Finite Group Databasebeta

← #10596133

Subgroups — #10596133

Order 576.

648 conjugacy classes of subgroups.

The subgroup lattice has 312 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C32×#123C211yesmaximal, characteristic, complemented
2#64148C211yesmaximal, characteristic
3#64145C211yesmaximal, characteristic, complemented
4C4×#638C211yesmaximal, characteristic, complemented
5#64133C211yesmaximal, characteristic, complemented
6#64107C211yesmaximal, characteristic, complemented
7#64105C211yesmaximal, characteristic, complemented
8#64097C211yesmaximal, characteristic, complemented
9#64096C211yesmaximal, characteristic, complemented
10#64094C211yesmaximal, characteristic, complemented
11#64094C211yesmaximal, characteristic, complemented
12#64092C211yesmaximal, characteristic, complemented
13#64088C211yesmaximal, characteristic, complemented
14#64087C211yesmaximal, characteristic, complemented
15#64086C211yesmaximal, characteristic, complemented
16#5662—34nomaximal
17C2×C32×D4C2211yescharacteristic
18C22×C32×C4C2211yescharacteristic
19C2×#638C2211yescharacteristic
20C2×#638C2211yescharacteristic
21C22×C3⋊Dic3C2211yescharacteristic
22C2×C4×C3⋊D3C2211yescharacteristic
23C2×#634C2211yescharacteristic
24C2×#634C2211yescharacteristic
25C2×#634C2211yescharacteristic
26C32×C4⋊C4C2211yescharacteristic
27C32×C4⋊C4C2211yescharacteristic
28C32×C22⋊C4C2211yescharacteristic
29C32×C22⋊C4C2211yescharacteristic
30C32×C22⋊C4C2211yescharacteristic
31#3757C2211yescharacteristic
32#3757C2211yescharacteristic
33#3757C2211yescharacteristic
34#3757C2211yescharacteristic
35#3757C2211yescharacteristic
36#3752C2211yescharacteristic
37#3752C2211yescharacteristic
38#3752C2211yescharacteristic
39#3752C2211yescharacteristic
40C4⋊(C3⋊Dic3)C2211yescharacteristic
41C4⋊(C3⋊Dic3)C2211yescharacteristic
42#3750C2211yescharacteristic
43#3750C2211yescharacteristic
44#3750C2211yescharacteristic
45#3750C2211yescharacteristic
46#3750C2211yescharacteristic
47#3750C2211yescharacteristic
48C4×C3⋊Dic3C2211yescharacteristic
49C4×C3⋊Dic3C2211yescharacteristic
50C4×C3⋊Dic3C2211yescharacteristic
51C4×C3⋊Dic3C2211yescharacteristic
52C3×#123D314yescomplemented
53#939—34no—
54#936—34no—
55C4×#67—34no—
56#924—34no—
57#898—34no—
58#896—34no—
59#888—34no—
60#887—34no—
61#885—34no—
62#885—34no—
63#883—34no—
64#879—34no—
65#878—34no—
66#877—34no—
67C23×C32C2311yescharacteristic
68C23×C32C2311yescharacteristic
69C22×C3⋊D3C2311yescharacteristic
70C32×D4—41no—
71C2×C32×C4—41no—
72C2×C32×C4C2311yescharacteristic
73C2×C32×C4C2311yescharacteristic
74C2×C32×C4C2311yescharacteristic
75C2×C32×C4C2311yescharacteristic
76C2×C32×C4C2311yescharacteristic
