Finite Group Databasebeta

← #80451

Subgroups — #80451

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#5726C211yesmaximal, characteristic, complemented
2#5563C211yesmaximal, characteristic, complemented
3C3×#461C211yesmaximal, characteristic, complemented
4#5172C212yesmaximal, complemented
5#5171C212yesmaximal, complemented
6#5079C211yesmaximal, characteristic, complemented
7C4×#213C211yesmaximal, characteristic, complemented
8#4842C211yesmaximal, characteristic, complemented
9#4834C211yesmaximal, characteristic, complemented
10#4815C212yesmaximal, complemented
11#4804C212yesmaximal, complemented
12#3303—31nomaximal, Sylow
13C2×#239C2212yes—
14C2×C3×SD16C2212yes—
15C3×C4⋊Q8C2211yescharacteristic
16C3×C4×D4C2211yescharacteristic
17#946C2212yes—
18#938C2212yes—
19C2×#213C2211yescharacteristic
20#930—21no—
21#926—21no—
22C2×D3×C8C2212yes—
23#895—21no—
24#894—22no—
25#892C2212yes—
26D3×C4⋊C4—21no—
27#885—22no—
28C4×D12C2211yescharacteristic
29C4×D12C2211yescharacteristic
30#869C2211yescharacteristic
31C3×C4⋊C8C2211yescharacteristic
32C3×#103C2212yes—
33#835C2212yes—
34#835C2212yes—
35#832C2211yescharacteristic
36#821C2212yes—
37(C2×D3)⋊C8C2212yes—
38#817C2212yes—
39#809C2211yescharacteristic
40#807C2211yescharacteristic
41C4×C3⋊C8C2211yescharacteristic
42#555—31no—
43#547—31no—
44#487—31no—
45#485—32no—
46#476—31no—
47#465—32no—
48#461D311yescharacteristic, complemented
49#449—32no—
50C4×D8—31no—
51C8×D4—31no—
52#416—32no—
53C2×C3×Q8C2312yes—
54C2×C3×D4C2311yescharacteristic
55C22×C3×C4—21no—
56C2×#67—22no—
57#239—24no—
58#239—42no—
59C2×D12—22no—
60C2×D12D411yescharacteristic
61C2×D12C2311yescharacteristic
62C2×C4×D3—21no—
63C2×C4×D3—22no—
64C2×C4×D3C2312yes—
65C2×Dic6—21no—
66C3×SD16—42no—
67C2×C3×C8C2312yes—
68C3×C4⋊C4—21no—
69C3×C4⋊C4—21no—
70C3×C4⋊C4C2311yescharacteristic
71C3×C22⋊C4—21no—
72C3×C42C2311yescharacteristic
73#217—22no—
74#213—22no—
75(C2×D3)⋊C4—22no—
76(C2×D3)⋊C4—22no—
77(C2×D3)⋊C4—21no—
78(C2×D3)⋊C4D412yes—
79C4⋊Dic3—21no—
80C4⋊Dic3D411yescharacteristic
81C4⋊Dic3C2311yescharacteristic
82C4⋊Dic3C2312yes—
83Dic3⋊C4—22no—
84C4×Dic3—22no—
85C2×C3⋊C8C2311yescharacteristic
86C2×C3⋊C8C2311yescharacteristic
87D3×C8—42no—
88C2×#41—32no—
89C2×SD16C2×D312yes—
90C2×D8—31no—
91C22×C8—32no—
92C4⋊Q8—31no—
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×D4C2×D311yescharacteristic
103#117—32no—
104#117—61no—
105C2×C4⋊C4—61no—
106#107—31no—
107#106—32no—
108C4⋊C8C2×D311yescharacteristic
109#103—32no—
110#103—32no—
111#103C2×D312yes—
112#102—31no—
113#102—32no—
114#102—31no—
115C22⋊C8—32no—
116C4×C8—31no—
117C23×C3—21no—
118C22×D3—21no—
119C22×D3C2×D412yes—
120C3×Q8—22no—
121C3×Q8—22no—
122C3×D4—21no—
123C3×D4—21no—
124C2×C3×C4—21no—
125C2×C3×C4—22no—
126C2×C3×C4—42no—
127C2×C3×C4C2×D411yescharacteristic
128C2×C3×C4C2×D411yescharacteristic
129C2×C3×C4C2411yescharacteristic
130#67—82no—
131C2×Dic3—22no—
132C2×Dic3—21no—
133C2×Dic3C2×D412yes—
