Finite Group Databasebeta

← #77276

Subgroups — #77276

Order 384.

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

normal subgroups only

This group has 430 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
1C2×#944C211yesmaximal, characteristic, complemented
2C2×#898C211yesmaximal, characteristic
3C3×#395C211yesmaximal, characteristic, complemented
4#5231C211yesmaximal, characteristic
5#4993C212yesmaximal
6#4982C212yesmaximal, complemented
7#4979C211yesmaximal, characteristic, complemented
8#4675C212yesmaximal
9#4673C211yesmaximal, characteristic
10#4658C212yesmaximal, complemented
11#4651C211yesmaximal, characteristic, complemented
12#2821—31nomaximal, Sylow
13C22×C3×Q8C2211yescharacteristic
14C22×Dic6C2211yescharacteristic
15C2×C3×C4⋊C4C2211yescharacteristic
16C2×C3×C4⋊C4C2212yes—
17#944—24no—
18C2×(C2×D3)⋊C4C2211yescharacteristic
19C2×(C2×D3)⋊C4C2211yescharacteristic
20C2×(C2×D3)⋊C4C2211yescharacteristic
21C2×(C2×D3)⋊C4C2212yes—
22C2×(C2×D3)⋊C4C2212yes—
23C2×C4⋊Dic3C2211yescharacteristic
24C2×Dic3⋊C4C2211yescharacteristic
25C2×Dic3⋊C4C2212yes—
26C2×Dic3⋊C4C2212yes—
27C2×Dic3⋊C4C2212yes—
28C2×C4×Dic3C2211yescharacteristic
29#898—24no—
30C3×#95C2211yescharacteristic
31C3×#95C2212yes—
32C3⋊#95C2212yes—
33C3⋊#95C2212yes—
34C3⋊#95C2211yescharacteristic
35C3⋊#95C2212yes—
36C3⋊#95C2212yes—
37C3⋊#95C2212yes—
38C2×C4⋊Q8—31no—
39C2×#126—31no—
40#400—32no—
41#399—32no—
42#398—32no—
43#397—32no—
44#395—31no—
45#395D311yescharacteristic, complemented
46#393—31no—
47#393—31no—
48#383—31no—
49C23×D3C2311yescharacteristic
50C2×C3×Q8—22no—
51C2×C3×Q8—42no—
52C22×C3×C4C2312yes—
53C22×C3×C4C2311yescharacteristic
54C22×C3×C4C2312yes—
55C22×C3×C4C2311yescharacteristic
56C22×C3×C4C2311yescharacteristic
57C22×Dic3C2312yes—
58C22×Dic3C2311yescharacteristic
59C22×Dic3C2312yes—
60C22×Dic3C2311yescharacteristic
61C22×Dic3C2311yescharacteristic
62C2×Dic6—22no—
63C2×Dic6—42no—
64C3×C4⋊C4—22no—
65C3×C4⋊C4—42no—
66(C2×D3)⋊C4—22no—
67(C2×D3)⋊C4—22no—
68(C2×D3)⋊C4—22no—
69(C2×D3)⋊C4—42no—
70(C2×D3)⋊C4—42no—
71C4⋊Dic3—22no—
72Dic3⋊C4—22no—
73Dic3⋊C4—24no—
74Dic3⋊C4—24no—
75Dic3⋊C4—42no—
76C4×Dic3D414yes—
77C22×Q8—31no—
78C22×Q8C2×D311yescharacteristic
79C4⋊Q8—64no—
80#126—64no—
81C2×C4⋊C4—31no—
82C2×C4⋊C4—31no—
83C2×C4⋊C4—32no—
84C2×C4⋊C4—32no—
85C2×C4⋊C4—32no—
86C2×C4⋊C4C2×D311yescharacteristic
87C2×C4⋊C4C2×D312yes—
88C2×C22⋊C4—31no—
89C2×C22⋊C4—31no—
90C2×C22⋊C4—31no—
91C2×C22⋊C4—32no—
92C2×C22⋊C4—32no—
93C2×C42—31no—
94#95—31no—
95#95—32no—
96#95—32no—
97#95—32no—
98#95—32no—
99#95—32no—
100#95C2×D311yescharacteristic
101#95C2×D312yes—
102C23×C3C2411yescharacteristic
103C22×D3—21no—
104C22×D3—22no—
105C22×D3—22no—
106C22×D3—21no—
107C22×D3—21no—
108C3×Q8—44no—
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—21no—
118C2×C3×C4—22no—
119C2×C3×C4—21no—
120C2×C3×C4—22no—
121C2×C3×C4C2×D412yes—
