Finite Group Databasebeta

← #78059

Subgroups — #78059

Order 384.

434 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 434 conjugacy classes of subgroups — too many to draw. Only its 133 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×#879C211yesmaximal, characteristic
3C3×#399C211yesmaximal, characteristic
4#4985C211yesmaximal, characteristic, complemented
5#4984C212yesmaximal, complemented
6#4983C212yesmaximal, complemented
7#4953C211yesmaximal, characteristic
8#4951C212yesmaximal
9#4948C212yesmaximal
10#4661C211yesmaximal, characteristic, complemented
11#4649C211yesmaximal, characteristic, complemented
12#2664—31nomaximal, Sylow
13C2×C3×C4⋊C4C2212yes—
14C2×C3×C4⋊C4C2211yescharacteristic
15C2×C3×C22⋊C4C2212yes—
16C2×C3×C22⋊C4C2211yescharacteristic
17C2×#217C2211yescharacteristic
18C2×#217C2211yescharacteristic
19C2×#217C2212yes—
20C2×C4⋊Dic3C2211yescharacteristic
21C2×C4⋊Dic3C2212yes—
22C2×C4⋊Dic3C2211yescharacteristic
23C2×C4⋊Dic3C2212yes—
24C2×Dic3⋊C4C2212yes—
25C2×Dic3⋊C4C2211yescharacteristic
26C2×Dic3⋊C4C2212yes—
27C2×Dic3⋊C4C2211yescharacteristic
28C2×C4×Dic3C2212yes—
29C2×C4×Dic3C2211yescharacteristic
30#890—24no—
31#879—24no—
32C3×#95C2211yescharacteristic
33C3⋊#95C2211yescharacteristic
34C3⋊#95C2212yes—
35C3⋊#95C2212yes—
36C3⋊#95C2212yes—
37C3⋊#95C2212yes—
38C2×#126—31no—
39C2×#125—31no—
40#400—31no—
41#399D311yescharacteristic
42#398—32no—
43#398—31no—
44#397—31no—
45#396—32no—
46#389—32no—
47#388—32no—
48C4×C4⋊C4—31no—
49C24×C3C2311yescharacteristic
50C22×C3×C4C2311yescharacteristic
51C22×C3×C4C2312yes—
52C22×C3×C4C2311yescharacteristic
53C22×C3×C4C2312yes—
54C22×Dic3C2312yes—
55C22×Dic3C2311yescharacteristic
56C22×Dic3C2312yes—
57C22×Dic3C2311yescharacteristic
58C22×Dic3C2311yescharacteristic
59C22×Dic3C2311yescharacteristic
60C3×C4⋊C4—22no—
61C3×C4⋊C4—24no—
62C3×C22⋊C4—22no—
63C3×C22⋊C4—24no—
64#217—22no—
65#217—22no—
66#217—42no—
67C4⋊Dic3—22no—
68C4⋊Dic3—22no—
69C4⋊Dic3—24no—
70C4⋊Dic3—42no—
71Dic3⋊C4—22no—
72Dic3⋊C4—24no—
73Dic3⋊C4—22no—
74Dic3⋊C4—24no—
75C4×Dic3—42no—
76C4×Dic3D414yes—
77#126—64no—
78#125—64no—
79C2×C4⋊C4—32no—
80C2×C4⋊C4—32no—
81C2×C4⋊C4—31no—
82C2×C4⋊C4—31no—
83C2×C4⋊C4—31no—
84C2×C4⋊C4—31no—
85C2×C4⋊C4—32no—
86C2×C4⋊C4—32no—
87C2×C4⋊C4C2×D311yescharacteristic
88C2×C4⋊C4C2×D312yes—
89C2×C22⋊C4—31no—
90C2×C22⋊C4—31no—
91C2×C22⋊C4—32no—
92C2×C22⋊C4C2×D311yescharacteristic
93C2×C22⋊C4C2×D312yes—
94C2×C42—31no—
95C2×C42—32no—
96#95—31no—
97#95—32no—
98#95—32no—
99#95—32no—
100#95—32no—
101#95C2×D311yescharacteristic
102C23×C3—21no—
103C23×C3—22no—
104C23×C3—21no—
105C23×C3—21no—
106C23×C3—22no—
107C23×C3C2411yescharacteristic
108C2×C3×C4—22no—
109C2×C3×C4—22no—
110C2×C3×C4—22no—
111C2×C3×C4—21no—
112C2×C3×C4—22no—
113C2×C3×C4—22no—
114C2×C3×C4—22no—
115C2×C3×C4—22no—
116C2×C3×C4C2×D412yes—
117C2×C3×C4#4114yes—
118C2×Dic3—22no—
119C2×Dic3—21no—
