Finite Group Databasebeta

← #10594939

Subgroups — #10594939

Order 576.

465 conjugacy classes of subgroups.

The subgroup lattice has 318 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#64133C211yesmaximal, characteristic, complemented
2C3×C4×#67C211yesmaximal, characteristic
3C3×#926C211yesmaximal, characteristic, complemented
4#63969C212yesmaximal
5#63956C212yesmaximal, complemented
6D3×C4⋊Dic3C211yesmaximal, characteristic, complemented
7#63877C212yesmaximal, complemented
8#63869C211yesmaximal, characteristic, complemented
9#63867C212yesmaximal, complemented
10#63857C211yesmaximal, characteristic, complemented
11#63844C211yesmaximal, characteristic, complemented
12#5541—31nomaximal
13#5529—31nomaximal
14C22×C32×C4C2211yescharacteristic
15C2×#634C2211yescharacteristic
16C2×C3×#67C2211yescharacteristic
17C2×C3×C4×D3C2211yescharacteristic
18C2×D3×Dic3C2212yes—
19#3757C2212yes—
20C4⋊(C3⋊Dic3)C2211yescharacteristic
21#3750C2212yes—
22C3×#217C2212yes—
23C3×#217C2211yescharacteristic
24C3×(C2×D3)⋊C4C2211yescharacteristic
25C3×C4⋊Dic3C2211yescharacteristic
26C3×C4⋊Dic3C2211yescharacteristic
27C3×Dic3⋊C4C2211yescharacteristic
28C3×C4×Dic3C2212yes—
29C3×C4×Dic3C2211yescharacteristic
30#3724C2212yes—
31#3724C2212yes—
32Dic3⋊Dic3C2212yes—
33Dic3⋊Dic3C2212yes—
34Dic3⋊Dic3C2212yes—
35(C2×D3)⋊Dic3C2212yes—
36Dic32C2212yes—
37C3×C4×D4—31no—
38C3×#117—31no—
39#935—32no—
40C4×#67D311yescharacteristic
41#926D311yescharacteristic, complemented
42C2×C4⋊Dic3—31no—
43#924—31no—
44#924—31no—
45#924—31no—
46#924—61no—
47#897—31no—
48D3×C4⋊C4—31no—
49#890—31no—
50#889—32no—
51#887—31no—
52#883—32no—
53#879—32no—
54#879—32no—
55#877—32no—
56#872—32no—
57#870—31no—
58#870—32no—
59C4×Dic6—31no—
60C23×C32C2311yescharacteristic
61C22×C3×D3C2311yescharacteristic
62C2×C32×C4—22no—
63C2×C32×C4C2311yescharacteristic
64C2×C32×C4C2311yescharacteristic
65C2×C3⋊Dic3C2312yes—
66C2×C3⋊Dic3C2312yes—
67#634—22no—
68C3×#67—22no—
69C2×C3×Dic3C2311yescharacteristic
70C2×C3×Dic3C2312yes—
71C2×C3×Dic3C2312yes—
72C2×C3×Dic3C2311yescharacteristic
73C2×C3×Dic3C2311yescharacteristic
74C3×C4×D3—22no—
75D3×Dic3—24no—
76#552—91noSylow
77C2×C3×D4—31no—
78C22×C3×C4—21no—
79C22×C3×C4—31no—
80C22×C3×C4C2×D311yescharacteristic
81C22×C3×C4C2×D311yescharacteristic
82C2×#67C2×D311yescharacteristic
83C22×Dic3—32no—
84C2×C4×D3—32no—
85C2×C4×D3C2×D311yescharacteristic
86C2×Dic6—31no—
87C2×Dic6—31no—
88C2×Dic6—61no—
89C3×C4⋊C4—31no—
90C3×C4⋊C4—31no—
91C3×C4⋊C4—31no—
92C3×C22⋊C4—31no—
93C3×C22⋊C4—32no—
