Finite Group Databasebeta

← #10594534

Subgroups — #10594534

Order 576.

491 conjugacy classes of subgroups; only the 136 normal subgroups are shown.

normal subgroups only

This group has 491 conjugacy classes of subgroups — too many to draw. Only its 136 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#64096C211yesmaximal, characteristic, complemented
2C3×#896C212yesmaximal, complemented
3#63902C211yesmaximal, characteristic
4#63902C211yesmaximal, characteristic
5C4×#625C211yesmaximal, characteristic
6#63896C212yesmaximal, complemented
7#63887C212yesmaximal, complemented
8D3×Dic3⋊C4C212yesmaximal, complemented
9#63873C211yesmaximal, characteristic, complemented
10#63870C211yesmaximal, characteristic, complemented
11#63870C211yesmaximal, characteristic, complemented
12#5636—32nomaximal
13C2×#634C2211yescharacteristic
14C2×C3×C4×D3C2212yes—
15C2×C3×Dic6C2212yes—
16C2×#625C2211yescharacteristic
17C2×D3×Dic3C2212yes—
18C2×D3×Dic3C2212yes—
19C32×C4⋊C4C2211yescharacteristic
20#3750C2211yescharacteristic
21#3750C2211yescharacteristic
22C4×C3⋊Dic3C2211yescharacteristic
23C4×C3⋊Dic3C2211yescharacteristic
24C4×C3⋊Dic3C2211yescharacteristic
25C3×(C2×D3)⋊C4C2212yes—
26C3×(C2×D3)⋊C4C2212yes—
27C3×Dic3⋊C4C2212yes—
28C3×Dic3⋊C4C2212yes—
29#3724C2211yescharacteristic
30#3724C2211yescharacteristic
31#3724C2211yescharacteristic
32#3724C2212yes—
33Dic3⋊Dic3C2212yes—
34Dic3⋊Dic3C2212yes—
35(C2×D3)⋊Dic3C2212yes—
36C3×C22⋊Q8—32no—
37#946—32no—
38#944—32no—
39C4×#67—32no—
40C2×Dic3⋊C4—32no—
41#896—32no—
42#896D312yescomplemented
43D3×C4⋊C4—32no—
44#889—32no—
45#889—32no—
46#887—32no—
47#887—61no—
48#67⋊C4—32no—
49#67⋊C4—32no—
50#878—32no—
51#878—32no—
52C22×C3×D3C2312yes—
53C2×C32×C4C2311yescharacteristic
54C2×C32×C4C2311yescharacteristic
55C2×C32×C4C2311yescharacteristic
56C2×C3⋊Dic3C2311yescharacteristic
57C2×C3⋊Dic3C2311yescharacteristic
58C2×C3⋊Dic3C2311yescharacteristic
59C2×C3⋊Dic3C2311yescharacteristic
60#634—22no—
61C2×C3×Dic3C2312yes—
62C2×C3×Dic3C2312yes—
63C2×C3×Dic3C2312yes—
64C3×C4×D3—42no—
65C3×Dic6—42no—
66#625Q814yescomplemented
67D3×Dic3—24no—
68D3×Dic3—42no—
69#554—91noSylow
70C2×C3×Q8—32no—
71C22×C3×C4—32no—
72C2×#67—32no—
73C22×Dic3—32no—
74C22×Dic3—32no—
75C2×C4×D3—32no—
76C2×C4×D3—32no—
77C2×C4×D3C2×D312yes—
78C2×Dic6—32no—
79C2×Dic6—61no—
80C2×Dic6C2×D312yes—
81C3×C4⋊C4—21no—
82C3×C4⋊C4—32no—
83C3×C4⋊C4—32no—
84C3×C4⋊C4C2×D312yes—
85C3×C22⋊C4—32no—
86C3×C22⋊C4—32no—
87#217—32no—
88(C2×D3)⋊C4—32no—
89(C2×D3)⋊C4C2×D312yes—
90(C2×D3)⋊C4C2×D312yes—
91C4⋊Dic3—32no—
92C4⋊Dic3—32no—
93Dic3⋊C4—32no—
94Dic3⋊C4—32no—
95Dic3⋊C4—32no—
96Dic3⋊C4—32no—
97Dic3⋊C4—32no—
98Dic3⋊C4—32no—
99Dic3⋊C4—32no—
100Dic3⋊C4—32no—
101Dic3⋊C4—32no—
102Dic3⋊C4—61no—
103Dic3⋊C4—61no—
104Dic3⋊C4C2×D312yes—
105Dic3⋊C4C2×D312yes—
106C4×Dic3—32no—
107C4×Dic3—32no—
108C4×Dic3—32no—
109C4×Dic3—61no—
110C4×Dic3—61no—
111C4×Dic3—61no—
112C22×C32C2411yescharacteristic
113C2×C3×D3—22no—
114C2×C3×D3—22no—
115C2×C3×D3C2×Q814yes—
116C32×C4—21no—
117C32×C4—21no—
118C32×C4—21no—
119C3⋊Dic3—21no—
120C3⋊Dic3—21no—
121C3⋊Dic3C2×Q812yes—
122C3⋊Dic3#4112yes—
123C3×Dic3—22no—
124C3×Dic3—22no—
125C3×Dic3—22no—
126C4⋊Q8—91no—
