Finite Group Databasebeta

← #10590581

Subgroups — #10590581

Order 576.

254 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 254 conjugacy classes of subgroups — too many to draw. Only its 86 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×D3×C4⋊C4C211yesmaximal, characteristic
2#63841C211yesmaximal, characteristic
3C2×#3711C211yesmaximal, characteristic
4#63634C211yesmaximal, characteristic, complemented
5C3×#808C211yesmaximal, characteristic, complemented
6#63575C211yesmaximal, characteristic, complemented
7#63574C211yesmaximal, characteristic, complemented
8#4964—31nomaximal
9#4861—31nomaximal
10C2×C3×C4×D3—21no—
11C2×C3×C4×D3C2211yescharacteristic
12C32×C4⋊C4C2211yescharacteristic
13C4⋊(C3⋊Dic3)C2211yescharacteristic
14C2×C3⋊(C3⋊C8)C2211yescharacteristic
15C3×C4⋊Dic3C2211yescharacteristic
16C3×C4⋊Dic3C2211yescharacteristic
17C3×Dic3⋊C4—21no—
18C2×C3×C3⋊C8C2211yescharacteristic
19(C2×D3)⋊Dic3—21no—
20Dic32—21no—
21#3711C414yescomplemented
22C2×C3×C4⋊C4—31no—
23#926—31no—
24C2×#208—31no—
25C2×ΓL(1,𝔽25)—31no—
26#892—31no—
27D3×C4⋊C4D311yescharacteristic
28C3×#106—31no—
29#818—31no—
30#808—31no—
31#808—31no—
32#808D311yescharacteristic, complemented
33#807—31no—
34#807—31no—
35#807—31no—
36#807—61no—
37C22×C3×D3D411yescharacteristic
38C2×C32×C4—21no—
39C2×C32×C4C2311yescharacteristic
40C2×C3⋊Dic3—21no—
41C2×C3×Dic3—21no—
42C2×C3×Dic3—21no—
43C2×C3×Dic3D411yescharacteristic
44C3×C4×D3—42no—
45C3×C4×D3C2×C412yes—
46C3×C4×D3Q812yes—
47C3⋊(C3⋊C8)C2×C412yescomplemented
48C3×C3⋊C8C2×C412yescomplemented
49#428—91noSylow
50C22×C3×C4—31no—
51C22×C3×C4—61no—
52C2×C4×D3—21no—
53C2×C4×D3C2×D311yescharacteristic
54C2×C3×C8—31no—
55C3×C4⋊C4—21no—
56C3×C4⋊C4—31no—
57C3×C4⋊C4—31no—
58C3×C4⋊C4—61no—
59C3×C4⋊C4C2×D311yescharacteristic
60C3×C4⋊C4C2×D311yescharacteristic
61#217—61no—
62(C2×D3)⋊C4—61no—
63C4⋊Dic3—31no—
64C4⋊Dic3—31no—
65C4⋊Dic3—61no—
66C4⋊Dic3C2×D311yescharacteristic
67C4⋊Dic3C2×D311yescharacteristic
68Dic3⋊C4—21no—
69C4×Dic3—61no—
70C4×Dic3—61no—
71#208—34no—
72C2×C3⋊C8—31no—
73C2×C3⋊C8—31no—
74C2×C3⋊C8—61no—
75C2×C3⋊C8C2×D311yescharacteristic
76ΓL(1,𝔽25)—34no—
77C22×C32C2×D411yescharacteristic
78C2×C3×D3—21no—
79C2×C3×D3—21no—
80C2×C3×D3C4⋊C412yes—
81C32×C4—41no—
82C32×C4C22×C411yescharacteristic
83C32×C4C2×Q811yescharacteristic
84C3⋊Dic3—41no—
85C3×Dic3—41no—
86C3×Dic3—41no—
87C3×Dic3C4⋊C412yes—
88C2×M16—91no—
89#117—91no—
90C2×C4⋊C4—31no—
91#107—91no—
92#107—91no—
93#106—31no—
94#106—91no—
95C23×C3—31no—
96C22×D3#6711yescharacteristic
97C2×C3×C4—21no—
98C2×C3×C4—21no—
99C2×C3×C4—21no—
100C2×C3×C4—31no—
101C2×C3×C4—32no—
102C2×C3×C4—32no—
103C2×C3×C4—41no—
104C2×C3×C4—61no—
105C2×C3×C4—61no—
106C2×C3×C4—122no—
107C2×C3×C4C22×D311yescharacteristic
108C2×C3×C4C22×D311yescharacteristic
109C2×Dic3—21no—
110C2×Dic3—21no—
