Finite Group Databasebeta

← #78050

Subgroups — #78050

Order 384.

500 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 500 conjugacy classes of subgroups — too many to draw. Only its 129 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
1C2×#887C211yesmaximal, characteristic, complemented
2C2×#885C211yesmaximal, characteristic
3C3×#399C211yesmaximal, characteristic, complemented
4#4993C211yesmaximal, characteristic, complemented
5#4990C212yesmaximal, complemented
6#4979C212yesmaximal, complemented
7#4959C212yesmaximal
8#4947C211yesmaximal, characteristic, complemented
9#4945C212yesmaximal, complemented
10#4673C211yesmaximal, characteristic, complemented
11#4651C211yesmaximal, characteristic, complemented
12#2632—31nomaximal, Sylow
13C22×#67C2211yescharacteristic
14C22×Dic6C2211yescharacteristic
15C22×Dic6C2211yescharacteristic
16C2×C3×C4⋊C4C2211yescharacteristic
17C2×C3×C4⋊C4C2212yes—
18C2×C3×C22⋊C4C2211yescharacteristic
19C2×C3×C22⋊C4C2212yes—
20C2×#217C2211yescharacteristic
21C2×#217C2212yes—
22C2×(C2×D3)⋊C4C2212yes—
23C2×(C2×D3)⋊C4C2211yescharacteristic
24C2×(C2×D3)⋊C4C2211yescharacteristic
25C2×(C2×D3)⋊C4C2212yes—
26C2×Dic3⋊C4C2212yes—
27C2×Dic3⋊C4C2212yes—
28C2×C4×Dic3C2212yes—
29C2×C4×Dic3C2211yescharacteristic
30#887—24no—
31#885—24no—
32C3×#95C2211yescharacteristic
33C3⋊#95C2212yes—
34C3⋊#95C2211yescharacteristic
35C3⋊#95C2212yes—
36C3⋊#95C2212yes—
37C3⋊#95C2212yes—
38C2×#124—31no—
39C2×C4×Q8—31no—
40#399—31no—
41#399D311yescharacteristic, complemented
42#395—32no—
43#394—32no—
44#393—31no—
45#393—31no—
46#388—32no—
47#388—32no—
48#383—31no—
49C24×C3C2311yescharacteristic
50C23×D3C2311yescharacteristic
51C22×C3×C4C2312yes—
52C22×C3×C4C2311yescharacteristic
53C22×C3×C4C2312yes—
54C22×C3×C4C2311yescharacteristic
55C2×#67—22no—
56C2×#67—42no—
57C22×Dic3C2312yes—
58C22×Dic3C2311yescharacteristic
59C22×Dic3C2311yescharacteristic
60C22×Dic3C2311yescharacteristic
61C22×Dic3C2312yes—
62C2×Dic6—22no—
63C2×Dic6—24no—
64C2×Dic6—22no—
65C2×Dic6—42no—
66C3×C4⋊C4—22no—
67C3×C4⋊C4—42no—
68C3×C22⋊C4—22no—
69C3×C22⋊C4—42no—
70#217—22no—
71#217—24no—
72(C2×D3)⋊C4—22no—
73(C2×D3)⋊C4—22no—
74(C2×D3)⋊C4—24no—
75(C2×D3)⋊C4—42no—
76Dic3⋊C4—24no—
77Dic3⋊C4—42no—
78C4×Dic3—24no—
79C4×Dic3D414yes—
80C22×Q8—31no—
81C22×Q8—31no—
82C22×D4—31no—
83#124—64no—
84C4×Q8—64no—
85C2×C4⋊C4—32no—
86C2×C4⋊C4—32no—
87C2×C4⋊C4C2×D311yescharacteristic
88C2×C4⋊C4C2×D312yes—
89C2×C22⋊C4—31no—
90C2×C22⋊C4—32no—
91C2×C22⋊C4—31no—
92C2×C22⋊C4—31no—
93C2×C22⋊C4—32no—
94C2×C22⋊C4—32no—
95C2×C22⋊C4C2×D311yescharacteristic
96C2×C22⋊C4C2×D312yes—
97C2×C42—31no—
98C2×C42—32no—
99#95—32no—
100#95—31no—
101#95—32no—
102#95—32no—
103#95—32no—
104#95C2×D311yescharacteristic
105C23×C3—21no—
106C23×C3—22no—
107C23×C3—22no—
108C23×C3—21no—
109C23×C3—21no—
110C23×C3C2411yescharacteristic
111C22×D3—21no—
112C22×D3—22no—
113C22×D3—22no—
114C22×D3—21no—
115C22×D3—21no—
116C2×C3×C4—22no—
117C2×C3×C4—22no—
118C2×C3×C4—21no—
119C2×C3×C4—22no—
120C2×C3×C4—22no—
121C2×C3×C4—22no—
122C2×C3×C4—22no—
123C2×C3×C4—22no—
124C2×C3×C4—22no—
125C2×C3×C4C2×D412yes—
126#67—44no—
127C2×Dic3—22no—
128C2×Dic3—22no—
129C2×Dic3—22no—
