Finite Group Databasebeta

← #64125⋊C3 #11721552

Subgroups — #64125⋊C3 #11721552

Order 864.

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

normal subgroups only

This group has 237 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#88722C211yesmaximal, characteristic, complemented
2#3830⋊C3C211yesmaximal, characteristic, complemented
3SD16×H(3,𝔽3)C211yesmaximal, characteristic, complemented
4#3756⋊C3C211yesmaximal, characteristic, complemented
5#3753⋊C3C211yesmaximal, characteristic, complemented
6#3744⋊C3C211yesmaximal, characteristic, complemented
7C8×(C3⋊D3)⋊C3C211yesmaximal, characteristic, complemented
8#64125C311yesmaximal, characteristic, complemented
9C3×#916—31nomaximal
10Q8×H(3,𝔽3)C2211yescharacteristic
11D4×H(3,𝔽3)C2211yescharacteristic
12#638⋊C3—21no—
13C2×(C3⋊Dic3)⋊C3—21no—
14C4⋊((C3⋊D3)⋊C3)—21no—
15C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
16C4×(C3⋊D3)⋊C3—21no—
17C4×(C3⋊D3)⋊C3C2211yescharacteristic
18#634⋊C3C2211yescharacteristic
19C8×H(3,𝔽3)C2211yescharacteristic
20(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
21#3832C2×C311yescharacteristic, complemented
22#3830C2×C311yescharacteristic, complemented
23C3×#239—31no—
24C3×#237—31no—
25C32×SD16—21no—
26C32×SD16D311yescharacteristic, complemented
27C32×SD16C2×C311yescharacteristic, complemented
28#3756C2×C311yescharacteristic, complemented
29#3753C2×C311yescharacteristic, complemented
30#3744C2×C311yescharacteristic, complemented
31C8×C3⋊D3C2×C311yescharacteristic, complemented
32C3×#216—31no—
33C3×#213—31no—
34C3×#204—31no—
35C3×D3×C8—31no—
36C22×H(3,𝔽3)—21no—
37C2×(C3⋊D3)⋊C3—21no—
38C2×(C3⋊D3)⋊C3D411yescharacteristic
39C4×H(3,𝔽3)—21no—
40C4×H(3,𝔽3)C2311yescharacteristic
41(C3⋊Dic3)⋊C3—21no—
42(C3⋊Dic3)⋊C3D411yescharacteristic
43C3×#135—91no—
44#916—31no—
45#916—91no—
46C32×Q8—21no—
47C32×Q8C2×D311yescharacteristic
48C32×Q8C22×C311yescharacteristic
49C32×D4—21no—
50C32×D4C2×D311yescharacteristic
51C32×D4C22×C311yescharacteristic
52#638—21no—
53C2×C3⋊Dic3—21no—
54C4⋊(C3⋊D3)—21no—
55C4⋊(C3⋊D3)C22×C311yescharacteristic
56C4×C3⋊D3—21no—
57C4×C3⋊D3C22×C311yescharacteristic
58#634C22×C311yescharacteristic
59C3×#67—61no—
60C2×C3×Dic3—61no—
61C3×D12—31no—
62C3×D12—61no—
63C3×C4×D3—31no—
64C3×C4×D3—61no—
65C3×Dic6—31no—
66C32×C8—21no—
67C32×C8C2×D311yescharacteristic
68C32×C8C22×C311yescharacteristic
69C3⋊(C3⋊C8)C22×C311yescharacteristic
70C3×C3⋊C8—31no—
71C2×H(3,𝔽3)—41no—
72C2×H(3,𝔽3)C2×D411yescharacteristic
73(C3⋊D3)⋊C3—21no—
74(C3⋊D3)⋊C3—41no—
75C3×#41—91no—
76C3×#41—91no—
77#239—31no—
78#239—91no—
79#237—31no—
80#237—91no—
81C3×Q16—91no—
82C3×SD16—31no—
83C3×SD16—31no—
84C3×SD16—61no—
85C3×SD16—91no—
86C3×SD16C3×D311yescharacteristic
87C3×D8—91no—
88C2×C3×C8—91no—
89#216—31no—
90#216—91no—
91#213—31no—
92#213—91no—
93#204—31no—
94#204—91no—
95D3×C8—31no—
96D3×C8—91no—
97C22×C32—21no—
98C22×C32—21no—
99C22×C32—41no—
100C2×C3⋊D3—21no—
101C2×C3⋊D3C3×D411yescharacteristic
102C2×C3×D3—31no—
103C2×C3×D3—61no—
104C32×C4—21no—
105C32×C4—21no—
106C32×C4—21no—
107C32×C4—41no—
108C32×C4C22×D311yescharacteristic
109C32×C4C23×C311yescharacteristic
110C3⋊Dic3—21no—
111C3⋊Dic3C3×D411yescharacteristic
112C3×Dic3—31no—
113C3×Dic3—61no—
114#135—91noSylow
