Finite Group Databasebeta

← #10594993

Subgroups — #10594993

Order 576.

502 conjugacy classes of subgroups.

The subgroup lattice has 326 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4⋊(C3⋊Dic3)C211yesmaximal, characteristic, complemented
2C3×C4×#67C211yesmaximal, characteristic
3C3×#924C211yesmaximal, characteristic, complemented
4#63982C212yesmaximal, complemented
5#67⋊Dic3C212yesmaximal
6#63918C211yesmaximal, characteristic, complemented
7#63893C212yesmaximal, complemented
8D3×C4⋊Dic3C211yesmaximal, characteristic, complemented
9#63856C211yesmaximal, characteristic, complemented
10#63845C212yesmaximal, complemented
11Dic3×Dic6C211yesmaximal, characteristic, complemented
12#5635—31nomaximal
13#5543—31nomaximal
14C22×C32×C4C2211yescharacteristic
15C22×C3⋊Dic3C2212yes—
16C2×C3×#67C2211yescharacteristic
17C2×C3×C4×D3C2211yescharacteristic
18C2×C3×Dic6C2211yescharacteristic
19C2×#627C2212yes—
20C2×D3×Dic3C2212yes—
21C4⋊(C3⋊Dic3)C2211yescharacteristic
22C4⋊(C3⋊Dic3)C2212yes—
23C4⋊(C3⋊Dic3)C2211yescharacteristic
24C3×#217C2212yes—
25C3×#217C2211yescharacteristic
26C3×(C2×D3)⋊C4C2211yescharacteristic
27C3×C4⋊Dic3C2211yescharacteristic
28C3×Dic3⋊C4C2212yes—
29C3×Dic3⋊C4C2211yescharacteristic
30C3×C4×Dic3C2211yescharacteristic
31#3724C2212yes—
32Dic3⋊Dic3C2212yes—
33Dic3⋊Dic3C2212yes—
34Dic3⋊Dic3C2212yes—
35(C2×D3)⋊Dic3C2212yes—
36Dic32C2212yes—
37C3×C22⋊Q8—31no—
38C3×C4×D4—31no—
39#946—31no—
40Q8×Dic3—31no—
41D4×Dic3—32no—
42C4×#67D311yescharacteristic
43C2×C4⋊Dic3—31no—
44C2×C4⋊Dic3—31no—
45C2×C4⋊Dic3—31no—
46C2×C4⋊Dic3—61no—
47#924—31no—
48#924D311yescharacteristic, complemented
49#897—32no—
50D3×C4⋊C4—31no—
51#890—32no—
52#889—32no—
53#888—31no—
54#67⋊C4—32no—
55#878—32no—
56#878—32no—
57#878—32no—
58#869—31no—
59C4×Dic6—31no—
60C23×C32C2311yescharacteristic
61C22×C3×D3C2311yescharacteristic
62C2×C32×C4—22no—
63C2×C32×C4C2311yescharacteristic
64C2×C32×C4C2311yescharacteristic
65C2×C3⋊Dic3—42no—
66C2×C3⋊Dic3C2312yes—
67C2×C3⋊Dic3C2312yes—
68C3×#67Q814yescomplemented
69C2×C3×Dic3C2312yes—
70C2×C3×Dic3C2311yescharacteristic
71C2×C3×Dic3C2311yescharacteristic
72C2×C3×Dic3C2312yes—
73C2×C3×Dic3C2311yescharacteristic
74C3×C4×D3—22no—
75C3×Dic6—22no—
76#627—42no—
77D3×Dic3—42no—
78#554—91noSylow
79C2×C3×Q8—31no—
80C2×C3×D4—31no—
81C22×C3×C4—21no—
82C22×C3×C4—31no—
83C22×C3×C4C2×D311yescharacteristic
84C22×C3×C4C2×D311yescharacteristic
85C2×#67C2×D311yescharacteristic
86C22×Dic3—32no—
87C22×Dic3—32no—
88C22×Dic3—32no—
89C22×Dic3—62no—
90C2×C4×D3—32no—
91C2×C4×D3C2×D311yescharacteristic
92C2×Dic6—32no—
93C2×Dic6—32no—
94C2×Dic6C2×D311yescharacteristic
95C3×C4⋊C4—31no—
