Finite Group Databasebeta

← #64120⋊C3 #11721536

Subgroups — #64120⋊C3 #11721536

Order 864.

260 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 61 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
1#3830⋊C3C211yesmaximal, characteristic, complemented
2D4×(C3⋊D3)⋊C3C211yesmaximal, characteristic, complemented
3D8×H(3,𝔽3)C211yesmaximal, characteristic, complemented
4#3753⋊C3C211yesmaximal, characteristic, complemented
5#3754⋊C3C211yesmaximal, characteristic, complemented
6#3744⋊C3C211yesmaximal, characteristic, complemented
7#3743⋊C3C211yesmaximal, characteristic, complemented
8#64120C311yesmaximal, characteristic, complemented
9C3×#911—31nomaximal
10C22×(C3⋊D3)⋊C3—21no—
11D4×H(3,𝔽3)C2211yescharacteristic
12D4×H(3,𝔽3)C2211yescharacteristic, complemented
13#638⋊C3—21no—
14#638⋊C3—21no—
15C2×(C3⋊Dic3)⋊C3—21no—
16C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
17C4×(C3⋊D3)⋊C3C2211yescharacteristic
18#634⋊C3C2211yescharacteristic, complemented
19C8×H(3,𝔽3)C2211yescharacteristic, complemented
20(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic, complemented
21#3830C2×C311yescharacteristic, complemented
22D4×C3⋊D3C2×C311yescharacteristic, complemented
23C3×#237—31no—
24C3×D3×D4—31no—
25C32×D8—21no—
26C32×D8D311yescharacteristic, complemented
27C32×D8C2×C311yescharacteristic, complemented
28#3754C2×C311yescharacteristic, complemented
29#3753C2×C311yescharacteristic, complemented
30#3744C2×C311yescharacteristic, complemented
31#3743C2×C311yescharacteristic, complemented
32C3×#214—31no—
33C3×#213—31no—
34C3×#204—31no—
35C3×ΓL(1,𝔽25)—31no—
36C22×H(3,𝔽3)—21no—
37C22×H(3,𝔽3)—21no—
38C2×(C3⋊D3)⋊C3—21no—
39C2×(C3⋊D3)⋊C3—42no—
40C2×(C3⋊D3)⋊C3D411yescharacteristic
41C4×H(3,𝔽3)C2311yescharacteristic
42(C3⋊Dic3)⋊C3—21no—
43(C3⋊Dic3)⋊C3D411yescharacteristic
44C3×AGL(1,ℤ/8)—91no—
45#911—31no—
46#911—91no—
47C22×C3⋊D3—21no—
48C22×C3×D3—61no—
49C32×D4—21no—
50C32×D4—21no—
51C32×D4C2×D311yescharacteristic
52C32×D4C2×D311yescharacteristic, complemented
53C32×D4C22×C311yescharacteristic
54C32×D4C22×C311yescharacteristic, complemented
55#638—21no—
56#638—21no—
57C2×C3⋊Dic3—21no—
58C4⋊(C3⋊D3)C22×C311yescharacteristic
59C4×C3⋊D3C22×C311yescharacteristic
60#634C22×C311yescharacteristic, complemented
61C3×#67—61no—
62C3×#67—61no—
63C2×C3×Dic3—61no—
64C3×D12—31no—
65C3×C4×D3—31no—
66C3×Dic6—31no—
67C32×C8—21no—
68C32×C8C2×D311yescharacteristic, complemented
69C32×C8C22×C311yescharacteristic, complemented
70C3⋊(C3⋊C8)C22×C311yescharacteristic, complemented
71C3×C3⋊C8—31no—
72C2×H(3,𝔽3)—41no—
73C2×H(3,𝔽3)—41no—
74C2×H(3,𝔽3)C2×D411yescharacteristic
75(C3⋊D3)⋊C3—21no—
76(C3⋊D3)⋊C3—41no—
77C3×#41—91no—
78C2×C3×D4—91no—
79#237—31no—
80#237—91no—
81D3×D4—31no—
82D3×D4—91no—
83C3×SD16—91no—
84C3×SD16—91no—
85C3×D8—31no—
86C3×D8—31no—
87C3×D8—61no—
88C3×D8—91no—
89C3×D8C3×D311yescharacteristic
90C3×M16—91no—
91#214—31no—
92#214—91no—
93#213—31no—
94#213—91no—
95#204—31no—
96#204—91no—
97ΓL(1,𝔽25)—31no—
98ΓL(1,𝔽25)—91no—
99C22×C32—21no—
100C22×C32—21no—
101C22×C32—21no—
102C22×C32—21no—
