Finite Group Databasebeta

← #10596019

Subgroups — #10596019

Order 576.

930 conjugacy classes of subgroups; only the 205 normal subgroups are shown.

normal subgroups only

This group has 930 conjugacy classes of subgroups — too many to draw. Only its 205 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
1C2×#3828C212yesmaximal, complemented
2C32×C4×D4C211yesmaximal, characteristic, complemented
3#64149C211yesmaximal, characteristic, complemented
4#64147C211yesmaximal, characteristic, complemented
5C4×#638C212yesmaximal, complemented
6#64104C211yesmaximal, characteristic, complemented
7#64097C211yesmaximal, characteristic, complemented
8#64094C212yesmaximal, complemented
9#64093C212yesmaximal, complemented
10C4×C4⋊(C3⋊D3)C211yesmaximal, characteristic, complemented
11C42×C3⋊D3C211yesmaximal, characteristic, complemented
12#5548—34nomaximal
13C2×C32×D4C2211yescharacteristic
14C22×C32×C4C2212yes—
15C2×#638C2212yes—
16C2×#638C2212yes—
17C2×#638C2212yes—
18#3828—28no—
19C2×C4⋊(C3⋊D3)C2211yescharacteristic
20C2×C4⋊(C3⋊D3)C2212yes—
21C2×C4×C3⋊D3C2211yescharacteristic
22C2×C4×C3⋊D3C2211yescharacteristic
23C2×C4×C3⋊D3C2212yes—
24C2×C4×C3⋊D3C2211yescharacteristic
25C2×#634C2212yes—
26C32×C4⋊C4C2211yescharacteristic
27C32×C22⋊C4C2212yes—
28C32×C42C2211yescharacteristic
29#3757C2212yes—
30#3752C2212yes—
31#3752C2212yes—
32C4⋊(C3⋊Dic3)C2211yescharacteristic
33#3750C2212yes—
34C4×C3⋊Dic3C2211yescharacteristic
35C4×C3⋊Dic3C2211yescharacteristic
36C4×C3⋊Dic3C2211yescharacteristic
37C2×#235—38no—
38C3×C4×D4D314yescomplemented
39#940—34no—
40#938—34no—
41C4×#67—38no—
42#895—34no—
43#888—34no—
44#885—38no—
45#884—38no—
46C4×D12—34no—
47C42×D3—34no—
48C23×C32C2312yes—
49C22×C3⋊D3C2312yes—
50C22×C3⋊D3C2311yescharacteristic
51C32×D4—22no—
52C2×C32×C4—24no—
53C2×C32×C4C2311yescharacteristic
54C2×C32×C4C2312yes—
55C2×C32×C4C2311yescharacteristic
56C2×C32×C4C2311yescharacteristic
57#638—24no—
58#638—24no—
59#638—24no—
60C2×C3⋊Dic3C2311yescharacteristic
61C2×C3⋊Dic3C2312yes—
62C2×C3⋊Dic3C2311yescharacteristic
63C2×C3⋊Dic3C2311yescharacteristic
64C4⋊(C3⋊D3)—24no—
65C4⋊(C3⋊D3)—22no—
66C4×C3⋊D3—24no—
67C4×C3⋊D3—22no—
68C4×C3⋊D3—22no—
69C4×C3⋊D3D414yescomplemented
70#634—24no—
71#532—91noSylow
72C2×C3×D4C2×D314yes—
73C22×C3×C4C2×D318yes—
74C2×#67—38no—
75C2×#67—38no—
76C2×#67—38no—
77#235—632no—
78C2×D12—34no—
79C2×D12—38no—
80C2×C4×D3—34no—
81C2×C4×D3—34no—
82C2×C4×D3—34no—
83C2×C4×D3—38no—
84C2×Dic6—38no—
85C3×C4⋊C4C2×D314yes—
86C3×C22⋊C4C2×D318yes—
87C3×C42C2×D314yes—
88#217—38no—
89(C2×D3)⋊C4—38no—
90(C2×D3)⋊C4—38no—
91C4⋊Dic3—34no—
92Dic3⋊C4—38no—
93C4×Dic3—34no—
94C4×Dic3—34no—
95C4×Dic3—34no—
96C22×C32—22no—
97C22×C32—22no—
98C22×C32—22no—
99C22×C32C2411yescharacteristic
100C2×C3⋊D3—21no—
101C2×C3⋊D3—21no—
102C2×C3⋊D3—22no—
103C2×C3⋊D3—22no—
104C2×C3⋊D3—22no—
105C2×C3⋊D3C2×D412yes—
106C32×C4—22no—
107C32×C4—21no—
108C32×C4C2×D412yes—
109C32×C4#4112yescomplemented
110C3⋊Dic3—22no—
111C3⋊Dic3—21no—
112C3⋊Dic3C2×D412yes—
113C3⋊Dic3#4112yescomplemented
114C2×#41—92no—
115C4⋊Q8—91no—
116#127—91no—
117#124—92no—
118#121—91no—
119#121—91no—
120#121—92no—
121C4×D4—92no—
122C4×D4—91no—
123C4×D4C3⋊D311yescharacteristic, complemented
124C2×C42—91no—
125C23×C3C22×D318yes—
126C22×D3—34no—
127C22×D3—38no—
