Finite Group Databasebeta

← #80353

Subgroups — #80353

Order 384.

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

normal subgroups only

This group has 408 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#5718C211yesmaximal, characteristic, complemented
2#5563C211yesmaximal, characteristic, complemented
3C3×#459C211yesmaximal, characteristic, complemented
4#5158C212yesmaximal, complemented
5#5157C212yesmaximal, complemented
6#5070C211yesmaximal, characteristic, complemented
7C4×#214C211yesmaximal, characteristic, complemented
8#4850C211yesmaximal, characteristic, complemented
9#4834C211yesmaximal, characteristic, complemented
10#4784C212yesmaximal, complemented
11#4772C212yesmaximal, complemented
12#3305—31nomaximal, Sylow
13C2×#237C2212yes—
14C2×C3×D8C2212yes—
15C3×#127C2211yescharacteristic
16C3×C4×D4C2211yescharacteristic
17#940—21no—
18#939—22no—
19#938C2212yes—
20#938C2212yes—
21D4×Dic3—21no—
22C2×#214C2211yescharacteristic
23#930—21no—
24#926—21no—
25C2×D3×C8C2212yes—
26#892C2212yes—
27#885—22no—
28C4×D12C2211yescharacteristic
29#869C2211yescharacteristic
30C4×Dic6C2211yescharacteristic
31C3×C4⋊C8C2211yescharacteristic
32C3×#102C2212yes—
33#832C2211yescharacteristic
34#832C2212yes—
35#832C2212yes—
36(C2×D3)⋊C8C2212yes—
37#818C2212yes—
38#816C2212yes—
39#810C2211yescharacteristic
40#808C2211yescharacteristic
41C4×C3⋊C8C2211yescharacteristic
42#555—31no—
43#550—31no—
44#486—31no—
45#482—32no—
46#477—31no—
47#466—32no—
48#459D311yescharacteristic, complemented
49#449—32no—
50C4×SD16—31no—
51C8×D4—31no—
52#416—32no—
53C2×C3×D4—21no—
54C2×C3×D4—21no—
55C2×C3×D4C2312yes—
56C2×C3×D4C2311yescharacteristic
57C22×C3×C4—21no—
58C2×#67—22no—
59C2×#67—22no—
60C2×#67—22no—
61C22×Dic3—22no—
62#237—24no—
63#237—42no—
64C2×D12D411yescharacteristic
65C2×C4×D3C2312yes—
66C2×Dic6—21no—
67C2×Dic6C2311yescharacteristic
68C3×D8—42no—
69C2×C3×C8C2312yes—
70C3×C4⋊C4C2311yescharacteristic
71C3×C22⋊C4—21no—
72C3×C42C2311yescharacteristic
73#217—22no—
74#217—22no—
75#214—22no—
76(C2×D3)⋊C4—22no—
77(C2×D3)⋊C4D412yes—
78C4⋊Dic3—21no—
79C4⋊Dic3D411yescharacteristic
80C4⋊Dic3C2311yescharacteristic
81C4⋊Dic3C2312yes—
82Dic3⋊C4—21no—
83C4×Dic3—21no—
84C4×Dic3—22no—
85C2×C3⋊C8C2311yescharacteristic
86C2×C3⋊C8C2311yescharacteristic
87D3×C8—42no—
88C2×#41—32no—
89C2×SD16—31no—
90C2×D8C2×D312yes—
91C22×C8—32no—
92C4⋊Q8—31no—
93#127—61no—
94#127C2×D311yescharacteristic
95#124—62no—
96#121—32no—
97#121—32no—
98#121—62no—
99#121—61no—
100C4×Q8—31no—
101C4×D4—31no—
102C4×D4—61no—
103C4×D4C2×D311yescharacteristic
104#117—32no—
105#117—61no—
106#107—32no—
107#106—31no—
108C4⋊C8C2×D311yescharacteristic
109#103—32no—
110#103—31no—
111#102—32no—
112#102—31no—
113#102—32no—
114#102C2×D312yes—
115C22⋊C8—32no—
116C4×C8—31no—
117C23×C3—21no—
118C23×C3—22no—
119C22×D3C2×D412yes—
120C3×D4—21no—
121C3×D4—22no—
122C3×D4—21no—
123C3×D4—22no—
124C3×D4—42no—
125C3×D4—81no—
126C2×C3×C4—21no—
127C2×C3×C4—42no—
128C2×C3×C4C2×D411yescharacteristic
129C2×C3×C4C2×D411yescharacteristic
