Finite Group Databasebeta

← #10593962

Subgroups — #10593962

Order 576.

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

normal subgroups only

This group has 426 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#64078C211yesmaximal, characteristic, complemented
2C3×#876C212yesmaximal, complemented
3#63913C211yesmaximal, characteristic
4#63896C212yesmaximal, complemented
5#63893C212yesmaximal, complemented
6#63891C212yesmaximal
7#63861C212yesmaximal, complemented
8#63845C212yesmaximal, complemented
9#63840C211yesmaximal, characteristic, complemented
10#5712—32nomaximal
11C2×#634C2211yescharacteristic
12C2×#634C2211yescharacteristic
13C2×#627C2211yescharacteristic
14C2×#625C2211yescharacteristic
15C2×D3×Dic3C2212yes—
16C32×C42C2211yescharacteristic
17C4⋊(C3⋊Dic3)C2212yes—
18C4⋊(C3⋊Dic3)C2212yes—
19C3×(C2×D3)⋊C4C2212yes—
20C3×(C2×D3)⋊C4C2212yes—
21C3×(C2×D3)⋊C4C2212yes—
22C3×Dic3⋊C4C2212yes—
23C3×Dic3⋊C4C2212yes—
24C3×Dic3⋊C4C2212yes—
25#3724C2212yes—
26Dic3⋊Dic3C2212yes—
27Dic3⋊Dic3C2212yes—
28(C2×D3)⋊Dic3C2212yes—
29(C2×D3)⋊Dic3C2212yes—
30Dic32C2211yescharacteristic
31Dic32C2211yescharacteristic
32C3×#126—32no—
33#898—32no—
34#897—32no—
35#896—32no—
36#890—32no—
37#889—32no—
38#888—32no—
39#886—32no—
40#885—32no—
41#883—32no—
42#879—32no—
43#878—32no—
44#878—32no—
45#876D312yescomplemented
46#869—32no—
47#869—61no—
48C22×C3×D3C2312yes—
49C2×C32×C4C2312yes—
50C2×C32×C4C2311yescharacteristic
51C2×C3⋊Dic3C2311yescharacteristic
52C2×C3⋊Dic3C2311yescharacteristic
53C2×C3⋊Dic3C2312yes—
54#634—41no—
55#634—41no—
56C2×C3×Dic3C2312yes—
57C2×C3×Dic3C2312yes—
58C2×C3×Dic3C2312yes—
59#627—41no—
60#625—41no—
61D3×Dic3—42no—
62#563—91noSylow
63C2×#67—32no—
64C22×Dic3—32no—
65C2×C4×D3—32no—
66C2×Dic6—32no—
67C2×Dic6—32no—
68C2×Dic6—32no—
69C2×Dic6—61no—
70C2×Dic6—61no—
71C3×C4⋊C4—32no—
72C3×C4⋊C4—32no—
73C3×C4⋊C4—32no—
74C3×C22⋊C4—32no—
75C3×C22⋊C4—32no—
76C3×C22⋊C4—32no—
77C3×C42—21no—
78C3×C42C2×D312yes—
79#217—32no—
80#217—32no—
81(C2×D3)⋊C4—32no—
82(C2×D3)⋊C4—32no—
83(C2×D3)⋊C4C2×D312yes—
84(C2×D3)⋊C4C2×D312yes—
85(C2×D3)⋊C4C2×D312yes—
86C4⋊Dic3—32no—
87C4⋊Dic3—32no—
88C4⋊Dic3—32no—
89C4⋊Dic3—32no—
90C4⋊Dic3—32no—
91C4⋊Dic3—32no—
92C4⋊Dic3—62no—
93C4⋊Dic3—62no—
94Dic3⋊C4—32no—
95Dic3⋊C4—32no—
96Dic3⋊C4—32no—
97Dic3⋊C4—32no—
98Dic3⋊C4C2×D312yes—
99Dic3⋊C4C2×D312yes—
100Dic3⋊C4C2×D312yes—
101C4×Dic3—32no—
102C4×Dic3—32no—
103C22×C32C2411yescharacteristic
104C2×C3×D3—22no—
105C2×C3×D3—22no—
106C2×C3×D3—22no—
107C32×C4—21no—
108C32×C4—22no—
109C3⋊Dic3—22no—
110C3⋊Dic3—21no—
111C3⋊Dic3—21no—
112C3×Dic3—22no—
113C3×Dic3—22no—
114C3×Dic3—22no—
115C4⋊Q8—91no—
116C4⋊Q8—91no—
117#126—32no—
118#126—92no—
119#125—92no—
120#124—91no—
121#123—92no—