128C3×D4—28no—
129C2×C3×C4—216no—
130C2×C3×C4C22×D314yes—
131C2×C3×C4C22×D318yes—
132C2×C3×C4C22×D314yes—
133C2×C3×C4C22×D314yes—
134#67—616no—
135#67—616no—
136#67—616no—
137C2×Dic3—34no—
138C2×Dic3—34no—
139C2×Dic3—34no—
140C2×Dic3—38no—
141D12—616no—
142D12—68no—
143C4×D3—316no—
144C4×D3—68no—
145C4×D3—68no—
146C4×D3—616no—
147Dic6—616no—
148C2×C32—42no—
149C2×C32C22×D411yescharacteristic
150C2×C32C2×#4111yescharacteristic
151C2×C32C2×#4111yescharacteristic
152C3⋊D3—22no—
153C3⋊D3—42no—
154#41—188no—
155C2×Q8—92no—
156C2×D4—92no—
157C2×D4—92no—
158C2×D4—92no—
159C2×D4—91no—
160C2×D4—92no—
161C2×D4C2×C3⋊D311yescharacteristic
162C22×C4—91no—
163C22×C4—91no—
164C22×C4—91no—
165C22×C4—92no—
166C22×C4C2×C3⋊D312yes—
167C4⋊C4—92no—
168C4⋊C4—91no—
169C4⋊C4C2×C3⋊D311yescharacteristic
170C22⋊C4—92no—
171C22⋊C4—92no—
172C22⋊C4—92no—
173C22⋊C4C2×C3⋊D312yes—
174C42—91no—
175C42—91no—
176C42—91no—
177C42C2×C3⋊D311yescharacteristic
178C22×C3—28no—
179C22×C3—28no—
180C22×C3—28no—
181C22×C3C23×D314yes—
182C2×D3—38no—
183C2×D3—64no—
184C2×D3—64no—
185C2×D3—68no—
186C2×D3—68no—
187C2×D3—68no—
188C3×C4—28no—
189C3×C4—24no—
190C3×C4#23518yescomplemented
191C3×C4D3×D418yes—
192Dic3—38no—
193Dic3—38no—
194Dic3—64no—
195Dic3—68no—
196C32#53211yescharacteristic, complemented, Sylow
197C23—91no—
198C23—92no—
199C23C22×C3⋊D312yes—
200Q8—184no—
201D4—22no—
202D4—184no—
203D4—184no—
204D4—184no—
205D4—184no—
206D4—182no—
207C2×C4—24no—
208C2×C4—91no—
209C2×C4—91no—
210C2×C4—94no—
211C2×C4—91no—
212C2×C4—92no—
213C2×C4—182no—
214C2×C4—182no—
215C2×C4—184no—
216C2×C4C22×C3⋊D311yescharacteristic
217C2×C4C22×C3⋊D312yes—
218C2×C4C22×C3⋊D311yescharacteristic
219C2×C4C22×C3⋊D311yescharacteristic
220C2×C3—48no—
221C2×C3C2×#23514yes—
222C2×C3C2×D3×D414yes—
223C2×C3D3×#4114yes—
224D3—68no—
225D3—128no—
226C22—22no—
227C22—22no—
228C22—22no—
229C22—92no—
230C22—181no—
231C22—181no—
232C22—182no—
233C22—182no—
234C22—182no—
235C22C23×C3⋊D311yescharacteristic
236C4—22no—
237C4—21no—
238C4—92no—
239C4—92no—
240C4—181no—
241C4—182no—
242C4#382812yescomplemented
243C4D4×C3⋊D312yes—
244C3#554814yescomplemented
245C2—42no—
246C2—182no—
247C2—362no—
248C2C2×#382811yescharacteristic
249C2C2×D4×C3⋊D311yescharacteristic
250C2#41×C3⋊D311yescharacteristic
subgroupG/H
centerC2×C4C22×C3⋊D3
commutator, socle, cosocleC22×C32C24
FrattiniC22C23×C3⋊D3
FittingC32×C4×D4C2
hypercenterC4×D4C3⋊D3
hypocenterC32#532
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C2×C4 ◁ C4×D4

Complements

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

NH ≅ G/Ntype
1. C2×#3828C2semidirect
2. C32×C4×D4C2semidirect
3. #64149C2semidirect
4. #64147C2semidirect
5. C4×#638C2semidirect
6. #64104C2semidirect
7. #64097C2semidirect
8. #64094C2semidirect
9. #64093C2semidirect
10. C4×C4⋊(C3⋊D3)C2semidirect
11. C42×C3⋊D3C2semidirect
38. C3×C4×D4D3semidirect
69. C4×C3⋊D3D4semidirect
109. C32×C4#41semidirect
113. C3⋊Dic3#41semidirect
123. C4×D4C3⋊D3semidirect
190. C3×C4#235semidirect
196. C32#532semidirect
242. C4#3828semidirect
244. C3#5548semidirect

Maximal subgroups

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

subgroupindex
C2×#38282
C32×C4×D42
#641492
#641472
C4×#6382
#641042
#640972
#640942
#640932
C4×C4⋊(C3⋊D3)2
C42×C3⋊D32
#55483