96C3×C4⋊C4—32no—
97C3×C4⋊C4—31no—
98C3×C22⋊C4—31no—
99C3×C22⋊C4—32no—
100C3×C22⋊C4—31no—
101C3×C42—31no—
102#217—32no—
103#217C2×D311yescharacteristic
104#217C2×D312yes—
105(C2×D3)⋊C4—32no—
106(C2×D3)⋊C4C2×D311yescharacteristic
107C4⋊Dic3—31no—
108C4⋊Dic3—31no—
109C4⋊Dic3—32no—
110C4⋊Dic3—31no—
111C4⋊Dic3—32no—
112C4⋊Dic3—31no—
113C4⋊Dic3—32no—
114C4⋊Dic3—32no—
115C4⋊Dic3—32no—
116C4⋊Dic3—61no—
117C4⋊Dic3—62no—
118C4⋊Dic3—61no—
119C4⋊Dic3C2×D311yescharacteristic
120Dic3⋊C4—32no—
121Dic3⋊C4—32no—
122Dic3⋊C4—32no—
123Dic3⋊C4—32no—
124Dic3⋊C4—32no—
125Dic3⋊C4C2×D312yes—
126Dic3⋊C4C2×D311yescharacteristic
127C4×Dic3—32no—
128C4×Dic3—32no—
129C4×Dic3C2×D311yescharacteristic
130C22×C32—21no—
131C22×C32—21no—
132C22×C32C2×Q812yes—
133C22×C32C2411yescharacteristic
134C2×C3×D3—21no—
135C2×C3×D3—21no—
136C2×C3×D3C2×Q812yes—
137C32×C4—21no—
138C32×C4#4112yes—
139C3⋊Dic3—22no—
140C3⋊Dic3—22no—
141C3×Dic3—22no—
142C3×Dic3—21no—
143C3×Dic3—22no—
144C3×Dic3—21no—
145C3×Dic3C2×Q812yes—
146C4⋊Q8—91no—
147#125—92no—
148C22⋊Q8—31no—
149C22⋊Q8—92no—
150C22⋊Q8—92no—
151C22⋊Q8—91no—
152C4×Q8—91no—
153C4×D4—31no—
154C4×D4—92no—
155C2×C4⋊C4—91no—
156C2×C4⋊C4—91no—
157C23×C3—21no—
158C23×C3—31no—
159C23×C3C22×D311yescharacteristic
160C23×C3C22×D311yescharacteristic
161C22×D3C22×D311yescharacteristic
162C3×Q8—62no—
163C3×D4—34no—
164C2×C3×C4—24no—
165C2×C3×C4—22no—
166C2×C3×C4—22no—
167C2×C3×C4—21no—
168C2×C3×C4—21no—
169C2×C3×C4—31no—
170C2×C3×C4—32no—
171C2×C3×C4—31no—
172C2×C3×C4—31no—
173C2×C3×C4—32no—
174C2×C3×C4—62no—
175C2×C3×C4C22×D311yescharacteristic
176C2×C3×C4C22×D311yescharacteristic
177C2×C3×C4C22×D311yescharacteristic
178C2×C3×C4C22×D311yescharacteristic
179#67Dic614yescomplemented
180C2×Dic3—32no—
181C2×Dic3—32no—
182C2×Dic3—32no—
183C2×Dic3—32no—
184C2×Dic3—62no—
185C2×Dic3—62no—
186C2×Dic3—122no—
187C2×Dic3—122no—
188C2×Dic3—122no—
189C2×Dic3—124no—
190C2×Dic3C22×D312yes—
191C2×Dic3C22×D311yescharacteristic
192C2×Dic3C22×D311yescharacteristic
193C2×Dic3C22×D312yes—
194C2×Dic3C22×D311yescharacteristic
195C4×D3—22no—
196C4×D3—122no—
197Dic6—22no—
198Dic6—122no—
199Dic6—122no—
200C2×C32—22no—
201C2×C32C22×Q811yescharacteristic
202C2×C32C2×#4111yescharacteristic
203C2×C32H(5,𝔽2)11yescharacteristic
204C3×D3—22no—
205C2×Q8—31no—
206C2×Q8—92no—
207C2×D4—31no—
208C22×C4—31no—
209C22×C4—92no—
210C22×C4—92no—
211C22×C4D3211yescharacteristic
212C4⋊C4—31no—
213C4⋊C4—31no—
214C4⋊C4—32no—
215C4⋊C4—92no—
216C4⋊C4—92no—
217C4⋊C4—92no—
218C4⋊C4—92no—
219C4⋊C4—91no—
220C4⋊C4—92no—
221C4⋊C4—91no—
222C22⋊C4—31no—
223C22⋊C4—32no—
224C22⋊C4—31no—
225C22⋊C4—92no—
226C42—31no—
227C42—92no—
