Finite Group Databasebeta

← #10594376

Subgroups — #10594376

Order 576.

594 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
1C2×D3×Dic6C212yesmaximal, complemented
2#64097C211yesmaximal, characteristic, complemented
3C3×#893C211yesmaximal, characteristic
4C3×#888C211yesmaximal, characteristic, complemented
5#63918C211yesmaximal, characteristic, complemented
6C4×#627C211yesmaximal, characteristic, complemented
7#63898C211yesmaximal, characteristic
8#63896C212yesmaximal, complemented
9Dic3×D12C211yesmaximal, characteristic
10#63887C212yesmaximal, complemented
11#63851C212yesmaximal, complemented
12#5727—31nomaximal
13D4×Dic6—31nomaximal
14C2×#634C2212yes—
15C2×C3×D12C2211yescharacteristic
16C2×C3×C4×D3C2212yes—
17C2×C3×Dic6C2212yes—
18C2×#627C2212yes—
19C2×#627C2211yescharacteristic
20C2×D3×Dic3C2212yes—
21C2×D3×Dic3C2212yes—
22D3×Dic6—28no—
23C32×C4⋊C4C2211yescharacteristic
24C4⋊(C3⋊Dic3)C2211yescharacteristic
25#3750C2212yes—
26C4×C3⋊Dic3C2211yescharacteristic
27C3×(C2×D3)⋊C4C2212yes—
28C3×C4⋊Dic3C2211yescharacteristic
29C3×Dic3⋊C4C2212yes—
30C3×C4×Dic3C2211yescharacteristic
31C3×C4×Dic3C2211yescharacteristic
32#3724C2211yescharacteristic
33Dic3⋊Dic3C2211yescharacteristic
34Dic3⋊Dic3C2211yescharacteristic
35Dic3⋊Dic3C2212yes—
36(C2×D3)⋊Dic3C2212yes—
37(C2×D3)⋊Dic3C2212yes—
38C2×D3×Q8—32no—
39C22×Dic6—32no—
40C3×C4⋊Q8—31no—
41C3×C4×D4—31no—
42#946—32no—
43D4×Dic3—31no—
44D4×Dic3—31no—
45#924—32no—
46#896—32no—
47#896—32no—
48#893—31no—
49#893D311yescharacteristic
50#888—31no—
51#888—31no—
52#888—31no—
53#888—61no—
54#888D311yescharacteristic, complemented
55#878—32no—
56#878—32no—
57C4×D12—31no—
58#869—31no—
59C4×Dic6—31no—
60C4×Dic6—31no—
61C22×C3×D3C2312yes—
62C2×C32×C4C2312yes—
63C2×C32×C4C2311yescharacteristic
64C2×C3⋊Dic3C2311yescharacteristic
65C2×C3⋊Dic3C2312yes—
66C2×C3⋊Dic3C2311yescharacteristic
67#634—24no—
68C2×C3×Dic3C2311yescharacteristic
69C2×C3×Dic3C2312yes—
70C2×C3×Dic3C2311yescharacteristic
71C2×C3×Dic3C2311yescharacteristic
72C2×C3×Dic3C2311yescharacteristic
73C3×D12Q814yescomplemented
74C3×C4×D3—24no—
75C3×Dic6—24no—
76#627—24no—
77#627D414yescomplemented
78D3×Dic3—24no—
79D3×Dic3—24no—
80D4×Q8—91noSylow
81C2×C3×Q8—32no—
82C2×C3×D4—31no—
83C22×C3×C4—32no—
84C22×Dic3—32no—
85C22×Dic3—32no—
86D3×Q8—68no—
87C2×D12C2×D311yescharacteristic
88C2×C4×D3—32no—
89C2×C4×D3—32no—
90C2×C4×D3C2×D312yes—
91C2×Dic6—31no—
92C2×Dic6—31no—
93C2×Dic6—32no—
94C2×Dic6—32no—
95C2×Dic6—32no—
96C2×Dic6—32no—
97C2×Dic6—62no—
98C2×Dic6—68no—
99C2×Dic6C2×D312yes—
100C3×C4⋊C4—21no—
101C3×C4⋊C4—31no—
