Finite Group Databasebeta

← #10596180

Subgroups — #10596180

Order 576.

714 conjugacy classes of subgroups.

The subgroup lattice has 330 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C32×#126C211yesmaximal, characteristic, complemented
2#64107C211yesmaximal, characteristic, complemented
3#64104C211yesmaximal, characteristic, complemented
4#64103C211yesmaximal, characteristic, complemented
5#64102C211yesmaximal, characteristic, complemented
6C4⋊C4×C3⋊D3C211yesmaximal, characteristic, complemented
7#64098C211yesmaximal, characteristic, complemented
8#64095C211yesmaximal, characteristic, complemented
9#64093C211yesmaximal, characteristic, complemented
10#64093C211yesmaximal, characteristic, complemented
11#64093C211yesmaximal, characteristic, complemented
12#64090C211yesmaximal, characteristic, complemented
13#64090C211yesmaximal, characteristic, complemented
14C4×C4⋊(C3⋊D3)C211yesmaximal, characteristic, complemented
15#64081C211yesmaximal, characteristic, complemented
16#5709—34nomaximal
17C2×#638C2211yescharacteristic
18C2×#638C2211yescharacteristic
19C2×#638C2211yescharacteristic
20C22×C3⋊Dic3C2211yescharacteristic
21C2×C4⋊(C3⋊D3)C2211yescharacteristic
22C2×C4⋊(C3⋊D3)C2211yescharacteristic
23C2×C4⋊(C3⋊D3)C2211yescharacteristic
24C2×C4×C3⋊D3C2211yescharacteristic
25C2×C4×C3⋊D3C2211yescharacteristic
26C2×C4×C3⋊D3C2211yescharacteristic
27C2×C4×C3⋊D3C2211yescharacteristic
28C2×C4×C3⋊D3C2211yescharacteristic
29C32×C4⋊C4C2211yescharacteristic
30C32×C4⋊C4C2211yescharacteristic
31C32×C4⋊C4C2211yescharacteristic
32C32×C22⋊C4C2211yescharacteristic
33C32×C22⋊C4C2211yescharacteristic
34C32×C22⋊C4C2211yescharacteristic
35C32×C42C2211yescharacteristic
36#3752C2211yescharacteristic
37#3752C2211yescharacteristic
38#3752C2211yescharacteristic
39#3752C2211yescharacteristic
40#3752C2211yescharacteristic
41#3752C2211yescharacteristic
42#3752C2211yescharacteristic
43C4⋊(C3⋊Dic3)C2211yescharacteristic
44C4⋊(C3⋊Dic3)C2211yescharacteristic
45#3750C2211yescharacteristic
46#3750C2211yescharacteristic
47#3750C2211yescharacteristic
48#3750C2211yescharacteristic
49#3750C2211yescharacteristic
50C4×C3⋊Dic3C2211yescharacteristic
51C4×C3⋊Dic3C2211yescharacteristic
52C3×#126D314yescomplemented
53#898—34no—
54#895—34no—
55#894—34no—
56#893—34no—
57D3×C4⋊C4—34no—
58#889—34no—
59#886—34no—
60#884—34no—
61#884—34no—
62#884—34no—
63#67⋊C4—34no—
64#67⋊C4—34no—
65C4×D12—34no—
66#872—34no—
67C23×C32C2311yescharacteristic
68C22×C3⋊D3C2311yescharacteristic
69C22×C3⋊D3C2311yescharacteristic
70C22×C3⋊D3C2311yescharacteristic
71C2×C32×C4C2311yescharacteristic
72C2×C32×C4C2311yescharacteristic
73C2×C32×C4C2311yescharacteristic
74C2×C32×C4C2311yescharacteristic
75C2×C32×C4C2311yescharacteristic
76C2×C32×C4C2311yescharacteristic
77#638—22no—
78#638—22no—
