Finite Group Databasebeta

← #10594497

Subgroups — #10594497

Order 576.

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

normal subgroups only

This group has 468 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#64099C211yesmaximal, characteristic, complemented
2C3×#894C212yesmaximal, complemented
3#63915C211yesmaximal, characteristic, complemented
4#63915C211yesmaximal, characteristic, complemented
5#63908C212yesmaximal, complemented
6#63893C212yesmaximal, complemented
7#63891C212yesmaximal, complemented
8#63885C212yesmaximal, complemented
9#63851C212yesmaximal, complemented
10#5667—32nomaximal
11C2×C3×D12C2212yes—
12C2×C3×C4×D3C2212yes—
13C2×#627C2211yescharacteristic
14C2×#627C2211yescharacteristic
15C2×#625C2211yescharacteristic
16C2×#625C2212yes—
17C2×#625C2211yescharacteristic
18C2×D3×Dic3C2212yes—
19C32×C4⋊C4C2211yescharacteristic
20C4⋊(C3⋊Dic3)C2211yescharacteristic
21C4⋊(C3⋊Dic3)C2212yes—
22#3750C2212yes—
23C4×C3⋊Dic3C2211yescharacteristic
24C3×(C2×D3)⋊C4C2212yes—
25C3×(C2×D3)⋊C4C2212yes—
26C3×(C2×D3)⋊C4C2212yes—
27C3×Dic3⋊C4C2212yes—
28Dic3⋊Dic3C2212yes—
29(C2×D3)⋊Dic3C2212yes—
30(C2×D3)⋊Dic3C2212yes—
31(C2×D3)⋊Dic3C2212yes—
32C3×#123—32no—
33#939—32no—
34#938—32no—
35#930—32no—
36#924—32no—
37#897—32no—
38#896—32no—
39#894D312yescomplemented
40#890—32no—
41#890—61no—
42#886—32no—
43#885—32no—
44#885—32no—
45#884—32no—
46#883—32no—
47#878—32no—
48C22×C3×D3C2312yes—
49C22×C3×D3C2312yes—
50C2×C32×C4C2312yes—
51C2×C32×C4C2311yescharacteristic
52C2×C3⋊Dic3C2311yescharacteristic
53C2×C3⋊Dic3C2311yescharacteristic
54C2×C3⋊Dic3C2312yes—
55C2×C3×Dic3C2312yes—
56C2×C3×Dic3C2312yes—
57C3×D12—42no—
58C3×C4×D3—42no—
59#627—41no—
60#627—41no—
61#625—42no—
62#625—41no—
63#625—41no—
64D3×Dic3—42no—
65#562—91noSylow
66C2×C3×D4—32no—
67C22×C3×C4—32no—
68C2×#67—32no—
69C2×#67—32no—
70C2×#67—32no—
71C2×#67—32no—
72C22×Dic3—32no—
73C2×D12C2×D312yes—
74C2×C4×D3—32no—
75C2×C4×D3C2×D312yes—
76C2×Dic6—32no—
77C2×Dic6—32no—
78C3×C4⋊C4—21no—
79C3×C4⋊C4—32no—
80C3×C4⋊C4C2×D312yes—
81C3×C22⋊C4—32no—
82C3×C22⋊C4—32no—
83C3×C22⋊C4—32no—
84#217—32no—
85#217—32no—
86#217—32no—
87(C2×D3)⋊C4—32no—
88(C2×D3)⋊C4—32no—
89(C2×D3)⋊C4—32no—
90(C2×D3)⋊C4C2×D312yes—
91(C2×D3)⋊C4C2×D312yes—
92(C2×D3)⋊C4C2×D312yes—
93C4⋊Dic3—32no—
94C4⋊Dic3—32no—
95C4⋊Dic3—32no—
96C4⋊Dic3—32no—
97C4⋊Dic3—62no—
98C4⋊Dic3—61no—
99Dic3⋊C4—32no—
100Dic3⋊C4—32no—
101Dic3⋊C4—32no—
102Dic3⋊C4—62no—
103Dic3⋊C4C2×D312yes—
104C4×Dic3—32no—
105C4×Dic3—61no—
106C22×C32C2411yescharacteristic
107C2×C3×D3—22no—
108C2×C3×D3—22no—
109C2×C3×D3—22no—
110C2×C3×D3—22no—
111C2×C3×D3—22no—
112C2×C3×D3—22no—
113C32×C4—21no—
114C32×C4—22no—
115C3⋊Dic3—22no—
116C3⋊Dic3—21no—
117C3⋊Dic3—21no—
118C3×Dic3—22no—
119C3×Dic3—22no—
120#125—91no—
121#124—91no—
122#124—91no—
123#123—32no—
124#123—92no—
125C22⋊Q8—92no—
126C22⋊Q8—92no—
127#121—92no—
128#121—92no—
129C23×C3—32no—
130C23×C3—32no—
131C22×D3C22×D312yes—
