Finite Group Databasebeta

← #73427

Subgroups — #73427

Order 384.

476 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 476 conjugacy classes of subgroups — too many to draw. Only its 145 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×#872C211yesmaximal, characteristic
2C2×#870C211yesmaximal, characteristic, complemented
3C3×#376C211yesmaximal, characteristic, complemented
4#4940C212yesmaximal
5#4928C212yesmaximal, complemented
6#4677C211yesmaximal, characteristic
7#4674C211yesmaximal, characteristic
8#4668C212yesmaximal
9#4661C211yesmaximal, characteristic, complemented
10#4659C211yesmaximal, characteristic, complemented
11#4653C212yesmaximal, complemented
12#2611—31nomaximal, Sylow
13C22×C4×D3C2211yescharacteristic
14C2×C3×C42C2212yes—
15C2×C3×C42C2211yescharacteristic
16C2×(C2×D3)⋊C4C2212yes—
17C2×(C2×D3)⋊C4C2212yes—
18C2×(C2×D3)⋊C4C2211yescharacteristic
19C2×(C2×D3)⋊C4C2211yescharacteristic
20C2×C4⋊Dic3C2212yes—
21C2×C4⋊Dic3C2211yescharacteristic
22C2×C4⋊Dic3C2212yes—
23C2×C4⋊Dic3C2211yescharacteristic
24C2×Dic3⋊C4C2212yes—
25C2×Dic3⋊C4C2211yescharacteristic
26C2×Dic3⋊C4C2211yescharacteristic
27C2×Dic3⋊C4C2212yes—
28C2×C4×Dic3C2211yescharacteristic
29#872—24no—
30#870—24no—
31C3×#95C2212yes—
32C3×#95C2211yescharacteristic
33C3×#95C2211yescharacteristic
34C3⋊#95C2212yes—
35C3⋊#95C2212yes—
36C3⋊#95C2212yes—
37C3⋊#95C2212yes—
38C2×#125—31no—
39C2×#117—31no—
40#400—31no—
41#399—31no—
42#398—31no—
43#397—31no—
44#389—32no—
45#388—32no—
46#387—32no—
47#385—32no—
48#376D311yescharacteristic, complemented
49C23×D3C2311yescharacteristic
50C22×C3×C4C2312yes—
51C22×C3×C4C2311yescharacteristic
52C22×C3×C4C2311yescharacteristic
53C22×C3×C4C2311yescharacteristic
54C22×C3×C4C2312yes—
55C22×Dic3C2311yescharacteristic
56C22×Dic3C2311yescharacteristic
57C22×Dic3C2312yes—
58C22×Dic3C2312yes—
59C22×Dic3C2311yescharacteristic
60C2×C4×D3—22no—
61C2×C4×D3—22no—
62C2×C4×D3—22no—
63C3×C42—24no—
64C3×C42D414yes—
65(C2×D3)⋊C4—24no—
66(C2×D3)⋊C4—22no—
67(C2×D3)⋊C4—22no—
68(C2×D3)⋊C4—24no—
69C4⋊Dic3—22no—
70C4⋊Dic3—22no—
71C4⋊Dic3—24no—
72C4⋊Dic3—24no—
73Dic3⋊C4—24no—
74Dic3⋊C4—24no—
75Dic3⋊C4—22no—
76Dic3⋊C4—22no—
77C4×Dic3—22no—
78C23×C4—31no—
79#125—64no—
80#117—64no—
81C2×C4⋊C4—31no—
82C2×C4⋊C4—31no—
83C2×C4⋊C4—32no—
84C2×C4⋊C4—32no—
85C2×C4⋊C4—31no—
86C2×C4⋊C4—31no—
87C2×C4⋊C4—32no—
88C2×C4⋊C4—32no—
89C2×C22⋊C4—31no—
90C2×C22⋊C4—31no—
91C2×C22⋊C4—32no—
92C2×C22⋊C4—32no—
93C2×C42—31no—
94C2×C42C2×D312yes—
95C2×C42C2×D311yescharacteristic
96#95—32no—
97#95—32no—
98#95—32no—
99#95—32no—
100#95C2×D312yes—
101#95C2×D311yescharacteristic
102#95C2×D311yescharacteristic
103C23×C3C2411yescharacteristic
104C22×D3—21no—
105C22×D3—22no—
106C22×D3—21no—
107C22×D3—22no—
108C22×D3#4112yes—
109C2×C3×C4—22no—
110C2×C3×C4—22no—
111C2×C3×C4—21no—
112C2×C3×C4—21no—
113C2×C3×C4—22no—
114C2×C3×C4—22no—
115C2×C3×C4—22no—
116C2×C3×C4—22no—
117C2×C3×C4—22no—
118C2×C3×C4C2×D412yes—
119C2×C3×C4C2×D412yes—
120C2×C3×C4C2×D412yes—
