Finite Group Databasebeta

← #10595505

Subgroups — #10595505

Order 576.

569 conjugacy classes of subgroups; only the 120 normal subgroups are shown.

normal subgroups only

This group has 569 conjugacy classes of subgroups — too many to draw. Only its 120 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#64162C211yesmaximal, characteristic, complemented
2C3×#953C212yesmaximal, complemented
3#63979C211yesmaximal, characteristic, complemented
4#63975C211yesmaximal, characteristic, complemented
5#63971C212yesmaximal, complemented
6#63970C212yesmaximal, complemented
7#63969C212yesmaximal, complemented
8#63958C212yesmaximal, complemented
9#63956C212yesmaximal, complemented
10#5595—32nomaximal
11C24×C32C2211yescharacteristic
12C2×#638C2212yes—
13C2×#638C2211yescharacteristic
14C2×C3×#67C2212yes—
15C2×C3×#67C2212yes—
16C2×C3×#67C2212yes—
17C2×#626C2212yescomplemented
18C2×#625C2211yescharacteristic, complemented
19C2×#624C2211yescharacteristic, complemented
20C2×D3×Dic3C2212yescomplemented
21#3757C2212yes—
22#3757C2211yescharacteristic
23C3×#217C2212yes—
24C3×#217C2212yes—
25C3×#217C2212yes—
26#3724C2212yescomplemented
27Dic3⋊Dic3C2212yescomplemented
28Dic3⋊Dic3C2212yescomplemented
29#3722C2211yescharacteristic, complemented
30(C2×D3)⋊Dic3C2212yescomplemented
31Dic32C2211yescharacteristic, complemented
32C3×#120—32no—
33#953—32no—
34#953—61no—
35#953D312yescomplemented
36#940—32no—
37#939—32no—
38#939—32no—
39#938—32no—
40#938—32no—
41#935—32no—
42#884—32no—
43#884—32no—
44#883—32no—
45#883—32no—
46#879—32no—
47#879—32no—
48C23×C32—42no—
49C23×C32—41no—
50C23×C32C2312yes—
51C23×C32C2311yescharacteristic
52C22×C3⋊D3C2311yescharacteristic
53C22×C3×D3C2312yes—
54#638—42no—
55#638—41no—
56C2×C3⋊Dic3C2311yescharacteristic
57C2×C3⋊Dic3C2312yes—
58C3×#67—42no—
59C3×#67—42no—
60C3×#67—42no—
61C2×C3×Dic3C2312yes—
62C2×C3×Dic3C2312yes—
63C2×C3×Dic3C2312yes—
64#626—42no—
65#625—41no—
66#624—41no—
67D3×Dic3—42no—
68#560—91noSylow
69C24×C3—21no—
70C24×C3C2×D312yes—
71C2×C3×D4—32no—
72C2×C3×D4—32no—
73C2×C3×D4—32no—
74C2×#67—32no—
75C2×#67—32no—
76C2×#67—32no—
77C2×#67—32no—
78C2×#67—32no—
79C2×#67—62no—
80C2×#67—61no—
81C2×#67C2×D312yes—
82C2×#67C2×D312yes—
83C2×#67C2×D312yes—
84C22×Dic3—32no—
85C2×D12—32no—
86C2×C4×D3—32no—
87C2×C4×D3—32no—
88C3×C22⋊C4—32no—
89C3×C22⋊C4—32no—
90C3×C22⋊C4—32no—
91#217—32no—
92#217—32no—
93#217—32no—
94#217—32no—
95#217—62no—
96#217—61no—
97#217C2×D312yes—
98#217C2×D312yes—
99#217C2×D312yes—
100(C2×D3)⋊C4—32no—
101(C2×D3)⋊C4—32no—
102C4⋊Dic3—32no—
103C4⋊Dic3—32no—
104Dic3⋊C4—32no—
105Dic3⋊C4—32no—
106Dic3⋊C4—32no—
107Dic3⋊C4—32no—
108C4×Dic3—32no—
109C22×C32—22no—
110C22×C32—21no—
111C22×C32—22no—
112C22×C32—22no—
113C22×C32—22no—
114C22×C32—44no—
115C22×C32C2411yescharacteristic
116C2×C3⋊D3—21no—
117C2×C3⋊D3—22no—
118C2×C3×D3—22no—
119C2×C3×D3—22no—
120C2×C3×D3—22no—
121C3⋊Dic3—22no—
122C3⋊Dic3—21no—
123C3×Dic3—22no—
124C3×Dic3—22no—
125C3×Dic3—22no—
126#127—91no—
127#126—92no—
128#123—91no—
129#123—92no—
130#121—92no—
131#121—92no—
132#121—92no—
133#120—32no—
134#120—91no—
135C23×C3—22no—
136C23×C3—21no—
137C23×C3—32no—
138C23×C3—42no—
139C23×C3—42no—
140C23×C3—42no—
141C23×C3—42no—
142C23×C3—44no—
143C23×C3C22×D312yes—
144C23×C3C22×D312yes—
145C23×C3C22×D312yes—
146C22×D3—32no—
147C22×D3—61no—
148C22×D3C22×D312yes—
149C3×D4—122no—
150C3×D4—122no—
151C3×D4—122no—
152C2×C3×C4—32no—
