Finite Group Databasebeta

← #10594507

Subgroups — #10594507

Order 576.

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

normal subgroups only

This group has 420 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#64099C211yesmaximal, characteristic, complemented
2C3×#898C212yesmaximal, complemented
3#63918C212yesmaximal, complemented
4#63915C211yesmaximal, characteristic
5#63899C212yesmaximal, complemented
6#63893C212yesmaximal, complemented
7#63891C212yesmaximal
8#63870C211yesmaximal, characteristic, complemented
9#63859C212yesmaximal, complemented
10#5712—32nomaximal
11C2×#627C2212yes—
12C2×#627C2211yescharacteristic
13C2×#625C2211yescharacteristic
14C2×D3×Dic3C2212yes—
15C32×C4⋊C4C2211yescharacteristic
16C4⋊(C3⋊Dic3)C2212yes—
17C4⋊(C3⋊Dic3)C2211yescharacteristic
18#3750C2212yes—
19C4×C3⋊Dic3C2211yescharacteristic
20C3×(C2×D3)⋊C4C2212yes—
21C3×(C2×D3)⋊C4C2212yes—
22C3×(C2×D3)⋊C4C2212yes—
23C3×C4⋊Dic3C2212yes—
24C3×Dic3⋊C4C2212yes—
25C3×C4×Dic3C2212yes—
26#3724C2211yescharacteristic
27#3724C2211yescharacteristic
28Dic3⋊Dic3C2212yes—
29Dic3⋊Dic3C2212yes—
30(C2×D3)⋊Dic3C2212yes—
31(C2×D3)⋊Dic3C2212yes—
32C3×#126—32no—
33#898D312yescomplemented
34#897—32no—
35#896—32no—
36#890—32no—
37#890—61no—
38#889—32no—
39#888—32no—
40#886—32no—
41#885—32no—
42#883—32no—
43#879—32no—
44#878—32no—
45#878—32no—
46#876—32no—
47#869—32no—
48C22×C3×D3C2312yes—
49C2×C32×C4C2312yes—
50C2×C32×C4C2311yescharacteristic
51C2×C3⋊Dic3C2311yescharacteristic
52C2×C3⋊Dic3C2312yes—
53C2×C3⋊Dic3C2311yescharacteristic
54C2×C3×Dic3C2312yes—
55C2×C3×Dic3C2312yes—
56C2×C3×Dic3C2312yes—
57#627—42no—
58#627—41no—
59#625—41no—
60D3×Dic3—42no—
61#563—91noSylow
62C2×#67—32no—
63C22×Dic3—32no—
64C2×C4×D3—32no—
65C2×Dic6—32no—
66C2×Dic6—32no—
67C2×Dic6—32no—
68C3×C4⋊C4—21no—
69C3×C4⋊C4—32no—
70C3×C4⋊C4—32no—
71C3×C4⋊C4C2×D312yes—
72C3×C22⋊C4—32no—
73C3×C22⋊C4—32no—
74C3×C22⋊C4—32no—
75C3×C42—32no—
76#217—32no—
77#217—32no—
78(C2×D3)⋊C4—32no—
79(C2×D3)⋊C4—32no—
80(C2×D3)⋊C4C2×D312yes—
81(C2×D3)⋊C4C2×D312yes—
82(C2×D3)⋊C4C2×D312yes—
83C4⋊Dic3—32no—
84C4⋊Dic3—32no—
85C4⋊Dic3—32no—
86C4⋊Dic3—32no—
87C4⋊Dic3—32no—
88C4⋊Dic3—61no—
89C4⋊Dic3—62no—
90C4⋊Dic3C2×D312yes—
91Dic3⋊C4—32no—
92Dic3⋊C4—32no—
93Dic3⋊C4—32no—
94Dic3⋊C4—32no—
95Dic3⋊C4—32no—
96Dic3⋊C4—32no—
97Dic3⋊C4—62no—
98Dic3⋊C4C2×D312yes—
99C4×Dic3—32no—
100C4×Dic3—61no—
101C4×Dic3C2×D312yes—
102C22×C32C2411yescharacteristic
103C2×C3×D3—22no—
104C2×C3×D3—22no—
105C2×C3×D3—22no—
106C32×C4—21no—
107C32×C4—22no—
108C3⋊Dic3—22no—
109C3⋊Dic3—21no—
110C3⋊Dic3—21no—
111C3×Dic3—22no—
112C3×Dic3—22no—
113C3×Dic3—22no—
114C4⋊Q8—92no—
115#126—32no—
116#126—92no—
117#125—91no—
118#125—91no—
119#124—91no—
120#123—92no—
