Finite Group Databasebeta

← #83571

Subgroups — #83571

Order 384.

408 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 408 conjugacy classes of subgroups — too many to draw. Only its 125 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
1C3×#550C211yesmaximal, characteristic, complemented
2#5563C211yesmaximal, characteristic, complemented
3#5238C212yesmaximal, complemented
4#5233C212yesmaximal, complemented
5#5072C211yesmaximal, characteristic, complemented
6#5070C211yesmaximal, characteristic, complemented
7#5038C212yesmaximal, complemented
8#5035C212yesmaximal, complemented
9C4×#215C211yesmaximal, characteristic, complemented
10#5021C211yesmaximal, characteristic, complemented
11D4×C3⋊C8C211yesmaximal, characteristic, complemented
12#3305—31nomaximal, Sylow
13C2×C3×#41C2212yes—
14C3×#127—21no—
15C3×#127C2211yescharacteristic
16C3×#121—22no—
17C3×#121C2212yes—
18C3×C4×Q8C2211yescharacteristic
19C3×C4×D4—21no—
20C3×C4×D4C2211yescharacteristic
21C2×#215C2211yescharacteristic
22#938—21no—
23C2×#213C2212yes—
24#930C2212yes—
25#926C2212yes—
26C22×C3⋊C8C2212yes—
27#892—21no—
28#885—22no—
29C4×D12C2211yescharacteristic
30#869C2211yescharacteristic
31#835C2211yescharacteristic
32#835C2212yes—
33#832C2212yes—
34#832C2212yes—
35#832C2212yes—
36#830C2212yes—
37#809C2211yescharacteristic
38#808C2211yescharacteristic
39#807C2212yes—
40C4⋊(C3⋊C8)C2211yescharacteristic
41C4×C3⋊C8C2211yescharacteristic
42#555—31no—
43#550D311yescharacteristic, complemented
44#486—31no—
45#482—32no—
46#477—31no—
47#466—32no—
48#459—31no—
49#449—32no—
50C4×SD16—31no—
51C8×D4—31no—
52#416—32no—
53C3×#41—24no—
54C3×#41—42no—
55C2×C3×Q8C2311yescharacteristic
56C2×C3×D4—22no—
57C2×C3×D4—22no—
58C2×C3×D4—21no—
59C2×C3×D4—21no—
60C2×C3×D4D411yescharacteristic
61C2×C3×D4C2312yes—
62C22×C3×C4—22no—
63C22×C3×C4C2312yes—
64C2×#67—22no—
65C2×D12C2311yescharacteristic
66C2×C4×D3—21no—
67C2×Dic6—21no—
68C3×C4⋊C4—21no—
69C3×C4⋊C4D411yescharacteristic
70C3×C4⋊C4C2311yescharacteristic
71C3×C22⋊C4—22no—
72C3×C22⋊C4D412yes—
73C3×C42—21no—
74C3×C42C2311yescharacteristic
75#217—22no—
76#215—22no—
77#213—42no—
78(C2×D3)⋊C4—21no—
79(C2×D3)⋊C4—22no—
80C4⋊Dic3—21no—
81C4⋊Dic3C2312yes—
82C4⋊Dic3C2311yescharacteristic
83C4×Dic3—22no—
84C2×C3⋊C8—42no—
85C2×C3⋊C8C2311yescharacteristic
86C2×C3⋊C8C2311yescharacteristic
87C2×C3⋊C8C2312yes—
88C2×#41C2×D312yes—
89C2×SD16—31no—
90C2×D8—32no—
91C22×C8—32no—
92C4⋊Q8—31no—
93#127—21no—
94#127C2×D311yescharacteristic
95#124—62no—
96#121—22no—
97#121—32no—
98#121—61no—
99#121C2×D312yes—
100C4×Q8C2×D311yescharacteristic
101C4×D4—21no—
102C4×D4—31no—
103C4×D4C2×D311yescharacteristic
104#117—32no—
105#117—61no—
106#107—32no—
107#106—31no—
108C4⋊C8—31no—
109#103—31no—
110#103—32no—
111#102—32no—
112#102—32no—
113#102—32no—
114#102—31no—
115C22⋊C8—32no—
116C4×C8—31no—
117C23×C3—22no—
118C23×C3C2×D412yes—
119C22×D3—21no—
120C3×Q8—21no—
121C3×Q8#4112yes—
122C3×D4—22no—
123C3×D4—22no—
124C3×D4—22no—
125C3×D4—44no—
126C3×D4—42no—
127C3×D4—81no—
128C3×D4—82no—
129C2×C3×C4—21no—
130C2×C3×C4—22no—
131C2×C3×C4—22no—
132C2×C3×C4—21no—
133C2×C3×C4—44no—
134C2×C3×C4C2×D412yes—
