Finite Group Databasebeta

← #11721112

Subgroups — #11721112

Order 864.

240 conjugacy classes of subgroups; only the 50 normal subgroups are shown.

normal subgroups only

This group has 240 conjugacy classes of subgroups — too many to draw. Only its 50 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
1C4×#1084C211yesmaximal, characteristic, complemented
2#3752⋊C3C211yesmaximal, characteristic
3#3750⋊C3C211yesmaximal, characteristic, complemented
4#88450C211yesmaximal, characteristic
5#88450C211yesmaximal, characteristic
6#88448C211yesmaximal, characteristic, complemented
7#88448C211yesmaximal, characteristic, complemented
8#63868—31nomaximal
9#63859—31nomaximal
10C22×(C3⋊D3)⋊C3C2211yescharacteristic
11C2×C4×H(3,𝔽3)C2211yescharacteristic
12C2×#1084C2211yescharacteristic
13C2×#1084C2211yescharacteristic
14C2×(C3⋊Dic3)⋊C3C2211yescharacteristic
15C2×(C3⋊Dic3)⋊C3C2211yescharacteristic
16C2×(C3⋊Dic3)⋊C3C2211yescharacteristic
17#3752D311yescharacteristic
18#3750D311yescharacteristic, complemented
19C3×(C2×D3)⋊C4—31no—
20C3×Dic3⋊C4—31no—
21C3×C4×Dic3—31no—
22C3×C4×Dic3—31no—
23C3×C4×Dic3—61no—
24#3724—31no—
25#3724—31no—
26Dic3⋊Dic3—31no—
27Dic3⋊Dic3—31no—
28#3722—31no—
29#3722—31no—
30(C2×D3)⋊Dic3—31no—
31(C2×D3)⋊Dic3—31no—
32C22×H(3,𝔽3)C2311yescharacteristic
33C2×(C3⋊D3)⋊C3—21no—
34C2×(C3⋊D3)⋊C3—21no—
35C2×(C3⋊D3)⋊C3—21no—
36C4×H(3,𝔽3)—21no—
37#1084—21no—
38#1084—21no—
39(C3⋊Dic3)⋊C3—21no—
40(C3⋊Dic3)⋊C3—21no—
41(C3⋊Dic3)⋊C3—21no—
42#898—91no—
43#879—91no—
44#876—91no—
45C22×C3⋊D3C2×D311yescharacteristic
46C22×C3×D3—31no—
47C2×C32×C4—21no—
48C2×C32×C4C2×D311yescharacteristic
49C2×C32×C4C2×D311yescharacteristic
50C2×C3⋊Dic3C2×D311yescharacteristic
51C2×C3⋊Dic3C2×D311yescharacteristic
52C2×C3⋊Dic3C2×D311yescharacteristic
53C2×C3×Dic3—31no—
54C2×C3×Dic3—31no—
55C2×C3×Dic3—31no—
56C2×C3×Dic3—31no—
57C2×C3×Dic3—31no—
58C2×C3×Dic3—31no—
59C2×C3×Dic3—31no—
60C2×C3×Dic3—61no—
61C2×C3×Dic3—61no—
62C2×H(3,𝔽3)#4111yescharacteristic
63C2×H(3,𝔽3)#4111yescharacteristic
64C2×H(3,𝔽3)#4111yescharacteristic
65(C3⋊D3)⋊C3—41no—
66C3×C4⋊C4—91no—
67C3×C22⋊C4—91no—
68C3×C42—91no—
69#217—91no—
70#217—91no—
71(C2×D3)⋊C4—31no—
72(C2×D3)⋊C4—91no—
73(C2×D3)⋊C4—91no—
74(C2×D3)⋊C4—91no—
75(C2×D3)⋊C4—91no—
76(C2×D3)⋊C4—91no—
77C4⋊Dic3—91no—
78C4⋊Dic3—91no—
79Dic3⋊C4—31no—
80Dic3⋊C4—91no—
81Dic3⋊C4—91no—
82Dic3⋊C4—91no—
83Dic3⋊C4—91no—
84Dic3⋊C4—181no—
85C4×Dic3—31no—
86C4×Dic3—31no—
87C4×Dic3—61no—
88C22×C32—21no—
89C22×C32C22×D311yescharacteristic
90C22×C32C22×D311yescharacteristic
91C2×C3⋊D3—21no—
92C2×C3⋊D3—21no—
93C2×C3⋊D3—21no—
94C2×C3×D3—61no—
95C2×C3×D3—61no—
96C2×C3×D3—61no—
97C32×C4—21no—
98C32×C4—22no—
99C32×C4—21no—
100C3⋊Dic3—21no—
101C3⋊Dic3—21no—
102C3⋊Dic3—21no—
103C3×Dic3—62no—
104C3×Dic3—62no—
105C3×Dic3—61no—
106C3×Dic3—61no—
107C3×Dic3—61no—
108C3×Dic3—61no—
109C3×Dic3—61no—
110C3×Dic3—61no—
111C3×Dic3—61no—
112#126—271noSylow
113H(3,𝔽3)#12611yescharacteristic, complemented, Sylow