130C2×Dic3—21no—
131C2×Dic3—21no—
132C2×Dic3—21no—
133C2×Dic3—22no—
134C2×Dic3—22no—
135C2×Dic3—22no—
136C2×Dic3—22no—
137C2×Dic3C2×D412yes—
138C2×Dic3C2×D412yes—
139C2×Dic3#4114yes—
140Dic6—28no—
141Dic6—44no—
142C24—31no—
143C24C22×D311yescharacteristic
144C2×Q8—64no—
145C2×Q8—62no—
146C2×Q8—62no—
147C2×Q8—122no—
148C2×D4—62no—
149C2×D4—122no—
150C22×C4—31no—
151C22×C4—32no—
152C22×C4—31no—
153C22×C4—32no—
154C22×C4—31no—
155C22×C4C22×D312yes—
156C22×C4C22×D311yescharacteristic
157C22×C4C22×D312yes—
158C22×C4C22×D311yescharacteristic
159C4⋊C4—22no—
160C4⋊C4—42no—
161C4⋊C4—64no—
162C4⋊C4—122no—
163C22⋊C4—22no—
164C22⋊C4—42no—
165C22⋊C4—62no—
166C22⋊C4—64no—
167C22⋊C4—62no—
168C22⋊C4—64no—
169C22⋊C4—62no—
170C22⋊C4—122no—
171C42—34no—
172C42—64no—
173C22×C3—41no—
174C22×C3—41no—
175C22×C3—41no—
176C22×C3—42no—
177C22×C3—42no—
178C22×C3C22×D411yescharacteristic
179C22×C3C2×#4112yes—
180C22×C3H(5,𝔽2)12yes—
181C22×C32−1+411yescharacteristic
182C22×C32−1+411yescharacteristic
183C2×D3—41no—
184C2×D3—41no—
185C2×D3—41no—
186C2×D3—42no—
187C2×D3—42no—
188C3×C4—22no—
189C3×C4—41no—
190C3×C4—42no—
191C3×C4—42no—
192Dic3—24no—
193Dic3—22no—
194Dic3—22no—
195Dic3—41no—
196Dic3—42no—
197C23—21no—
198C23—22no—
199C23—21no—
200C23—22no—
201C23—21no—
202C23—61no—
203C23—62no—
204C23—61no—
205C23—62no—
206C23—61no—
207C23C23×D311yescharacteristic
208Q8—68no—
209Q8—124no—
210D4—124no—
211C2×C4—22no—
212C2×C4—22no—
213C2×C4—21no—
214C2×C4—22no—
215C2×C4—22no—
216C2×C4—22no—
217C2×C4—22no—
218C2×C4—22no—
219C2×C4—22no—
220C2×C4—32no—
221C2×C4—32no—
222C2×C4—34no—
223C2×C4—62no—
224C2×C4—62no—
225C2×C4—61no—
226C2×C4—62no—
227C2×C4—61no—
228C2×C4—61no—
229C2×C4—62no—
230C2×C4—62no—
231C2×C4—62no—
232C2×C4—62no—
233C2×C4D3×D412yes—
234C2×C3—81no—
235C2×C3#52511yescharacteristic
236C2×C3#53511yescharacteristic
237C2×C3#53611yescharacteristic
238C2×C3#54212yes—
239C2×C3#54212yes—
240D3—81no—
241C22—42no—
242C22—41no—
243C22—42no—
244C22—41no—
245C22—41no—
246C22—122no—
247C22—121no—
248C22—122no—
249C22—121no—
250C22—121no—
251C22C2×D3×D411yescharacteristic
252C22#100412yes—
253C22#100711yescharacteristic
254C22D3×#4112yes—
255C22#101011yescharacteristic
256C4—22no—
257C4—42no—
258C4—41no—
259C4—42no—
260C4—64no—
261C4—62no—
262C4—62no—
263C4—122no—
264C4—121no—
265C3#263211yescharacteristic, complemented, Sylow
266C2—81no—
267C2—241no—
268C2#562811yescharacteristic
269C2#564012yes—
270C2#565611yescharacteristic
271C2#566212yes—
272C2#571511yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC3×#399C2
prime radicalC22×#67C22
hypercenter#399D3
hypocenterC3#2632
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #399

Complements

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

NH ≅ G/Ntype
1. C2×#887C2semidirect
3. C3×#399C2semidirect
4. #4993C2semidirect
5. #4990C2semidirect
6. #4979C2semidirect
8. #4947C2semidirect
9. #4945C2semidirect
10. #4673C2semidirect
11. #4651C2semidirect
41. #399D3semidirect
265. C3#2632semidirect

Maximal subgroups

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

subgroupindex
C2×#8872
C2×#8852
C3×#3992
#49932
#49902
#49792
#49592
#49472
#49452
#46732
#46512
#26323