94C3×C22⋊C4—31no—
95C3×C42—31no—
96C3×C42—32no—
97#217—32no—
98#217—32no—
99#217—32no—
100#217—62no—
101#217C2×D311yescharacteristic
102#217C2×D312yes—
103(C2×D3)⋊C4—32no—
104(C2×D3)⋊C4C2×D311yescharacteristic
105C4⋊Dic3—31no—
106C4⋊Dic3—31no—
107C4⋊Dic3—32no—
108C4⋊Dic3—32no—
109C4⋊Dic3—32no—
110C4⋊Dic3—61no—
111C4⋊Dic3C2×D311yescharacteristic
112C4⋊Dic3C2×D311yescharacteristic
113Dic3⋊C4—32no—
114Dic3⋊C4—32no—
115Dic3⋊C4—32no—
116Dic3⋊C4—32no—
117Dic3⋊C4—32no—
118Dic3⋊C4—32no—
119Dic3⋊C4—32no—
120Dic3⋊C4—32no—
121Dic3⋊C4—32no—
122Dic3⋊C4—62no—
123Dic3⋊C4C2×D311yescharacteristic
124C4×Dic3—32no—
125C4×Dic3—32no—
126C4×Dic3C2×D312yes—
127C4×Dic3C2×D311yescharacteristic
128C22×C32—21no—
129C22×C32—21no—
130C22×C32—21no—
131C22×C32C2411yescharacteristic
132C2×C3×D3—21no—
133C2×C3×D3—21no—
134C2×C3×D3#4112yes—
135C32×C4—21no—
136C32×C4#4112yes—
137C3⋊Dic3—22no—
138C3⋊Dic3—22no—
139C3×Dic3—22no—
140C3×Dic3—21no—
141C3×Dic3—21no—
142C3×Dic3—21no—
143C3×Dic3—22no—
144#126—92no—
145#125—92no—
146#125—91no—
147#123—92no—
148C22⋊Q8—91no—
149C22⋊Q8—91no—
150C4×Q8—91no—
151C4×D4—31no—
152#117—31no—
153#117—92no—
154C2×C4⋊C4—91no—
155C23×C3—21no—
156C23×C3—31no—
157C23×C3C22×D311yescharacteristic
158C23×C3C22×D311yescharacteristic
159C22×D3C22×D311yescharacteristic
160C3×D4—62no—
161C2×C3×C4—22no—
162C2×C3×C4—22no—
163C2×C3×C4—21no—
164C2×C3×C4—21no—
165C2×C3×C4—31no—
166C2×C3×C4—32no—
167C2×C3×C4—31no—
168C2×C3×C4—31no—
169C2×C3×C4—32no—
170C2×C3×C4—42no—
171C2×C3×C4—62no—
172C2×C3×C4C22×D311yescharacteristic
173C2×C3×C4C22×D311yescharacteristic
174C2×C3×C4C22×D311yescharacteristic
175C2×C3×C4C22×D311yescharacteristic
176#67—22no—
177C2×Dic3—32no—
178C2×Dic3—32no—
179C2×Dic3—32no—
180C2×Dic3—32no—
181C2×Dic3—62no—
182C2×Dic3—62no—
183C2×Dic3—64no—
184C2×Dic3C22×D311yescharacteristic
185C2×Dic3C22×D312yes—
186C2×Dic3C22×D312yes—
187C2×Dic3C22×D311yescharacteristic
188C2×Dic3C22×D311yescharacteristic
189C4×D3—22no—
190C4×D3—64no—
191Dic6—62no—
192Dic6—62no—
193Dic6—122no—
194C2×C32—41no—
195C2×C32C2×#4111yescharacteristic
196C2×C32C2×#4111yescharacteristic
197C2×C322−1+411yescharacteristic
198C3×D3—22no—
199C2×Q8—91no—
200C2×D4—31no—
201C22×C4—31no—
202C22×C4—92no—
203C22×C4D3211yescharacteristic
204C4⋊C4—31no—
205C4⋊C4—31no—
206C4⋊C4—31no—
207C4⋊C4—92no—
208C4⋊C4—92no—
209C4⋊C4—92no—
210C4⋊C4—92no—
211C4⋊C4—92no—
212C4⋊C4—91no—
213C4⋊C4—92no—
214C22⋊C4—32no—
215C22⋊C4—31no—
216C22⋊C4—31no—
217C22⋊C4—92no—
218C22⋊C4—92no—
219C42—32no—
220C42—31no—
