Finite Group Databasebeta

← #10595508

Subgroups — #10595508

Order 576.

566 conjugacy classes of subgroups; only the 128 normal subgroups are shown.

normal subgroups only

This group has 566 conjugacy classes of subgroups — too many to draw. Only its 128 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
1C2×#3757C211yesmaximal, characteristic, complemented
2C3×#953C212yesmaximal, complemented
3#63982C211yesmaximal, characteristic, complemented
4#63980C211yesmaximal, characteristic, complemented
5#67⋊Dic3C212yesmaximal, complemented
6#63970C212yesmaximal, complemented
7#63969C212yesmaximal, complemented
8#63959C212yesmaximal, complemented
9#63956C212yesmaximal, complemented
10#5593—32nomaximal
11C24×C32C2211yescharacteristic
12C22×C3⋊Dic3C2211yescharacteristic
13C22×C3⋊Dic3C2211yescharacteristic
14C2×C3×#67C2212yes—
15C2×C3×#67C2212yes—
16C2×C3×#67C2212yes—
17C2×#627C2211yescharacteristic, complemented
18C2×#625C2211yescharacteristic, complemented
19C2×D3×Dic3C2212yescomplemented
20#3757C2212yes—
21#3757C2212yes—
22C3×#217C2212yes—
23C3×#217C2212yes—
24C3×#217C2212yes—
25#3724C2211yescharacteristic, complemented
26#3724C2211yescharacteristic, complemented
27Dic3⋊Dic3C2212yescomplemented
28Dic3⋊Dic3C2212yescomplemented
29(C2×D3)⋊Dic3C2212yescomplemented
30(C2×D3)⋊Dic3C2212yescomplemented
31Dic32C2212yescomplemented
32C3×#120—32no—
33#953D312yescomplemented
34C2×#217—32no—
35C2×#217—61no—
36#939—32no—
37#939—32no—
38#938—32no—
39#936—32no—
40#935—32no—
41D4×Dic3—32no—
42#885—32no—
43#883—32no—
44#67⋊C4—32no—
45#879—32no—
46#879—32no—
47#878—32no—
48C23×C32—22no—
49C23×C32—41no—
50C23×C32—41no—
51C23×C32C2312yes—
52C23×C32C2311yescharacteristic
53C22×C3×D3C2312yes—
54C2×C3⋊Dic3—41no—
55C2×C3⋊Dic3—41no—
56C2×C3⋊Dic3C2312yes—
57C2×C3⋊Dic3C2311yescharacteristic
58C2×C3⋊Dic3C2311yescharacteristic
59C3×#67—24no—
60C3×#67—42no—
61C3×#67—42no—
62C2×C3×Dic3C2312yes—
63C2×C3×Dic3C2312yes—
64C2×C3×Dic3C2312yes—
65#627—41no—
66#625—41no—
67D3×Dic3—42no—
68#538—91noSylow
69C24×C3—21no—
70C24×C3C2×D312yes—
71C2×C3×D4—32no—
72C2×C3×D4—32no—
73C2×C3×D4—32no—
74C2×#67—32no—
75C2×#67C2×D312yes—
76C2×#67C2×D312yes—
77C2×#67C2×D312yes—
78C22×Dic3—32no—
79C22×Dic3—32no—
80C22×Dic3—32no—
81C22×Dic3—61no—
82C22×Dic3—61no—
83C2×C4×D3—32no—
84C2×Dic6—32no—
85C3×C22⋊C4—32no—
86C3×C22⋊C4—32no—
87C3×C22⋊C4—32no—
88#217—32no—
89#217—32no—
90#217—32no—
91#217—32no—
92#217—32no—
93#217—32no—
94#217—62no—
95#217—62no—
96#217C2×D312yes—
97#217C2×D312yes—
98#217C2×D312yes—
99(C2×D3)⋊C4—32no—
100(C2×D3)⋊C4—32no—
101C4⋊Dic3—32no—
102C4⋊Dic3—32no—
103Dic3⋊C4—32no—
104Dic3⋊C4—32no—
105Dic3⋊C4—32no—
106Dic3⋊C4—32no—
107C4×Dic3—32no—
108C4×Dic3—32no—
109C22×C32—21no—
110C22×C32—21no—
111C22×C32—22no—
112C22×C32—22no—
113C22×C32—22no—
114C22×C32—44no—
115C22×C32#4112yes—
116C22×C32C2411yescharacteristic
117C2×C3×D3—22no—
118C2×C3×D3—22no—
119C2×C3×D3—22no—
120C3⋊Dic3—22no—
121C3⋊Dic3—21no—
122C3⋊Dic3—21no—
123C3×Dic3—22no—
124C3×Dic3—22no—
125C3×Dic3—22no—
126#126—92no—
127#124—92no—
128#123—92no—
129C22⋊Q8—91no—
130#121—92no—
131#121—91no—
132#120—32no—
133C4×D4—92no—
134C2×C22⋊C4—91no—
135C23×C3—24no—
136C23×C3—24no—
137C23×C3—21no—
138C23×C3—22no—
139C23×C3—32no—
140C23×C3—42no—
141C23×C3—42no—
142C23×C3—42no—
143C23×C3—42no—
144C23×C3C22×D312yes—
145C23×C3C22×D312yes—
146C23×C3C22×D312yes—
147C22×D3C22×D312yes—
148C3×D4—64no—
149C3×D4—122no—
150C3×D4—122no—
151C2×C3×C4—32no—
152C2×C3×C4—32no—