228C22×C3—22no—
229C22×C3—22no—
230C22×C3—21no—
231C22×C3—21no—
232C22×C3—21no—
233C22×C3—21no—
234C22×C3—21no—
235C22×C3—22no—
236C22×C3—32no—
237C22×C3—61no—
238C22×C3—61no—
239C22×C3C2×Dic612yes—
240C22×C3D3×Q812yes—
241C22×C3C23×D311yescharacteristic
242C22×C3C23×D311yescharacteristic
243C2×D3—21no—
244C2×D3—21no—
245C2×D3C2×Dic612yes—
246C3×C4—22no—
247C3×C4—21no—
248C3×C4—21no—
249C3×C4—22no—
250C3×C4—32no—
251C3×C4—61no—
252C3×C4—62no—
253C3×C4—61no—
254C3×C4—62no—
255C3×C4#23712yes—
256C3×C4#23712yes—
257Dic3—22no—
258Dic3—21no—
259Dic3—21no—
260Dic3—22no—
261Dic3—62no—
262Dic3—62no—
263Dic3—62no—
264Dic3—62no—
265Dic3—122no—
266Dic3—122no—
267Dic3C2×Dic612yes—
268C32#55411yescharacteristic, complemented, Sylow
269C23—31no—
270C23C2×D3211yescharacteristic
271Q8—62no—
272Q8—362no—
273D4—34no—
274C2×C4—22no—
275C2×C4—31no—
276C2×C4—31no—
277C2×C4—32no—
278C2×C4—32no—
279C2×C4—31no—
280C2×C4—62no—
281C2×C4—92no—
282C2×C4—92no—
283C2×C4—362no—
284C2×C4—362no—
285C2×C4C2×D3211yescharacteristic
286C2×C4C2×D3211yescharacteristic
287C2×C3—24no—
288C2×C3—22no—
289C2×C3—22no—
290C2×C3—21no—
291C2×C3—21no—
292C2×C3—21no—
293C2×C3—62no—
294C2×C3C22×Dic611yescharacteristic
295C2×C3C2×#23711yescharacteristic
296C2×C3C2×#23711yescharacteristic
297C2×C3C2×D3×Q811yescharacteristic
298C2×C3#100911yescharacteristic
299C2×C3#100911yescharacteristic
300D3—22no—
301C22—21no—
302C22—21no—
303C22—32no—
304C22—61no—
305C22—61no—
306C22D3×Dic612yes—
307C22C22×D3211yescharacteristic
308C4—21no—
309C4—32no—
310C4—62no—
311C4—61no—
312C4—61no—
313C4—62no—
314C4—182no—
315C4—182no—
316C4#380512yes—
317C3—21no—
318C3#554311yescharacteristic, complemented
319C3#563511yescharacteristic, complemented
320C2—22no—
321C2—62no—
322C2C2×D3×Dic611yescharacteristic
323C2#6430811yescharacteristic
324C2C2×#380511yescharacteristic
subgroupG/H
center, FrattiniC22C22×D32
commutator, socle, cosocleC22×C32C24
FittingC22×C32×C4C22
prime radicalC2×C3×#67C22
hypercenterC22×C4D32
hypocenterC32#554
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
1. C2×C4⋊(C3⋊Dic3)C2semidirect
3. C3×#924C2semidirect
4. #63982C2semidirect
6. #63918C2semidirect
7. #63893C2semidirect
8. D3×C4⋊Dic3C2semidirect
9. #63856C2semidirect
10. #63845C2semidirect
11. Dic3×Dic6C2semidirect
48. #924D3semidirect
68. C3×#67Q8semidirect
179. #67Dic6semidirect
268. C32#554semidirect
318. C3#5543semidirect
319. C3#5635semidirect

Maximal subgroups

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

subgroupindex
C2×C4⋊(C3⋊Dic3)2
C3×C4×#672
C3×#9242
#639822
#67⋊Dic32
#639182
#638932
D3×C4⋊Dic32
#638562
#638452
Dic3×Dic62
#56353
#55433