111C2×Dic3—61no—
112C2×Dic3—61no—
113C2×Dic3—121no—
114C2×Dic3#6711yescharacteristic
115C4×D3—42no—
116C4×D3Dic612yes—
117C4×D3C4×D312yes—
118C3×C8—32no—
119C3⋊C8—32no—
120C3⋊C8—32no—
121C3⋊C8—62no—
122C3⋊C8C4×D312yescomplemented
123C2×C32C2×C4⋊C411yescharacteristic
124C2×C32AGL(1,ℤ/8)11yescharacteristic
125C2×C32#13711yescharacteristic
126C3×D3—22no—
127C22×C4—31no—
128C22×C4—61no—
129M16—94no—
130C2×C8—31no—
131C2×C8—91no—
132C4⋊C4—31no—
133C4⋊C4—31no—
134C4⋊C4—61no—
135C4⋊C4—91no—
136C4⋊C4D3211yescharacteristic
137C22⋊C4—181no—
138C42—181no—
139C22×C3—21no—
140C22×C3—32no—
141C22×C3—61no—
142C22×C3—61no—
143C22×C3D3×D411yescharacteristic
144C22×C3C2×#6711yescharacteristic
145C2×D3—21no—
146C2×D3—21no—
147C2×D3Dic3⋊C412yes—
148C3×C4—21no—
149C3×C4—21no—
150C3×C4—32no—
151C3×C4—41no—
152C3×C4—41no—
153C3×C4—81no—
154C3×C4—121no—
155C3×C4—121no—
156C3×C4C2×Dic611yescharacteristic
157C3×C4C2×C4×D311yescharacteristic
158C3×C4C2×C4×D311yescharacteristic
159C3×C4D3×Q811yescharacteristic
160Dic3—41no—
161Dic3—41no—
162Dic3—121no—
163Dic3—121no—
164Dic3—241no—
165Dic3Dic3⋊C412yes—
166C32#42811yescharacteristic, complemented, Sylow
167C23—31no—
168C2×C4—21no—
169C2×C4—31no—
170C2×C4—32no—
171C2×C4—32no—
172C2×C4—61no—
173C2×C4—61no—
174C2×C4—122no—
175C2×C4—181no—
176C2×C4C2×D3211yescharacteristic
177C8—32no—
178C8—92no—
179C2×C3—21no—
180C2×C3—21no—
181C2×C3—21no—
182C2×C3—62no—
183C2×C3D3×C4⋊C411yescharacteristic
184C2×C3#91411yescharacteristic
185C2×C3#91511yescharacteristic
186C2×C3C2×Dic3⋊C411yescharacteristic
187C2×C3#93211yescharacteristic
188C2×C3#94211yescharacteristic
189D3—22no—
190C22—32no—
191C22—61no—
192C22—61no—
193C22D3×#6711yescharacteristic
194C4—32no—
195C4—41no—
196C4—121no—
197C4—121no—
198C4—361no—
199C4D3×Dic611yescharacteristic
200C4C4×D3211yescharacteristic
201C3—21no—
202C3#486111yescharacteristic, complemented
203C3#496411yescharacteristic, complemented
204C2—62no—
205C2D3×Dic3⋊C411yescharacteristic
206C2#6393411yescharacteristic
207C2#6394611yescharacteristic
subgroupG/H
centerC22D3×#67
commutatorC32×C4C22×C4
FrattiniC2×C4C2×D32
FittingC32×C4⋊C4C22
socleC22×C32C2×D4
prime radicalC22×C3×D3D4
cosocleC2×C32×C4C23
hypercenterC4⋊C4D32
hypocenterC32#428
LCSG ▷ C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C2×C4 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
4. #63634C2semidirect
5. C3×#808C2semidirect
6. #63575C2semidirect
7. #63574C2semidirect
21. #3711C4semidirect
32. #808D3semidirect
47. C3⋊(C3⋊C8)C2×C4semidirect
48. C3×C3⋊C8C2×C4semidirect
122. C3⋊C8C4×D3semidirect
166. C32#428semidirect
202. C3#4861semidirect
203. C3#4964semidirect

Maximal subgroups

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

subgroupindex
C3×D3×C4⋊C42
#638412
C2×#37112
#636342
C3×#8082
#635752
#635742
#49643
#48613