122C22⋊Q8—92no—
123C22⋊Q8—92no—
124C23×C3—32no—
125C22×D3C22×D312yes—
126C2×C3×C4—22no—
127C2×C3×C4—21no—
128C2×C3×C4—32no—
129C2×C3×C4—32no—
130C2×C3×C4—32no—
131C2×C3×C4C22×D312yes—
132C2×C3×C4C22×D312yes—
133C2×C3×C4C22×D312yes—
134#67—122no—
135C2×Dic3—32no—
136C2×Dic3—32no—
137C2×Dic3—32no—
138C2×Dic3—32no—
139C2×Dic3—61no—
140C2×Dic3—61no—
141C2×Dic3—62no—
142C2×Dic3—122no—
143C2×Dic3C22×D312yes—
144C2×Dic3C22×D312yes—
145C2×Dic3C22×D312yes—
146C4×D3—122no—
147Dic6—122no—
148Dic6—122no—
149Dic6—122no—
150Dic6—122no—
151Dic6—122no—
152C2×C32H(5,𝔽2)11yescharacteristic
153C2×C322−1+412yes—
154C3×D3—42no—
155C2×Q8—91no—
156C2×Q8—91no—
157C2×Q8—91no—
158C2×D4—91no—
159C22×C4—92no—
160C4⋊C4—32no—
161C4⋊C4—32no—
162C4⋊C4—32no—
163C4⋊C4—92no—
164C4⋊C4—92no—
165C4⋊C4—92no—
166C4⋊C4—92no—
167C4⋊C4—92no—
168C22⋊C4—32no—
169C22⋊C4—32no—
170C22⋊C4—32no—
171C22⋊C4—92no—
172C22⋊C4—92no—
173C42—91no—
174C42—91no—
175C42D3211yescharacteristic
176C22×C3—21no—
177C22×C3—62no—
178C22×C3—62no—
179C22×C3—62no—
180C22×C3C23×D312yes—
181C2×D3—22no—
182C2×D3—22no—
183C2×D3—22no—
184C3×C4—22no—
185C3×C4—24no—
186C3×C4—22no—
187C3×C4—22no—
188C3×C4—22no—
189C3×C4—62no—
190C3×C4—62no—
191C3×C4—62no—
192Dic3—22no—
193Dic3—22no—
194Dic3—22no—
195Dic3—62no—
196Dic3—62no—
197Dic3—62no—
198Dic3—62no—
199Dic3—122no—
200Dic3—121no—
201Dic3—121no—
202C32#56311yescharacteristic, complemented, Sylow
203C23—32no—
204Q8—361no—
205Q8—361no—
206Q8—361no—
207D4—361no—
208C2×C4—32no—
209C2×C4—32no—
210C2×C4—32no—
211C2×C4—91no—
212C2×C4—91no—
213C2×C4—92no—
214C2×C4—362no—
215C2×C4C2×D3212yes—
216C2×C4C2×D3211yescharacteristic
217C2×C3—22no—
218C2×C3—21no—
219C2×C3—122no—
220C2×C3#100912yes—
221C2×C3#101012yes—
222C2×C3#101012yes—
223D3—42no—
224C22—62no—
225C22—62no—
226C22—62no—
227C22C22×D3211yescharacteristic
228C4—22no—
229C4—21no—
230C4—62no—
231C4—62no—
232C4—62no—
233C4—182no—
234C4—181no—
235C4—181no—
236C3—21no—
237C3#571212yescomplemented
238C2—122no—
239C2#6429812yes—
240C2#6430811yescharacteristic
subgroupG/H
center, FrattiniC22C22×D32
commutator, socle, cosocleC22×C32C24
FittingC32×C42C22
prime radicalC2×#625C22
hypercenterC42D32
hypocenterC32#563
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C42

Complements

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

NH ≅ G/Ntype
1. #64078C2semidirect
2. C3×#876C2semidirect
4. #63896C2semidirect
5. #63893C2semidirect
7. #63861C2semidirect
8. #63845C2semidirect
9. #63840C2semidirect
45. #876D3semidirect
202. C32#563semidirect
237. C3#5712semidirect

Maximal subgroups

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

subgroupindex
#640782
C3×#8762
#639132
#638962
#638932
#638912
#638612
#638452
#638402
#57123