Finite Group Databasebeta

← #81618

Subgroups — #81618

Order 384.

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

normal subgroups only

This group has 420 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#5726C211yesmaximal, characteristic, complemented
3C3×#487C211yesmaximal, characteristic, complemented
4#5087C211yesmaximal, characteristic, complemented
5#5083C211yesmaximal, characteristic, complemented
6#4812C212yesmaximal, complemented
7#4811C212yesmaximal, complemented
8#4810C212yesmaximal, complemented
9#4803C212yesmaximal, complemented
10C4×D24C211yesmaximal, characteristic, complemented
11#4688C211yesmaximal, characteristic, complemented
12#3330—31nomaximal, Sylow
13C2×#239—21no—
14C2×#239C2212yes—
15C3×C4⋊Q8C2211yescharacteristic
16C3×C4⋊Q8C2211yescharacteristic
17#946—21no—
18#946C2212yes—
19C2×#215C2212yes—
20C2×D24C2211yescharacteristic
21C2×ΓL(1,𝔽25)C2212yes—
22#895—21no—
23#895C2212yes—
24#894—22no—
25#894—22no—
26D3×C4⋊C4—21no—
27D3×C4⋊C4C2212yes—
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#807C2212yes—
41C4⋊(C3⋊C8)C2211yescharacteristic
42#547—31no—
43#547—31no—
44#487D311yescharacteristic, complemented
45#484—32no—
46#474—31no—
47#468—32no—
48#461—31no—
49#449—32no—
50C4×D8—31no—
51#436—31no—
52#417—32no—
53C2×C3×Q8—21no—
54C2×C3×Q8C2312yes—
55#239—24no—
56#239—42no—
57#239—44no—
58C2×D12—22no—
59C2×D12—22no—
60C2×D12D411yescharacteristic
61C2×D12C2311yescharacteristic
62C2×D12C2311yescharacteristic
63C2×C4×D3—22no—
64C2×C4×D3—21no—
65C2×C4×D3—21no—
66C2×C4×D3—22no—
67C2×C4×D3C2312yes—
68C2×C3×C8C2311yescharacteristic
69C2×C3×C8C2311yescharacteristic
70C3×C4⋊C4—21no—
71C3×C4⋊C4—21no—
72C3×C4⋊C4—21no—
73C3×C4⋊C4C2312yes—
74C3×C42C2311yescharacteristic
75#215—42no—
76(C2×D3)⋊C4—22no—
77(C2×D3)⋊C4—21no—
78(C2×D3)⋊C4—21no—
79(C2×D3)⋊C4—22no—
80(C2×D3)⋊C4D412yes—
81C4⋊Dic3D411yescharacteristic
82C4⋊Dic3C2311yescharacteristic
83C4⋊Dic3C2311yescharacteristic
84Dic3⋊C4—22no—
85Dic3⋊C4—22no—
86C2×C3⋊C8C2312yes—
87D24—22no—
88ΓL(1,𝔽25)—42no—
89C2×#41—32no—
90C2×#41—61no—
91C2×SD16—32no—
92C2×D8—31no—
93C2×M16—32no—
94C4⋊Q8C2×D311yescharacteristic
95C4⋊Q8C2×D311yescharacteristic
96#123—62no—
97#123—62no—
98C22⋊Q8—32no—
99C22⋊Q8—61no—
100#121—32no—
101#121—61no—
102C4×D4—31no—
103C4×D4—31no—
104C4×D4—31no—
105C2×C4⋊C4—32no—
106C2×C4⋊C4—61no—
107#107—31no—
108#107—32no—
109C4⋊C8—31no—
110#103—32no—
111#103C2×D312yes—
112#103C2×D312yes—
113#102—31no—
114#102—31no—
115#102—32no—
116C22⋊C8—32no—
117C4×C8C2×D311yescharacteristic
118C22×D3—21no—
119C22×D3—21no—
120C22×D3C2×D412yes—
121C3×Q8—22no—
122C3×Q8—22no—
123C3×Q8—42no—
124C2×C3×C4—22no—
125C2×C3×C4—22no—
126C2×C3×C4C2×D411yescharacteristic
127C2×C3×C4C2×D411yescharacteristic
128C2×C3×C4C2411yescharacteristic
129C2×Dic3—21no—
130C2×Dic3—21no—
131C2×Dic3C2×D412yes—
132D12—22no—
133D12—21no—
134D12—21no—
135D12—21no—
136D12—44no—
