Finite Group Databasebeta

← #10594571

Subgroups — #10594571

Order 576.

470 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 470 conjugacy classes of subgroups — too many to draw. Only its 122 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#64101C211yesmaximal, characteristic, complemented
2C3×#898C212yesmaximal, complemented
3#63919C211yesmaximal, characteristic, complemented
4C4×#627C211yesmaximal, characteristic, complemented
5#63915C211yesmaximal, characteristic, complemented
6#63912C211yesmaximal, characteristic
7#63902C211yesmaximal, characteristic, complemented
8#63893C212yesmaximal, complemented
9#63890C212yesmaximal, complemented
10#63865C211yesmaximal, characteristic, complemented
11#63852C212yesmaximal, complemented
12#5710—32nomaximal
13C2×C4×C3⋊D3C2211yescharacteristic
14C2×#627C2211yescharacteristic
15C2×#627C2212yes—
16C2×#626C2212yes—
17C2×#625C2211yescharacteristic
18C2×D3×Dic3C2212yes—
19C32×C4⋊C4C2211yescharacteristic
20#3752C2211yescharacteristic
21#3752C2211yescharacteristic
22C4⋊(C3⋊Dic3)C2211yescharacteristic
23C4×C3⋊Dic3C2211yescharacteristic
24C4×C3⋊Dic3C2211yescharacteristic
25C3×(C2×D3)⋊C4C2212yes—
26C3×(C2×D3)⋊C4C2212yes—
27C3×(C2×D3)⋊C4C2212yes—
28C3×C4⋊Dic3C2212yes—
29C3×Dic3⋊C4C2212yes—
30C3×C4×Dic3C2212yes—
31#3724C2211yescharacteristic
32#3724C2211yescharacteristic
33Dic3⋊Dic3C2212yes—
34#3722C2212yes—
35(C2×D3)⋊Dic3C2212yes—
36C3×#126—32no—
37#898D312yescomplemented
38#897—32no—
39#896—32no—
40#894—32no—
41#892—32no—
42#892—61no—
43#888—32no—
44#886—32no—
45#885—32no—
46#885—32no—
47#884—32no—
48#67⋊C4—32no—
49#878—32no—
50#875—32no—
51C4×Dic6—32no—
52C22×C3⋊D3C2311yescharacteristic
53C22×C3×D3C2312yes—
54C2×C32×C4C2311yescharacteristic
55C2×C32×C4C2311yescharacteristic
56C2×C32×C4C2311yescharacteristic
57C2×C3⋊Dic3C2311yescharacteristic
58C2×C3⋊Dic3C2311yescharacteristic
59C2×C3⋊Dic3C2311yescharacteristic
60C4×C3⋊D3—22no—
61C2×C3×Dic3C2312yes—
62C2×C3×Dic3C2312yes—
63C2×C3×Dic3C2312yes—
64#627—22no—
65#627—42no—
66#626—42no—
67#625—22no—
68D3×Dic3—42no—
69#542—91noSylow
70C2×#67—32no—
71C2×#67—32no—
72C22×Dic3—32no—
73C2×D12—32no—
74C2×C4×D3—32no—
75C2×C4×D3—32no—
76C2×C4×D3—61no—
77C2×Dic6—32no—
78C2×Dic6—32no—
79C2×Dic6—32no—
80C3×C4⋊C4—21no—
81C3×C4⋊C4—32no—
82C3×C4⋊C4—32no—
83C3×C4⋊C4C2×D312yes—
84C3×C22⋊C4—32no—
85C3×C22⋊C4—32no—
86C3×C22⋊C4—32no—
87C3×C42—32no—
88#217—32no—
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—61no—
95(C2×D3)⋊C4—61no—
96(C2×D3)⋊C4C2×D312yes—
97(C2×D3)⋊C4C2×D312yes—
98(C2×D3)⋊C4C2×D312yes—
99C4⋊Dic3—32no—
100C4⋊Dic3—32no—
101C4⋊Dic3—61no—
102C4⋊Dic3C2×D312yes—
103Dic3⋊C4—32no—
104Dic3⋊C4—32no—
105Dic3⋊C4—32no—
106Dic3⋊C4C2×D312yes—
107C4×Dic3—32no—
108C4×Dic3—32no—
109C4×Dic3—61no—
110C4×Dic3—61no—
111C4×Dic3C2×D312yes—
112C22×C32C2411yescharacteristic
113C2×C3⋊D3—21no—
114C2×C3⋊D3—21no—
115C2×C3⋊D3—21no—
116C2×C3×D3—22no—
117C2×C3×D3—22no—
118C2×C3×D3—22no—
119C32×C4—21no—
120C32×C4—21no—
121C32×C4—21no—
122C3⋊Dic3—21no—
123C3⋊Dic3—21no—
124C3⋊Dic3#4112yes—
125C3×Dic3—22no—
126C3×Dic3—22no—
127C3×Dic3—22no—
128C4⋊Q8—91no—
129#126—32no—
130#124—92no—
131#124—91no—
132#123—92no—
133C22⋊Q8—92no—
134C22⋊Q8—91no—
135#121—91no—