77#638—22no—
78#638—41no—
79C2×C3⋊Dic3—22no—
80C2×C3⋊Dic3C2311yescharacteristic
81C2×C3⋊Dic3C2311yescharacteristic
82C2×C3⋊Dic3C2311yescharacteristic
83C2×C3⋊Dic3C2311yescharacteristic
84C2×C3⋊Dic3C2311yescharacteristic
85C2×C3⋊Dic3C2311yescharacteristic
86C2×C3⋊Dic3C2311yescharacteristic
87C4×C3⋊D3—41no—
88#634—22no—
89#634—41no—
90#634—41no—
91#542—91noSylow
92C2×C3×D4C2×D314yes—
93C22×C3×C4C2×D314yes—
94C2×#67—34no—
95C2×#67—34no—
96C22×Dic3—34no—
97C2×C4×D3—34no—
98C2×Dic6—34no—
99C2×Dic6—34no—
100C2×Dic6—34no—
101C3×C4⋊C4C2×D314yes—
102C3×C4⋊C4C2×D314yes—
103C3×C22⋊C4C2×D314yes—
104C3×C22⋊C4C2×D314yes—
105C3×C22⋊C4C2×D314yes—
106#217—34no—
107#217—34no—
108#217—34no—
109#217—34no—
110#217—34no—
111(C2×D3)⋊C4—34no—
112(C2×D3)⋊C4—34no—
113(C2×D3)⋊C4—34no—
114(C2×D3)⋊C4—34no—
115C4⋊Dic3—34no—
116C4⋊Dic3—34no—
117Dic3⋊C4—34no—
118Dic3⋊C4—34no—
119Dic3⋊C4—34no—
120Dic3⋊C4—34no—
121Dic3⋊C4—34no—
122Dic3⋊C4—34no—
123C4×Dic3—34no—
124C4×Dic3—34no—
125C4×Dic3—34no—
126C4×Dic3—34no—
127C22×C32—21no—
128C22×C32—21no—
129C22×C32—21no—
130C22×C32—21no—
131C22×C32—21no—
132C22×C32—21no—
133C22×C32C2411yescharacteristic
134C2×C3⋊D3—21no—
135C2×C3⋊D3—21no—
136C2×C3⋊D3—21no—
137C32×C4—21no—
138C32×C4—21no—
139C32×C4—21no—
140C32×C4—21no—
141C32×C4—21no—
142C3⋊Dic3—21no—
143C3⋊Dic3—21no—
144C3⋊Dic3—21no—
145C3⋊Dic3—21no—
146C3⋊Dic3—21no—
147C3⋊Dic3—21no—
148C3⋊Dic3#4112yes—
149C4⋊Q8—91no—
150#126—91no—
151#126—91no—
152#124—91no—
153#124—91no—
154#124—91no—
155#123—91no—
156#123C3⋊D311yescharacteristic, complemented
157C22⋊Q8—91no—
158C22⋊Q8—91no—
159C22⋊Q8—91no—
160#121—91no—
161C4×Q8—91no—
162C4×D4—91no—
163#117—91no—
164C23×C3C22×D314yes—
165C23×C3C22×D314yes—
166C22×D3—34no—
167C3×D4—44no—
168C2×C3×C4—44no—
169C2×C3×C4C22×D314yes—
170C2×C3×C4C22×D314yes—
171C2×C3×C4C22×D314yes—
172C2×C3×C4C22×D314yes—
173C2×C3×C4C22×D314yes—
174#67—68no—
175#67—124no—
176C2×Dic3—34no—
177C2×Dic3—34no—
178C2×Dic3—34no—
179C2×Dic3—34no—
180C2×Dic3—34no—
181C2×Dic3—34no—
182C2×Dic3—34no—
183C2×Dic3—68no—
184C4×D3—124no—
185Dic6—68no—
186Dic6—124no—
187Dic6—124no—
188C2×C32—41no—
189C2×C32—41no—
190C2×C32C2×#4111yescharacteristic
191C2×C32H(5,𝔽2)11yescharacteristic
192C2×C322−1+411yescharacteristic
193C3⋊D3—41no—
194C2×Q8—91no—
195C2×Q8—91no—
196C2×Q8—91no—
197C2×D4—91no—
198C2×D4—91no—
199C2×D4C2×C3⋊D311yescharacteristic
200C22×C4—91no—
201C22×C4—91no—
202C22×C4C2×C3⋊D311yescharacteristic
203C4⋊C4—91no—
204C4⋊C4—91no—
205C4⋊C4—91no—
206C4⋊C4—91no—
207C4⋊C4—91no—
208C4⋊C4—91no—
209C4⋊C4—91no—
210C4⋊C4—91no—
211C4⋊C4C2×C3⋊D311yescharacteristic
212C4⋊C4C2×C3⋊D311yescharacteristic
213C22⋊C4—91no—
214C22⋊C4—91no—
215C22⋊C4—91no—