103C22×C32—41no—
104C22×C32—41no—
105C2×C3⋊D3—21no—
106C2×C3⋊D3—42no—
107C2×C3⋊D3C3×D411yescharacteristic
108C2×C3×D3—31no—
109C2×C3×D3—61no—
110C2×C3×D3—122no—
111C32×C4—21no—
112C32×C4C22×D311yescharacteristic
113C32×C4C23×C311yescharacteristic
114C3⋊Dic3—21no—
115C3⋊Dic3C3×D411yescharacteristic
116C3×Dic3—31no—
117C3×Dic3—61no—
118AGL(1,ℤ/8)—91noSylow
119H(3,𝔽3)AGL(1,ℤ/8)11yescharacteristic, complemented, Sylow
120C23×C3—181no—
121C22×D3—61no—
122C22×D3—181no—
123C3×Q8—91no—
124C3×D4—31no—
125C3×D4—31no—
126C3×D4—31no—
127C3×D4—31no—
128C3×D4—61no—
129C3×D4—61no—
130C3×D4—91no—
131C3×D4—181no—
132C3×D4—181no—
133C3×D4C2×C3×D311yescharacteristic
134C3×D4C2×C3×D311yescharacteristic
135C2×C3×C4—91no—
136C2×C3×C4—181no—
137#67—61no—
138#67—61no—
139#67—181no—
140#67—181no—
141C2×Dic3—61no—
142C2×Dic3—181no—
143D12—31no—
144D12—91no—
145C4×D3—31no—
146C4×D3—91no—
147Dic6—31no—
148Dic6—91no—
149C3×C8—31no—
150C3×C8—31no—
151C3×C8—61no—
152C3×C8—91no—
153C3×C8C2×C3×D311yescharacteristic
154C3⋊C8—31no—
155C3⋊C8—91no—
156C2×C32—21no—
157C2×C32—41no—
158C2×C32—41no—
159C2×C32—41no—
160C2×C32—41no—
161C2×C32—81no—
162C2×C32—81no—
163C2×C32D3×D411yescharacteristic
164C2×C32C2×C3×D411yescharacteristic
165C3⋊D3—21no—
166C3⋊D3—41no—
167C3×D3—61no—
168C3×D3—121no—
169#41—91no—
170C2×D4—91no—
171SD16—91no—
172SD16—91no—
173D8—91no—
174D8(C3⋊D3)⋊C311yescharacteristic, complemented
175M16—91no—
176C22×C3—21no—
177C22×C3—21no—
178C22×C3—61no—
179C22×C3—61no—
180C22×C3—61no—
181C22×C3—61no—
182C22×C3—91no—
183C22×C3—121no—
184C22×C3—121no—
185C22×C3—181no—
186C22×C3—362no—
187C2×D3—31no—
188C2×D3—61no—
189C2×D3—91no—
190C2×D3—122no—
191C2×D3—181no—
192C2×D3—362no—
193C3×C4—31no—
194C3×C4—31no—
195C3×C4—61no—
196C3×C4—91no—
197C3×C4—181no—
198C3×C4C22×C3×D311yescharacteristic
199Dic3—31no—
200Dic3—61no—
201Dic3—91no—
202Dic3—181no—
203C32—21no—
204C32#91111yescharacteristic, complemented
205C32C3×AGL(1,ℤ/8)11yescharacteristic, complemented
206C23—181no—
207Q8—91no—
208D4—91no—
209D4—181no—
210D4—181no—
211D4C2×(C3⋊D3)⋊C311yescharacteristic
212D4C2×(C3⋊D3)⋊C311yescharacteristic, complemented
213C2×C4—91no—
214C2×C4—181no—
215C8—91no—
216C8C2×(C3⋊D3)⋊C311yescharacteristic, complemented
217C2×C3—31no—
218C2×C3—31no—
219C2×C3—41no—
220C2×C3—41no—
221C2×C3—61no—
222C2×C3—121no—
223C2×C3—121no—
224C2×C3—121no—
225C2×C3—121no—
226C2×C3—181no—
227C2×C3—241no—
228C2×C3—241no—
229C2×C3—361no—
230C2×C3C3×D3×D411yescharacteristic
231D3—61no—
232D3—121no—
233D3—181no—
234D3—361no—
235C22—21no—
236C22—21no—
237C22—91no—
238C22—181no—
239C22—362no—
240C4—91no—
241C4—181no—
242C4C22×(C3⋊D3)⋊C311yescharacteristic
243C3—31no—
244C3—31no—
245C3—61no—
246C3C3×#91111yescharacteristic
247C2—41no—
248C2—41no—
249C2—181no—
250C2—361no—
251C2D4×(C3⋊D3)⋊C311yescharacteristic
subgroupG/H
centerC2D4×(C3⋊D3)⋊C3
commutator, cosocleC32×C4C23×C3
FrattiniC3×C4C22×C3×D3
FittingD8×H(3,𝔽3)C2
socleC2×C3C3×D3×D4
hypercenterD8(C3⋊D3)⋊C3
hypocenterC32C3×AGL(1,ℤ/8)
LCSG ▷ C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2 ◁ C4 ◁ D8