137D12—44no—
138D12#4112yescomplemented
139C4×D3—22no—
140C4×D3—22no—
141C4×D3—44no—
142C4×D3—44no—
143C4×D3—42no—
144C4×D3—42no—
145C3×C8—21no—
146C3×C8—21no—
147C3⋊C8—22no—
148#41—64no—
149#41—122no—
150#41—124no—
151C2×Q8—21no—
152C2×Q8C22×D312yes—
153C2×D4—31no—
154C2×D4—31no—
155C2×D4—31no—
156C2×D4—62no—
157C2×D4—62no—
158C22×C4—32no—
159C22×C4—62no—
160C22×C4—62no—
161C22×C4—61no—
162C22×C4—61no—
163SD16—122no—
164D8—62no—
165M16—122no—
166C2×C8—32no—
167C2×C8C22×D311yescharacteristic
168C2×C8C22×D311yescharacteristic
169C4⋊C4—21no—
170C4⋊C4—21no—
171C4⋊C4—21no—
172C4⋊C4—31no—
173C4⋊C4—31no—
174C4⋊C4—31no—
175C4⋊C4—62no—
176C4⋊C4—62no—
177C4⋊C4C22×D312yes—
178C22⋊C4—32no—
179C22⋊C4—62no—
180C22⋊C4—62no—
181C22⋊C4—61no—
182C22⋊C4—61no—
183C42C22×D311yescharacteristic
184C22×C3C22×D411yescharacteristic
185C2×D3—22no—
186C2×D3—22no—
187C2×D3—22no—
188C2×D3—22no—
189C2×D3—41no—
190C2×D3—41no—
191C2×D3—41no—
192C2×D3—41no—
193C2×D3—41no—
194C3×C4—21no—
195C3×C4—42no—
196C3×C4—42no—
197C3×C4AGL(1,ℤ/8)12yes—
198C3×C4C2×#4111yescharacteristic
199C3×C4H(5,𝔽2)11yescharacteristic
200Dic3—22no—
201Dic3—41no—
202Dic3—41no—
203C23—32no—
204C23—61no—
205C23—61no—
206Q8—22no—
207Q8—22no—
208Q8—42no—
209D4—32no—
210D4—62no—
211D4—61no—
212D4—61no—
213D4—61no—
214D4—124no—
215D4—124no—
216C2×C4—22no—
217C2×C4—22no—
218C2×C4—32no—
219C2×C4—62no—
220C2×C4—62no—
221C2×C4—61no—
222C2×C4—61no—
223C2×C4—124no—
224C2×C4—124no—
225C2×C4—122no—
226C2×C4—122no—
227C2×C4D3×D411yescharacteristic
228C2×C4D3×D411yescharacteristic
229C2×C4C23×D311yescharacteristic
230C8—21no—
231C8—21no—
232C8—62no—
233C2×C3#54611yescharacteristic
234C2×C3C2×AGL(1,ℤ/8)11yescharacteristic
235C2×C3#57811yescharacteristic
236D3—42no—
237D3—42no—
238D3—81no—
239C22—62no—
240C22—62no—
241C22—62no—
242C22—62no—
243C22—121no—
244C22—121no—
245C22—121no—
246C22—121no—
247C22—121no—
248C22C2×D3×D411yescharacteristic
249C4—21no—
250C4—42no—
251C4—42no—
252C4—62no—
253C4—121no—
254C4—121no—
255C4#91412yes—
256C4D3×#4111yescharacteristic
257C4#100911yescharacteristic
258C3#333011yescharacteristic, complemented, Sylow
259C2—122no—
260C2—122no—
261C2—241no—
262C2#567611yescharacteristic
263C2C2×#91411yescharacteristic
264C2#576611yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#487C2
socleC22×C3C22×D4
hypercenter#487D3
hypocenterC3#3330
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. #5726C2semidirect
2. #5726C2semidirect
3. C3×#487C2semidirect
4. #5087C2semidirect
5. #5083C2semidirect
6. #4812C2semidirect
7. #4811C2semidirect
8. #4810C2semidirect
9. #4803C2semidirect
10. C4×D24C2semidirect
11. #4688C2semidirect
44. #487D3semidirect
138. D12#41semidirect
258. C3#3330semidirect

Maximal subgroups

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

subgroupindex
#57262
C3×#4872
#50872
#50832
#48122
#48112
#48102
#48032
C4×D242
#46882
#33303