216C22⋊C4—91no—
217C22⋊C4—91no—
218C22⋊C4—91no—
219C22⋊C4—91no—
220C22⋊C4—91no—
221C22⋊C4—91no—
222C22⋊C4C2×C3⋊D311yescharacteristic
223C22⋊C4C2×C3⋊D311yescharacteristic
224C22⋊C4C2×C3⋊D311yescharacteristic
225C42—91no—
226C42—91no—
227C42—91no—
228C42—91no—
229C22×C3—24no—
230C22×C3—24no—
231C22×C3—24no—
232C22×C3—24no—
233C22×C3—24no—
234C22×C3—24no—
235C22×C3C23×D314yes—
236C2×D3—64no—
237C2×D3—64no—
238C2×D3—64no—
239C3×C4—24no—
240C3×C4—24no—
241C3×C4—24no—
242C3×C4—24no—
243C3×C4—24no—
244Dic3—38no—
245Dic3—64no—
246Dic3—64no—
247Dic3—64no—
248Dic3—64no—
249Dic3—64no—
250Dic3—64no—
251C32#54211yescharacteristic, complemented, Sylow
252C23—91no—
253C23C22×C3⋊D311yescharacteristic
254C23C22×C3⋊D311yescharacteristic
255Q8—182no—
256Q8—361no—
257Q8—361no—
258D4—41no—
259D4—182no—
260D4—361no—
261C2×C4—41no—
262C2×C4—91no—
263C2×C4—91no—
264C2×C4—91no—
265C2×C4—91no—
266C2×C4—91no—
267C2×C4—91no—
268C2×C4—91no—
269C2×C4—182no—
270C2×C4—361no—
271C2×C4C22×C3⋊D311yescharacteristic
272C2×C4C22×C3⋊D311yescharacteristic
273C2×C4C22×C3⋊D311yescharacteristic
274C2×C4C22×C3⋊D311yescharacteristic
275C2×C4C22×C3⋊D311yescharacteristic
276C2×C3—44no—
277C2×C3—44no—
278C2×C3#100414yes—
279C2×C3D3×#4114yes—
280C2×C3#101014yes—
281D3—124no—
282C22—21no—
283C22—21no—
284C22—21no—
285C22—21no—
286C22—21no—
287C22—21no—
288C22—181no—
289C22—181no—
290C22—181no—
291C22C23×C3⋊D311yescharacteristic
292C4—21no—
293C4—21no—
294C4—21no—
295C4—21no—
296C4—21no—
297C4—92no—
298C4—181no—
299C4—181no—
300C4—181no—
301C4—181no—
302C4—181no—
303C4—181no—
304C3#566214yescomplemented
305C2—41no—
306C2—41no—
307C2—361no—
308C2#6436111yescharacteristic
309C2#41×C3⋊D311yescharacteristic
310C2#6436711yescharacteristic
subgroupG/H
center, FrattiniC22C23×C3⋊D3
commutator, socle, cosocleC22×C32C24
FittingC32×#123C2
prime radical#64148C2
hypercenter#123C3⋊D3
hypocenterC32#542
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ #123

Complements

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

NH ≅ G/Ntype
1. C32×#123C2semidirect
3. #64145C2semidirect
4. C4×#638C2semidirect
5. #64133C2semidirect
6. #64107C2semidirect
7. #64105C2semidirect
8. #64097C2semidirect
9. #64096C2semidirect
10. #64094C2semidirect
11. #64094C2semidirect
12. #64092C2semidirect
13. #64088C2semidirect
14. #64087C2semidirect
15. #64086C2semidirect
52. C3×#123D3semidirect
156. #123C3⋊D3semidirect
251. C32#542semidirect
304. C3#5662semidirect

Maximal subgroups

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

subgroupindex
C32×#1232
#641482
#641452
C4×#6382
#641332
#641072
#641052
#640972
#640962
#640942
#640922
#640882
#640872
#640862
#56623