Finite Group Databasebeta

← #10594452

Subgroups — #10594452

Order 576.

484 conjugacy classes of subgroups.

The subgroup lattice has 305 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#64099C211yesmaximal, characteristic, complemented
2C3×#893C211yesmaximal, characteristic
3C3×#890C211yesmaximal, characteristic, complemented
4#63918C211yesmaximal, characteristic, complemented
5C4×#627C211yesmaximal, characteristic, complemented
6#63899C212yesmaximal, complemented
7#63898C211yesmaximal, characteristic
8#63893C212yesmaximal, complemented
9Dic3×D12C211yesmaximal, characteristic
10D3×C4⋊Dic3C212yesmaximal, complemented
11#63851C212yesmaximal, complemented
12#5690—31nomaximal
13#5543—31nomaximal
14C2×C3×D12C2211yescharacteristic
15C2×C3×C4×D3C2212yes—
16C2×#627C2211yescharacteristic
17C2×#627C2212yes—
18C2×D3×Dic3C2212yes—
19C2×D3×Dic3C2212yes—
20C32×C4⋊C4C2211yescharacteristic
21C4⋊(C3⋊Dic3)C2212yes—
22C4⋊(C3⋊Dic3)C2211yescharacteristic
23#3750C2212yes—
24C4×C3⋊Dic3C2211yescharacteristic
25C3×(C2×D3)⋊C4C2212yes—
26C3×C4⋊Dic3C2212yes—
27C3×C4⋊Dic3C2211yescharacteristic
28C3×Dic3⋊C4C2212yes—
29C3×C4×Dic3C2211yescharacteristic
30C3×C4×Dic3C2211yescharacteristic
31#3724C2211yescharacteristic
32Dic3⋊Dic3C2212yes—
33Dic3⋊Dic3C2211yescharacteristic
34Dic3⋊Dic3C2211yescharacteristic
35(C2×D3)⋊Dic3C2212yes—
36(C2×D3)⋊Dic3C2212yes—
37C3×#125—31no—
38C3×C4×D4—31no—
39D4×Dic3—31no—
40D4×Dic3—31no—
41C2×C4⋊Dic3—32no—
42#924—32no—
43#897—32no—
44#896—32no—
45#896—32no—
46#893—31no—
47#893D311yescharacteristic
48D3×C4⋊C4—32no—
49#890—31no—
50#890—31no—
51#890—61no—
52#890D311yescharacteristic, complemented
53#888—31no—
54#878—32no—
55#878—32no—
56C4×D12—31no—
57#869—31no—
58C4×Dic6—31no—
59C4×Dic6—31no—
60C22×C3×D3C2312yes—
61C2×C32×C4C2312yes—
62C2×C32×C4C2311yescharacteristic
63C2×C3⋊Dic3C2311yescharacteristic
64C2×C3⋊Dic3C2312yes—
65C2×C3⋊Dic3C2311yescharacteristic
66C2×C3×Dic3C2311yescharacteristic
67C2×C3×Dic3C2312yes—
68C2×C3×Dic3C2311yescharacteristic
69C2×C3×Dic3C2311yescharacteristic
70C2×C3×Dic3C2311yescharacteristic
71C3×D12Q814yescomplemented
72C3×C4×D3—24no—
73#627—22no—
74#627—42no—
75D3×Dic3—42no—
76D3×Dic3—42no—
77#554—91noSylow
78C2×C3×D4—31no—
79C22×C3×C4—32no—
80C22×Dic3—32no—
81C22×Dic3—32no—
82C2×D12C2×D311yescharacteristic
83C2×C4×D3—32no—
84C2×C4×D3—32no—
85C2×C4×D3C2×D312yes—
86C2×Dic6—31no—
87C2×Dic6—31no—
88C2×Dic6—32no—
89C2×Dic6—32no—