115H(3,𝔽3)#13511yescharacteristic, complemented, Sylow
116C3×Q8—31no—
117C3×Q8—31no—
118C3×Q8—61no—
119C3×Q8—91no—
120C3×Q8C2×C3×D311yescharacteristic
121C3×D4—31no—
122C3×D4—31no—
123C3×D4—61no—
124C3×D4—91no—
125C3×D4—181no—
126C3×D4—181no—
127C3×D4C2×C3×D311yescharacteristic
128C2×C3×C4—91no—
129C2×C3×C4—181no—
130C2×C3×C4—181no—
131#67—61no—
132#67—181no—
133C2×Dic3—61no—
134C2×Dic3—181no—
135D12—31no—
136D12—61no—
137D12—91no—
138D12—181no—
139C4×D3—31no—
140C4×D3—61no—
141C4×D3—91no—
142C4×D3—181no—
143Dic6—31no—
144Dic6—91no—
145C3×C8—31no—
146C3×C8—31no—
147C3×C8—61no—
148C3×C8—91no—
149C3×C8C2×C3×D311yescharacteristic
150C3⋊C8—31no—
151C3⋊C8—91no—
152C2×C32—21no—
153C2×C32—41no—
154C2×C32—41no—
155C2×C32—81no—
156C2×C32D3×D411yescharacteristic
157C2×C32C2×C3×D411yescharacteristic
158C3⋊D3—21no—
159C3⋊D3—41no—
160C3×D3—61no—
161C3×D3—121no—
162#41—91no—
163#41—91no—
164Q16—91no—
165SD16—91no—
166SD16(C3⋊D3)⋊C311yescharacteristic, complemented
167D8—91no—
168C2×C8—91no—
169C22×C3—21no—
170C22×C3—61no—
171C22×C3—61no—
172C22×C3—91no—
173C22×C3—121no—
174C22×C3—181no—
175C2×D3—31no—
176C2×D3—61no—
177C2×D3—91no—
178C2×D3—181no—
179C3×C4—21no—
180C3×C4—31no—
181C3×C4—31no—
182C3×C4—61no—
183C3×C4—61no—
184C3×C4—61no—
185C3×C4—91no—
186C3×C4—121no—
187C3×C4—181no—
188C3×C4C22×C3×D311yescharacteristic
189Dic3—31no—
190Dic3—61no—
191Dic3—91no—
192Dic3—181no—
193C32—21no—
194C32#91611yescharacteristic, complemented
195C32C3×#13511yescharacteristic, complemented
196Q8—91no—
197Q8C2×(C3⋊D3)⋊C311yescharacteristic
198D4—91no—
199D4—181no—
200D4—181no—
201D4C2×(C3⋊D3)⋊C311yescharacteristic
202C2×C4—91no—
203C2×C4—181no—
204C2×C4—181no—
205C8—91no—
206C8C2×(C3⋊D3)⋊C311yescharacteristic
207C2×C3—31no—
208C2×C3—31no—
209C2×C3—41no—
210C2×C3—61no—
211C2×C3—121no—
212C2×C3—121no—
213C2×C3—181no—
214C2×C3—241no—
215C2×C3—361no—
216C2×C3C3×D3×D411yescharacteristic
217D3—61no—
218D3—121no—
219D3—181no—
220D3—361no—
221C22—21no—
222C22—91no—
223C22—181no—
224C4—21no—
225C4—91no—
226C4—181no—
227C4C22×(C3⋊D3)⋊C311yescharacteristic
228C3—31no—
229C3—31no—
230C3—61no—
231C3C3×#91611yescharacteristic
232C2—41no—
233C2—181no—
234C2—361no—
235C2D4×(C3⋊D3)⋊C311yescharacteristic
subgroupG/H
centerC2D4×(C3⋊D3)⋊C3
commutator, cosocleC32×C4C23×C3
FrattiniC3×C4C22×C3×D3
FittingSD16×H(3,𝔽3)C2
socleC2×C3C3×D3×D4
hypercenterSD16(C3⋊D3)⋊C3
hypocenterC32C3×#135
LCSG ▷ C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2 ◁ C4 ◁ SD16

Complements

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

NH ≅ G/Ntype
1. #88722C2semidirect
2. #3830⋊C3C2semidirect
3. SD16×H(3,𝔽3)C2semidirect
4. #3756⋊C3C2semidirect
5. #3753⋊C3C2semidirect
6. #3744⋊C3C2semidirect
7. C8×(C3⋊D3)⋊C3C2semidirect
8. #64125C3semidirect
21. #3832C2×C3semidirect
22. #3830C2×C3semidirect
26. C32×SD16D3semidirect
27. C32×SD16C2×C3semidirect
28. #3756C2×C3semidirect
29. #3753C2×C3semidirect
30. #3744C2×C3semidirect
31. C8×C3⋊D3C2×C3semidirect
115. H(3,𝔽3)#135semidirect
166. SD16(C3⋊D3)⋊C3semidirect
194. C32#916semidirect
195. C32C3×#135semidirect

Maximal subgroups

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

subgroupindex
#887222
#3830⋊C32
SD16×H(3,𝔽3)2
#3756⋊C32
#3753⋊C32
#3744⋊C32
C8×(C3⋊D3)⋊C32
#641253
C3×#9163