102C3×C4⋊C4—32no—
103C3×C4⋊C4C2×D311yescharacteristic
104C3×C4⋊C4C2×D311yescharacteristic
105C3×C22⋊C4—32no—
106C3×C42—31no—
107C3×C42—31no—
108#217—32no—
109#217—32no—
110(C2×D3)⋊C4—32no—
111(C2×D3)⋊C4—32no—
112(C2×D3)⋊C4C2×D312yes—
113C4⋊Dic3—31no—
114C4⋊Dic3—31no—
115C4⋊Dic3—31no—
116C4⋊Dic3—31no—
117C4⋊Dic3—32no—
118C4⋊Dic3—61no—
119C4⋊Dic3C2×D311yescharacteristic
120Dic3⋊C4—32no—
121Dic3⋊C4—32no—
122Dic3⋊C4—31no—
123Dic3⋊C4—31no—
124Dic3⋊C4—31no—
125Dic3⋊C4—31no—
126Dic3⋊C4—32no—
127Dic3⋊C4—62no—
128Dic3⋊C4C2×D312yes—
129C4×Dic3—31no—
130C4×Dic3—31no—
131C4×Dic3—61no—
132C4×Dic3C2×D311yescharacteristic
133C4×Dic3C2×D311yescharacteristic
134C22×C32C2411yescharacteristic
135C2×C3×D3—22no—
136C2×C3×D3—22no—
137C2×C3×D3C2×Q814yescomplemented
138C32×C4—22no—
139C32×C4C2×Q812yescomplemented
140C3⋊Dic3—21no—
141C3⋊Dic3—22no—
142C3⋊Dic3C2×D412yes—
143C3×Dic3—22no—
144C3×Dic3—21no—
145C3×Dic3—21no—
146C3×Dic3C2×D412yes—
147C3×Dic3C2×D412yes—
148C22×Q8—92no—
149C4⋊Q8—31no—
150C4⋊Q8—91no—
151C4⋊Q8—91no—
152C22⋊Q8—92no—
153C22⋊Q8—92no—
154C22⋊Q8—92no—
155C4×Q8—91no—
156C4×D4—31no—
157C4×D4—91no—
158C4×D4—91no—
159C23×C3—32no—
160C22×D3C22×D312yes—
161C3×Q8—64no—
162C3×D4—34no—
163C2×C3×C4—21no—
164C2×C3×C4—22no—
165C2×C3×C4—31no—
166C2×C3×C4—31no—
167C2×C3×C4—31no—
168C2×C3×C4—31no—
169C2×C3×C4—32no—
170C2×C3×C4—64no—
171C2×C3×C4C22×D312yes—
172C2×C3×C4C22×D312yes—
173C2×C3×C4C22×D311yescharacteristic
174C2×C3×C4C22×D311yescharacteristic
175C2×Dic3—31no—
176C2×Dic3—31no—
177C2×Dic3—31no—
178C2×Dic3—31no—
179C2×Dic3—32no—
180C2×Dic3—32no—
181C2×Dic3—61no—
182C2×Dic3—61no—
183C2×Dic3—62no—
184C2×Dic3—64no—
185C2×Dic3—64no—
186C2×Dic3C22×D311yescharacteristic
187C2×Dic3C22×D311yescharacteristic
188C2×Dic3C22×D312yes—
189C2×Dic3C22×D311yescharacteristic
190C2×Dic3C22×D311yescharacteristic
191D12Dic614yescomplemented
192C4×D3—24no—
193C4×D3—64no—
194C4×D3—64no—
195Dic6—24no—
196Dic6—34no—
197Dic6—34no—
198Dic6—64no—
199Dic6—64no—
200Dic6—64no—
201Dic6—64no—
202Dic6—124no—
203C2×C32C22×D411yescharacteristic
204C2×C32C22×Q811yescharacteristic
205C2×C322−1+411yescharacteristic
206C3×D3—24no—
207C2×Q8—32no—
208C2×Q8—91no—
209C2×Q8—92no—
210C2×Q8—92no—
211C2×Q8—188no—
212C2×D4—31no—
213C22×C4—32no—
214C22×C4—92no—
215C22×C4—92no—
216C4⋊C4—31no—
217C4⋊C4—32no—
218C4⋊C4—91no—
219C4⋊C4—91no—
220C4⋊C4—91no—
221C4⋊C4—92no—
222C4⋊C4—91no—
223C4⋊C4—92no—
224C4⋊C4D3211yescharacteristic
225C22⋊C4—32no—
226C22⋊C4—92no—
227C22⋊C4—92no—
228C42—31no—
229C42—31no—
230C42—91no—
231C22×C3—21no—
232C22×C3—34no—
233C22×C3—62no—
