Finite Group Databasebeta

← #80985

Subgroups — #80985

Order 384.

420 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 420 conjugacy classes of subgroups — too many to draw. Only its 117 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#5555C211yesmaximal, characteristic, complemented
3C3×#474C211yesmaximal, characteristic, complemented
4#5093C211yesmaximal, characteristic, complemented
5C4×#213C211yesmaximal, characteristic, complemented
6#4886C212yesmaximal, complemented
7#4884C212yesmaximal, complemented
8#4841C211yesmaximal, characteristic, complemented
9#4838C211yesmaximal, characteristic, complemented
10#4781C212yesmaximal, complemented
11#4771C212yesmaximal, complemented
12#3330—31nomaximal, Sylow
13C2×#239—21no—
14C2×#237C2212yes—
15C3×C4⋊Q8C2211yescharacteristic
16C3×C4×D4C2211yescharacteristic
17#946—21no—
18C2×#213C2211yescharacteristic
19#930—21no—
20C2×C4⋊Dic3—21no—
21C2×#204C2212yes—
22C2×ΓL(1,𝔽25)C2212yes—
23#897C2212yes—
24#895C2212yes—
25#894—22no—
26D3×C4⋊C4C2212yes—
27#886—22no—
28C4×D12C2211yescharacteristic
29C4×D12C2211yescharacteristic
30#869C2211yescharacteristic
31C3×#107C2212yes—
32C3×C4⋊C8C2211yescharacteristic
33C3×#102C2212yes—
34#832C2211yescharacteristic
35(C2×D3)⋊C8C2212yes—
36#816C2212yes—
37#810C2212yes—
38#810C2212yes—
39#809C2211yescharacteristic
40#807C2211yescharacteristic
41C4×C3⋊C8C2211yescharacteristic
42#547—31no—
43#547—31no—
44#487—31no—
45#484—32no—
46#474D311yescharacteristic, complemented
47#468—32no—
48#461—31no—
49#449—32no—
50C4×D8—31no—
51#436—31no—
52#417—32no—
53C2×C3×Q8—21no—
54C2×C3×D4C2311yescharacteristic
55C22×C3×C4—21no—
56C2×#67—22no—
57C22×Dic3—22no—
58#239—44no—
59#237—24no—
60#237—42no—
61C2×D12—22no—
62C2×D12D411yescharacteristic
63C2×D12C2311yescharacteristic
64C2×C4×D3—22no—
65C2×C4×D3—21no—
66C2×C4×D3C2312yes—
67C2×Dic6C2312yes—
68C2×C3×C8C2312yes—
69C3×C4⋊C4—21no—
70C3×C4⋊C4C2311yescharacteristic
71C3×C4⋊C4C2312yes—
72C3×C22⋊C4—21no—
73C3×C42C2311yescharacteristic
74#213—22no—
75(C2×D3)⋊C4—22no—
76(C2×D3)⋊C4—21no—
77(C2×D3)⋊C4—22no—
78(C2×D3)⋊C4D412yes—
79C4⋊Dic3—21no—
80C4⋊Dic3—21no—
81C4⋊Dic3—22no—
82C4⋊Dic3D411yescharacteristic
83C4⋊Dic3C2311yescharacteristic
84Dic3⋊C4—22no—
85C2×C3⋊C8C2311yescharacteristic
86C2×C3⋊C8C2311yescharacteristic
87#204—42no—
88ΓL(1,𝔽25)—42no—
89C2×#41—32no—
90C2×#41—61no—
91C2×SD16—32no—
92C2×D8—31no—
93C2×M16—32no—
94C4⋊Q8—31no—
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×D4C2×D311yescharacteristic
105C2×C4⋊C4—32no—
106C2×C4⋊C4—61no—
107#107—31no—
108#107C2×D312yes—
109C4⋊C8C2×D311yescharacteristic
110#103—32no—
111#103—32no—
112#103—32no—
113#102—31no—
114#102—31no—
115#102C2×D312yes—
116C22⋊C8—32no—
117C4×C8—31no—
118C23×C3—21no—
119C22×D3—21no—
120C22×D3C2×D412yes—
121C3×Q8—42no—
122C3×D4—21no—
123C3×D4#4112yescomplemented
124C2×C3×C4—22no—
125C2×C3×C4—21no—
126C2×C3×C4—42no—
127C2×C3×C4C2×D411yescharacteristic
128C2×C3×C4C2×D411yescharacteristic
129C2×C3×C4C2411yescharacteristic
130#67—44no—
131C2×Dic3—21no—
132C2×Dic3—22no—
133C2×Dic3—44no—
