Finite Group Databasebeta

← #10596295

Subgroups — #10596295

Order 576.

876 conjugacy classes of subgroups; only the 233 normal subgroups are shown.

normal subgroups only

This group has 876 conjugacy classes of subgroups — too many to draw. Only its 233 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×C32×#41C211yesmaximal, characteristic, complemented
2C2×#3830C211yesmaximal, characteristic, complemented
3#64155C211yesmaximal, characteristic, complemented
4#64153C211yesmaximal, characteristic, complemented
5#64148C212yesmaximal, complemented
6#64144C212yesmaximal, complemented
7D4×C3⋊Dic3C211yesmaximal, characteristic, complemented
8#64138C212yesmaximal, complemented
9C4×#638C211yesmaximal, characteristic, complemented
10#64133C211yesmaximal, characteristic, complemented
11C2×#3750C212yesmaximal, complemented
12#5824—34nomaximal
13C32×#41—24no—
14C2×C32×Q8C2211yescharacteristic
15C2×C32×D4C2212yes—
16C2×C32×D4C2211yescharacteristic
17C22×C32×C4C2212yes—
18C22×C32×C4C2211yescharacteristic
19C2×#638C2212yes—
20C2×#638C2211yescharacteristic
21C22×C3⋊Dic3C2212yes—
22C22×C3⋊Dic3C2212yes—
23#3830—24no—
24C2×C4×C3⋊D3C2211yescharacteristic
25C2×#634C2211yescharacteristic
26#3757C2212yes—
27#3757C2211yescharacteristic
28#3757C2212yes—
29#3752C2211yescharacteristic
30#3752C2212yes—
31C4⋊(C3⋊Dic3)C2211yescharacteristic
32#3750C2211yescharacteristic
33#3750C2212yes—
34#3750C2212yes—
35#3750C2212yes—
36#3750C2212yes—
37C4×C3⋊Dic3C2211yescharacteristic
38C2×C3×#41D314yescomplemented
39C2×#237—34no—
40#946—34no—
41#944—34no—
42#939—38no—
43#935—38no—
44D4×Dic3—34no—
45#929—38no—
46C4×#67—34no—
47#924—34no—
48C2×Dic3⋊C4—38no—
49C23×C32C2311yescharacteristic
50C23×C32C2312yes—
51C22×C3⋊D3C2311yescharacteristic
52C32×Q8—22no—
53C32×D4—24no—
54C32×D4D414yescomplemented
55C2×C32×C4—24no—
56C2×C32×C4—22no—
57C2×C32×C4C2311yescharacteristic
58C2×C32×C4C2312yes—
59C2×C32×C4C2311yescharacteristic
60#638—24no—
61#638—22no—
62C2×C3⋊Dic3—24no—
63C2×C3⋊Dic3—24no—
64C2×C3⋊Dic3C2311yescharacteristic
65C2×C3⋊Dic3C2312yes—
66C2×C3⋊Dic3C2311yescharacteristic
67C2×C3⋊Dic3C2312yes—
68C2×C3⋊Dic3C2311yescharacteristic
69C4×C3⋊D3—22no—
70#634—22no—
71#547—91noSylow
72C3×#41—216no—
73C2×C3×Q8C2×D314yes—
74C2×C3×D4C2×D318yes—
75C2×C3×D4C2×D314yes—
76C22×C3×C4C2×D318yes—
77C22×C3×C4C2×D314yes—
78C2×#67—34no—
79C2×#67—38no—
80C22×Dic3—38no—
81C22×Dic3—38no—
82#237—616no—
83C2×C4×D3—34no—
84C2×Dic6—34no—
85#217—38no—
86#217—34no—
87#217—38no—
88(C2×D3)⋊C4—34no—
89(C2×D3)⋊C4—38no—
90C4⋊Dic3—34no—
91Dic3⋊C4—34no—
92Dic3⋊C4—38no—
93Dic3⋊C4—38no—
94Dic3⋊C4—38no—
95Dic3⋊C4—38no—
96C4×Dic3—34no—
97C22×C32—22no—
98C22×C32—22no—
99C22×C32—21no—
100C22×C32—21no—
101C22×C32—21no—
102C22×C32C2×D414yes—
103C22×C32C2411yescharacteristic
104C2×C3⋊D3—21no—
105C2×C3⋊D3—21no—
106C2×C3⋊D3—21no—
107C32×C4—21no—
108C32×C4—22no—
109C32×C4C2×D412yes—
110C3⋊Dic3—22no—
111C3⋊Dic3—21no—
112C3⋊Dic3—22no—
113C3⋊Dic3—21no—
114C3⋊Dic3#4112yescomplemented
115C2×#41—91no—
116C2×#41C3⋊D311yescharacteristic, complemented
117C4⋊Q8—91no—
118#123—92no—
119#123—92no—
120C22⋊Q8—91no—
121C22⋊Q8—91no—
122#121—92no—
123C4×D4—91no—
124C4×D4—91no—
125C2×C4⋊C4—92no—
126C23×C3C22×D314yes—
127C23×C3C22×D318yes—