120C2×Dic3—21no—
121C2×Dic3—21no—
122C2×Dic3—22no—
123C2×Dic3—21no—
124C2×Dic3—22no—
125C2×Dic3—22no—
126C2×Dic3—21no—
127C2×Dic3—22no—
128C2×Dic3—22no—
129C2×Dic3—21no—
130C2×Dic3—22no—
131C2×Dic3—22no—
132C2×Dic3C2×D412yes—
133C2×Dic3C2×D412yes—
134C24C22×D311yescharacteristic
135C22×C4—31no—
136C22×C4—31no—
137C22×C4—32no—
138C22×C4—31no—
139C22×C4—31no—
140C22×C4—32no—
141C22×C4C22×D311yescharacteristic
142C22×C4C22×D312yes—
143C22×C4C22×D311yescharacteristic
144C22×C4C22×D312yes—
145C4⋊C4—22no—
146C4⋊C4—24no—
147C4⋊C4—64no—
148C4⋊C4—64no—
149C4⋊C4—62no—
150C4⋊C4—62no—
151C4⋊C4—62no—
152C4⋊C4—62no—
153C4⋊C4—64no—
154C4⋊C4—122no—
155C22⋊C4—22no—
156C22⋊C4—24no—
157C22⋊C4—62no—
158C22⋊C4—62no—
159C22⋊C4—122no—
160C42—34no—
161C42—122no—
162C22×C3—41no—
163C22×C3—41no—
164C22×C3—41no—
165C22×C3—42no—
166C22×C3—42no—
167C22×C3C22×D411yescharacteristic
168C22×C3C2×#4112yes—
169C22×C3H(5,𝔽2)11yescharacteristic
170C22×C32−1+411yescharacteristic
171C22×C32−1+412yes—
172C3×C4—22no—
173C3×C4—24no—
174C3×C4—41no—
175C3×C4—42no—
176Dic3—22no—
177Dic3—22no—
178Dic3—41no—
179Dic3—42no—
180Dic3—41no—
181Dic3—42no—
182C23—21no—
183C23—22no—
184C23—21no—
185C23—21no—
186C23—22no—
187C23C23×D311yescharacteristic
188C2×C4—22no—
189C2×C4—22no—
190C2×C4—22no—
191C2×C4—21no—
192C2×C4—22no—
193C2×C4—22no—
194C2×C4—22no—
195C2×C4—22no—
196C2×C4—32no—
197C2×C4—32no—
198C2×C4—62no—
199C2×C4—61no—
200C2×C4—61no—
201C2×C4—61no—
202C2×C4—61no—
203C2×C4—62no—
204C2×C4—62no—
205C2×C4—61no—
206C2×C4—62no—
207C2×C4—62no—
208C2×C4—62no—
209C2×C4—61no—
210C2×C4—62no—
211C2×C4—62no—
212C2×C4D3×D412yes—
213C2×C4#23714yes—
214C2×C3—81no—
215C2×C3#53211yescharacteristic
216C2×C3#53611yescharacteristic
217C2×C3#53711yescharacteristic
218C2×C3#53912yes—
219C2×C3#54112yes—
220C22—41no—
221C22—42no—
222C22—41no—
223C22—42no—
224C22—41no—
225C22C2×D3×D411yescharacteristic
226C22C2×#23712yes—
227C22#100711yescharacteristic
228C22#100911yescharacteristic
229C22#101012yes—
230C4—22no—
231C4—24no—
232C4—41no—
233C4—42no—
234C4—62no—
235C4—62no—
236C4—121no—
237C4—122no—
238C4—121no—
239C4—122no—
240C3#266411yescharacteristic, complemented, Sylow
241C2—81no—
242C2#562512yes—
243C2#562911yescharacteristic
244C2#564811yescharacteristic
245C2#566512yes—
246C2#571611yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC3×#399C2
prime radicalC24×C3C23
hypercenter#399D3
hypocenterC3#2664
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #399

Complements

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

NH ≅ G/Ntype
1. C2×#890C2semidirect
4. #4985C2semidirect
5. #4984C2semidirect
6. #4983C2semidirect
10. #4661C2semidirect
11. #4649C2semidirect
240. C3#2664semidirect

Maximal subgroups

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

subgroupindex
C2×#8902
C2×#8792
C3×#3992
#49852
#49842
#49832
#49532
#49512
#49482
#46612
#46492
#26643