79#638—41no—
80C2×C3⋊Dic3—41no—
81C2×C3⋊Dic3C2311yescharacteristic
82C2×C3⋊Dic3C2311yescharacteristic
83C2×C3⋊Dic3C2311yescharacteristic
84C2×C3⋊Dic3C2311yescharacteristic
85C2×C3⋊Dic3C2311yescharacteristic
86C4⋊(C3⋊D3)—22no—
87C4⋊(C3⋊D3)—22no—
88C4⋊(C3⋊D3)—41no—
89C4×C3⋊D3—22no—
90C4×C3⋊D3—22no—
91C4×C3⋊D3—41no—
92C4×C3⋊D3—41no—
93C4×C3⋊D3—41no—
94#553—91noSylow
95C2×#67—34no—
96C2×#67—34no—
97C2×#67—34no—
98C22×Dic3—34no—
99C2×D12—34no—
100C2×D12—34no—
101C2×D12—34no—
102C2×C4×D3—34no—
103C2×C4×D3—34no—
104C2×C4×D3—34no—
105C2×C4×D3—34no—
106C2×C4×D3—34no—
107C3×C4⋊C4C2×D314yes—
108C3×C4⋊C4C2×D314yes—
109C3×C4⋊C4C2×D314yes—
110C3×C22⋊C4C2×D314yes—
111C3×C22⋊C4C2×D314yes—
112C3×C22⋊C4C2×D314yes—
113C3×C42C2×D314yes—
114(C2×D3)⋊C4—34no—
115(C2×D3)⋊C4—34no—
116(C2×D3)⋊C4—34no—
117(C2×D3)⋊C4—34no—
118(C2×D3)⋊C4—34no—
119(C2×D3)⋊C4—34no—
120(C2×D3)⋊C4—34no—
121C4⋊Dic3—34no—
122C4⋊Dic3—34no—
123Dic3⋊C4—34no—
124Dic3⋊C4—34no—
125Dic3⋊C4—34no—
126Dic3⋊C4—34no—
127Dic3⋊C4—34no—
128C4×Dic3—34no—
129C4×Dic3—34no—
130C22×C32—21no—
131C22×C32—21no—
132C22×C32—21no—
133C22×C32C2411yescharacteristic
134C2×C3⋊D3—21no—
135C2×C3⋊D3—21no—
136C2×C3⋊D3—21no—
137C2×C3⋊D3—21no—
138C2×C3⋊D3—21no—
139C2×C3⋊D3—21no—
140C2×C3⋊D3—21no—
141C2×C3⋊D3—21no—
142C2×C3⋊D3#4112yes—
143C32×C4—21no—
144C32×C4—21no—
145C32×C4—21no—
146C32×C4—21no—
147C32×C4—21no—
148C32×C4—21no—
149C3⋊Dic3—21no—
150C3⋊Dic3—21no—
151C3⋊Dic3—21no—
152C3⋊Dic3—21no—
153C3⋊Dic3#4112yes—
154#126—91no—
155#126C3⋊D311yescharacteristic, complemented
156#125—91no—
157#123—91no—
158#123—91no—
159#121—91no—
160#121—91no—
161#121—91no—
162#121—91no—
163C4×D4—91no—
164C4×D4—91no—
165C4×D4—91no—
166C4×D4—91no—
167#117—91no—
168C2×C4⋊C4—91no—
169C23×C3C22×D314yes—
170C22×D3—34no—
171C22×D3—34no—
172C22×D3—34no—
173C2×C3×C4C22×D314yes—
174C2×C3×C4C22×D314yes—
175C2×C3×C4C22×D314yes—
176C2×C3×C4C22×D314yes—
177C2×C3×C4C22×D314yes—
178C2×C3×C4C22×D314yes—
179#67—68no—
180#67—68no—
181#67—124no—
182C2×Dic3—34no—
183C2×Dic3—34no—
184C2×Dic3—34no—
185C2×Dic3—34no—
186C2×Dic3—34no—
187C2×Dic3—124no—
188D12—68no—
189D12—68no—
190D12—124no—
191C4×D3—68no—
192C4×D3—68no—
193C4×D3—124no—
194C4×D3—124no—
195C4×D3—124no—
196C2×C32—41no—
197C2×C32C2×#4111yescharacteristic
198C2×C32C2×#4111yescharacteristic
199C2×C32H(5,𝔽2)11yescharacteristic
200C3⋊D3—22no—
201C3⋊D3—41no—
202C3⋊D3—41no—
203C2×D4—91no—
204C2×D4—91no—
205C2×D4—91no—
206C2×D4—91no—
207C2×D4—91no—
208C2×D4—91no—
209C22×C4—91no—
210C22×C4—91no—
211C22×C4—91no—
212C22×C4—91no—
213C22×C4—91no—
214C22×C4—91no—
215C4⋊C4—91no—
216C4⋊C4—91no—
217C4⋊C4—91no—
218C4⋊C4—91no—
219C4⋊C4—91no—
220C4⋊C4—91no—
221C4⋊C4—91no—
222C4⋊C4C2×C3⋊D311yescharacteristic
223C4⋊C4C2×C3⋊D311yescharacteristic
224C4⋊C4C2×C3⋊D311yescharacteristic
225C22⋊C4—91no—
226C22⋊C4—91no—