128C22×D3—34no—
129C3×Q8—28no—
130C3×D4—216no—
131C3×D4#67116yescomplemented
132C2×C3×C4—216no—
133C2×C3×C4—28no—
134C2×C3×C4C22×D314yes—
135C2×C3×C4C22×D318yes—
136C2×C3×C4C22×D314yes—
137#67—68no—
138#67—616no—
139C2×Dic3—34no—
140C2×Dic3—34no—
141C2×Dic3—38no—
142C2×Dic3—34no—
143C2×Dic3—38no—
144C2×Dic3—616no—
145C2×Dic3—616no—
146C4×D3—68no—
147Dic6—68no—
148C2×C32—24no—
149C2×C32—41no—
150C2×C32C22×D411yescharacteristic
151C2×C32C2×#4111yescharacteristic
152C2×C322−1+411yescharacteristic
153C3⋊D3—41no—
154#41—24no—
155#41—184no—
156C2×Q8—91no—
157C2×Q8C2×C3⋊D311yescharacteristic
158C2×D4—91no—
159C2×D4—92no—
160C2×D4C2×C3⋊D312yes—
161C2×D4C2×C3⋊D311yescharacteristic
162C22×C4—92no—
163C22×C4—92no—
164C22×C4—91no—
165C22×C4C2×C3⋊D312yes—
166C22×C4C2×C3⋊D311yescharacteristic
167C4⋊C4—91no—
168C4⋊C4—92no—
169C4⋊C4—92no—
170C4⋊C4—92no—
171C4⋊C4—91no—
172C4⋊C4—92no—
173C22⋊C4—92no—
174C22⋊C4—91no—
175C22⋊C4—92no—
176C22⋊C4—91no—
177C22⋊C4—92no—
178C42—91no—
179C22×C3—28no—
180C22×C3—28no—
181C22×C3—24no—
182C22×C3—24no—
183C22×C3—24no—
184C22×C3C2×#67116yes—
185C22×C3C23×D314yes—
186C2×D3—64no—
187C2×D3—64no—
188C2×D3—64no—
189C3×C4—24no—
190C3×C4—28no—
191C3×C4C2×#6718yes—
192Dic3—38no—
193Dic3—68no—
194Dic3—64no—
195Dic3—68no—
196Dic3—64no—
197C32#54711yescharacteristic, complemented, Sylow
198C23—91no—
199C23C22×C3⋊D311yescharacteristic
200C23C22×C3⋊D312yes—
201Q8—22no—
202Q8—182no—
203D4—24no—
204D4—182no—
205D4—184no—
206D4#63814yescomplemented
207C2×C4—24no—
208C2×C4—22no—
209C2×C4—91no—
210C2×C4—91no—
211C2×C4—92no—
212C2×C4—91no—
213C2×C4—92no—
214C2×C4—184no—
215C2×C4—184no—
216C2×C4—182no—
217C2×C4C22×C3⋊D311yescharacteristic
218C2×C4C22×C3⋊D312yes—
219C2×C4C22×C3⋊D311yescharacteristic
220C2×C3—216no—
221C2×C3—44no—
222C2×C3D3×#4114yes—
223C2×C3#101014yes—
224C2×C3C22×#6714yes—
225D3—124no—
226C22—22no—
227C22—22no—
228C22—21no—
229C22—21no—
230C22—21no—
231C22—181no—
232C22—181no—
233C22—181no—
234C22C2×#63814yes—
235C22C23×C3⋊D311yescharacteristic
236C4—21no—
237C4—22no—
238C4—92no—
239C4—182no—
240C4—181no—
241C4—181no—
242C4—182no—
243C4C2×#63812yes—
244C3#582414yescomplemented
245C2—24no—
246C2—41no—
247C2—361no—
248C2#41×C3⋊D311yescharacteristic
249C2#6436711yescharacteristic
250C2C22×#63811yescharacteristic
subgroupG/H
center, FrattiniC22C23×C3⋊D3
commutator, socle, cosocleC22×C32C24
FittingC2×C32×#41C2
hypercenterC2×#41C3⋊D3
hypocenterC32#547
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C2×#41

Complements

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

NH ≅ G/Ntype
1. C2×C32×#41C2semidirect
2. C2×#3830C2semidirect
3. #64155C2semidirect
4. #64153C2semidirect
5. #64148C2semidirect
6. #64144C2semidirect
7. D4×C3⋊Dic3C2semidirect
8. #64138C2semidirect
9. C4×#638C2semidirect
10. #64133C2semidirect
11. C2×#3750C2semidirect
38. C2×C3×#41D3semidirect
54. C32×D4D4semidirect
114. C3⋊Dic3#41semidirect
116. C2×#41C3⋊D3semidirect
131. C3×D4#67semidirect
197. C32#547semidirect
206. D4#638semidirect
244. C3#5824semidirect

Maximal subgroups

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

subgroupindex
C2×C32×#412
C2×#38302
#641552
#641532
#641482
#641442
D4×C3⋊Dic32
#641382
C4×#6382
#641332
C2×#37502
#58243