Finite Group Databasebeta

← #10594808

Subgroups — #10594808

Order 576.

481 conjugacy classes of subgroups.

The subgroup lattice has 310 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#64133C211yesmaximal, characteristic
2C3×#924C211yesmaximal, characteristic
3C2×C3×C4×Dic3C211yesmaximal, characteristic
4#63964C212yesmaximal
5#63955C212yesmaximal
6#63918C211yesmaximal, characteristic, complemented
7#63867C212yesmaximal, complemented
8#63866C211yesmaximal, characteristic, complemented
9#63844C211yesmaximal, characteristic, complemented
10Dic3×Dic6C211yesmaximal, characteristic, complemented
11#63837C212yesmaximal, complemented
12#5622—31nomaximal
13#5470—31nomaximal
14C22×C32×C4C2211yescharacteristic
15C2×#634C2211yescharacteristic
16C22×C3×Dic3C2211yescharacteristic
17C22×C3×Dic3C2211yescharacteristic
18C2×C3×Dic6C2211yescharacteristic
19C2×#627C2212yes—
20#3757C2212yes—
21C4⋊(C3⋊Dic3)C2211yescharacteristic
22#3750C2212yes—
23C3×#217C2212yes—
24C3×C4⋊Dic3C2211yescharacteristic
25C3×Dic3⋊C4C2212yes—
26C3×C4×Dic3C2211yescharacteristic
27C3×C4×Dic3C2211yescharacteristic
28C3×C4×Dic3C2211yescharacteristic
29C3×C4×Dic3C2211yescharacteristic
30#3724C2212yes—
31Dic3⋊Dic3C2212yes—
32Dic3⋊Dic3C2212yes—
33Dic3⋊Dic3C2212yes—
34Dic3⋊Dic3C2212yes—
35Dic32C2212yes—
36Dic32C2212yes—
37C3×C22⋊Q8—31no—
38C2×C3×C42—31no—
39Q8×Dic3—31no—
40#944—31no—
41#926—32no—
42#924—31no—
43#924—31no—
44#924—32no—
45#924—61no—
46#924D311yescharacteristic
47C2×C4×Dic3D311yescharacteristic
48#889—32no—
49#888—31no—
50#887—32no—
51#887—31no—
52#878—32no—
53#877—32no—
54#870—32no—
55#869—31no—
56#869—31no—
57C4×Dic6—31no—
58C4×Dic6—31no—
59C4×Dic6—32no—
60C23×C32C2311yescharacteristic
61C2×C32×C4—22no—
62C2×C32×C4C2311yescharacteristic
63C2×C32×C4C2311yescharacteristic
64C2×C3⋊Dic3C2312yes—
65C2×C3⋊Dic3C2312yes—
66#634—22no—
67C2×C3×Dic3—22no—
68C2×C3×Dic3Q814yescomplemented
69C2×C3×Dic3C2312yes—
70C2×C3×Dic3C2312yes—
71C2×C3×Dic3C2311yescharacteristic
72C2×C3×Dic3C2311yescharacteristic
73C2×C3×Dic3C2311yescharacteristic
74C2×C3×Dic3C2311yescharacteristic
75C3×Dic6—22no—
76#627—24no—
77#533—91noSylow
78C2×C3×Q8—31no—
79C22×C3×C4—21no—
80C22×C3×C4—31no—
81C22×C3×C4—31no—
82C22×C3×C4C2×D311yescharacteristic
83C22×C3×C4C2×D311yescharacteristic
84C22×Dic3C2×D311yescharacteristic
85C22×Dic3C2×D311yescharacteristic
86C2×Dic6—31no—
87C2×Dic6—31no—
88C2×Dic6—32no—
89C2×Dic6—32no—
90C2×Dic6—61no—
91C2×Dic6C2×D311yescharacteristic
