Finite Group Databasebeta

← #10594569

Subgroups — #10594569

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×#896C212yesmaximal, complemented
3#63915C211yesmaximal, characteristic, complemented
4#63912C211yesmaximal, characteristic
5#63907C212yesmaximal, complemented
6C4×#625C211yesmaximal, characteristic, complemented
7#63873C211yesmaximal, characteristic, complemented
8#63865C211yesmaximal, characteristic, complemented
9#63861C212yesmaximal, complemented
10#63850C212yesmaximal, complemented
11#63838C211yesmaximal, characteristic, complemented
12#5640—32nomaximal
13C2×C4×C3⋊D3C2211yescharacteristic
14C2×C3×C4×D3C2212yes—
15C2×C3×Dic6C2212yes—
16C2×#627C2211yescharacteristic
17C2×#626C2212yes—
18C2×#625C2211yescharacteristic
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×Dic3⋊C4C2212yes—
28C3×Dic3⋊C4C2212yes—
29#3724C2211yescharacteristic
30#3724C2212yes—
31#3724C2211yescharacteristic
32#3722C2212yes—
33(C2×D3)⋊Dic3C2212yes—
34(C2×D3)⋊Dic3C2212yes—
35Dic32C2212yes—
36C3×C22⋊Q8—32no—
37#947—32no—
38#944—32no—
39#929—32no—
40C4×#67—32no—
41#898—32no—
42#896—32no—
43#896D312yescomplemented
44#894—32no—
45#892—32no—
46#892—61no—
47#887—32no—
48#885—32no—
49#885—32no—
50#884—32no—
51#879—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—
64C3×C4×D3—42no—
65C3×Dic6—42no—
66#627—22no—
67#626—42no—
68#625—22no—
69#542—91noSylow
70C2×C3×Q8—32no—
71C22×C3×C4—32no—
72C2×#67—32no—
73C2×#67—32no—
74C2×D12—32no—
75C2×C4×D3—32no—
76C2×C4×D3—61no—
77C2×C4×D3C2×D312yes—
78C2×Dic6—32no—
79C2×Dic6C2×D312yes—
80C3×C4⋊C4—21no—
81C3×C4⋊C4—32no—
82C3×C4⋊C4—32no—
83C3×C4⋊C4C2×D312yes—
84C3×C22⋊C4—32no—
85C3×C22⋊C4—32no—
86#217—32no—
87#217—32no—
88(C2×D3)⋊C4—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—
98C4⋊Dic3—32no—
99C4⋊Dic3—61no—
100Dic3⋊C4—32no—
101Dic3⋊C4—32no—
102Dic3⋊C4—32no—
103Dic3⋊C4—32no—
104Dic3⋊C4C2×D312yes—
105Dic3⋊C4C2×D312yes—
106C4×Dic3—32no—
107C4×Dic3—32no—
108C4×Dic3—32no—
109C4×Dic3—32no—
110C4×Dic3—61no—
111C4×Dic3—61no—
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—92no—
130#124—91no—
131#124—92no—
132#123—92no—
133C22⋊Q8—32no—
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—
143C3×Q8—122no—
144C2×C3×C4—21no—
145C2×C3×C4—21no—
146C2×C3×C4—21no—
147C2×C3×C4—32no—
148C2×C3×C4—32no—
149C2×C3×C4—32no—
150C2×C3×C4—122no—
151C2×C3×C4C22×D312yes—
152C2×C3×C4C22×D312yes—
153C2×C3×C4C22×D312yes—
154#67—64no—
155#67—122no—
156C2×Dic3—32no—
157C2×Dic3—32no—
158C2×Dic3—32no—
159C2×Dic3—61no—
160C2×Dic3—61no—
161C2×Dic3—61no—
162C2×Dic3C22×D312yes—
163C2×Dic3C22×D312yes—
164C2×Dic3C22×D312yes—
165D12—122no—
166C4×D3—42no—
167C4×D3—64no—
168C4×D3—64no—
169Dic6—42no—
170Dic6—64no—
171C2×C32C2×#4111yescharacteristic
172C2×C32H(5,𝔽2)11yescharacteristic
173C2×C322−1+411yescharacteristic
174C3⋊D3—41no—
175C3×D3—42no—
176C2×Q8—32no—
177C2×Q8—91no—
178C2×D4—92no—
179C2×D4—91no—
180C22×C4—32no—
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—92no—
192C22⋊C4—92no—
193C22⋊C4—92no—
194C22⋊C4—91no—
195C22⋊C4—91no—
196C42—92no—
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—122no—
235Q8—182no—
236D4—182no—
237D4—362no—
238C2×C4—32no—
239C2×C4—32no—
240C2×C4—32no—
241C2×C4—91no—
242C2×C4—91no—
243C2×C4—91no—
244C2×C4—122no—
245C2×C4—182no—
246C2×C4C2×D3211yescharacteristic
247C2×C4C2×D3211yescharacteristic
248C2×C4C2×D3211yescharacteristic
249C2×C3—21no—
250C2×C3—21no—
251C2×C3—21no—
252C2×C3—122no—
253C2×C3#100412yes—
254C2×C3#100712yes—
255C2×C3D3×#4112yes—
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#564012yescomplemented
277C2—122no—
278C2—361no—
279C2#6430611yescharacteristic
280C2#6431311yescharacteristic
281C2#6431611yescharacteristic
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×#896C2semidirect
3. #63915C2semidirect
5. #63907C2semidirect
6. C4×#625C2semidirect
7. #63873C2semidirect
8. #63865C2semidirect
9. #63861C2semidirect
10. #63850C2semidirect
11. #63838C2semidirect
43. #896D3semidirect
231. C32#542semidirect
276. C3#5640semidirect

Maximal subgroups

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

subgroupindex
#641012
C3×#8962
#639152
#639122
#639072
C4×#6252
#638732
#638652
#638612
#638502
#638382
#56403