Finite Group Databasebeta

← #77537

Subgroups — #77537

Order 384.

476 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 476 conjugacy classes of subgroups — too many to draw. Only its 135 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×#890C211yesmaximal, characteristic, complemented
2C2×#877C211yesmaximal, characteristic
3C3×#397C211yesmaximal, characteristic
4#4983C211yesmaximal, characteristic, complemented
5#4951C212yesmaximal
6#4949C211yesmaximal, characteristic
7#4946C212yesmaximal
8#4659C211yesmaximal, characteristic, complemented
9#4652C212yesmaximal, complemented
10#4648C211yesmaximal, characteristic, complemented
11#4647C212yesmaximal, complemented
12#2611—31nomaximal, Sylow
13C23×Dic3C2211yescharacteristic
14C2×C3×C4⋊C4C2211yescharacteristic
15C2×C3×C22⋊C4C2212yes—
16C2×C3×C22⋊C4C2211yescharacteristic
17C2×#217C2212yes—
18C2×#217C2211yescharacteristic
19C2×C4⋊Dic3C2212yes—
20C2×C4⋊Dic3C2211yescharacteristic
21C2×Dic3⋊C4C2211yescharacteristic
22C2×Dic3⋊C4C2212yes—
23C2×Dic3⋊C4C2212yes—
24C2×Dic3⋊C4C2212yes—
25C2×Dic3⋊C4C2211yescharacteristic
26C2×C4×Dic3C2212yes—
27C2×C4×Dic3C2211yescharacteristic
28C2×C4×Dic3C2211yescharacteristic
29#890—24no—
30#877—24no—
31C3×#95C2212yes—
32C3×#95C2211yescharacteristic
33C3⋊#95C2212yes—
34C3⋊#95C2212yes—
35C3⋊#95C2211yescharacteristic
36C3⋊#95C2212yes—
37C3⋊#95C2212yes—
38C2×#125—31no—
39C2×#117—31no—
40#400—31no—
41#399—31no—
42#398—31no—
43#397D311yescharacteristic
44#389—32no—
45#388—32no—
46#387—32no—
47#385—32no—
48#376—31no—
49C24×C3C2311yescharacteristic
50C22×C3×C4C2312yes—
51C22×C3×C4C2312yes—
52C22×C3×C4C2311yescharacteristic
53C22×C3×C4C2311yescharacteristic
54C22×Dic3—22no—
55C22×Dic3—22no—
56C22×Dic3—22no—
57C22×Dic3C2311yescharacteristic
58C22×Dic3C2312yes—
59C22×Dic3C2311yescharacteristic
60C22×Dic3C2311yescharacteristic
61C22×Dic3C2312yes—
62C22×Dic3C2311yescharacteristic
63C3×C4⋊C4—22no—
64C3×C22⋊C4—24no—
65C3×C22⋊C4—22no—
66#217—22no—
67#217—24no—
68C4⋊Dic3—22no—
69C4⋊Dic3—24no—
70Dic3⋊C4—22no—
71Dic3⋊C4—22no—
72Dic3⋊C4—24no—
73Dic3⋊C4—24no—
74Dic3⋊C4—24no—
75C4×Dic3—22no—
76C4×Dic3—24no—
77C4×Dic3D414yes—
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—32no—
87C2×C4⋊C4—32no—
88C2×C4⋊C4C2×D311yescharacteristic
89C2×C22⋊C4—31no—
90C2×C22⋊C4—32no—
91C2×C22⋊C4C2×D311yescharacteristic
92C2×C22⋊C4C2×D312yes—
93C2×C42—31no—
94C2×C42—31no—
95C2×C42—32no—
96#95—32no—
97#95—31no—
98#95—32no—
99#95—32no—
100#95—32no—
101#95C2×D311yescharacteristic
102#95C2×D312yes—
103C23×C3—21no—
104C23×C3—22no—
105C23×C3—21no—
106C23×C3—22no—
107C23×C3#4112yes—
108C23×C3C2411yescharacteristic
109C2×C3×C4—22no—
110C2×C3×C4—21no—
111C2×C3×C4—22no—
112C2×C3×C4—22no—
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×Dic3—28no—
120C2×Dic3—21no—
121C2×Dic3—22no—
122C2×Dic3—22no—
123C2×Dic3—21no—
124C2×Dic3—21no—
125C2×Dic3—21no—