121C2×C3×C4#4114yes—
122C2×Dic3—21no—
123C2×Dic3—21no—
124C2×Dic3—21no—
125C2×Dic3—21no—
126C2×Dic3—22no—
127C2×Dic3—22no—
128C2×Dic3—22no—
129C2×Dic3—22no—
130C2×Dic3—22no—
131C2×Dic3—22no—
132C2×Dic3—22no—
133C2×Dic3—22no—
134C2×Dic3#4112yes—
135C4×D3—28no—
136C24—31no—
137C22×C4—31no—
138C22×C4—31no—
139C22×C4—31no—
140C22×C4—32no—
141C22×C4—32no—
142C22×C4—62no—
143C22×C4—62no—
144C22×C4—62no—
145C22×C4C22×D312yes—
146C22×C4C22×D311yescharacteristic
147C22×C4C22×D311yescharacteristic
148C22×C4C22×D311yescharacteristic
149C22×C4C22×D312yes—
150C4⋊C4—64no—
151C4⋊C4—62no—
152C4⋊C4—62no—
153C4⋊C4—64no—
154C4⋊C4—64no—
155C4⋊C4—62no—
156C4⋊C4—62no—
157C4⋊C4—64no—
158C22⋊C4—64no—
159C22⋊C4—62no—
160C22⋊C4—62no—
161C22⋊C4—64no—
162C42—24no—
163C42—62no—
164C42D1214yes—
165C22×C3C22×D411yescharacteristic
166C22×C3C2×#4112yes—
167C22×C3C2×#4111yescharacteristic
168C22×C3C2×#4111yescharacteristic
169C22×C32−1+412yes—
170C2×D3—22no—
171C2×D3—24no—
172C2×D3—41no—
173C2×D3—41no—
174C2×D3—42no—
175C3×C4—22no—
176C3×C4—22no—
177C3×C4—22no—
178C3×C4—24no—
179C3×C4—42no—
180Dic3—22no—
181Dic3—42no—
182Dic3—41no—
183Dic3—42no—
184Dic3—41no—
185C23—32no—
186C23—61no—
187C23—62no—
188C23—61no—
189C23—62no—
190C23C23×D311yescharacteristic
191C2×C4—22no—
192C2×C4—22no—
193C2×C4—21no—
194C2×C4—21no—
195C2×C4—22no—
196C2×C4—22no—
197C2×C4—22no—
198C2×C4—22no—
199C2×C4—22no—
200C2×C4—32no—
201C2×C4—68no—
202C2×C4—61no—
203C2×C4—61no—
204C2×C4—61no—
205C2×C4—61no—
206C2×C4—62no—
207C2×C4—62no—
208C2×C4—62no—
209C2×C4—62no—
210C2×C4—62no—
211C2×C4—62no—
212C2×C4—62no—
213C2×C4—62no—
214C2×C4C2×D1212yes—
215C2×C4C2×D1212yes—
216C2×C4C2×D1212yes—
217C2×C4#23514yes—
218C2×C3#52511yescharacteristic
219C2×C3#53211yescharacteristic
220C2×C3#53611yescharacteristic
221C2×C3#55212yes—
222C2×C3#55212yes—
223D3—42no—
224C22—62no—
225C22—64no—
226C22—121no—
227C22—121no—
228C22—122no—
229C22C22×D1211yescharacteristic
230C22C2×#23512yes—
231C22D3×#4111yescharacteristic
232C22D3×#4111yescharacteristic
233C22#101012yes—
234C4—22no—
235C4—22no—
236C4—22no—
237C4—24no—
238C4—42no—
239C4—62no—
240C4—121no—
241C4—122no—
242C4—122no—
243C4—121no—
244C3#261111yescharacteristic, complemented, Sylow
245C2—122no—
246C2#547711yescharacteristic
247C2#552411yescharacteristic
248C2#552611yescharacteristic
249C2#552912yes—
250C2#553112yes—
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC3×#376C2
prime radicalC23×D3C23
hypercenter#376D3
hypocenterC3#2611
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #376

Complements

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

NH ≅ G/Ntype
2. C2×#870C2semidirect
3. C3×#376C2semidirect
5. #4928C2semidirect
9. #4661C2semidirect
10. #4659C2semidirect
11. #4653C2semidirect
48. #376D3semidirect
244. C3#2611semidirect

Maximal subgroups

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

subgroupindex
C2×#8722
C2×#8702
C3×#3762
#49402
#49282
#46772
#46742
#46682
#46612
#46592
#46532
#26113