90C3×C4⋊C4—21no—
91C3×C4⋊C4—31no—
92C3×C4⋊C4—32no—
93C3×C4⋊C4—32no—
94C3×C4⋊C4C2×D311yescharacteristic
95C3×C4⋊C4C2×D311yescharacteristic
96C3×C22⋊C4—32no—
97C3×C42—31no—
98C3×C42—31no—
99#217—32no—
100#217—32no—
101(C2×D3)⋊C4—32no—
102(C2×D3)⋊C4—32no—
103(C2×D3)⋊C4C2×D312yes—
104C4⋊Dic3—31no—
105C4⋊Dic3—31no—
106C4⋊Dic3—32no—
107C4⋊Dic3—32no—
108C4⋊Dic3—31no—
109C4⋊Dic3—31no—
110C4⋊Dic3—32no—
111C4⋊Dic3—62no—
112C4⋊Dic3—61no—
113C4⋊Dic3C2×D311yescharacteristic
114C4⋊Dic3C2×D312yes—
115Dic3⋊C4—32no—
116Dic3⋊C4—32no—
117Dic3⋊C4—31no—
118Dic3⋊C4—31no—
119Dic3⋊C4—31no—
120Dic3⋊C4—31no—
121Dic3⋊C4—32no—
122Dic3⋊C4—62no—
123Dic3⋊C4C2×D312yes—
124C4×Dic3—31no—
125C4×Dic3—31no—
126C4×Dic3—61no—
127C4×Dic3C2×D311yescharacteristic
128C4×Dic3C2×D311yescharacteristic
129C22×C32C2411yescharacteristic
130C2×C3×D3—22no—
131C2×C3×D3—22no—
132C2×C3×D3C2×Q814yes—
133C32×C4—22no—
134C32×C4C2×Q812yes—
135C3⋊Dic3—21no—
136C3⋊Dic3—21no—
137C3⋊Dic3—22no—
138C3×Dic3—21no—
139C3×Dic3—21no—
140C3×Dic3—22no—
141C3×Dic3—21no—
142C3×Dic3#4112yes—
143C4⋊Q8—91no—
144#125—31no—
145#125—91no—
146C22⋊Q8—92no—
147C22⋊Q8—92no—
148C22⋊Q8—92no—
149C4×Q8—91no—
150C4×D4—31no—
151C4×D4—91no—
152C4×D4—91no—
153C2×C4⋊C4—92no—
154C23×C3—32no—
155C22×D3C22×D312yes—
156C3×D4—34no—
157C2×C3×C4—21no—
158C2×C3×C4—22no—
159C2×C3×C4—31no—
160C2×C3×C4—31no—
161C2×C3×C4—31no—
162C2×C3×C4—32no—
163C2×C3×C4—31no—
164C2×C3×C4—64no—
165C2×C3×C4C22×D312yes—
166C2×C3×C4C22×D312yes—
167C2×C3×C4C22×D311yescharacteristic
168C2×C3×C4C22×D311yescharacteristic
169C2×Dic3—31no—
170C2×Dic3—31no—
171C2×Dic3—31no—
172C2×Dic3—31no—
173C2×Dic3—32no—
174C2×Dic3—32no—
175C2×Dic3—61no—
176C2×Dic3—61no—
177C2×Dic3—62no—
178C2×Dic3—122no—
179C2×Dic3—122no—
180C2×Dic3C22×D311yescharacteristic
181C2×Dic3C22×D311yescharacteristic
182C2×Dic3C22×D311yescharacteristic
183C2×Dic3C22×D312yes—
184C2×Dic3C22×D311yescharacteristic
185D12Dic614yescomplemented
186C4×D3—24no—
187C4×D3—122no—
188C4×D3—122no—
189Dic6—62no—
190Dic6—62no—
191Dic6—122no—
192Dic6—122no—
193C2×C32C22×Q811yescharacteristic
194C2×C32C2×#4111yescharacteristic
195C2×C32H(5,𝔽2)11yescharacteristic
196C3×D3—24no—
197C2×Q8—91no—
198C2×Q8—92no—
199C2×D4—31no—
200C22×C4—32no—
201C22×C4—92no—
202C22×C4—92no—
203C4⋊C4—32no—
204C4⋊C4—31no—
205C4⋊C4—32no—
206C4⋊C4—91no—
207C4⋊C4—92no—
208C4⋊C4—91no—
209C4⋊C4—91no—
210C4⋊C4—92no—
211C4⋊C4—91no—
212C4⋊C4—92no—
213C4⋊C4D3211yescharacteristic