153C2×C3×C4—32no—
154#67—24no—
155#67—42no—
156#67—42no—
157#67—122no—
158C2×Dic3—32no—
159C2×Dic3—32no—
160C2×Dic3—32no—
161C2×Dic3—32no—
162C2×Dic3—62no—
163C2×Dic3—61no—
164C2×Dic3—61no—
165C2×Dic3—122no—
166C2×Dic3—122no—
167C2×Dic3—122no—
168C2×Dic3—122no—
169C2×Dic3—122no—
170C2×Dic3C22×D312yes—
171C2×Dic3C22×D312yes—
172C2×Dic3C22×D312yes—
173C4×D3—122no—
174Dic6—122no—
175C2×C32—22no—
176C2×C32—42no—
177C2×C32C2×#4111yescharacteristic
178C2×C32H(5,𝔽2)11yescharacteristic
179C2×C32H(5,𝔽2)11yescharacteristic
180C3×D3—42no—
181C24D3211yescharacteristic
182C2×Q8—91no—
183C2×D4—32no—
184C2×D4—32no—
185C2×D4—32no—
186C2×D4—91no—
187C22×C4—92no—
188C22×C4—91no—
189C22×C4—91no—
190C4⋊C4—92no—
191C4⋊C4—92no—
192C4⋊C4—91no—
193C4⋊C4—91no—
194C22⋊C4—32no—
195C22⋊C4—32no—
196C22⋊C4—32no—
197C22⋊C4—92no—
198C22⋊C4—92no—
199C22⋊C4—92no—
200C22⋊C4—92no—
201C42—92no—
202C22×C3—22no—
203C22×C3—22no—
204C22×C3—22no—
205C22×C3—22no—
206C22×C3—24no—
207C22×C3—22no—
208C22×C3—22no—
209C22×C3—22no—
210C22×C3—22no—
211C22×C3—22no—
212C22×C3—22no—
213C22×C3—21no—
214C22×C3—22no—
215C22×C3—42no—
216C22×C3—42no—
217C22×C3—48no—
218C22×C3—48no—
219C22×C3—62no—
220C22×C3—62no—
221C22×C3—62no—
222C22×C3#23714yes—
223C22×C3C23×D312yes—
224C2×D3—22no—
225C2×D3—22no—
226C2×D3—22no—
227C3×C4—62no—
228C3×C4—62no—
229C3×C4—62no—
230Dic3—22no—
231Dic3—22no—
232Dic3—22no—
233Dic3—62no—
234Dic3—62no—
235Dic3—62no—
236Dic3—62no—
237Dic3—122no—
238Dic3—121no—
239Dic3—121no—
240C32#53811yescharacteristic, complemented, Sylow
241C23—22no—
242C23—32no—
243C23—41no—
244C23—41no—
245C23C2×D3212yes—
246C23C2×D3211yescharacteristic
247Q8—361no—
248D4—64no—
249D4—122no—
250D4—122no—
251D4—361no—
252C2×C4—32no—
253C2×C4—32no—
254C2×C4—32no—
255C2×C4—92no—
256C2×C4—91no—
257C2×C4—91no—
258C2×C4—362no—
259C2×C4—361no—
260C2×C4—361no—
261C2×C3—24no—
262C2×C3—24no—
263C2×C3—21no—
264C2×C3—21no—
265C2×C3—21no—
266C2×C3—44no—
267C2×C3—42no—
268C2×C3—42no—
269C2×C3—122no—
270C2×C3C2×#23712yes—
271C2×C3#100412yes—
272C2×C3#100412yes—
273D3—42no—
274C22—21no—
275C22—21no—
276C22—22no—
277C22—22no—
278C22—22no—
279C22—44no—
280C22—62no—
281C22—62no—
282C22—62no—
283C22#380512yes—
284C22C22×D3211yescharacteristic
285C4—62no—
286C4—62no—
287C4—62no—
288C4—182no—
289C4—181no—
290C4—181no—
291C3—21no—
292C3#559312yescomplemented
293C2—22no—
294C2—42no—
295C2—122no—
296C2C2×#380511yescharacteristic
297C2#6433011yescharacteristic
298C2#6433011yescharacteristic
subgroupG/H
center, FrattiniC22C22×D32
commutator, socle, cosocleC22×C32C24
FittingC24×C32C22
hypercenterC24D32
hypocenterC32#538
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C24

Complements

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

NH ≅ G/Ntype
1. C2×#3757C2semidirect
2. C3×#953C2semidirect
3. #63982C2semidirect
4. #63980C2semidirect
5. #67⋊Dic3C2semidirect
6. #63970C2semidirect
7. #63969C2semidirect
8. #63959C2semidirect
9. #63956C2semidirect
17. C2×#627C22semidirect
18. C2×#625C22semidirect
19. C2×D3×Dic3C22semidirect
25. #3724C22semidirect
26. #3724C22semidirect
27. Dic3⋊Dic3C22semidirect
28. Dic3⋊Dic3C22semidirect
29. (C2×D3)⋊Dic3C22semidirect
30. (C2×D3)⋊Dic3C22semidirect
31. Dic32C22semidirect
33. #953D3semidirect
240. C32#538semidirect
292. C3#5593semidirect

Maximal subgroups

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

subgroupindex
C2×#37572
C3×#9532
#639822
#639802
#67⋊Dic32
#639702
#639692
#639592
#639562
#55933