Finite Group Databasebeta

← #73793

Subgroups — #73793

Order 384.

428 conjugacy classes of subgroups.

The subgroup lattice has 272 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×C4×C4⋊C4C211yesmaximal, characteristic, complemented
2#4987C211yesmaximal, characteristic
3#4987C211yesmaximal, characteristic
4#4975C211yesmaximal, characteristic, complemented
5#4972C211yesmaximal, characteristic, complemented
6#4940C212yesmaximal
7C4×(C2×D3)⋊C4C211yesmaximal, characteristic
8#4933C212yesmaximal, complemented
9#4928C211yesmaximal, characteristic, complemented
10#4676C212yesmaximal
11#4658C212yesmaximal, complemented
12#2497—31nomaximal, Sylow
13C22×C4×D3C2211yescharacteristic
14C2×C3×C4⋊C4C2211yescharacteristic
15C2×C3×C4⋊C4C2211yescharacteristic
16C2×C3×C42C2211yescharacteristic
17C2×C3×C42C2212yes—
18C2×(C2×D3)⋊C4C2211yescharacteristic
19C2×(C2×D3)⋊C4C2212yes—
20C2×(C2×D3)⋊C4C2212yes—
21C2×(C2×D3)⋊C4C2211yescharacteristic
22C2×C4⋊Dic3C2211yescharacteristic
23C2×C4⋊Dic3C2211yescharacteristic
24C2×Dic3⋊C4C2212yes—
25C2×Dic3⋊C4C2212yes—
26C2×Dic3⋊C4C2212yes—
27C2×Dic3⋊C4C2212yes—
28C2×C4×Dic3C2211yescharacteristic
29C3×#95C2212yes—
30C3⋊#95C2211yescharacteristic
31C3⋊#95C2212yes—
32C3⋊#95C2212yes—
33C3⋊#95C2212yes—
34C3⋊#95C2212yes—
35C3⋊#95C2211yescharacteristic
36#400—32no—
37#396—32no—
38#389—31no—
39#389—31no—
40#388—32no—
41#387—32no—
42#384—31no—
43#380—31no—
44#380—31no—
45C4×C4⋊C4D311yescharacteristic, complemented
46C4×C22⋊C4—31no—
47C23×D3C2311yescharacteristic
48C22×C3×C4C2312yes—
49C22×C3×C4C2311yescharacteristic
50C22×C3×C4C2312yes—
51C22×C3×C4C2311yescharacteristic
52C22×C3×C4C2311yescharacteristic
53C22×Dic3C2311yescharacteristic
54C22×Dic3C2312yes—
55C22×Dic3C2312yes—
56C22×Dic3C2311yescharacteristic
57C22×Dic3C2311yescharacteristic
58C2×C4×D3—22no—
59C2×C4×D3—22no—
60C2×C4×D3—22no—
61C3×C4⋊C4—22no—
62C3×C4⋊C4—22no—
63C3×C42—22no—
64C3×C42—42no—
65(C2×D3)⋊C4—22no—
66(C2×D3)⋊C4—42no—
67(C2×D3)⋊C4—42no—
68(C2×D3)⋊C4Q814yes—
69C4⋊Dic3—22no—
70C4⋊Dic3—22no—
71Dic3⋊C4—24no—
72Dic3⋊C4—42no—
73Dic3⋊C4—42no—
74Dic3⋊C4—42no—
75C4×Dic3—22no—
76C23×C4—31no—
77C2×C4⋊C4—31no—
78C2×C4⋊C4—31no—
79C2×C4⋊C4—32no—
80C2×C4⋊C4—32no—
81C2×C4⋊C4—32no—
82C2×C4⋊C4—32no—
83C2×C4⋊C4C2×D311yescharacteristic
84C2×C4⋊C4C2×D311yescharacteristic
85C2×C22⋊C4—31no—
86C2×C22⋊C4—31no—
87C2×C22⋊C4—32no—
88C2×C22⋊C4—32no—
89C2×C42—31no—
90C2×C42C2×D311yescharacteristic
91C2×C42C2×D312yes—
92#95—31no—
93#95—31no—
94#95—32no—
95#95—32no—
96#95—32no—
97#95—32no—
98#95C2×D312yes—
99C23×C3C2411yescharacteristic
100C22×D3—21no—
101C22×D3—21no—
102C22×D3—21no—
103C22×D3—21no—
104C22×D3—21no—
105C22×D3—21no—
106C22×D3C2×Q812yes—
107C2×C3×C4—21no—
108C2×C3×C4—21no—
109C2×C3×C4—22no—
110C2×C3×C4—21no—
111C2×C3×C4—21no—
112C2×C3×C4—22no—
113C2×C3×C4—22no—
114C2×C3×C4—21no—
115C2×C3×C4—22no—
116C2×C3×C4—22no—
117C2×C3×C4—22no—
118C2×C3×C4C2×Q812yes—
119C2×C3×C4#4112yes—
120C2×C3×C4#4112yes—
121C2×C3×C4#4112yes—
122C2×Dic3—21no—
123C2×Dic3—21no—
124C2×Dic3—21no—
125C2×Dic3—21no—
126C2×Dic3—22no—
127C2×Dic3—22no—
128C2×Dic3—22no—
129C2×Dic3—21no—
130C2×Dic3—21no—
131C2×Dic3—21no—
132C2×Dic3—22no—
133C2×Dic3—22no—
134C2×Dic3—22no—
135C2×Dic3C2×Q812yes—
136C2×Dic3#4112yes—
137C4×D3—44no—
