Finite Group Databasebeta

← #10595367

Subgroups — #10595367

Order 576.

478 conjugacy classes of subgroups.

The subgroup lattice has 307 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#64153C211yesmaximal, characteristic, complemented
2C3×#946C211yesmaximal, characteristic
3C3×Q8×Dic3C211yesmaximal, characteristic, complemented
4#63919C211yesmaximal, characteristic, complemented
5#63899C212yesmaximal
6#63877C212yesmaximal
7C4×D3×Dic3C211yesmaximal, characteristic
8#63857C211yesmaximal, characteristic
9#63847C212yesmaximal, complemented
10#63844C211yesmaximal, characteristic, complemented
11#63837C212yesmaximal, complemented
12#5622—31nomaximal
13#5578—31nomaximal
14C2×C32×Q8C2211yescharacteristic
15C2×#634C2211yescharacteristic
16C2×C3×C4×D3C2211yescharacteristic
17C2×#627C2212yes—
18C2×D3×Dic3C2211yescharacteristic
19C2×D3×Dic3C2211yescharacteristic
20#3750C2212yes—
21#3750C2212yes—
22C4×C3⋊Dic3C2211yescharacteristic
23C3×(C2×D3)⋊C4C2212yes—
24C3×C4⋊Dic3C2212yes—
25C3×C4⋊Dic3C2211yescharacteristic
26C3×C4⋊Dic3C2211yescharacteristic
27C3×Dic3⋊C4C2212yes—
28C3×C4×Dic3C2212yes—
29C3×C4×Dic3C2211yescharacteristic
30#3724C2212yes—
31#3724C2212yes—
32Dic3⋊Dic3C2212yes—
33(C2×D3)⋊Dic3C2212yes—
34Dic32C2211yescharacteristic
35Dic32C2212yes—
36Dic32C2211yescharacteristic
37C3×C22⋊Q8—31no—
38C3×C4×Q8—31no—
39#946D311yescharacteristic
40Q8×Dic3D311yescharacteristic, complemented
41#944—31no—
42#944—31no—
43#944—61no—
44#924—31no—
45C2×C4×Dic3—31no—
46#897—31no—
47#896—32no—
48#889—32no—
49#889—32no—
50#888—31no—
51#888—31no—
52#887—32no—
53#887—31no—
54#878—32no—
55#877—32no—
56#872—32no—
57C42×D3—31no—
58C4×Dic6—31no—
59C4×Dic6—32no—
60C22×C3×D3C2311yescharacteristic
61C32×Q8—22no—
62C2×C32×C4C2312yes—
63C2×C32×C4C2311yescharacteristic
64C2×C3⋊Dic3C2312yes—
65C2×C3⋊Dic3C2311yescharacteristic
66C2×C3⋊Dic3C2311yescharacteristic
67#634—22no—
68C2×C3×Dic3C2311yescharacteristic
69C2×C3×Dic3C2311yescharacteristic
70C2×C3×Dic3C2312yes—
71C2×C3×Dic3C2312yes—
72C2×C3×Dic3C2311yescharacteristic
73C3×C4×D3—22no—
74#627—24no—
75D3×Dic3—22no—
76D3×Dic3Q814yescomplemented
77#533—91noSylow
78C2×C3×Q8—21no—
79C2×C3×Q8C2×D311yescharacteristic
80C2×C3×Q8C2×D311yescharacteristic
81C22×C3×C4—31no—
82C22×Dic3—31no—
83C22×Dic3—31no—
84C2×C4×D3—31no—
85C2×C4×D3—31no—
86C2×C4×D3C2×D311yescharacteristic
87C2×Dic6—31no—
88C2×Dic6—31no—
