Finite Group Databasebeta

← #80455

Subgroups — #80455

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#5551C211yesmaximal, characteristic, complemented
3C3×#461C211yesmaximal, characteristic, complemented
4#5174C212yesmaximal, complemented
5#5172C212yesmaximal, complemented
6#5079C211yesmaximal, characteristic, complemented
7C4×#213C211yesmaximal, characteristic, complemented
8#4846C211yesmaximal, characteristic, complemented
9#4838C211yesmaximal, characteristic, complemented
10#4811C212yesmaximal, complemented
11#4803C212yesmaximal, complemented
12#3330—31nomaximal, Sylow
13C2×#239C2212yes—
14C2×#235—21no—
15C2×C3×SD16C2212yes—
16C3×C4⋊Q8C2211yescharacteristic
17C3×C4×D4C2211yescharacteristic
18#946C2212yes—
19#938C2212yes—
20C2×#213C2211yescharacteristic
21#924—21no—
22C2×ΓL(1,𝔽25)C2212yes—
23#895—21no—
24#894—22no—
25D3×C4⋊C4—21no—
26D3×C4⋊C4C2212yes—
27#883—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#818C2212yes—
39#809C2211yescharacteristic
40#807C2211yescharacteristic
41C4×C3⋊C8C2211yescharacteristic
42#547—31no—
43#547—31no—
44#487—31no—
45#484—32no—
46#474—31no—
47#468—32no—
48#461D311yescharacteristic, complemented
49#449—32no—
50C4×D8—31no—
51#436—31no—
52#417—32no—
53C2×C3×Q8C2312yes—
54C2×C3×D4C2311yescharacteristic
55C22×C3×C4—21no—
56C2×#67—22no—
57#239—24no—
58#239—42no—
59#235—44no—
60C2×D12—22no—
61C2×D12D411yescharacteristic
62C2×D12C2311yescharacteristic
63C2×C4×D3—22no—
64C2×C4×D3—21no—
65C2×C4×D3—22no—
66C2×C4×D3C2312yes—
67C2×Dic6—21no—
68C3×SD16—42no—
69C2×C3×C8C2312yes—
70C3×C4⋊C4—21no—
71C3×C4⋊C4—21no—
72C3×C4⋊C4C2311yescharacteristic
73C3×C22⋊C4—21no—
74C3×C42C2311yescharacteristic
75#217—22no—
76#213—22no—
77(C2×D3)⋊C4—22no—
78(C2×D3)⋊C4—21no—
79(C2×D3)⋊C4D412yes—
80C4⋊Dic3—21no—
81C4⋊Dic3D411yescharacteristic
82C4⋊Dic3C2311yescharacteristic
83C4⋊Dic3C2312yes—
84Dic3⋊C4—22no—
85Dic3⋊C4—22no—
86C2×C3⋊C8C2311yescharacteristic
87C2×C3⋊C8C2311yescharacteristic
88ΓL(1,𝔽25)—42no—
89C2×#41—32no—
90C2×#41—61no—
91C2×SD16C2×D312yes—
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—32no—
108#107—31no—
109C4⋊C8C2×D311yescharacteristic
110#103—32no—
111#103—32no—
112#103C2×D312yes—
113#102—31no—
114#102—32no—
115#102—31no—
116C22⋊C8—32no—
117C4×C8—31no—
118C23×C3—21no—
119C22×D3—21no—
120C22×D3C2×D412yes—
121C3×Q8—22no—
122C3×Q8—22no—
123C3×D4—21no—
124C3×D4—21no—
125C2×C3×C4—21no—
126C2×C3×C4—22no—
127C2×C3×C4—42no—
128C2×C3×C4C2×D411yescharacteristic
129C2×C3×C4C2×D411yescharacteristic
130C2×C3×C4C2411yescharacteristic
131#67—44no—
132C2×Dic3—21no—
133C2×Dic3—22no—
134C2×Dic3C2×D412yes—
135D12—22no—
136D12—21no—
137D12—44no—
138D12#4112yescomplemented
139C4×D3—22no—
140C4×D3—22no—
141C4×D3—44no—
142C4×D3—44no—
143C4×D3—42no—
144Dic6—42no—
145C3×C8—22no—
146C3⋊C8—21no—
147C3⋊C8—21no—
148#41—64no—
149#41—124no—
150#41—122no—
151C2×Q8—61no—
152C2×Q8C22×D312yes—
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—42no—
164D8—62no—
165M16—122no—
166C2×C8—31no—
167C2×C8—31no—
168C2×C8C22×D312yes—
169C4⋊C4—21no—
170C4⋊C4—21no—
171C4⋊C4—31no—
172C4⋊C4—32no—
173C4⋊C4—31no—
174C4⋊C4—62no—
175C4⋊C4—62no—
176C4⋊C4—61no—
177C4⋊C4C22×D311yescharacteristic
178C22⋊C4—21no—
179C22⋊C4—32no—
180C22⋊C4—62no—
181C22⋊C4—62no—
182C22⋊C4—61no—
183C42C22×D311yescharacteristic
184C22×C3—41no—
185C22×C3—41no—
186C22×C3—41no—
187C22×C3C22×D411yescharacteristic
188C2×D3—22no—
189C2×D3—22no—
190C2×D3—22no—
191C2×D3—22no—
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—22no—
207Q8—22no—
208Q8—122no—
209D4—21no—
210D4—21no—
211D4—32no—
212D4—62no—
213D4—61no—
214D4—124no—
215D4—124no—
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—81no—
234C2×C3#54611yescharacteristic
235C2×C3C2×AGL(1,ℤ/8)11yescharacteristic
236C2×C3#57811yescharacteristic
237D3—42no—
238D3—42no—
239C22—41no—
240C22—41no—
241C22—41no—
242C22—62no—
243C22—62no—
244C22—62no—
245C22—62no—
246C22—121no—
247C22—121no—
248C22C2×D3×D411yescharacteristic
249C4—21no—
250C4—41no—
251C4—42no—
252C4—62no—
253C4—122no—
254C4—121no—
255C4#91412yes—
256C4#100411yescharacteristic
257C4D3×#4111yescharacteristic
258C3#333011yescharacteristic, complemented, Sylow
259C2—81no—
260C2—122no—
261C2—122no—
262C2#561211yescharacteristic
263C2C2×#91411yescharacteristic
264C2#577911yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#461C2
socleC22×C3C22×D4
hypercenter#461D3
hypocenterC3#3330
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. #5551C2semidirect
3. C3×#461C2semidirect
4. #5174C2semidirect
5. #5172C2semidirect
6. #5079C2semidirect
7. C4×#213C2semidirect
8. #4846C2semidirect
9. #4838C2semidirect
10. #4811C2semidirect
11. #4803C2semidirect
48. #461D3semidirect
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
#55512
C3×#4612
#51742
#51722
#50792
C4×#2132
#48462
#48382
#48112
#48032
#33303