126C2×Dic3—22no—
127C2×Dic3—21no—
128C2×Dic3—22no—
129C2×Dic3—22no—
130C2×Dic3—22no—
131C2×Dic3—22no—
132C2×Dic3C2×D412yes—
133C2×Dic3C2×D412yes—
134C2×Dic3#4114yes—
135C2×Dic3#4112yes—
136C24C22×D311yescharacteristic
137C22×C4—32no—
138C22×C4—31no—
139C22×C4—31no—
140C22×C4—31no—
141C22×C4—31no—
142C22×C4—32no—
143C22×C4—62no—
144C22×C4—62no—
145C22×C4—62no—
146C22×C4C22×D312yes—
147C22×C4C22×D312yes—
148C22×C4C22×D311yescharacteristic
149C22×C4C22×D311yescharacteristic
150C4⋊C4—22no—
151C4⋊C4—62no—
152C4⋊C4—64no—
153C4⋊C4—64no—
154C4⋊C4—64no—
155C4⋊C4—64no—
156C4⋊C4—62no—
157C4⋊C4—62no—
158C22⋊C4—24no—
159C22⋊C4—22no—
160C22⋊C4—62no—
161C22⋊C4—64no—
162C42—34no—
163C42—62no—
164C42—64no—
165C22×C3—22no—
166C22×C3—24no—
167C22×C3—41no—
168C22×C3—41no—
169C22×C3—42no—
170C22×C3C22×D411yescharacteristic
171C22×C3C2×#4111yescharacteristic
172C22×C3C2×#4111yescharacteristic
173C22×C3C2×#4112yes—
174C22×C32−1+412yes—
175C3×C4—22no—
176C3×C4—42no—
177C3×C4—41no—
178C3×C4—42no—
179Dic3—22no—
180Dic3—22no—
181Dic3—22no—
182Dic3—24no—
183Dic3—42no—
184Dic3—41no—
185C23—21no—
186C23—22no—
187C23—21no—
188C23—22no—
189C23#23712yes—
190C23C23×D311yescharacteristic
191C2×C4—22no—
192C2×C4—21no—
193C2×C4—22no—
194C2×C4—22no—
195C2×C4—22no—
196C2×C4—22no—
197C2×C4—22no—
198C2×C4—22no—
199C2×C4—22no—
200C2×C4—32no—
201C2×C4—32no—
202C2×C4—34no—
203C2×C4—32no—
204C2×C4—68no—
205C2×C4—61no—
206C2×C4—61no—
207C2×C4—61no—
208C2×C4—62no—
209C2×C4—62no—
210C2×C4—61no—
211C2×C4—62no—
212C2×C4—61no—
213C2×C4—62no—
214C2×C4—62no—
215C2×C4—62no—
216C2×C4—62no—
217C2×C4D3×D412yes—
218C2×C3—42no—
219C2×C3#52511yescharacteristic
220C2×C3#53211yescharacteristic
221C2×C3#53611yescharacteristic
222C2×C3#55212yes—
223C2×C3#55212yes—
224C22—24no—
225C22—22no—
226C22—41no—
227C22—41no—
228C22—42no—
229C22C2×D3×D411yescharacteristic
230C22C2×#23711yescharacteristic
231C22D3×#4112yes—
232C22D3×#4111yescharacteristic
233C22#101012yes—
234C4—22no—
235C4—42no—
236C4—41no—
237C4—42no—
238C4—62no—
239C4—64no—
240C4—62no—
241C4—62no—
242C4—121no—
243C4—122no—
244C3#261111yescharacteristic, complemented, Sylow
245C2—42no—
246C2#560611yescharacteristic
247C2#564811yescharacteristic
248C2#565012yes—
249C2#566811yescharacteristic
250C2#570812yes—
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC3×#397C2
prime radicalC24×C3C23
hypercenter#397D3
hypocenterC3#2611
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #397

Complements

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

NH ≅ G/Ntype
1. C2×#890C2semidirect
4. #4983C2semidirect
8. #4659C2semidirect
9. #4652C2semidirect
10. #4648C2semidirect
11. #4647C2semidirect
244. C3#2611semidirect

Maximal subgroups

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

subgroupindex
C2×#8902
C2×#8772
C3×#3972
#49832
#49512
#49492
#49462
#46592
#46522
#46482
#46472
#26113