138C24—31no—
139C22×C4—31no—
140C22×C4—31no—
141C22×C4—31no—
142C22×C4—32no—
143C22×C4—32no—
144C22×C4—62no—
145C22×C4—62no—
146C22×C4—62no—
147C22×C4C22×D312yes—
148C22×C4C22×D311yescharacteristic
149C22×C4C22×D312yes—
150C22×C4C22×D311yescharacteristic
151C22×C4C22×D311yescharacteristic
152C4⋊C4—22no—
153C4⋊C4—22no—
154C4⋊C4—64no—
155C4⋊C4—62no—
156C4⋊C4—62no—
157C4⋊C4—122no—
158C4⋊C4—122no—
159C4⋊C4—122no—
160C22⋊C4—34no—
161C22⋊C4—62no—
162C22⋊C4—122no—
163C22⋊C4—122no—
164C42—22no—
165C42—42no—
166C42—62no—
167C22×C3C22×Q811yescharacteristic
168C22×C3C2×#4111yescharacteristic
169C22×C3C2×#4111yescharacteristic
170C22×C3C2×#4111yescharacteristic
171C22×C3C2×#4111yescharacteristic
172C22×C3H(5,𝔽2)11yescharacteristic
173C22×C3H(5,𝔽2)11yescharacteristic
174C2×D3—22no—
175C2×D3—22no—
176C2×D3—22no—
177C2×D3—41no—
178C2×D3—41no—
179C2×D3—41no—
180C2×D3—41no—
181C3×C4—22no—
182C3×C4—42no—
183C3×C4—42no—
184C3×C4—41no—
185C3×C4#12614yes—
186Dic3—22no—
187Dic3—22no—
188Dic3—41no—
189Dic3—42no—
190Dic3—42no—
191C23—32no—
192C23—61no—
193C23—61no—
194C23—61no—
195C23—61no—
196C23—61no—
197C23—61no—
198C23C23×D311yescharacteristic
199C2×C4—21no—
200C2×C4—21no—
201C2×C4—21no—
202C2×C4—22no—
203C2×C4—22no—
204C2×C4—22no—
205C2×C4—22no—
206C2×C4—21no—
207C2×C4—22no—
208C2×C4—22no—
209C2×C4—21no—
210C2×C4—32no—
211C2×C4—32no—
212C2×C4—61no—
213C2×C4—61no—
214C2×C4—61no—
215C2×C4—61no—
216C2×C4—62no—
217C2×C4—62no—
218C2×C4—61no—
219C2×C4—61no—
220C2×C4—61no—
221C2×C4—62no—
222C2×C4—62no—
223C2×C4—62no—
224C2×C4—62no—
225C2×C4—124no—
226C2×C4#23512yes—
227C2×C4#23512yes—
228C2×C4#23512yes—
229C2×C4D3×Q812yes—
230C2×C3C2×#12611yescharacteristic
231C2×C3#53311yescharacteristic
232C2×C3#54011yescharacteristic
233C2×C3#55311yescharacteristic
234C2×C3#55411yescharacteristic
235C2×C3#55411yescharacteristic
236C2×C3#55511yescharacteristic
237D3—42no—
238C22—62no—
239C22—62no—
240C22—62no—
241C22—121no—
242C22—121no—
243C22—121no—
244C22—121no—
245C22C2×#23511yescharacteristic
246C22C2×#23511yescharacteristic
247C22C2×#23511yescharacteristic
248C22#100411yescharacteristic
249C22C2×D3×Q811yescharacteristic
250C22D3×#4111yescharacteristic
251C22#100911yescharacteristic
252C4—22no—
253C4—41no—
254C4—42no—
255C4—42no—
256C4—62no—
257C4—62no—
258C4—121no—
259C4—122no—
260C4—122no—
261C4#87614yes—
262C3#249711yescharacteristic, complemented, Sylow
263C2—122no—
264C2C2×#87611yescharacteristic
265C2#551111yescharacteristic
266C2#553211yescharacteristic
267C2#555811yescharacteristic
268C2#556211yescharacteristic
269C2#557811yescharacteristic
270C2#557911yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC3×C4×C4⋊C4C2
prime radicalC23×D3C23
hypercenterC4×C4⋊C4D3
hypocenterC3#2497
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ C4×C4⋊C4

Complements

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

NH ≅ G/Ntype
1. C3×C4×C4⋊C4C2semidirect
4. #4975C2semidirect
5. #4972C2semidirect
8. #4933C2semidirect
9. #4928C2semidirect
11. #4658C2semidirect
45. C4×C4⋊C4D3semidirect
262. C3#2497semidirect

Maximal subgroups

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

subgroupindex
C3×C4×C4⋊C42
#49872
#49752
#49722
#49402
C4×(C2×D3)⋊C42
#49332
#49282
#46762
#46582
#24973