122C2×Dic3—21no—
123C2×Dic3—22no—
124C2×Dic3—21no—
125C2×Dic3—22no—
126C2×Dic3—21no—
127C2×Dic3—22no—
128C2×Dic3—22no—
129C2×Dic3—22no—
130C2×Dic3—22no—
131C2×Dic3—22no—
132C2×Dic3—22no—
133C2×Dic3C2×D412yes—
134C2×Dic3C2×D412yes—
135Dic6—44no—
136C24—31no—
137C2×Q8—22no—
138C2×Q8—42no—
139C2×Q8—62no—
140C2×Q8—122no—
141C22×C4—31no—
142C22×C4—31no—
143C22×C4—32no—
144C22×C4—31no—
145C22×C4—32no—
146C22×C4C22×D312yes—
147C22×C4C22×D311yescharacteristic
148C22×C4C22×D312yes—
149C22×C4C22×D311yescharacteristic
150C22×C4C22×D311yescharacteristic
151C4⋊C4—22no—
152C4⋊C4—42no—
153C4⋊C4—62no—
154C4⋊C4—62no—
155C4⋊C4—64no—
156C4⋊C4—64no—
157C4⋊C4—122no—
158C22⋊C4—62no—
159C22⋊C4—62no—
160C22⋊C4—62no—
161C22⋊C4—122no—
162C22⋊C4—122no—
163C42—34no—
164C22×C3C22×D411yescharacteristic
165C22×C3H(5,𝔽2)12yes—
166C22×C32−1+412yes—
167C22×C32−1+411yescharacteristic
168C22×C32−1+411yescharacteristic
169C2×D3—41no—
170C2×D3—41no—
171C2×D3—41no—
172C2×D3—42no—
173C2×D3—42no—
174C3×C4—22no—
175C3×C4—41no—
176C3×C4—42no—
177C3×C4—41no—
178C3×C4—42no—
179Dic3—22no—
180Dic3—22no—
181Dic3—42no—
182Dic3—41no—
183Dic3—42no—
184C23—61no—
185C23—62no—
186C23—62no—
187C23—61no—
188C23—61no—
189C23C23×D311yescharacteristic
190Q8—44no—
191Q8—124no—
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—22no—
201C2×C4—21no—
202C2×C4—22no—
203C2×C4—21no—
204C2×C4—32no—
205C2×C4—32no—
206C2×C4—62no—
207C2×C4—61no—
208C2×C4—61no—
209C2×C4—61no—
210C2×C4—62no—
211C2×C4—62no—
212C2×C4—62no—
213C2×C4—62no—
214C2×C4—62no—
215C2×C4—62no—
216C2×C4—62no—
217C2×C4D3×D412yes—
218C2×C3#53511yescharacteristic
219C2×C3#53611yescharacteristic
220C2×C3#53611yescharacteristic
221C2×C3#56312yes—
222C2×C3#56312yes—
223D3—81no—
224C22—122no—
225C22—121no—
226C22—122no—
227C22—121no—
228C22—121no—
229C22C2×D3×D411yescharacteristic
230C22#100412yes—
231C22#100712yes—
232C22#100711yescharacteristic
233C22#101011yescharacteristic
234C4—22no—
235C4—42no—
236C4—42no—
237C4—41no—
238C4—41no—
239C4—62no—
240C4—62no—
241C4—121no—
242C4—122no—
243C4—122no—
244C3#282111yescharacteristic, complemented, Sylow
245C2—241no—
246C2#559211yescharacteristic
247C2#562811yescharacteristic
248C2#564612yes—
249C2#572411yescharacteristic
250C2#573512yes—
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC3×#395C2
prime radicalC23×D3C23
hypercenter#395D3
hypocenterC3#2821
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #395

Complements

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

NH ≅ G/Ntype
1. C2×#944C2semidirect
3. C3×#395C2semidirect
6. #4982C2semidirect
7. #4979C2semidirect
10. #4658C2semidirect
11. #4651C2semidirect
45. #395D3semidirect
244. C3#2821semidirect

Maximal subgroups

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

subgroupindex
C2×#9442
C2×#8982
C3×#3952
#52312
#49932
#49822
#49792
#46752
#46732
#46582
#46512
#28213