92C3×C4⋊C4—31no—
93C3×C4⋊C4—32no—
94C3×C22⋊C4—32no—
95C3×C42—31no—
96C3×C42—31no—
97C3×C42—31no—
98C3×C42—31no—
99#217—32no—
100#217—32no—
101#217—62no—
102#217C2×D312yes—
103C4⋊Dic3—31no—
104C4⋊Dic3—31no—
105C4⋊Dic3—32no—
106C4⋊Dic3—32no—
107C4⋊Dic3—32no—
108C4⋊Dic3—32no—
109C4⋊Dic3—61no—
110C4⋊Dic3C2×D311yescharacteristic
111Dic3⋊C4—32no—
112Dic3⋊C4—32no—
113Dic3⋊C4—32no—
114Dic3⋊C4—32no—
115Dic3⋊C4—32no—
116Dic3⋊C4—32no—
117Dic3⋊C4—32no—
118Dic3⋊C4—32no—
119Dic3⋊C4—62no—
120Dic3⋊C4C2×D312yes—
121C4×Dic3—32no—
122C4×Dic3—32no—
123C4×Dic3—32no—
124C4×Dic3—32no—
125C4×Dic3C2×D311yescharacteristic
126C4×Dic3C2×D311yescharacteristic
127C4×Dic3C2×D311yescharacteristic
128C4×Dic3C2×D311yescharacteristic
129C22×C32—21no—
130C22×C32—21no—
131C22×C32C2×Q812yes—
132C22×C32C2411yescharacteristic
133C32×C4—21no—
134C32×C4#4112yes—
135C3⋊Dic3—22no—
136C3⋊Dic3—22no—
137C3×Dic3—22no—
138C3×Dic3—22no—
139C3×Dic3—21no—
140C3×Dic3C2×Q812yes—
141C3×Dic3C2×Q812yes—
142C3×Dic3#4112yes—
143C4⋊Q8—91no—
144C4⋊Q8—91no—
145#125—92no—
146C22⋊Q8—31no—
147C22⋊Q8—92no—
148C22⋊Q8—91no—
149C4×Q8—92no—
150C4×Q8—91no—
151C4×Q8—91no—
152#117—92no—
153C2×C42—31no—
154C23×C3—21no—
155C23×C3C22×D311yescharacteristic
156C23×C3C22×D311yescharacteristic
157C3×Q8—62no—
158C2×C3×C4—22no—
159C2×C3×C4—22no—
160C2×C3×C4—21no—
161C2×C3×C4—21no—
162C2×C3×C4—34no—
163C2×C3×C4—31no—
164C2×C3×C4—31no—
165C2×C3×C4—31no—
166C2×C3×C4—32no—
167C2×C3×C4—32no—
168C2×C3×C4—31no—
169C2×C3×C4—42no—
170C2×C3×C4—62no—
171C2×C3×C4C22×D311yescharacteristic
172C2×C3×C4C22×D311yescharacteristic
173C2×C3×C4C22×D311yescharacteristic
174C2×C3×C4C22×D311yescharacteristic
175C2×Dic3—22no—
176C2×Dic3—32no—
177C2×Dic3—32no—
178C2×Dic3—32no—
179C2×Dic3—32no—
180C2×Dic3—62no—
181C2×Dic3—62no—
182C2×Dic3Dic614yescomplemented
183C2×Dic3C22×D312yes—
184C2×Dic3C22×D312yes—
185C2×Dic3C22×D311yescharacteristic
186C2×Dic3C22×D311yescharacteristic
187C2×Dic3C22×D311yescharacteristic
188C2×Dic3C22×D311yescharacteristic
189Dic6—22no—
190Dic6—64no—
191Dic6—62no—
192Dic6—62no—
193Dic6—64no—
194Dic6—122no—
195C2×C32—22no—
196C2×C32C22×Q811yescharacteristic
197C2×C32C2×#4111yescharacteristic
198C2×C32C2×#4111yescharacteristic
199C2×Q8—31no—
200C2×Q8—92no—
201C2×Q8—91no—
202C22×C4—31no—
203C22×C4—31no—
204C22×C4D3211yescharacteristic
205C4⋊C4—32no—
206C4⋊C4—31no—
207C4⋊C4—92no—
208C4⋊C4—92no—
209C4⋊C4—92no—
210C4⋊C4—92no—
211C4⋊C4—92no—
212C4⋊C4—92no—
213C4⋊C4—91no—
214C22⋊C4—32no—
215C22⋊C4—92no—
