Finite Group Databasebeta

← #10591451

Subgroups — #10591451

Order 576.

285 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 285 conjugacy classes of subgroups — too many to draw. Only its 64 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
1C3×#938C211yesmaximal, characteristic, complemented
2C2×C3×#213C211yesmaximal, characteristic, complemented
3#63908C211yesmaximal, characteristic, complemented
4C2×#3711C211yesmaximal, characteristic, complemented
5#63659C211yesmaximal, characteristic, complemented
6#63573C211yesmaximal, characteristic, complemented
7#63562C211yesmaximal, characteristic, complemented
8#5159—31nomaximal
9#5039—31nomaximal
10C2×C32×D4C2211yescharacteristic
11C2×C3×#67—21no—
12C2×C3×D12C2211yescharacteristic
13C2×C3×C4×D3C2211yescharacteristic
14C2×#625—21no—
15C4⋊(C3⋊Dic3)C2211yescharacteristic
16C2×C3⋊(C3⋊C8)C2211yescharacteristic
17C3×#217—21no—
18C3×#213—22no—
19C3×C4⋊Dic3C2211yescharacteristic
20C2×C3×C3⋊C8C2211yescharacteristic
21(C2×D3)⋊Dic3—21no—
22#3711—22no—
23C2×C3×D8—31no—
24C3×#121—31no—
25#938—31no—
26#938D311yescharacteristic, complemented
27C2×#213D311yescharacteristic, complemented
28#930—31no—
29C2×#208—31no—
30C2×ΓL(1,𝔽25)—31no—
31#832—31no—
32#832—31no—
33#832—31no—
34#832—61no—
35#817—31no—
36#809—31no—
37#808—31no—
38C23×C32—21no—
39C22×C3×D3—21no—
40C22×C3×D3D411yescharacteristic
41C32×D4—21no—
42C32×D4—21no—
43C2×C32×C4C2311yescharacteristic
44C2×C3⋊Dic3—21no—
45C3×#67—42no—
46C2×C3×Dic3—21no—
47C2×C3×Dic3D411yescharacteristic
48C3×D12—21no—
49C3×D12—21no—
50C3×C4×D3—21no—
51C3×C4×D3—21no—
52#625—42no—
53C3⋊(C3⋊C8)—21no—
54C3×C3⋊C8D412yescomplemented
55#469—91noSylow
56C2×C3×D4—21no—
57C2×C3×D4—31no—
58C2×C3×D4—61no—
59C2×C3×D4C2×D311yescharacteristic
60C2×C3×D4C2×D311yescharacteristic
61C22×C3×C4—31no—
62C2×#67—21no—
63C2×#67—61no—
64C2×#67—61no—
65C2×D12C2×D311yescharacteristic
66C2×C4×D3C2×D311yescharacteristic
67C3×D8—62no—
68C2×C3×C8—31no—
69C3×C4⋊C4—31no—
70C3×C22⋊C4—61no—
71#217—21no—
72#217—61no—
73#213—22no—
74(C2×D3)⋊C4—61no—
75C4⋊Dic3—31no—
76C4⋊Dic3—31no—
77C4⋊Dic3—61no—
78C4⋊Dic3C2×D311yescharacteristic
79#208—62no—
80C2×C3⋊C8—31no—
81C2×C3⋊C8—31no—
82C2×C3⋊C8—61no—
83C2×C3⋊C8C2×D311yescharacteristic
84ΓL(1,𝔽25)—62no—
85C22×C32—41no—
86C22×C32—41no—
87C22×C32—41no—
88C22×C32C2×D411yescharacteristic
89C2×C3×D3—21no—
90C2×C3×D3—21no—
91C2×C3×D3—21no—
92C2×C3×D3—41no—
93C2×C3×D3—41no—
94C2×C3×D3—41no—
95C32×C4C2×D411yescharacteristic
96C32×C4#4111yescharacteristic
97C3⋊Dic3—41no—
98C3×Dic3—21no—
99C3×Dic3—41no—
100C2×D8—31no—
101C2×M16—91no—
102#121—31no—
103#121—91no—
104#106—91no—
105#102—91no—
106#102—91no—
107C23×C3—21no—
108C23×C3—21no—
109C23×C3—31no—
110C23×C3—41no—
111C23×C3—61no—
112C22×D3—21no—
113C22×D3#6711yescharacteristic
114C3×D4—22no—
115C3×D4—21no—
116C3×D4—21no—
117C3×D4—21no—
118C3×D4—21no—
119C3×D4—41no—
120C3×D4—61no—
121C3×D4—61no—
122C3×D4—122no—
123C2×C3×C4—21no—
124C2×C3×C4—31no—
125C2×C3×C4—61no—
126C2×C3×C4—61no—
127C2×C3×C4—61no—
128C2×C3×C4C22×D311yescharacteristic
129C2×C3×C4C22×D311yescharacteristic
130#67—42no—
131#67—122no—
132#67—122no—
133C2×Dic3—21no—