221C42—92no—
222C22×C3—22no—
223C22×C3—21no—
224C22×C3—21no—
225C22×C3—21no—
226C22×C3—21no—
227C22×C3—21no—
228C22×C3—21no—
229C22×C3—21no—
230C22×C3—32no—
231C22×C3—41no—
232C22×C3—41no—
233C22×C3—61no—
234C22×C3—61no—
235C22×C3C23×D311yescharacteristic
236C22×C3C23×D311yescharacteristic
237C2×D3—21no—
238C2×D3—21no—
239C2×D3#23512yes—
240C3×C4—21no—
241C3×C4—21no—
242C3×C4—22no—
243C3×C4—41no—
244C3×C4—62no—
245C3×C4—61no—
246C3×C4—61no—
247C3×C4—61no—
248C3×C4—62no—
249C3×C4#23512yes—
250C3×C4#23712yes—
251Dic3—21no—
252Dic3—21no—
253Dic3—22no—
254Dic3—22no—
255Dic3—21no—
256Dic3—62no—
257Dic3—62no—
258Dic3—62no—
259Dic3—62no—
260Dic3—122no—
261Dic3—122no—
262C32#55211yescharacteristic, complemented, Sylow
263C23—31no—
264C23C2×D3211yescharacteristic
265Q8—182no—
266D4—62no—
267C2×C4—22no—
268C2×C4—31no—
269C2×C4—32no—
270C2×C4—32no—
271C2×C4—31no—
272C2×C4—31no—
273C2×C4—62no—
274C2×C4—92no—
275C2×C4—92no—
276C2×C4—184no—
277C2×C4C2×D3211yescharacteristic
278C2×C4C2×D3211yescharacteristic
279C2×C3—21no—
280C2×C3—21no—
281C2×C3—21no—
282C2×C3—42no—
283C2×C3—41no—
284C2×C3—41no—
285C2×C3—62no—
286C2×C3C2×#23511yescharacteristic
287C2×C3C2×#23511yescharacteristic
288C2×C3C2×#23711yescharacteristic
289C2×C3D3×#4111yescharacteristic
290C2×C3#101011yescharacteristic
291C2×C3#101011yescharacteristic
292D3—22no—
293C22—21no—
294C22—21no—
295C22—21no—
296C22—32no—
297C22—61no—
298C22—61no—
299C22C22×D3211yescharacteristic
300C4—21no—
301C4—61no—
302C4—62no—
303C4—62no—
304C4—61no—
305C4—61no—
306C4—182no—
307C4—182no—
308C4#380412yes—
309C3—21no—
310C3#552911yescharacteristic, complemented
311C3#554111yescharacteristic, complemented
312C2—41no—
313C2—62no—
314C2#6429811yescharacteristic
315C2D3×#23511yescharacteristic
316C2C2×#380411yescharacteristic
subgroupG/H
center, FrattiniC22C22×D32
commutator, socle, cosocleC22×C32C24
FittingC22×C32×C4C22
prime radicalC2×C3×#67C22
hypercenterC22×C4D32
hypocenterC32#552
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
1. #64133C2semidirect
3. C3×#926C2semidirect
5. #63956C2semidirect
6. D3×C4⋊Dic3C2semidirect
7. #63877C2semidirect
8. #63869C2semidirect
9. #63867C2semidirect
10. #63857C2semidirect
11. #63844C2semidirect
41. #926D3semidirect
262. C32#552semidirect
310. C3#5529semidirect
311. C3#5541semidirect

Maximal subgroups

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

subgroupindex
#641332
C3×C4×#672
C3×#9262
#639692
#639562
D3×C4⋊Dic32
#638772
#638692
#638672
#638572
#638442
#55413
#55293