Finite Group Databasebeta

← #10591423

Subgroups — #10591423

Order 576.

237 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 237 conjugacy classes of subgroups — too many to draw. Only its 60 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
2#63841C211yesmaximal, characteristic, complemented
3#63659C211yesmaximal, characteristic, complemented
4C3×#832C211yesmaximal, characteristic, complemented
5#63575C211yesmaximal, characteristic, complemented
6#63574C211yesmaximal, characteristic, complemented
7(C2×D3)⋊(C3⋊C8)C211yesmaximal, characteristic, complemented
8#5035—31nomaximal
9#4784—31nomaximal
10C2×C32×D4C2211yescharacteristic
11C2×C3×#67—21no—
12C2×C3×C4×D3C2211yescharacteristic
13C4⋊(C3⋊Dic3)C2211yescharacteristic
14C2×C3⋊(C3⋊C8)C2211yescharacteristic
15C3×#217—21no—
16C3×C4⋊Dic3C2211yescharacteristic
17C3×C4⋊Dic3C2211yescharacteristic
18C2×C3×C3⋊C8C2211yescharacteristic
19(C2×D3)⋊Dic3—21no—
20Dic32—21no—
21C3×#121—31no—
22#938D311yescharacteristic
23#926—31no—
24#892—31no—
25C3×#102—31no—
26#832—31no—
27#832—31no—
28#832—61no—
29#832D311yescharacteristic, complemented
30#830—31no—
31(C2×D3)⋊C8—31no—
32#818—31no—
33#808—31no—
34#808—31no—
35#807—31no—
36C23×C32—21no—
37C22×C3×D3D411yescharacteristic
38C32×D4—21no—
39C32×D4—21no—
40C2×C32×C4C2311yescharacteristic
41C2×C3⋊Dic3—21no—
42C3×#67—42no—
43C2×C3×Dic3—21no—
44C2×C3×Dic3—21no—
45C2×C3×Dic3D411yescharacteristic
46C3×C4×D3—21no—
47C3×C4×D3—21no—
48C3⋊(C3⋊C8)—21no—
49C3×C3⋊C8—21no—
50#482—91noSylow
51C2×C3×D4—21no—
52C2×C3×D4—61no—
53C2×C3×D4C2×D311yescharacteristic
54C2×C3×D4C2×D311yescharacteristic
55C22×C3×C4—31no—
56C2×#67—21no—
57C2×C4×D3C2×D311yescharacteristic
58C2×C3×C8—31no—
59C3×C4⋊C4—31no—
60C3×C4⋊C4—31no—
61C3×C22⋊C4—61no—
62#217—21no—
63#217—61no—
64(C2×D3)⋊C4—61no—
65C4⋊Dic3—31no—
66C4⋊Dic3—31no—
67C4⋊Dic3—61no—
68C4⋊Dic3C2×D311yescharacteristic
69C4⋊Dic3C2×D311yescharacteristic
70C4×Dic3—61no—
71C4×Dic3—61no—
72C2×C3⋊C8—31no—
73C2×C3⋊C8—31no—
74C2×C3⋊C8—61no—
75C2×C3⋊C8C2×D311yescharacteristic
76C22×C32—41no—
77C22×C32—41no—
78C22×C32—41no—
79C22×C32C2×D411yescharacteristic
80C2×C3×D3—21no—
81C2×C3×D3—21no—
82C2×C3×D3—21no—
83C32×C4#4111yescharacteristic
84C32×C4#4111yescharacteristic
85C3⋊Dic3—41no—
86C3×Dic3—21no—
87C3×Dic3—41no—
88C3×Dic3—41no—
89#121—31no—
90#117—91no—
91#107—91no—
92#106—91no—
93#102—31no—
94#102—91no—
95C22⋊C8—91no—
96C23×C3—21no—
97C23×C3—21no—
98C23×C3—31no—
99C23×C3—41no—
100C22×D3#6711yescharacteristic
101C3×D4—22no—
102C3×D4—21no—
103C3×D4—21no—
104C3×D4—21no—
105C3×D4—21no—
106C3×D4—41no—
107C3×D4—122no—
108C2×C3×C4—21no—
109C2×C3×C4—31no—
110C2×C3×C4—61no—
111C2×C3×C4—61no—
112C2×C3×C4—61no—
113C2×C3×C4—61no—
114C2×C3×C4C22×D311yescharacteristic
115C2×C3×C4C22×D311yescharacteristic
116#67—42no—
117C2×Dic3—21no—
