8—62no—
161C2×C8—32no—
162C2×C8—122no—
163C2×C8C22×D311yescharacteristic
164C2×C8C22×D311yescharacteristic
165C4⋊C4—21no—
166C4⋊C4—21no—
167C4⋊C4—21no—
168C4⋊C4—31no—
169C4⋊C4—31no—
170C4⋊C4—31no—
171C4⋊C4—62no—
172C4⋊C4C22×D312yes—
173C22⋊C4—32no—
174C22⋊C4—62no—
175C22⋊C4—62no—
176C22⋊C4—62no—
177C22⋊C4—61no—
178C22⋊C4—61no—
179C42—62no—
180C42C22×D311yescharacteristic
181C22×C3C22×D411yescharacteristic
182C2×D3—22no—
183C2×D3—22no—
184C2×D3—22no—
185C2×D3—22no—
186C2×D3—41no—
187C2×D3—41no—
188C2×D3—41no—
189C2×D3—41no—
190C2×D3—41no—
191C3×C4—21no—
192C3×C4—42no—
193C3×C4—42no—
194C3×C4#13512yes—
195C3×C4C2×#4111yescharacteristic
196C3×C4H(5,𝔽2)11yescharacteristic
197Dic3—22no—
198Dic3—41no—
199Dic3—41no—
200C23—32no—
201C23—61no—
202C23—61no—
203Q8—22no—
204Q8—22no—
205Q8—81no—
206D4—32no—
207D4—62no—
208D4—61no—
209D4—61no—
210D4—61no—
211D4—124no—
212D4—242no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—32no—
216C2×C4—62no—
217C2×C4—62no—
218C2×C4—61no—
219C2×C4—61no—
220C2×C4—124no—
221C2×C4—122no—
222C2×C4—122no—
223C2×C4D3×D411yescharacteristic
224C2×C4D3×D411yescharacteristic
225C2×C4C23×D311yescharacteristic
226C8—21no—
227C8—21no—
228C8—62no—
229C2×C3#54611yescharacteristic
230C2×C3C2×#13511yescharacteristic
231C2×C3#57711yescharacteristic
232D3—42no—
233D3—42no—
234D3—81no—
235C22—62no—
236C22—62no—
237C22—62no—
238C22—62no—
239C22—121no—
240C22—121no—
241C22—121no—
242C22—121no—
243C22—121no—
244C22C2×D3×D411yescharacteristic
245C4—21no—
246C4—42no—
247C4—42no—
248C4—62no—
249C4—121no—
250C4—121no—
251C4#91912yes—
252C4D3×#4111yescharacteristic
253C4#100911yescharacteristic
254C3#330311yescharacteristic, complemented, Sylow
255C2—122no—
256C2—122no—
257C2—241no—
258C2#567611yescharacteristic
259C2#576211yescharacteristic
260C2C2×#91911yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#487C2
socleC22×C3C22×D4
hypercenter#487D3
hypocenterC3#3303
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ #487

Complements

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

NH ≅ G/Ntype
1. #5734C2semidirect
2. #5726C2semidirect
3. C3×#487C2semidirect
4. #5084C2semidirect
5. #5083C2semidirect
6. #4814C2semidirect
7. #4810C2semidirect
8. #4807C2semidirect
9. #4804C2semidirect
10. C4×D24C2semidirect
11. C8×D12C2semidirect
44. #487D3semidirect
254. C3#3303semidirect

Maximal subgroups

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

subgroupindex
#57342
#57262
C3×#4872
#50842
#50832
#48142
#48102
#48072
#48042
C4×D242
C8×D122
#33033