Finite Group Databasebeta

← #10595030

Subgroups — #10595030

Order 576.

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

normal subgroups only

This group has 511 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
1#64138C211yesmaximal, characteristic, complemented
2C3×C4×#67C212yesmaximal, complemented
3#63979C211yesmaximal, characteristic
4#63975C211yesmaximal, characteristic, complemented
5#63959C212yesmaximal, complemented
6#63905C212yesmaximal, complemented
7#63904C212yesmaximal, complemented
8#63852C212yesmaximal, complemented
9#63837C212yesmaximal, complemented
10#5559—32nomaximal
11C22×C32×C4C2211yescharacteristic
12C2×#638C2211yescharacteristic
13C2×C3×#67C2212yes—
14C2×C3×C4×D3C2212yes—
15C2×#627C2212yes—
16C2×#626C2212yes—
17C2×#625C2211yescharacteristic
18C2×#624C2211yescharacteristic
19#3757C2211yescharacteristic
20#3752C2212yes—
21#3750C2212yes—
22C3×#217C2212yes—
23C3×(C2×D3)⋊C4C2212yes—
24C3×Dic3⋊C4C2212yes—
25C3×C4×Dic3C2212yes—
26Dic3⋊Dic3C2212yes—
27#3722C2211yescharacteristic
28(C2×D3)⋊Dic3C2212yes—
29(C2×D3)⋊Dic3C2212yes—
30Dic32C2211yescharacteristic
31Dic32C2212yes—
32C3×C4×D4—32no—
33#940—32no—
34#936—32no—
35#929—32no—
36#929—32no—
37#929—61no—
38C4×#67—32no—
39C4×#67D312yescomplemented
40#893—32no—
41#887—32no—
42#885—32no—
43#885—32no—
44#883—32no—
45#883—32no—
46#875—32no—
47C4×Dic6—32no—
48C23×C32C2311yescharacteristic
49C22×C3⋊D3C2311yescharacteristic
50C22×C3×D3C2312yes—
51C2×C32×C4—41no—
52C2×C32×C4C2312yes—
53#638—41no—
54C2×C3⋊Dic3C2312yes—
55C2×C3⋊Dic3C2311yescharacteristic
56C3×#67—24no—
57C2×C3×Dic3C2312yes—
58C2×C3×Dic3C2312yes—
59C2×C3×Dic3C2312yes—
60C3×C4×D3—42no—
61#627—24no—
62#626—24no—
63#625—41no—
64#624—41no—
65#559—91noSylow
66C2×C3×D4—32no—
67C22×C3×C4—21no—
68C22×C3×C4—32no—
69C22×C3×C4C2×D312yes—
70C2×#67—32no—
71C2×#67—32no—
72C2×#67—32no—
73C2×#67—61no—
74C2×#67C2×D312yes—
75C2×D12—32no—
76C2×C4×D3—32no—
77C2×C4×D3C2×D312yes—
78C2×Dic6—32no—
79C2×Dic6—32no—
80C3×C4⋊C4—32no—
81C3×C22⋊C4—32no—
82C3×C22⋊C4—32no—
83C3×C42—32no—
84#217—32no—
85#217—32no—
86#217—32no—
87#217—61no—
88#217C2×D312yes—
89(C2×D3)⋊C4—32no—
90(C2×D3)⋊C4—32no—
91(C2×D3)⋊C4—32no—
92(C2×D3)⋊C4—32no—
93(C2×D3)⋊C4—32no—
94(C2×D3)⋊C4—62no—
95(C2×D3)⋊C4C2×D312yes—
96C4⋊Dic3—32no—
97Dic3⋊C4—32no—
98Dic3⋊C4—32no—
99Dic3⋊C4—32no—
100Dic3⋊C4—62no—
101Dic3⋊C4C2×D312yes—
102C4×Dic3—32no—
103C4×Dic3—32no—
104C4×Dic3—32no—
105C4×Dic3C2×D312yes—
106C22×C32—21no—
107C22×C32—22no—
108C22×C32C2411yescharacteristic
109C2×C3⋊D3—22no—
110C2×C3⋊D3—21no—
111C2×C3×D3—22no—
112C2×C3×D3—22no—
113C2×C3×D3—22no—
114C32×C4—22no—
115C3⋊Dic3—22no—
116C3⋊Dic3—21no—
117C3×Dic3—22no—
118C3×Dic3—22no—
119C3×Dic3#4114yes—
120#127—91no—
121#124—92no—
122#124—92no—
123#123—91no—
124#123—92no—
125#123—91no—
126C4×Q8—92no—
127C4×D4—32no—
128C4×D4—92no—
129C23×C3—21no—
130C23×C3—32no—
131C23×C3C22×D312yes—
132C22×D3—32no—
133C22×D3—61no—