121C22⋊Q8—92no—
122C22⋊Q8—92no—
123C23×C3—32no—
124C22×D3C22×D312yes—
125C2×C3×C4—21no—
126C2×C3×C4—22no—
127C2×C3×C4—32no—
128C2×C3×C4—32no—
129C2×C3×C4—32no—
130C2×C3×C4C22×D312yes—
131C2×C3×C4C22×D312yes—
132C2×C3×C4C22×D312yes—
133#67—122no—
134C2×Dic3—32no—
135C2×Dic3—32no—
136C2×Dic3—32no—
137C2×Dic3—32no—
138C2×Dic3—61no—
139C2×Dic3—61no—
140C2×Dic3—62no—
141C2×Dic3—122no—
142C2×Dic3C22×D312yes—
143C2×Dic3C22×D312yes—
144C2×Dic3C22×D312yes—
145C4×D3—122no—
146Dic6—122no—
147Dic6—122no—
148Dic6—122no—
149C2×C32H(5,𝔽2)11yescharacteristic
150C2×C322−1+411yescharacteristic
151C2×C322−1+411yescharacteristic
152C3×D3—42no—
153C2×Q8—92no—
154C2×Q8—91no—
155C2×D4—91no—
156C22×C4—92no—
157C4⋊C4—32no—
158C4⋊C4—32no—
159C4⋊C4—91no—
160C4⋊C4—92no—
161C4⋊C4—92no—
162C4⋊C4—91no—
163C4⋊C4—91no—
164C4⋊C4—92no—
165C4⋊C4—92no—
166C4⋊C4D3211yescharacteristic
167C22⋊C4—32no—
168C22⋊C4—32no—
169C22⋊C4—32no—
170C22⋊C4—92no—
171C22⋊C4—92no—
172C42—32no—
173C42—91no—
174C22×C3—21no—
175C22×C3—62no—
176C22×C3—62no—
177C22×C3—62no—
178C22×C3C23×D312yes—
179C2×D3—22no—
180C2×D3—22no—
181C2×D3—22no—
182C3×C4—22no—
183C3×C4—22no—
184C3×C4—22no—
185C3×C4—22no—
186C3×C4—42no—
187C3×C4—62no—
188C3×C4—62no—
189C3×C4—62no—
190Dic3—22no—
191Dic3—22no—
192Dic3—22no—
193Dic3—62no—
194Dic3—62no—
195Dic3—62no—
196Dic3—62no—
197Dic3—121no—
198Dic3—122no—
199Dic3—121no—
200C32#56311yescharacteristic, complemented, Sylow
201C23—32no—
202Q8—362no—
203Q8—361no—
204D4—361no—
205C2×C4—32no—
206C2×C4—32no—
207C2×C4—32no—
208C2×C4—91no—
209C2×C4—91no—
210C2×C4—92no—
211C2×C4—362no—
212C2×C4C2×D3212yes—
213C2×C4C2×D3211yescharacteristic
214C2×C3—21no—
215C2×C3—21no—
216C2×C3—21no—
217C2×C3—122no—
218C2×C3#100912yes—
219C2×C3#101012yes—
220C2×C3#101012yes—
221D3—42no—
222C22—62no—
223C22—62no—
224C22—62no—
225C22C22×D3211yescharacteristic
226C4—21no—
227C4—22no—
228C4—62no—
229C4—62no—
230C4—62no—
231C4—181no—
232C4—182no—
233C4—181no—
234C3—21no—
235C3#571212yescomplemented
236C2—122no—
237C2#6429811yescharacteristic
238C2#6430911yescharacteristic
239C2#6432011yescharacteristic
subgroupG/H
center, FrattiniC22C22×D32
commutator, socle, cosocleC22×C32C24
FittingC32×C4⋊C4C22
prime radicalC2×#625C22
hypercenterC4⋊C4D32
hypocenterC32#563
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
1. #64099C2semidirect
2. C3×#898C2semidirect
3. #63918C2semidirect
5. #63899C2semidirect
6. #63893C2semidirect
8. #63870C2semidirect
9. #63859C2semidirect
33. #898D3semidirect
200. C32#563semidirect
235. C3#5712semidirect

Maximal subgroups

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

subgroupindex
#640992
C3×#8982
#639182
#639152
#638992
#638932
#638912
#638702
#638592
#57123