227C22⋊C4—91no—
228C22⋊C4—91no—
229C22⋊C4—91no—
230C22⋊C4—91no—
231C22⋊C4—91no—
232C22⋊C4C2×C3⋊D311yescharacteristic
233C22⋊C4C2×C3⋊D311yescharacteristic
234C22⋊C4C2×C3⋊D311yescharacteristic
235C42—91no—
236C42—91no—
237C42C2×C3⋊D311yescharacteristic
238C22×C3—24no—
239C22×C3—24no—
240C22×C3—24no—
241C22×C3C23×D314yes—
242C2×D3—38no—
243C2×D3—64no—
244C2×D3—64no—
245C2×D3—64no—
246C2×D3—64no—
247C2×D3—64no—
248C2×D3—64no—
249C2×D3—64no—
250C2×D3—64no—
251C3×C4—24no—
252C3×C4—24no—
253C3×C4—24no—
254C3×C4—24no—
255C3×C4—24no—
256C3×C4—24no—
257Dic3—38no—
258Dic3—64no—
259Dic3—64no—
260Dic3—64no—
261Dic3—64no—
262C32#55311yescharacteristic, complemented, Sylow
263C23—91no—
264C23—91no—
265C23—91no—
266C23C22×C3⋊D311yescharacteristic
267D4—182no—
268D4—182no—
269D4—182no—
270D4—182no—
271D4—361no—
272D4—361no—
273C2×C4—91no—
274C2×C4—91no—
275C2×C4—91no—
276C2×C4—91no—
277C2×C4—91no—
278C2×C4—182no—
279C2×C4—182no—
280C2×C4—361no—
281C2×C4—361no—
282C2×C4—361no—
283C2×C4—361no—
284C2×C4C22×C3⋊D311yescharacteristic
285C2×C4C22×C3⋊D311yescharacteristic
286C2×C4C22×C3⋊D311yescharacteristic
287C2×C4C22×C3⋊D311yescharacteristic
288C2×C4C22×C3⋊D311yescharacteristic
289C2×C4C22×C3⋊D311yescharacteristic
290C2×C3—44no—
291C2×C3D3×#4114yes—
292C2×C3D3×#4114yes—
293C2×C3#100914yes—
294D3—68no—
295D3—124no—
296D3—124no—
297C22—21no—
298C22—21no—
299C22—21no—
300C22—92no—
301C22—181no—
302C22—181no—
303C22—181no—
304C22—181no—
305C22—181no—
306C22—181no—
307C22—181no—
308C22—181no—
309C22C23×C3⋊D311yescharacteristic
310C4—21no—
311C4—21no—
312C4—21no—
313C4—21no—
314C4—21no—
315C4—21no—
316C4—92no—
317C4—181no—
318C4—181no—
319C4—181no—
320C4—181no—
321C3#570914yescomplemented
322C2—41no—
323C2—182no—
324C2—361no—
325C2—361no—
326C2#41×C3⋊D311yescharacteristic
327C2#41×C3⋊D311yescharacteristic
328C2#6436611yescharacteristic
subgroupG/H
center, FrattiniC22C23×C3⋊D3
commutator, socle, cosocleC22×C32C24
FittingC32×#126C2
hypercenter#126C3⋊D3
hypocenterC32#553
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ #126

Complements

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

NH ≅ G/Ntype
1. C32×#126C2semidirect
2. #64107C2semidirect
3. #64104C2semidirect
4. #64103C2semidirect
5. #64102C2semidirect
6. C4⋊C4×C3⋊D3C2semidirect
7. #64098C2semidirect
8. #64095C2semidirect
9. #64093C2semidirect
10. #64093C2semidirect
11. #64093C2semidirect
12. #64090C2semidirect
13. #64090C2semidirect
14. C4×C4⋊(C3⋊D3)C2semidirect
15. #64081C2semidirect
52. C3×#126D3semidirect
155. #126C3⋊D3semidirect
262. C32#553semidirect
321. C3#5709semidirect

Maximal subgroups

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

subgroupindex
C32×#1262
#641072
#641042
#641032
#641022
C4⋊C4×C3⋊D32
#640982
#640952
#640932
#640902
C4×C4⋊(C3⋊D3)2
#640812
#57093