132C22×D3C22×D312yes—
133C3×D4—122no—
134C2×C3×C4—21no—
135C2×C3×C4—22no—
136C2×C3×C4—32no—
137C2×C3×C4—32no—
138C2×C3×C4—122no—
139C2×C3×C4C22×D312yes—
140C2×C3×C4C22×D312yes—
141C2×C3×C4C22×D312yes—
142#67—122no—
143#67—122no—
144#67—122no—
145#67—122no—
146C2×Dic3—32no—
147C2×Dic3—32no—
148C2×Dic3—32no—
149C2×Dic3—32no—
150C2×Dic3—61no—
151C2×Dic3—61no—
152C2×Dic3—62no—
153C2×Dic3—122no—
154C2×Dic3C22×D312yes—
155C2×Dic3C22×D312yes—
156D12—42no—
157C4×D3—42no—
158C4×D3—122no—
159Dic6—122no—
160Dic6—122no—
161C2×C32H(5,𝔽2)11yescharacteristic
162C2×C32H(5,𝔽2)11yescharacteristic
163C2×C322−1+411yescharacteristic
164C3×D3—42no—
165C3×D3—42no—
166C2×Q8—91no—
167C2×Q8—91no—
168C2×D4—32no—
169C2×D4—92no—
170C2×D4—91no—
171C2×D4—91no—
172C22×C4—32no—
173C22×C4—92no—
174C4⋊C4—32no—
175C4⋊C4—92no—
176C4⋊C4—92no—
177C4⋊C4—92no—
178C4⋊C4—91no—
179C4⋊C4D3211yescharacteristic
180C22⋊C4—32no—
181C22⋊C4—32no—
182C22⋊C4—32no—
183C22⋊C4—92no—
184C22⋊C4—92no—
185C22⋊C4—92no—
186C42—91no—
187C22×C3—21no—
188C22×C3—62no—
189C22×C3—62no—
190C22×C3—62no—
191C22×C3—62no—
192C22×C3—62no—
193C22×C3—62no—
194C22×C3C23×D312yes—
195C2×D3—22no—
196C2×D3—22no—
197C2×D3—22no—
198C2×D3—22no—
199C2×D3—22no—
200C2×D3—22no—
201C3×C4—22no—
202C3×C4—22no—
203C3×C4—22no—
204C3×C4—22no—
205C3×C4—42no—
206C3×C4—62no—
207C3×C4—62no—
208Dic3—22no—
209Dic3—22no—
210Dic3—62no—
211Dic3—62no—
212Dic3—62no—
213Dic3—62no—
214Dic3—121no—
215Dic3—122no—
216Dic3—121no—
217C32#56211yescharacteristic, complemented, Sylow
218C23—32no—
219C23—32no—
220Q8—361no—
221Q8—361no—
222D4—122no—
223D4—361no—
224D4—362no—
225D4—361no—
226C2×C4—32no—
227C2×C4—32no—
228C2×C4—91no—
229C2×C4—91no—
230C2×C4—92no—
231C2×C4—122no—
232C2×C4—362no—
233C2×C4C2×D3211yescharacteristic
234C2×C4C2×D3212yes—
235C2×C3—21no—
236C2×C3—21no—
237C2×C3—21no—
238C2×C3—122no—
239C2×C3—122no—
240C2×C3#100412yes—
241C2×C3#100912yes—
242C2×C3#101012yes—
243D3—42no—
244D3—42no—
245C22—62no—
246C22—62no—
247C22—62no—
248C22—62no—
249C22—62no—
250C22—62no—
251C22C22×D3211yescharacteristic
252C4—21no—
253C4—22no—
254C4—62no—
255C4—62no—
256C4—181no—
257C4—182no—
258C4—181no—
259C3—21no—
260C3#566712yescomplemented
261C2—122no—
262C2—122no—
263C2#6429811yescharacteristic
264C2#6431311yescharacteristic
265C2#6432011yescharacteristic
subgroupG/H
center, FrattiniC22C22×D32
commutator, socle, cosocleC22×C32C24
FittingC32×C4⋊C4C22
hypercenterC4⋊C4D32
hypocenterC32#562
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
1. #64099C2semidirect
2. C3×#894C2semidirect
3. #63915C2semidirect
4. #63915C2semidirect
5. #63908C2semidirect
6. #63893C2semidirect
7. #63891C2semidirect
8. #63885C2semidirect
9. #63851C2semidirect
39. #894D3semidirect
217. C32#562semidirect
260. C3#5667semidirect

Maximal subgroups

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

subgroupindex
#640992
C3×#8942
#639152
#639082
#638932
#638912
#638852
#638512
#56673