134C22×D3C22×D312yes—
135C3×D4—64no—
136C2×C3×C4—22no—
137C2×C3×C4—32no—
138C2×C3×C4—32no—
139C2×C3×C4—32no—
140C2×C3×C4—42no—
141C2×C3×C4—42no—
142C2×C3×C4—122no—
143C2×C3×C4C22×D312yes—
144C2×C3×C4C22×D312yes—
145#67—24no—
146#67—64no—
147#67—122no—
148#67—122no—
149#67—122no—
150C2×Dic3—32no—
151C2×Dic3—32no—
152C2×Dic3—32no—
153C2×Dic3—61no—
154C2×Dic3—62no—
155C2×Dic3C22×D312yes—
156C2×Dic3C22×D312yes—
157C2×Dic3C22×D312yes—
158D12—64no—
159C4×D3—42no—
160C4×D3—122no—
161Dic6—64no—
162Dic6—64no—
163C2×C32—41no—
164C2×C32C2×#4112yes—
165C2×C32H(5,𝔽2)11yescharacteristic
166C3⋊D3—41no—
167C3×D3—42no—
168C2×Q8—92no—
169C2×D4—32no—
170C2×D4—92no—
171C2×D4—91no—
172C2×D4—91no—
173C22×C4—32no—
174C22×C4—91no—
175C22×C4D3211yescharacteristic
176C4⋊C4—32no—
177C4⋊C4—92no—
178C4⋊C4—92no—
179C22⋊C4—32no—
180C22⋊C4—32no—
181C22⋊C4—91no—
182C22⋊C4—92no—
183C22⋊C4—92no—
184C22⋊C4—91no—
185C22⋊C4—92no—
186C42—32no—
187C42—92no—
188C42—91no—
189C22×C3—22no—
190C22×C3—22no—
191C22×C3—22no—
192C22×C3—22no—
193C22×C3—21no—
194C22×C3—42no—
195C22×C3—62no—
196C22×C3—62no—
197C22×C3—62no—
198C22×C3C23×D312yes—
199C2×D3—22no—
200C2×D3—22no—
201C2×D3—22no—
202C2×D3—62no—
203C2×D3—62no—
204C2×D3—62no—
205C2×D3—121no—
206C2×D3—122no—
207C3×C4—22no—
208C3×C4—22no—
209C3×C4—34no—
210C3×C4—42no—
211C3×C4—62no—
212C3×C4—62no—
213Dic3—22no—
214Dic3—22no—
215Dic3—62no—
216Dic3—62no—
217Dic3—62no—
218Dic3—121no—
219Dic3—122no—
220Dic3#23514yes—
221C32#55911yescharacteristic, complemented, Sylow
222C23—32no—
223C23—91no—
224C23C2×D3211yescharacteristic
225Q8—184no—
226D4—64no—
227D4—184no—
228D4—361no—
229D4—361no—
230C2×C4—32no—
231C2×C4—32no—
232C2×C4—32no—
233C2×C4—41no—
234C2×C4—92no—
235C2×C4—91no—
236C2×C4—122no—
237C2×C4—361no—
238C2×C4C2×D3212yes—
239C2×C3—22no—
240C2×C3—21no—
241C2×C3—42no—
242C2×C3—42no—
243C2×C3—122no—
244C2×C3C2×#23512yes—
245C2×C3#100412yes—
246C2×C3D3×#4112yes—
247D3—42no—
248D3—122no—
249D3—241no—
250C22—22no—
251C22—21no—
252C22—62no—
253C22—62no—
254C22—62no—
255C22—182no—
256C22—181no—
257C22C22×D3211yescharacteristic
258C4—22no—
259C4—34no—
260C4—62no—
261C4—62no—
262C4—182no—
263C4—181no—
264C3—21no—
265C3#555912yescomplemented
266C2—41no—
267C2—122no—
268C2—361no—
269C2D3×#23512yes—
270C2#6433011yescharacteristic
subgroupG/H
center, FrattiniC22C22×D32
commutator, socle, cosocleC22×C32C24
FittingC22×C32×C4C22
prime radical#63979C2
hypercenterC22×C4D32
hypocenterC32#559
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
1. #64138C2semidirect
2. C3×C4×#67C2semidirect
4. #63975C2semidirect
5. #63959C2semidirect
6. #63905C2semidirect
7. #63904C2semidirect
8. #63852C2semidirect
9. #63837C2semidirect
39. C4×#67D3semidirect
221. C32#559semidirect
265. C3#5559semidirect

Maximal subgroups

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

subgroupindex
#641382
C3×C4×#672
#639792
#639752
#639592
#639052
#639042
#638522
#638372
#55593