135C2×C3×C4C2×D411yescharacteristic
136C2×C3×C4C2×D411yescharacteristic
137C2×C3×C4C2411yescharacteristic
138#67—82no—
139C2×Dic3—21no—
140C2×Dic3—22no—
141D12—21no—
142D12—21no—
143C4×D3—42no—
144Dic6—81no—
145C3⋊C8—21no—
146C3⋊C8—21no—
147C3⋊C8—22no—
148#41—24no—
149#41—42no—
150C2×Q8—61no—
151C2×Q8C22×D311yescharacteristic
152C2×D4—22no—
153C2×D4—22no—
154C2×D4—21no—
155C2×D4—21no—
156C2×D4—31no—
157C2×D4—62no—
158C2×D4#6711yescharacteristic
159C2×D4C22×D312yes—
160C22×C4—22no—
161C22×C4—61no—
162C22×C4C22×D312yes—
163SD16—62no—
164D8—122no—
165C2×C8—31no—
166C2×C8—31no—
167C2×C8—32no—
168C2×C8—122no—
169C4⋊C4—21no—
170C4⋊C4—31no—
171C4⋊C4—32no—
172C4⋊C4—61no—
173C4⋊C4#6711yescharacteristic
174C4⋊C4C22×D311yescharacteristic
175C22⋊C4—22no—
176C22⋊C4—62no—
177C22⋊C4—62no—
178C22⋊C4—61no—
179C22⋊C4#6712yes—
180C42—21no—
181C42—62no—
182C42C22×D311yescharacteristic
183C22×C3—22no—
184C22×C3—22no—
185C22×C3—22no—
186C22×C3—42no—
187C22×C3—42no—
188C22×C3—42no—
189C22×C3C22×D411yescharacteristic
190C2×D3—41no—
191C2×D3—41no—
192C2×D3—41no—
193C3×C4—22no—
194C3×C4—22no—
195C3×C4—21no—
196C3×C4—41no—
197C3×C4#13512yes—
198C3×C4C2×#4111yescharacteristic
199C3×C4H(5,𝔽2)11yescharacteristic
200Dic3—41no—
201Dic3—42no—
202C23—22no—
203C23—61no—
204C23C2×#6712yes—
205Q8—21no—
206Q8—241no—
207Q8#23712yes—
208D4—22no—
209D4—22no—
210D4—22no—
211D4—44no—
212D4—42no—
213D4—61no—
214D4—61no—
215D4—81no—
216D4—82no—
217D4—242no—
218C2×C4—21no—
219C2×C4—22no—
220C2×C4—22no—
221C2×C4—21no—
222C2×C4—44no—
223C2×C4—61no—
224C2×C4—62no—
225C2×C4—122no—
226C2×C4C2×#6712yes—
227C2×C4C2×#6711yescharacteristic
228C2×C4C2×#6711yescharacteristic
229C2×C4C23×D311yescharacteristic
230C8—61no—
231C8—61no—
232C8—62no—
233C2×C3—42no—
234C2×C3—82no—
235C2×C3#54611yescharacteristic
236C2×C3C2×#13511yescharacteristic
237C2×C3#57611yescharacteristic
238D3—81no—
239C22—22no—
240C22—22no—
241C22—22no—
242C22—42no—
243C22—42no—
244C22—42no—
245C22—121no—
246C22—121no—
247C22—121no—
248C22C22×#6711yescharacteristic
249C4—22no—
250C4—22no—
251C4—21no—
252C4—41no—
253C4—121no—
254C4—122no—
255C4#95012yes—
256C4C2×#23711yescharacteristic
257C4#100411yescharacteristic
258C3#330511yescharacteristic, complemented, Sylow
259C2—42no—
260C2—82no—
261C2—241no—
262C2#580611yescharacteristic
263C2C2×#95011yescharacteristic
264C2#583511yescharacteristic
subgroupG/H
centerC22C22×#67
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#550C2
socleC22×C3C22×D4
hypercenter#550D3
hypocenterC3#3305
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C4×D4 ◁ #550

Complements

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

NH ≅ G/Ntype
1. C3×#550C2semidirect
2. #5563C2semidirect
3. #5238C2semidirect
4. #5233C2semidirect
5. #5072C2semidirect
6. #5070C2semidirect
7. #5038C2semidirect
8. #5035C2semidirect
9. C4×#215C2semidirect
10. #5021C2semidirect
11. D4×C3⋊C8C2semidirect
43. #550D3semidirect
258. C3#3305semidirect

Maximal subgroups

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

subgroupindex
C3×#5502
#55632
#52382
#52332
#50722
#50702
#50382
#50352
C4×#2152
#50212
D4×C3⋊C82
#33053