136C4×Q8—91no—
137C4×D4—91no—
138#117—91no—
139C23×C3—32no—
140C22×D3—32no—
141C22×D3—61no—
142C22×D3C22×D312yes—
143C2×C3×C4—21no—
144C2×C3×C4—21no—
145C2×C3×C4—21no—
146C2×C3×C4—32no—
147C2×C3×C4—32no—
148C2×C3×C4—32no—
149C2×C3×C4C22×D312yes—
150C2×C3×C4C22×D312yes—
151C2×C3×C4C22×D312yes—
152#67—64no—
153#67—122no—
154C2×Dic3—32no—
155C2×Dic3—32no—
156C2×Dic3—32no—
157C2×Dic3—61no—
158C2×Dic3—61no—
159C2×Dic3—61no—
160C2×Dic3—122no—
161C2×Dic3C22×D312yes—
162C2×Dic3C22×D312yes—
163C2×Dic3C22×D312yes—
164D12—122no—
165C4×D3—64no—
166C4×D3—64no—
167C4×D3—122no—
168Dic6—64no—
169Dic6—122no—
170Dic6—122no—
171C2×C32C2×#4111yescharacteristic
172C2×C32H(5,𝔽2)11yescharacteristic
173C2×C322−1+411yescharacteristic
174C3⋊D3—41no—
175C3×D3—42no—
176C2×Q8—91no—
177C2×Q8—92no—
178C2×D4—92no—
179C2×D4—91no—
180C22×C4—92no—
181C22×C4—91no—
182C4⋊C4—32no—
183C4⋊C4—32no—
184C4⋊C4—91no—
185C4⋊C4—91no—
186C4⋊C4—92no—
187C4⋊C4—91no—
188C4⋊C4D3211yescharacteristic
189C22⋊C4—32no—
190C22⋊C4—32no—
191C22⋊C4—32no—
192C22⋊C4—92no—
193C22⋊C4—92no—
194C22⋊C4—91no—
195C22⋊C4—91no—
196C42—32no—
197C42—91no—
198C42—91no—
199C22×C3—21no—
200C22×C3—62no—
201C22×C3—62no—
202C22×C3—62no—
203C22×C3C23×D312yes—
204C2×D3—22no—
205C2×D3—22no—
206C2×D3—22no—
207C2×D3—62no—
208C2×D3—62no—
209C2×D3—62no—
210C2×D3—62no—
211C2×D3—121no—
212C2×D3—121no—
213C3×C4—22no—
214C3×C4—22no—
215C3×C4—22no—
216C3×C4—22no—
217C3×C4—41no—
218C3×C4—41no—
219C3×C4—62no—
220C3×C4—62no—
221C3×C4—62no—
222Dic3—22no—
223Dic3—22no—
224Dic3—22no—
225Dic3—34no—
226Dic3—62no—
227Dic3—62no—
228Dic3—62no—
229Dic3—121no—
230Dic3—121no—
231C32#54211yescharacteristic, complemented, Sylow
232C23—32no—
233C23—91no—
234Q8—182no—
235Q8—362no—
236D4—182no—
237D4—362no—
238C2×C4—32no—
239C2×C4—32no—
240C2×C4—32no—
241C2×C4—91no—
242C2×C4—91no—
243C2×C4—91no—
244C2×C4—182no—
245C2×C4—362no—
246C2×C4C2×D3211yescharacteristic
247C2×C4C2×D3211yescharacteristic
248C2×C4C2×D3211yescharacteristic
249C2×C3—21no—
250C2×C3—21no—
251C2×C3—21no—
252C2×C3—122no—
253C2×C3D3×#4112yes—
254C2×C3#100912yes—
255C2×C3#101012yes—
256D3—42no—
257D3—122no—
258D3—122no—
259C22—62no—
260C22—62no—
261C22—62no—
262C22—181no—
263C22—181no—
264C22—181no—
265C22C22×D3211yescharacteristic
266C4—21no—
267C4—21no—
268C4—21no—
269C4—62no—
270C4—62no—
271C4—62no—
272C4—92no—
273C4—181no—
274C4—181no—
275C3—21no—
276C3#571012yescomplemented
277C2—122no—
278C2—361no—
279C2#6430611yescharacteristic
280C2#6430911yescharacteristic
281C2#6432011yescharacteristic
subgroupG/H
center, FrattiniC22C22×D32
commutator, socle, cosocleC22×C32C24
FittingC32×C4⋊C4C22
prime radical#63912C2
hypercenterC4⋊C4D32
hypocenterC32#542
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
1. #64101C2semidirect
2. C3×#898C2semidirect
3. #63919C2semidirect
4. C4×#627C2semidirect
5. #63915C2semidirect
7. #63902C2semidirect
8. #63893C2semidirect
9. #63890C2semidirect
10. #63865C2semidirect
11. #63852C2semidirect
37. #898D3semidirect
231. C32#542semidirect
276. C3#5710semidirect

Maximal subgroups

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

subgroupindex
#641012
C3×#8982
#639192
C4×#6272
#639152
#639122
#639022
#638932
#638902
#638652
#638522
#57103