Finite Group Databasebeta

← #10591440

Subgroups — #10591440

Order 576.

265 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 265 conjugacy classes of subgroups — too many to draw. Only its 84 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×D4×Dic3C211yesmaximal, characteristic
2C2×C3×#214C211yesmaximal, characteristic
3Dic3×Dic6C211yesmaximal, characteristic, complemented
4#63659C211yesmaximal, characteristic
5#63575C211yesmaximal, characteristic, complemented
6#63565C211yesmaximal, characteristic, complemented
7#63554C211yesmaximal, characteristic, complemented
8#5164—31nomaximal
9#5018—31nomaximal
10C2×C32×D4C2211yescharacteristic
11C22×C3×Dic3—21no—
12C2×C3×Dic6C2211yescharacteristic
13C4⋊(C3⋊Dic3)C2211yescharacteristic
14C2×C3⋊(C3⋊C8)C2211yescharacteristic
15C3×#217—21no—
16C3×#214C414yescomplemented
17C3×C4⋊Dic3C2211yescharacteristic
18C3×C4×Dic3C2211yescharacteristic
19C2×C3×C3⋊C8C2211yescharacteristic
20Dic3⋊Dic3—21no—
21Dic32—21no—
22C2×C3×SD16—31no—
23C3×C4×D4—31no—
24Q8×Dic3—31no—
25D4×Dic3D311yescharacteristic
26C2×#214D311yescharacteristic
27C4×Dic6—31no—
28#835—31no—
29#832—31no—
30#832—31no—
31#832—61no—
32#818—31no—
33#815—31no—
34#810—31no—
35#807—31no—
36#803—31no—
37C23×C32—21no—
38C32×D4—21no—
39C32×D4C2×C412yes—
40C2×C32×C4C2311yescharacteristic
41C2×C3⋊Dic3—21no—
42C2×C3×Dic3—21no—
43C2×C3×Dic3—21no—
44C2×C3×Dic3—42no—
45C2×C3×Dic3D411yescharacteristic
46C2×C3×Dic3D411yescharacteristic
47C3×Dic6—21no—
48C3×Dic6C2×C412yescomplemented
49C3⋊(C3⋊C8)—21no—
50C3×C3⋊C8C2×C412yescomplemented
51#440—91noSylow
52C2×C3×Q8—31no—
53C2×C3×D4—21no—
54C2×C3×D4C2×D311yescharacteristic
55C2×C3×D4C2×D311yescharacteristic
56C22×C3×C4—61no—
57C22×Dic3—21no—
58C2×Dic6C2×D311yescharacteristic
59C3×SD16—34no—
60C2×C3×C8—31no—
61C3×C4⋊C4—31no—
62C3×C22⋊C4—61no—
63C3×C42—31no—
64#217—21no—
65#214Dic314yescomplemented
66C4⋊Dic3—31no—
67C4⋊Dic3—31no—
68C4⋊Dic3—61no—
69C4⋊Dic3—61no—
70C4⋊Dic3C2×D311yescharacteristic
71Dic3⋊C4—61no—
72C4×Dic3—61no—
73C4×Dic3—61no—
74C4×Dic3C2×D311yescharacteristic
75C2×C3⋊C8—31no—
76C2×C3⋊C8—31no—
77C2×C3⋊C8—61no—
78C2×C3⋊C8C2×D311yescharacteristic
79C22×C32—22no—
80C22×C32—41no—
81C22×C32—41no—
82C22×C32C2×D411yescharacteristic
83C32×C4C22×C411yescharacteristic
84C32×C4#4111yescharacteristic
85C3⋊Dic3—41no—
86C3×Dic3—22no—
87C3×Dic3—21no—
88C3×Dic3—21no—
89C3×Dic3—41no—
90C2×SD16—31no—
91C4×Q8—91no—
92C4×D4—31no—
93#107—91no—
94#103—91no—
95#102—91no—
96#97—91no—
97C23×C3—21no—
98C23×C3—21no—
99C23×C3—41no—
100C3×Q8—32no—
101C3×Q8—61no—
102C3×D4—21no—
103C3×D4—21no—
104C3×D4—22no—
105C3×D4—41no—
106C3×D4C4×D312yes—
107C3×D4C2×Dic312yes—
108C2×C3×C4—21no—
109C2×C3×C4—31no—
110C2×C3×C4—31no—
111C2×C3×C4—61no—
112C2×C3×C4—61no—
113C2×C3×C4—122no—
114C2×C3×C4C22×D311yescharacteristic
115C2×C3×C4C22×D311yescharacteristic
116C2×Dic3—21no—
117C2×Dic3—21no—