234C22×C3—62no—
235C22×C3C23×D311yescharacteristic
236C22×C3C23×D311yescharacteristic
237C2×D3—22no—
238C2×D3—22no—
239C2×D3C2×Dic614yescomplemented
240C3×C4—22no—
241C3×C4—22no—
242C3×C4—22no—
243C3×C4—32no—
244C3×C4—32no—
245C3×C4—42no—
246C3×C4—62no—
247C3×C4—61no—
248C3×C4—61no—
249C3×C4C2×Dic612yescomplemented
250C3×C4D3×Q812yescomplemented
251Dic3—22no—
252Dic3—21no—
253Dic3—21no—
254Dic3—32no—
255Dic3—32no—
256Dic3—62no—
257Dic3—62no—
258Dic3—62no—
259Dic3—62no—
260Dic3—61no—
261Dic3—61no—
262Dic3—122no—
263Dic3D3×D412yes—
264Dic3D3×D412yes—
265C32D4×Q811yescharacteristic, complemented, Sylow
266C23—32no—
267Q8—64no—
268Q8—94no—
269Q8—184no—
270Q8—184no—
271D4—34no—
272C2×C4—31no—
273C2×C4—31no—
274C2×C4—31no—
275C2×C4—31no—
276C2×C4—32no—
277C2×C4—64no—
278C2×C4—91no—
279C2×C4—91no—
280C2×C4—92no—
281C2×C4—184no—
282C2×C4—184no—
283C2×C4C2×D3212yes—
284C2×C4C2×D3211yescharacteristic
285C2×C3—21no—
286C2×C3—21no—
287C2×C3—21no—
288C2×C3—64no—
289C2×C3C22×Dic611yescharacteristic
290C2×C3C2×D3×D411yescharacteristic
291C2×C3C2×D3×D411yescharacteristic
292C2×C3C2×D3×Q811yescharacteristic
293C2×C3#100711yescharacteristic
294C2×C3#101011yescharacteristic
295D3—24no—
296C22—34no—
297C22—62no—
298C22—62no—
299C22C22×D3211yescharacteristic
300C4—22no—
301C4—32no—
302C4—32no—
303C4—62no—
304C4—61no—
305C4—61no—
306C4—92no—
307C4—182no—
308C4—181no—
309C4D3×Dic612yescomplemented
310C3—21no—
311C3D4×Dic611yescharacteristic, complemented
312C3#572711yescharacteristic, complemented
313C2—64no—
314C2C2×D3×Dic611yescharacteristic
315C2D32×D411yescharacteristic
316C2#6431511yescharacteristic
subgroupG/H
center, FrattiniC22C22×D32
commutator, socle, cosocleC22×C32C24
FittingC32×C4⋊C4C22
prime radicalC2×C3×D12C22
hypercenterC4⋊C4D32
hypocenterC32D4×Q8
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
1. C2×D3×Dic6C2semidirect
2. #64097C2semidirect
4. C3×#888C2semidirect
5. #63918C2semidirect
6. C4×#627C2semidirect
8. #63896C2semidirect
10. #63887C2semidirect
11. #63851C2semidirect
54. #888D3semidirect
73. C3×D12Q8semidirect
77. #627D4semidirect
137. C2×C3×D3C2×Q8semidirect
139. C32×C4C2×Q8semidirect
191. D12Dic6semidirect
239. C2×D3C2×Dic6semidirect
249. C3×C4C2×Dic6semidirect
250. C3×C4D3×Q8semidirect
265. C32D4×Q8semidirect
309. C4D3×Dic6semidirect
311. C3D4×Dic6semidirect
312. C3#5727semidirect

Maximal subgroups

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

subgroupindex
C2×D3×Dic62
#640972
C3×#8932
C3×#8882
#639182
C4×#6272
#638982
#638962
Dic3×D122
#638872
#638512
#57273
D4×Dic63