Complements

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

NH ≅ G/Ntype
1. #3830⋊C3C2semidirect
2. D4×(C3⋊D3)⋊C3C2semidirect
3. D8×H(3,𝔽3)C2semidirect
4. #3753⋊C3C2semidirect
5. #3754⋊C3C2semidirect
6. #3744⋊C3C2semidirect
7. #3743⋊C3C2semidirect
8. #64120C3semidirect
12. D4×H(3,𝔽3)C22semidirect
18. #634⋊C3C22semidirect
19. C8×H(3,𝔽3)C22semidirect
20. (C3⋊(C3⋊C8))⋊C3C22semidirect
21. #3830C2×C3semidirect
22. D4×C3⋊D3C2×C3semidirect
26. C32×D8D3semidirect
27. C32×D8C2×C3semidirect
28. #3754C2×C3semidirect
29. #3753C2×C3semidirect
30. #3744C2×C3semidirect
31. #3743C2×C3semidirect
52. C32×D4C2×D3semidirect
54. C32×D4C22×C3semidirect
60. #634C22×C3semidirect
68. C32×C8C2×D3semidirect
69. C32×C8C22×C3semidirect
70. C3⋊(C3⋊C8)C22×C3semidirect
119. H(3,𝔽3)AGL(1,ℤ/8)semidirect
174. D8(C3⋊D3)⋊C3semidirect
204. C32#911semidirect
205. C32C3×AGL(1,ℤ/8)semidirect
212. D4C2×(C3⋊D3)⋊C3semidirect
216. C8C2×(C3⋊D3)⋊C3semidirect

Maximal subgroups

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

subgroupindex
#3830⋊C32
D4×(C3⋊D3)⋊C32
D8×H(3,𝔽3)2
#3753⋊C32
#3754⋊C32
#3744⋊C32
#3743⋊C32
#641203
C3×#9113