130C2×C3×C4C2411yescharacteristic
131#67—44no—
132#67—82no—
133#67—82no—
134C2×Dic3—21no—
135C2×Dic3—21no—
136C2×Dic3—22no—
137C2×Dic3—44no—
138C2×Dic3C2×D412yes—
139D12—22no—
140C4×D3—22no—
141C4×D3—22no—
142Dic6—21no—
143Dic6—81no—
144Dic6#4112yes—
145C3×C8—22no—
146C3⋊C8—21no—
147C3⋊C8—21no—
148#41—64no—
149#41—122no—
150C2×Q8—31no—
151C2×Q8—61no—
152C2×D4—21no—
153C2×D4—21no—
154C2×D4—31no—
155C2×D4—62no—
156C2×D4—62no—
157C2×D4—62no—
158C2×D4C22×D312yes—
159C2×D4C22×D311yescharacteristic
160C22×C4—21no—
161C22×C4—32no—
162C22×C4—62no—
163SD16—62no—
164D8—42no—
165C2×C8—31no—
166C2×C8—31no—
167C2×C8—122no—
168C2×C8C22×D312yes—
169C4⋊C4—31no—
170C4⋊C4—32no—
171C4⋊C4—31no—
172C4⋊C4—61no—
173C4⋊C4—61no—
174C4⋊C4C22×D311yescharacteristic
175C22⋊C4—21no—
176C22⋊C4—32no—
177C22⋊C4—62no—
178C22⋊C4—62no—
179C22⋊C4—62no—
180C42—62no—
181C42—61no—
182C42C22×D311yescharacteristic
183C22×C3—42no—
184C22×C3—41no—
185C22×C3—41no—
186C22×C3—42no—
187C22×C3—41no—
188C22×C3—42no—
189C22×C3C22×D411yescharacteristic
190C2×D3—22no—
191C2×D3—22no—
192C2×D3—22no—
193C3×C4—21no—
194C3×C4—41no—
195C3×C4#13512yes—
196C3×C4C2×#4111yescharacteristic
197C3×C4H(5,𝔽2)11yescharacteristic
198Dic3—22no—
199Dic3—22no—
200Dic3—42no—
201Dic3—41no—
202C23—21no—
203C23—22no—
204C23—32no—
205Q8—32no—
206Q8—61no—
207Q8—241no—
208D4—21no—
209D4—22no—
210D4—21no—
211D4—22no—
212D4—42no—
213D4—62no—
214D4—81no—
215D4—124no—
216D4—242no—
217D4—242no—
218C2×C4—21no—
219C2×C4—32no—
220C2×C4—42no—
221C2×C4—61no—
222C2×C4—62no—
223C2×C4—62no—
224C2×C4—61no—
225C2×C4—62no—
226C2×C4—124no—
227C2×C4D3×D411yescharacteristic
228C2×C4D3×D411yescharacteristic
229C2×C4C23×D311yescharacteristic
230C8—22no—
231C8—61no—
232C8—61no—
233C2×C3—82no—
234C2×C3—81no—
235C2×C3#54611yescharacteristic
236C2×C3C2×#13511yescharacteristic
237C2×C3#57611yescharacteristic
238D3—42no—
239C22—41no—
240C22—42no—
241C22—42no—
242C22—41no—
243C22—42no—
244C22—41no—
245C22—62no—
246C22—62no—
247C22—62no—
248C22C2×D3×D411yescharacteristic
249C4—21no—
250C4—41no—
251C4—62no—
252C4—62no—
253C4—121no—
254C4—122no—
255C4#91212yes—
256C4#100411yescharacteristic
257C4D3×#4111yescharacteristic
258C3#330511yescharacteristic, complemented, Sylow
259C2—81no—
260C2—82no—
261C2—122no—
262C2#561211yescharacteristic
263C2C2×#91211yescharacteristic
264C2#577411yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#459C2
socleC22×C3C22×D4
hypercenter#459D3
hypocenterC3#3305
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ #459

Complements

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

NH ≅ G/Ntype
1. #5718C2semidirect
2. #5563C2semidirect
3. C3×#459C2semidirect
4. #5158C2semidirect
5. #5157C2semidirect
6. #5070C2semidirect
7. C4×#214C2semidirect
8. #4850C2semidirect
9. #4834C2semidirect
10. #4784C2semidirect
11. #4772C2semidirect
48. #459D3semidirect
258. C3#3305semidirect

Maximal subgroups

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

subgroupindex
#57182
#55632
C3×#4592
#51582
#51572
#50702
C4×#2142
#48502
#48342
#47842
#47722
#33053