114C23×C3—91no—
115C22×D3—31no—
116C22×D3—91no—
117C2×C3×C4—31no—
118C2×C3×C4—61no—
119C2×C3×C4—91no—
120C2×C3×C4—91no—
121C2×C3×C4—91no—
122C2×C3×C4—91no—
123C2×C3×C4—91no—
124C2×C3×C4—121no—
125C2×C3×C4D3211yescharacteristic
126C2×Dic3—31no—
127C2×Dic3—31no—
128C2×Dic3—31no—
129C2×Dic3—31no—
130C2×Dic3—31no—
131C2×Dic3—31no—
132C2×Dic3—31no—
133C2×Dic3—61no—
134C2×Dic3—61no—
135C2×Dic3—91no—
136C2×Dic3—181no—
137C2×Dic3—181no—
138C2×C32—21no—
139C2×C32—21no—
140C2×C32—21no—
141C2×C32#23511yescharacteristic
142C2×C32#23511yescharacteristic
143C2×C32#23711yescharacteristic
144C2×C32#23711yescharacteristic
145C2×C32#23711yescharacteristic
146C2×C32#23911yescharacteristic
147C3⋊D3—41no—
148C3×D3—121no—
149C4⋊C4—91no—
150C4⋊C4—271no—
151C4⋊C4—271no—
152C22⋊C4—91no—
153C22⋊C4—271no—
154C22⋊C4—271no—
155C42—91no—
156C22×C3—31no—
157C22×C3—61no—
158C22×C3—121no—
159C22×C3—181no—
160C22×C3—181no—
161C22×C3—181no—
162C22×C3C2×D3211yescharacteristic
163C2×D3—61no—
164C2×D3—61no—
165C2×D3—61no—
166C2×D3—181no—
167C2×D3—181no—
168C2×D3—181no—
169C3×C4—21no—
170C3×C4—61no—
171C3×C4—121no—
172C3×C4—122no—
173C3×C4—181no—
174C3×C4—181no—
175C3×C4—181no—
176C3×C4—181no—
177C3×C4—181no—
178Dic3—61no—
179Dic3—61no—
180Dic3—61no—
181Dic3—61no—
182Dic3—62no—
183Dic3—62no—
184Dic3—61no—
185Dic3—61no—
186Dic3—61no—
187Dic3—181no—
188Dic3—181no—
189Dic3—361no—
190C32—21no—
191C32#87911yescharacteristic, complemented
192C32#89811yescharacteristic, complemented
193C23—91no—
194C2×C4—91no—
195C2×C4—91no—
196C2×C4—91no—
197C2×C4—91no—
198C2×C4—91no—
199C2×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
200C2×C3—31no—
201C2×C3—31no—
202C2×C3—31no—
203C2×C3—61no—
204C2×C3—61no—
205C2×C3—61no—
206C2×C3—121no—
207C2×C3—121no—
208C2×C3—121no—
209C2×C3—361no—
210C2×C3#379611yescharacteristic
211C2×C3#379811yescharacteristic
212C2×C3#380511yescharacteristic
213D3—121no—
214D3—361no—
215C22—181no—
216C22—181no—
217C22—181no—
218C22C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
219C4—21no—
220C4—181no—
221C4—181no—
222C4—181no—
223C4—181no—
224C4—181no—
225C3—31no—
226C3—61no—
227C3—121no—
228C3#6386211yescharacteristic
229C2—361no—
230C2#8864911yescharacteristic
231C2#8865011yescharacteristic
232C2#8867211yescharacteristic
subgroupG/H
centerC22C2×((C3⋊D3)⋊C3)⋊C2
commutator, cosocleC22×H(3,𝔽3)C23
Frattini, socleC22×C3C2×D32
FittingC2×C4×H(3,𝔽3)C22
prime radicalC22×(C3⋊D3)⋊C3C22
hypercenterC2×C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#126
LCSG ▷ C22×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
1. C4×#1084C2semidirect
3. #3750⋊C3C2semidirect
6. #88448C2semidirect
7. #88448C2semidirect
18. #3750D3semidirect
113. H(3,𝔽3)#126semidirect
191. C32#879semidirect
192. C32#898semidirect

Maximal subgroups

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

subgroupindex
C4×#10842
#3752⋊C32
#3750⋊C32
#884502
#884482
#638683
#638593