214C22⋊C4—32no—
215C22⋊C4—92no—
216C22⋊C4—92no—
217C42—31no—
218C42—31no—
219C42—91no—
220C22×C3—21no—
221C22×C3—34no—
222C22×C3—62no—
223C22×C3—62no—
224C22×C3C23×D311yescharacteristic
225C22×C3C23×D311yescharacteristic
226C2×D3—22no—
227C2×D3—22no—
228C2×D3C2×Dic614yes—
229C3×C4—22no—
230C3×C4—22no—
231C3×C4—22no—
232C3×C4—32no—
233C3×C4—42no—
234C3×C4—61no—
235C3×C4—61no—
236C3×C4—62no—
237C3×C4—61no—
238C3×C4C2×Dic612yes—
239C3×C4D3×Q812yes—
240Dic3—21no—
241Dic3—21no—
242Dic3—22no—
243Dic3—21no—
244Dic3—61no—
245Dic3—61no—
246Dic3—62no—
247Dic3—62no—
248Dic3—61no—
249Dic3—61no—
250Dic3—121no—
251Dic3—121no—
252Dic3—122no—
253Dic3#23712yes—
254C32#55411yescharacteristic, complemented, Sylow
255C23—32no—
256Q8—182no—
257Q8—362no—
258D4—34no—
259C2×C4—31no—
260C2×C4—31no—
261C2×C4—31no—
262C2×C4—31no—
263C2×C4—32no—
264C2×C4—64no—
265C2×C4—91no—
266C2×C4—91no—
267C2×C4—92no—
268C2×C4—362no—
269C2×C4—362no—
270C2×C4C2×D3212yes—
271C2×C4C2×D3211yescharacteristic
272C2×C3—21no—
273C2×C3—21no—
274C2×C3—21no—
275C2×C3—64no—
276C2×C3C22×Dic611yescharacteristic
277C2×C3C2×#23711yescharacteristic
278C2×C3C2×D3×Q811yescharacteristic
279C2×C3D3×#4111yescharacteristic
280C2×C3#100911yescharacteristic
281C2×C3#100911yescharacteristic
282D3—24no—
283C22—34no—
284C22—62no—
285C22—62no—
286C22C22×D3211yescharacteristic
287C4—22no—
288C4—32no—
289C4—61no—
290C4—61no—
291C4—62no—
292C4—61no—
293C4—181no—
294C4—181no—
295C4—182no—
296C4D3×Dic612yes—
297C3—21no—
298C3#554311yescharacteristic, complemented
299C3#569011yescharacteristic, complemented
300C2—64no—
301C2C2×D3×Dic611yescharacteristic
302C2D3×#23711yescharacteristic
303C2#6432011yescharacteristic
subgroupG/H
center, FrattiniC22C22×D32
commutator, socle, cosocleC22×C32C24
FittingC32×C4⋊C4C22
prime radicalC2×C3×D12C22
hypercenterC4⋊C4D32
hypocenterC32#554
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
1. #64099C2semidirect
3. C3×#890C2semidirect
4. #63918C2semidirect
5. C4×#627C2semidirect
6. #63899C2semidirect
8. #63893C2semidirect
10. D3×C4⋊Dic3C2semidirect
11. #63851C2semidirect
52. #890D3semidirect
71. C3×D12Q8semidirect
185. D12Dic6semidirect
254. C32#554semidirect
298. C3#5543semidirect
299. C3#5690semidirect

Maximal subgroups

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

subgroupindex
#640992
C3×#8932
C3×#8902
#639182
C4×#6272
#638992
#638982
#638932
Dic3×D122
D3×C4⋊Dic32
#638512
#56903
#55433