134D12—22no—
135D12—21no—
136D12—44no—
137D12#4112yes—
138C4×D3—22no—
139C4×D3—22no—
140C4×D3—44no—
141C4×D3—42no—
142Dic6—81no—
143C3×C8—22no—
144C3⋊C8—21no—
145C3⋊C8—21no—
146#41—64no—
147#41—122no—
148C2×Q8—61no—
149C2×Q8C22×D312yes—
150C2×D4—31no—
151C2×D4—31no—
152C2×D4—62no—
153C2×D4—62no—
154C2×D4C22×D311yescharacteristic
155C22×C4—21no—
156C22×C4—32no—
157C22×C4—62no—
158C22×C4—61no—
159SD16—42no—
160D8—62no—
161C2×C8—31no—
162C2×C8—31no—
163C2×C8—122no—
164C2×C8C22×D312yes—
165C4⋊C4—21no—
166C4⋊C4—21no—
167C4⋊C4—31no—
168C4⋊C4—32no—
169C4⋊C4—31no—
170C4⋊C4—62no—
171C4⋊C4—61no—
172C4⋊C4C22×D311yescharacteristic
173C22⋊C4—21no—
174C22⋊C4—32no—
175C22⋊C4—62no—
176C22⋊C4—62no—
177C22⋊C4—62no—
178C22⋊C4—61no—
179C42—62no—
180C42C22×D311yescharacteristic
181C22×C3—41no—
182C22×C3—41no—
183C22×C3—41no—
184C22×C3C22×D411yescharacteristic
185C2×D3—22no—
186C2×D3—22no—
187C2×D3—22no—
188C2×D3—22no—
189C2×D3—41no—
190C2×D3—41no—
191C3×C4—21no—
192C3×C4—42no—
193C3×C4—41no—
194C3×C4#13512yes—
195C3×C4C2×#4111yescharacteristic
196C3×C4H(5,𝔽2)11yescharacteristic
197Dic3—22no—
198Dic3—41no—
199Dic3—42no—
200C23—21no—
201C23—32no—
202C23—61no—
203Q8—22no—
204Q8—22no—
205Q8—241no—
206D4—21no—
207D4—21no—
208D4—32no—
209D4—62no—
210D4—61no—
211D4—124no—
212D4—242no—
213C2×C4—21no—
214C2×C4—22no—
215C2×C4—32no—
216C2×C4—42no—
217C2×C4—62no—
218C2×C4—62no—
219C2×C4—62no—
220C2×C4—61no—
221C2×C4—124no—
222C2×C4—122no—
223C2×C4D3×D411yescharacteristic
224C2×C4D3×D411yescharacteristic
225C2×C4C23×D311yescharacteristic
226C8—22no—
227C8—61no—
228C8—61no—
229C2×C3—81no—
230C2×C3#54611yescharacteristic
231C2×C3C2×#13511yescharacteristic
232C2×C3#57711yescharacteristic
233D3—42no—
234D3—42no—
235C22—41no—
236C22—41no—
237C22—41no—
238C22—62no—
239C22—62no—
240C22—62no—
241C22—62no—
242C22—121no—
243C22—121no—
244C22C2×D3×D411yescharacteristic
245C4—21no—
246C4—41no—
247C4—42no—
248C4—62no—
249C4—121no—
250C4—122no—
251C4#91612yes—
252C4#100411yescharacteristic
253C4D3×#4111yescharacteristic
254C3#330311yescharacteristic, complemented, Sylow
255C2—81no—
256C2—122no—
257C2—122no—
258C2#561211yescharacteristic
259C2C2×#91611yescharacteristic
260C2#577811yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#461C2
socleC22×C3C22×D4
hypercenter#461D3
hypocenterC3#3303
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ #461

Complements

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

NH ≅ G/Ntype
1. #5726C2semidirect
2. #5563C2semidirect
3. C3×#461C2semidirect
4. #5172C2semidirect
5. #5171C2semidirect
6. #5079C2semidirect
7. C4×#213C2semidirect
8. #4842C2semidirect
9. #4834C2semidirect
10. #4815C2semidirect
11. #4804C2semidirect
48. #461D3semidirect
254. C3#3303semidirect

Maximal subgroups

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

subgroupindex
#57262
#55632
C3×#4612
#51722
#51712
#50792
C4×#2132
#48422
#48342
#48152
#48042
#33033