89C2×Dic6—32no—
90C2×Dic6—32no—
91C2×Dic6—61no—
92C3×C4⋊C4—31no—
93C3×C4⋊C4—32no—
94C3×C4⋊C4—31no—
95C3×C4⋊C4—32no—
96C3×C22⋊C4—32no—
97C3×C42—31no—
98C3×C42—32no—
99#217—32no—
100(C2×D3)⋊C4—32no—
101(C2×D3)⋊C4C2×D312yes—
102C4⋊Dic3—32no—
103C4⋊Dic3C2×D311yescharacteristic
104C4⋊Dic3C2×D312yes—
105C4⋊Dic3C2×D311yescharacteristic
106Dic3⋊C4—32no—
107Dic3⋊C4—32no—
108Dic3⋊C4—32no—
109Dic3⋊C4—32no—
110Dic3⋊C4—32no—
111Dic3⋊C4—32no—
112Dic3⋊C4—32no—
113Dic3⋊C4—32no—
114Dic3⋊C4—32no—
115Dic3⋊C4—62no—
116Dic3⋊C4—62no—
117Dic3⋊C4C2×D312yes—
118C4×Dic3—31no—
119C4×Dic3—31no—
120C4×Dic3—31no—
121C4×Dic3—32no—
122C4×Dic3—31no—
123C4×Dic3—32no—
124C4×Dic3—31no—
125C4×Dic3—31no—
126C4×Dic3—61no—
127C4×Dic3C2×D312yes—
128C4×Dic3C2×D311yescharacteristic
129C22×C32C2411yescharacteristic
130C2×C3×D3—21no—
131C2×C3×D3—21no—
132C2×C3×D3C2×Q812yes—
133C32×C4—22no—
134C32×C4#4112yes—
135C3⋊Dic3—22no—
136C3⋊Dic3C2×Q812yes—
137C3⋊Dic3#4112yes—
138C3×Dic3—21no—
139C3×Dic3—22no—
140C3×Dic3—22no—
141C3×Dic3—21no—
142C3×Dic3C2×Q812yes—
143C4⋊Q8—91no—
144C4⋊Q8—91no—
145#125—92no—
146C22⋊Q8—31no—
147C22⋊Q8—92no—
148C22⋊Q8—91no—
149C4×Q8—31no—
150C4×Q8—91no—
151C4×Q8—92no—
152#117—92no—
153C2×C42—91no—
154C23×C3—31no—
155C22×D3C22×D311yescharacteristic
156C3×Q8—22no—
157C3×Q8—22no—
158C3×Q8—42no—
159C2×C3×C4—21no—
160C2×C3×C4—22no—
161C2×C3×C4—31no—
162C2×C3×C4—31no—
163C2×C3×C4—31no—
164C2×C3×C4—32no—
165C2×C3×C4—32no—
166C2×C3×C4—62no—
167C2×C3×C4C22×D312yes—
168C2×C3×C4C22×D312yes—
169C2×C3×C4C22×D311yescharacteristic
170C2×C3×C4C22×D311yescharacteristic
171C2×Dic3—31no—
172C2×Dic3—31no—
173C2×Dic3—31no—
174C2×Dic3—31no—
175C2×Dic3—34no—
176C2×Dic3—32no—
177C2×Dic3—32no—
178C2×Dic3—61no—
179C2×Dic3—61no—
180C2×Dic3—62no—
181C2×Dic3—62no—
182C2×Dic3C22×D311yescharacteristic
183C2×Dic3C22×D311yescharacteristic
184C2×Dic3C22×D312yes—
185C2×Dic3C22×D312yes—
186C2×Dic3C22×D311yescharacteristic
187C4×D3—22no—
188C4×D3—34no—
189C4×D3—62no—
190Dic6—64no—
191Dic6—64no—
192Dic6—62no—
193Dic6—62no—
194Dic6—122no—
195C2×C32C22×Q811yescharacteristic
196C2×C32C2×#4111yescharacteristic
197C2×C32C2×#4111yescharacteristic
198C3×D3—22no—
199C2×Q8—92no—
200C2×Q8—91no—
201C2×Q8D3211yescharacteristic
202C22×C4—31no—
203C22×C4—91no—
204C22×C4—91no—
205C4⋊C4—32no—
206C4⋊C4—31no—
207C4⋊C4—32no—
208C4⋊C4—31no—
209C4⋊C4—92no—
210C4⋊C4—92no—
211C4⋊C4—92no—
212C4⋊C4—92no—