118C2×Dic3—21no—
119C2×Dic3—61no—
120C2×Dic3—61no—
121C2×Dic3—121no—
122C2×Dic3#6711yescharacteristic
123C4×D3—21no—
124C4×D3—21no—
125C3×C8—61no—
126C3⋊C8—21no—
127C3⋊C8—61no—
128C3⋊C8—61no—
129C3⋊C8—121no—
130C2×C32—81no—
131C2×C32#12311yescharacteristic
132C2×C32#13511yescharacteristic
133C2×C32AGL(1,ℤ/8)11yescharacteristic
134C3×D3—41no—
135C2×D4—61no—
136C2×D4D3211yescharacteristic
137C22×C4—31no—
138C2×C8—31no—
139C2×C8—91no—
140C4⋊C4—31no—
141C4⋊C4—31no—
142C4⋊C4—91no—
143C22⋊C4—61no—
144C22⋊C4—181no—
145C42—181no—
146C22×C3—21no—
147C22×C3—42no—
148C22×C3—41no—
149C22×C3—41no—
150C22×C3—41no—
151C22×C3—41no—
152C22×C3—41no—
153C22×C3—41no—
154C22×C3—61no—
155C22×C3—61no—
156C22×C3—61no—
157C22×C3—81no—
158C22×C3—81no—
159C22×C3D3×D411yescharacteristic
160C22×C3C2×#6711yescharacteristic
161C2×D3—21no—
162C2×D3—21no—
163C2×D3—21no—
164C3×C4—21no—
165C3×C4—21no—
166C3×C4—61no—
167C3×C4—121no—
168C3×C4—121no—
169C3×C4#23511yescharacteristic
170C3×C4#23711yescharacteristic
171C3×C4#23711yescharacteristic
172C3×C4#23711yescharacteristic
173Dic3—21no—
174Dic3—41no—
175Dic3—41no—
176Dic3—121no—
177Dic3—121no—
178Dic3—241no—
179C32#48211yescharacteristic, complemented, Sylow
180C23—21no—
181C23—31no—
182D4—21no—
183D4—21no—
184D4—122no—
185C2×C4—31no—
186C2×C4—61no—
187C2×C4—61no—
188C2×C4—61no—
189C2×C4—61no—
190C2×C4—181no—
191C2×C4C2×D3211yescharacteristic
192C8—61no—
193C8—181no—
194C2×C3—21no—
195C2×C3—21no—
196C2×C3—21no—
197C2×C3—82no—
198C2×C3—81no—
199C2×C3—81no—
200C2×C3—121no—
201C2×C3#88311yescharacteristic
202C2×C3#91211yescharacteristic
203C2×C3#91411yescharacteristic
204C2×C3#93211yescharacteristic
205C2×C3#93511yescharacteristic
206C2×C3#95011yescharacteristic
207D3—41no—
208C22—41no—
209C22—41no—
210C22—41no—
211C22—61no—
212C22—61no—
213C22—61no—
214C22D3×#6711yescharacteristic
215C4—61no—
216C4—121no—
217C4—121no—
218C4—361no—
219C4#380411yescharacteristic
220C4#380511yescharacteristic
221C3—21no—
222C3#478411yescharacteristic, complemented
223C3#503511yescharacteristic, complemented
224C2—81no—
225C2—121no—
226C2#6393311yescharacteristic
227C2#6393411yescharacteristic
228C2#6396911yescharacteristic
subgroupG/H
centerC22D3×#67
commutator, cosocleC2×C32×C4C23
FrattiniC2×C4C2×D32
FittingC2×C32×D4C22
socleC22×C32C2×D4
prime radicalC3×#938C2
hypercenterC2×D4D32
hypocenterC32#482
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
2. #63841C2semidirect
3. #63659C2semidirect
4. C3×#832C2semidirect
5. #63575C2semidirect
6. #63574C2semidirect
7. (C2×D3)⋊(C3⋊C8)C2semidirect
29. #832D3semidirect
179. C32#482semidirect
222. C3#4784semidirect
223. C3#5035semidirect

Maximal subgroups

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

subgroupindex
C3×#9382
#638412
#636592
C3×#8322
#635752
#635742
(C2×D3)⋊(C3⋊C8)2
#50353
#47843