216C42—31no—
217C42—31no—
218C42—31no—
219C42—31no—
220C42—92no—
221C42—92no—
222C22×C3—21no—
223C22×C3—21no—
224C22×C3—21no—
225C22×C3—21no—
226C22×C3—21no—
227C22×C3—22no—
228C22×C3—41no—
229C22×C3—41no—
230C22×C3C2×Dic612yes—
231C22×C3D3×Q812yes—
232C22×C3C23×D311yescharacteristic
233C22×C3C23×D311yescharacteristic
234C3×C4—21no—
235C3×C4—21no—
236C3×C4—22no—
237C3×C4—32no—
238C3×C4—32no—
239C3×C4—32no—
240C3×C4—41no—
241C3×C4—62no—
242C3×C4—62no—
243C3×C4—61no—
244C3×C4#23512yes—
245C3×C4#23712yes—
246Dic3—21no—
247Dic3—22no—
248Dic3—22no—
249Dic3—62no—
250Dic3—62no—
251Dic3—62no—
252Dic3—62no—
253Dic3—122no—
254Dic3—122no—
255Dic3C2×Dic612yes—
256Dic3C2×Dic612yes—
257Dic3#23512yes—
258C32#53311yescharacteristic, complemented, Sylow
259C23C2×D3211yescharacteristic
260Q8—62no—
261Q8—184no—
262Q8—182no—
263C2×C4—22no—
264C2×C4—31no—
265C2×C4—31no—
266C2×C4—31no—
267C2×C4—34no—
268C2×C4—31no—
269C2×C4—32no—
270C2×C4—32no—
271C2×C4—62no—
272C2×C4—92no—
273C2×C4—92no—
274C2×C4C2×D3211yescharacteristic
275C2×C4C2×D3211yescharacteristic
276C2×C3—22no—
277C2×C3—22no—
278C2×C3—21no—
279C2×C3—21no—
280C2×C3—21no—
281C2×C3—42no—
282C2×C3C22×Dic611yescharacteristic
283C2×C3C2×#23511yescharacteristic
284C2×C3C2×#23511yescharacteristic
285C2×C3C2×#23711yescharacteristic
286C2×C3C2×D3×Q811yescharacteristic
287C2×C3D3×#4111yescharacteristic
288C22—21no—
289C22—21no—
290C22D3×Dic612yes—
291C22C22×D3211yescharacteristic
292C4—21no—
293C4—32no—
294C4—32no—
295C4—32no—
296C4—61no—
297C4—62no—
298C4—62no—
299C4—182no—
300C4—182no—
301C4#380412yes—
302C3—21no—
303C3#547011yescharacteristic, complemented
304C3#562211yescharacteristic, complemented
305C2—22no—
306C2C2×D3×Dic611yescharacteristic
307C2D3×#23511yescharacteristic
308C2C2×#380411yescharacteristic
subgroupG/H
center, FrattiniC22C22×D32
commutator, socle, cosocleC22×C32C24
FittingC22×C32×C4C22
prime radicalC23×C32C23
hypercenterC22×C4D32
hypocenterC32#533
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
6. #63918C2semidirect
7. #63867C2semidirect
8. #63866C2semidirect
9. #63844C2semidirect
10. Dic3×Dic6C2semidirect
11. #63837C2semidirect
68. C2×C3×Dic3Q8semidirect
182. C2×Dic3Dic6semidirect
258. C32#533semidirect
303. C3#5470semidirect
304. C3#5622semidirect

Maximal subgroups

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

subgroupindex
#641332
C3×#9242
C2×C3×C4×Dic32
#639642
#639552
#639182
#638672
#638662
#638442
Dic3×Dic62
#638372
#56223
#54703