134C2×Dic3C2×D412yes—
135D12—22no—
136D12—21no—
137D12—21no—
138D12—44no—
139C4×D3—22no—
140C4×D3—22no—
141C4×D3—44no—
142C4×D3—42no—
143Dic6—22no—
144Dic6—22no—
145C3×C8—22no—
146C3⋊C8—21no—
147C3⋊C8—21no—
148#41—64no—
149#41—122no—
150#41—124no—
151C2×Q8—21no—
152C2×Q8—32no—
153C2×D4—31no—
154C2×D4—31no—
155C2×D4—62no—
156C2×D4—62no—
157C2×D4C22×D311yescharacteristic
158C22×C4—21no—
159C22×C4—32no—
160C22×C4—62no—
161C22×C4—62no—
162C22×C4—61no—
163SD16—122no—
164D8—62no—
165M16—122no—
166C2×C8—31no—
167C2×C8—31no—
168C2×C8C22×D312yes—
169C4⋊C4—21no—
170C4⋊C4—31no—
171C4⋊C4—31no—
172C4⋊C4—62no—
173C4⋊C4—62no—
174C4⋊C4—61no—
175C4⋊C4—61no—
176C4⋊C4C22×D311yescharacteristic
177C4⋊C4C22×D312yes—
178C22⋊C4—21no—
179C22⋊C4—32no—
180C22⋊C4—62no—
181C22⋊C4—62no—
182C22⋊C4—61no—
183C42C22×D311yescharacteristic
184C22×C3—22no—
185C22×C3—41no—
186C22×C3—41no—
187C22×C3C22×D411yescharacteristic
188C2×D3—22no—
189C2×D3—22no—
190C2×D3—22no—
191C2×D3—41no—
192C2×D3—41no—
193C2×D3—41no—
194C3×C4—21no—
195C3×C4—42no—
196C3×C4—41no—
197C3×C4AGL(1,ℤ/8)12yes—
198C3×C4C2×#4111yescharacteristic
199C3×C4H(5,𝔽2)11yescharacteristic
200Dic3—22no—
201Dic3—42no—
202Dic3—41no—
203C23—21no—
204C23—32no—
205C23—61no—
206Q8—42no—
207Q8—62no—
208Q8—62no—
209D4—21no—
210D4—62no—
211D4—61no—
212D4—61no—
213D4—124no—
214D4—124no—
215D4#23912yescomplemented
216C2×C4—21no—
217C2×C4—22no—
218C2×C4—32no—
219C2×C4—42no—
220C2×C4—62no—
221C2×C4—62no—
222C2×C4—61no—
223C2×C4—62no—
224C2×C4—124no—
225C2×C4—124no—
226C2×C4—122no—
227C2×C4D3×D411yescharacteristic
228C2×C4D3×D411yescharacteristic
229C2×C4C23×D311yescharacteristic
230C8—22no—
231C8—61no—
232C8—61no—
233C2×C3—42no—
234C2×C3#54611yescharacteristic
235C2×C3C2×AGL(1,ℤ/8)11yescharacteristic
236C2×C3#57811yescharacteristic
237D3—42no—
238D3—81no—
239C22—22no—
240C22—41no—
241C22—41no—
242C22—62no—
243C22—62no—
244C22—62no—
245C22—121no—
246C22—121no—
247C22—121no—
248C22C2×D3×D411yescharacteristic
249C4—21no—
250C4—41no—
251C4—42no—
252C4—62no—
253C4—121no—
254C4—122no—
255C4#91112yes—
256C4C2×#23911yescharacteristic
257C4#100911yescharacteristic
258C3#333011yescharacteristic, complemented, Sylow
259C2—42no—
260C2—122no—
261C2—241no—
262C2#563011yescharacteristic
263C2C2×#91111yescharacteristic
264C2#577911yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#474C2
socleC22×C3C22×D4
hypercenter#474D3
hypocenterC3#3330
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ #474

Complements

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

NH ≅ G/Ntype
1. #5726C2semidirect
2. #5555C2semidirect
3. C3×#474C2semidirect
4. #5093C2semidirect
5. C4×#213C2semidirect
6. #4886C2semidirect
7. #4884C2semidirect
8. #4841C2semidirect
9. #4838C2semidirect
10. #4781C2semidirect
11. #4771C2semidirect
46. #474D3semidirect
123. C3×D4#41semidirect
215. D4#239semidirect
258. C3#3330semidirect

Maximal subgroups

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

subgroupindex
#57262
#55552
C3×#4742
#50932
C4×#2132
#48862
#48842
#48412
#48382
#47812
#47712
#33303