153C2×C3×C4—32no—
154C2×C3×C4—32no—
155#67—42no—
156#67—42no—
157#67—42no—
158#67—122no—
159#67—122no—
160#67—122no—
161#67—122no—
162#67—122no—
163#67—124no—
164#67—122no—
165C2×Dic3—32no—
166C2×Dic3—32no—
167C2×Dic3—32no—
168C2×Dic3—61no—
169C2×Dic3—62no—
170C2×Dic3—122no—
171C2×Dic3C22×D312yes—
172C2×Dic3C22×D312yes—
173C2×Dic3C22×D312yes—
174D12—122no—
175C4×D3—122no—
176C4×D3—122no—
177C2×C32—42no—
178C2×C32—41no—
179C2×C32H(5,𝔽2)12yes—
180C2×C32H(5,𝔽2)11yescharacteristic
181C3⋊D3—41no—
182C3×D3—42no—
183C24D3211yescharacteristic
184C2×D4—32no—
185C2×D4—32no—
186C2×D4—32no—
187C2×D4—92no—
188C2×D4—91no—
189C2×D4—92no—
190C2×D4—91no—
191C22×C4—91no—
192C22×C4—92no—
193C4⋊C4—92no—
194C4⋊C4—92no—
195C4⋊C4—92no—
196C22⋊C4—32no—
197C22⋊C4—32no—
198C22⋊C4—32no—
199C22⋊C4—91no—
200C22⋊C4—92no—
201C22⋊C4—92no—
202C22⋊C4—91no—
203C42—91no—
204C22×C3—22no—
205C22×C3—24no—
206C22×C3—22no—
207C22×C3—22no—
208C22×C3—22no—
209C22×C3—22no—
210C22×C3—22no—
211C22×C3—22no—
212C22×C3—22no—
213C22×C3—22no—
214C22×C3—22no—
215C22×C3—21no—
216C22×C3—42no—
217C22×C3—42no—
218C22×C3—42no—
219C22×C3—48no—
220C22×C3—48no—
221C22×C3—62no—
222C22×C3—62no—
223C22×C3—62no—
224C22×C3C23×D312yes—
225C2×D3—22no—
226C2×D3—22no—
227C2×D3—22no—
228C2×D3—62no—
229C2×D3—62no—
230C2×D3—62no—
231C2×D3—121no—
232C2×D3—122no—
233C3×C4—62no—
234C3×C4—62no—
235C3×C4—62no—
236Dic3—22no—
237Dic3—22no—
238Dic3—22no—
239Dic3—62no—
240Dic3—62no—
241Dic3—62no—
242Dic3—122no—
243Dic3—121no—
244C32#56011yescharacteristic, complemented, Sylow
245C23—32no—
246C23—41no—
247C23—42no—
248C23—91no—
249C23C2×D3212yes—
250C23C2×D3211yescharacteristic
251D4—122no—
252D4—122no—
253D4—122no—
254D4—362no—
255D4—362no—
256D4—361no—
257D4—361no—
258C2×C4—32no—
259C2×C4—32no—
260C2×C4—32no—
261C2×C4—91no—
262C2×C4—92no—
263C2×C4—361no—
264C2×C4—362no—
265C2×C3—21no—
266C2×C3—22no—
267C2×C3—44no—
268C2×C3—42no—
269C2×C3—42no—
270C2×C3—42no—
271C2×C3—42no—
272C2×C3—122no—
273C2×C3#100412yes—
274C2×C3#100412yes—
275C2×C3#100412yes—
276D3—42no—
277D3—122no—
278D3—241no—
279C22—22no—
280C22—22no—
281C22—22no—
282C22—21no—
283C22—22no—
284C22—44no—
285C22—62no—
286C22—62no—
287C22—62no—
288C22—181no—
289C22—182no—
290C22C22×D3211yescharacteristic
291C4—62no—
292C4—62no—
293C4—62no—
294C4—182no—
295C4—181no—
296C3—21no—
297C3#559512yescomplemented
298C2—42no—
299C2—41no—
300C2—122no—
301C2—361no—
302C2#6433011yescharacteristic
303C2#6433012yes—
subgroupG/H
center, FrattiniC22C22×D32
commutator, socle, cosocleC22×C32C24
FittingC24×C32C22
hypercenterC24D32
hypocenterC32#560
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C24

Complements

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

NH ≅ G/Ntype
1. #64162C2semidirect
2. C3×#953C2semidirect
3. #63979C2semidirect
4. #63975C2semidirect
5. #63971C2semidirect
6. #63970C2semidirect
7. #63969C2semidirect
8. #63958C2semidirect
9. #63956C2semidirect
17. C2×#626C22semidirect
18. C2×#625C22semidirect
19. C2×#624C22semidirect
20. C2×D3×Dic3C22semidirect
26. #3724C22semidirect
27. Dic3⋊Dic3C22semidirect
28. Dic3⋊Dic3C22semidirect
29. #3722C22semidirect
30. (C2×D3)⋊Dic3C22semidirect
31. Dic32C22semidirect
35. #953D3semidirect
244. C32#560semidirect
297. C3#5595semidirect

Maximal subgroups

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

subgroupindex
#641622
C3×#9532
#639792
#639752
#639712
#639702
#639692
#639582
#639562
#55953