213C4⋊C4—92no—
214C22⋊C4—32no—
215C22⋊C4—92no—
216C42—31no—
217C42—32no—
218C42—91no—
219C42—91no—
220C42—92no—
221C42—91no—
222C22×C3—21no—
223C22×C3—32no—
224C22×C3—61no—
225C22×C3—61no—
226C22×C3C23×D311yescharacteristic
227C22×C3C23×D311yescharacteristic
228C2×D3—21no—
229C2×D3—21no—
230C2×D3D3×Q812yes—
231C3×C4—22no—
232C3×C4—22no—
233C3×C4—22no—
234C3×C4—32no—
235C3×C4—42no—
236C3×C4—61no—
237C3×C4—62no—
238C3×C4—62no—
239C3×C4—61no—
240C3×C4#23512yes—
241C3×C4#23712yes—
242Dic3—21no—
243Dic3—22no—
244Dic3—22no—
245Dic3—21no—
246Dic3—32no—
247Dic3—32no—
248Dic3—32no—
249Dic3—32no—
250Dic3—62no—
251Dic3—62no—
252Dic3—62no—
253Dic3—62no—
254Dic3—122no—
255Dic3D3×Q812yes—
256C32#53311yescharacteristic, complemented, Sylow
257C23—31no—
258Q8—22no—
259Q8—184no—
260Q8—182no—
261C2×C4—31no—
262C2×C4—31no—
263C2×C4—32no—
264C2×C4—31no—
265C2×C4—32no—
266C2×C4—62no—
267C2×C4—94no—
268C2×C4—91no—
269C2×C4—91no—
270C2×C4—92no—
271C2×C4—182no—
272C2×C4C2×D3212yes—
273C2×C4C2×D3211yescharacteristic
274C2×C3—21no—
275C2×C3—21no—
276C2×C3—21no—
277C2×C3—62no—
278C2×C3C2×#23511yescharacteristic
279C2×C3C2×#23711yescharacteristic
280C2×C3C2×D3×Q811yescharacteristic
281C2×C3C2×D3×Q811yescharacteristic
282C2×C3D3×#4111yescharacteristic
283C2×C3D3×#4111yescharacteristic
284D3—22no—
285C22—32no—
286C22—61no—
287C22—61no—
288C22C22×D3211yescharacteristic
289C4—22no—
290C4—32no—
291C4—61no—
292C4—62no—
293C4—62no—
294C4—61no—
295C4—92no—
296C4—92no—
297C4—182no—
298C4#380412yes—
299C3—21no—
300C3#557811yescharacteristic, complemented
301C3#562211yescharacteristic, complemented
302C2—62no—
303C2D32×Q811yescharacteristic
304C2#6431911yescharacteristic
305C2C2×#380411yescharacteristic
subgroupG/H
center, FrattiniC22C22×D32
commutator, socle, cosocleC22×C32C24
FittingC2×C32×Q8C22
prime radicalC22×C3×D3C23
hypercenterC2×Q8D32
hypocenterC32#533
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
1. #64153C2semidirect
3. C3×Q8×Dic3C2semidirect
4. #63919C2semidirect
9. #63847C2semidirect
10. #63844C2semidirect
11. #63837C2semidirect
40. Q8×Dic3D3semidirect
76. D3×Dic3Q8semidirect
256. C32#533semidirect
300. C3#5578semidirect
301. C3#5622semidirect

Maximal subgroups

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

subgroupindex
#641532
C3×#9462
C3×Q8×Dic32
#639192
#638992
#638772
C4×D3×Dic32
#638572
#638472
#638442
#638372
#56223
#55783