118C2×Dic3—42no—
119C2×Dic3—61no—
120C2×Dic3—61no—
121C2×Dic3—121no—
122C2×Dic3#6711yescharacteristic
123C2×Dic3#6711yescharacteristic
124Dic6—21no—
125Dic6C2×Dic312yescomplemented
126C3×C8—32no—
127C3⋊C8—61no—
128C3⋊C8—61no—
129C3⋊C8—121no—
130C3⋊C8C2×Dic312yescomplemented
131C2×C32—42no—
132C2×C32C4×D411yescharacteristic
133C2×C32AGL(1,ℤ/8)11yescharacteristic
134C2×C32#13711yescharacteristic
135C2×Q8—31no—
136C2×D4D3211yescharacteristic
137C22×C4—61no—
138SD16—34no—
139C2×C8—31no—
140C2×C8—91no—
141C4⋊C4—31no—
142C4⋊C4—91no—
143C4⋊C4—181no—
144C22⋊C4—61no—
145C42—31no—
146C42—181no—
147C22×C3—22no—
148C22×C3—22no—
149C22×C3—21no—
150C22×C3—42no—
151C22×C3—41no—
152C22×C3—41no—
153C22×C3—41no—
154C22×C3—41no—
155C22×C3—81no—
156C22×C3—81no—
157C22×C3D3×D411yescharacteristic
158C22×C3C2×#6711yescharacteristic
159C3×C4—21no—
160C3×C4—21no—
161C3×C4—61no—
162C3×C4—61no—
163C3×C4—62no—
164C3×C4—121no—
165C3×C4C2×C4×D311yescharacteristic
166C3×C4#23511yescharacteristic
167C3×C4#23711yescharacteristic
168C3×C4C22×Dic311yescharacteristic
169Dic3—21no—
170Dic3—22no—
171Dic3—21no—
172Dic3—41no—
173Dic3—121no—
174Dic3—121no—
175Dic3—241no—
176C32#44011yescharacteristic, complemented, Sylow
177C23—21no—
178Q8—32no—
179Q8—61no—
180D4—21no—
181D4D3×Dic312yes—
182C2×C4—31no—
183C2×C4—31no—
184C2×C4—61no—
185C2×C4—61no—
186C2×C4—122no—
187C2×C4—181no—
188C2×C4C2×D3211yescharacteristic
189C8—32no—
190C8—181no—
191C2×C3—21no—
192C2×C3—21no—
193C2×C3—21no—
194C2×C3—42no—
195C2×C3—42no—
196C2×C3—82no—
197C2×C3#91411yescharacteristic
198C2×C3#91511yescharacteristic
199C2×C3C4×#6711yescharacteristic
200C2×C3#93211yescharacteristic
201C2×C3D4×Dic311yescharacteristic
202C2×C3#95111yescharacteristic
203C22—22no—
204C22—41no—
205C22—41no—
206C22D3×#6711yescharacteristic
207C4—61no—
208C4—61no—
209C4—62no—
210C4—121no—
211C4—361no—
212C4C2×D3×Dic311yescharacteristic
213C4#380411yescharacteristic
214C3—21no—
215C3#501811yescharacteristic, complemented
216C3#516411yescharacteristic, complemented
217C2—42no—
218C2#6392911yescharacteristic
219C2#6393411yescharacteristic
220C2Dic3×#6711yescharacteristic
subgroupG/H
centerC22D3×#67
commutatorC32×C4C22×C4
FrattiniC2×C4C2×D32
Fitting, prime radicalC2×C32×D4C22
socleC22×C32C2×D4
cosocleC2×C32×C4C23
hypercenterC2×D4D32
hypocenterC32#440
LCSG ▷ 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
3. Dic3×Dic6C2semidirect
5. #63575C2semidirect
6. #63565C2semidirect
7. #63554C2semidirect
16. C3×#214C4semidirect
48. C3×Dic6C2×C4semidirect
50. C3×C3⋊C8C2×C4semidirect
65. #214Dic3semidirect
125. Dic6C2×Dic3semidirect
130. C3⋊C8C2×Dic3semidirect
176. C32#440semidirect
215. C3#5018semidirect
216. C3#5164semidirect

Maximal subgroups

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

subgroupindex
C3×D4×Dic32
C2×C3×#2142
Dic3×Dic62
#636592
#635752
#635652
#635542
#51643
#50183