134C2×Dic3—61no—
135C2×Dic3—61no—
136C2×Dic3—121no—
137C2×Dic3#6711yescharacteristic
138D12—21no—
139D12—21no—
140C4×D3—21no—
141C4×D3—21no—
142C3×C8—32no—
143C3⋊C8—61no—
144C3⋊C8—61no—
145C3⋊C8—121no—
146C3⋊C8#6712yescomplemented
147C2×C32—81no—
148C2×C32#12111yescharacteristic
149C2×C32AGL(1,ℤ/8)11yescharacteristic
150C2×C32AGL(1,ℤ/8)11yescharacteristic
151C3×D3—41no—
152C3×D3—81no—
153C2×D4—31no—
154C2×D4—61no—
155C2×D4—181no—
156C2×D4D3211yescharacteristic
157C22×C4—31no—
158D8—62no—
159M16—182no—
160C2×C8—31no—
161C2×C8—91no—
162C4⋊C4—31no—
163C4⋊C4—91no—
164C22⋊C4—61no—
165C22⋊C4—181no—
166C22×C3—21no—
167C22×C3—42no—
168C22×C3—41no—
169C22×C3—41no—
170C22×C3—41no—
171C22×C3—41no—
172C22×C3—41no—
173C22×C3—41no—
174C22×C3—61no—
175C22×C3—61no—
176C22×C3—61no—
177C22×C3—81no—
178C22×C3—81no—
179C22×C3—121no—
180C22×C3—121no—
181C22×C3—121no—
182C22×C3D3×D411yescharacteristic
183C22×C3C2×#6711yescharacteristic
184C2×D3—21no—
185C2×D3—21no—
186C2×D3—21no—
187C2×D3—41no—
188C2×D3—41no—
189C2×D3—41no—
190C3×C4—21no—
191C3×C4—21no—
192C3×C4—61no—
193C3×C4—121no—
194C3×C4D3×D411yescharacteristic
195C3×C4#23711yescharacteristic
196C3×C4#23711yescharacteristic
197C3×C4C2×#6711yescharacteristic
198Dic3—21no—
199Dic3—41no—
200Dic3—121no—
201Dic3—121no—
202Dic3—241no—
203C32#46911yescharacteristic, complemented, Sylow
204C23—21no—
205C23—31no—
206C23—61no—
207D4—21no—
208D4—21no—
209D4—61no—
210D4—61no—
211D4—122no—
212D4—362no—
213C2×C4—31no—
214C2×C4—61no—
215C2×C4—61no—
216C2×C4—61no—
217C2×C4—181no—
218C2×C4C2×D3211yescharacteristic
219C8—32no—
220C8—181no—
221C2×C3—21no—
222C2×C3—21no—
223C2×C3—21no—
224C2×C3—82no—
225C2×C3—81no—
226C2×C3—81no—
227C2×C3—121no—
228C2×C3—241no—
229C2×C3#91111yescharacteristic
230C2×C3#91111yescharacteristic
231C2×C3#93211yescharacteristic
232C2×C3#93811yescharacteristic
233C2×C3#93911yescharacteristic
234C2×C3#94911yescharacteristic
235D3—41no—
236D3—81no—
237C22—41no—
238C22—41no—
239C22—41no—
240C22—61no—
241C22—61no—
242C22—61no—
243C22—121no—
244C22—121no—
245C22—121no—
246C22D3×#6711yescharacteristic
247C4—61no—
248C4—121no—
249C4—361no—
250C4#380511yescharacteristic
251C4D3×#6711yescharacteristic
252C3—21no—
253C3#503911yescharacteristic, complemented
254C3#515911yescharacteristic, complemented
255C2—81no—
256C2—121no—
257C2—241no—
258C2#6392511yescharacteristic
259C2#6393211yescharacteristic
260C2#6397011yescharacteristic
subgroupG/H
centerC22D3×#67
commutator, cosocleC2×C32×C4C23
FrattiniC2×C4C2×D32
FittingC2×C32×D4C22
socleC22×C32C2×D4
hypercenterC2×D4D32
hypocenterC32#469
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
1. C3×#938C2semidirect
2. C2×C3×#213C2semidirect
3. #63908C2semidirect
4. C2×#3711C2semidirect
5. #63659C2semidirect
6. #63573C2semidirect
7. #63562C2semidirect
26. #938D3semidirect
27. C2×#213D3semidirect
54. C3×C3⋊C8D4semidirect
146. C3⋊C8#67semidirect
203. C32#469semidirect
253. C3#5039semidirect
254. C3#5159semidirect

Maximal subgroups

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

subgroupindex
C3×#9382
C2×C3×#2132
#639082
C2×#37112
#636592
#635732
#635622
#51593
#50393