127#125—91no—
128#125—91no—
129C22⋊Q8—32no—
130C22⋊Q8—92no—
131C22⋊Q8—92no—
132C4×Q8—91no—
133C4×D4—91no—
134C4×D4—91no—
135C4×D4—91no—
136C2×C4⋊C4—92no—
137C23×C3—32no—
138C22×D3C22×D312yes—
139C3×Q8—122no—
140C2×C3×C4—21no—
141C2×C3×C4—21no—
142C2×C3×C4—21no—
143C2×C3×C4—32no—
144C2×C3×C4—32no—
145C2×C3×C4—32no—
146C2×C3×C4—122no—
147C2×C3×C4C22×D312yes—
148C2×C3×C4C22×D312yes—
149C2×C3×C4C22×D312yes—
150#67—38no—
151C2×Dic3—32no—
152C2×Dic3—32no—
153C2×Dic3—32no—
154C2×Dic3—32no—
155C2×Dic3—61no—
156C2×Dic3—61no—
157C2×Dic3—61no—
158C2×Dic3—61no—
159C2×Dic3—64no—
160C2×Dic3—122no—
161C2×Dic3C22×D312yes—
162C2×Dic3C22×D312yes—
163C2×Dic3C22×D312yes—
164C4×D3—42no—
165C4×D3—64no—
166C4×D3—122no—
167Dic6—42no—
168Dic6—64no—
169Dic6—64no—
170C2×C32C22×Q811yescharacteristic
171C2×C32C2×#4111yescharacteristic
172C2×C32H(5,𝔽2)11yescharacteristic
173C3×D3—24no—
174C2×Q8—32no—
175C2×Q8—91no—
176C2×D4—91no—
177C22×C4—32no—
178C22×C4—92no—
179C22×C4—92no—
180C4⋊C4—32no—
181C4⋊C4—32no—
182C4⋊C4—92no—
183C4⋊C4—92no—
184C4⋊C4—91no—
185C4⋊C4—91no—
186C4⋊C4—91no—
187C4⋊C4—92no—
188C4⋊C4—91no—
189C4⋊C4—91no—
190C4⋊C4D3211yescharacteristic
191C22⋊C4—32no—
192C22⋊C4—32no—
193C22⋊C4—92no—
194C42—91no—
195C42—91no—
196C42—91no—
197C22×C3—21no—
198C22×C3—34no—
199C22×C3—62no—
200C22×C3—62no—
201C22×C3C23×D312yes—
202C2×D3—22no—
203C2×D3—22no—
204C2×D3D3×Q814yes—
205C3×C4—22no—
206C3×C4—22no—
207C3×C4—22no—
208C3×C4—22no—
209C3×C4—41no—
210C3×C4—41no—
211C3×C4—62no—
212C3×C4—62no—
213C3×C4—62no—
214Dic3—22no—
215Dic3—22no—
216Dic3—22no—
217Dic3—34no—
218Dic3—34no—
219Dic3—62no—
220Dic3—62no—
221Dic3—62no—
222Dic3—62no—
223Dic3—62no—
224Dic3—121no—
225C32#55411yescharacteristic, complemented, Sylow
226C23—32no—
227Q8—122no—
228Q8—182no—
229D4—94no—
230C2×C4—32no—
231C2×C4—32no—
232C2×C4—32no—
233C2×C4—91no—
234C2×C4—91no—
235C2×C4—91no—
236C2×C4—91no—
237C2×C4—122no—
238C2×C4—184no—
239C2×C4—362no—
240C2×C4C2×D3211yescharacteristic
241C2×C4C2×D3211yescharacteristic
242C2×C4C2×D3211yescharacteristic
243C2×C3—21no—
244C2×C3—21no—
245C2×C3—21no—
246C2×C3—64no—
247C2×C3#100412yes—
248C2×C3C2×D3×Q812yes—
249C2×C3D3×#4112yes—
250D3—24no—
251C22—34no—
252C22—62no—
253C22—62no—
254C22C22×D3211yescharacteristic
255C4—21no—
256C4—21no—
257C4—21no—
258C4—62no—
259C4—62no—
260C4—62no—
261C4—92no—
262C4—92no—
263C4—181no—
264C4—181no—
265C3—21no—
266C3#563612yescomplemented
267C2—64no—
268C2#6430611yescharacteristic
269C2#6431311yescharacteristic
270C2D32×Q811yescharacteristic
subgroupG/H
center, FrattiniC22C22×D32
commutator, socle, cosocleC22×C32C24
FittingC32×C4⋊C4C22
prime radicalC2×#625C22
hypercenterC4⋊C4D32
hypocenterC32#554
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
1. #64096C2semidirect
2. C3×#896C2semidirect
6. #63896C2semidirect
7. #63887C2semidirect
8. D3×Dic3⋊C4C2semidirect
9. #63873C2semidirect
10. #63870C2semidirect
11. #63870C2semidirect
42. #896D3semidirect
66. #625Q8semidirect
225. C32#554semidirect
266. C3#5636semidirect

Maximal subgroups

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

subgroupindex
#640962
C3×#8962
#639022
C4×#6252
#638962
#638872
D3×Dic3⋊C42
#638732
#638702
#56363