Finite Group Databasebeta

← #11721121

Subgroups — #11721121

Order 864.

296 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 296 conjugacy classes of subgroups — too many to draw. Only its 64 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
1C2×((C3⋊D3)⋊C3)⋊C4C211yesmaximal, characteristic
2C4×#1084C211yesmaximal, characteristic, complemented
3#3752⋊C3C211yesmaximal, characteristic
4#3750⋊C3C211yesmaximal, characteristic, complemented
5#88450C211yesmaximal, characteristic
6#88448C211yesmaximal, characteristic, complemented
7#88447C211yesmaximal, characteristic, complemented
8#63877—31nomaximal
9#63836—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((C3⋊D3)⋊C3)⋊C4C414yescomplemented
18C2×#624—31no—
19C2×D3×Dic3—31no—
20#3752D311yescharacteristic
21#3750D311yescharacteristic, complemented
22C3×(C2×D3)⋊C4—31no—
23C3×Dic3⋊C4—31no—
24C3×C4×Dic3—31no—
25C3×C4×Dic3—31no—
26C3×C4×Dic3—61no—
27#3724—31no—
28Dic3⋊Dic3—31no—
29#3722—31no—
30(C2×D3)⋊Dic3—31no—
31Dic32—31no—
32Dic32—31no—
33C22×H(3,𝔽3)C2311yescharacteristic
34C2×(C3⋊D3)⋊C3—21no—
35C2×(C3⋊D3)⋊C3—21no—
36C2×(C3⋊D3)⋊C3C2×C412yes—
37C4×H(3,𝔽3)—21no—
38#1084—21no—
39#1084C2×C412yes—
40(C3⋊Dic3)⋊C3—21no—
41(C3⋊Dic3)⋊C3—21no—
42(C3⋊Dic3)⋊C3C2×C412yes—
43#892—91no—
44#877—91no—
45#872—91no—
46C22×C3⋊D3C2×D311yescharacteristic
47C22×C3×D3—31no—
48C2×C32×C4—21no—
49C2×C32×C4C2×D311yescharacteristic
50C2×C32×C4C2×D311yescharacteristic
51C2×C3⋊Dic3C2×D311yescharacteristic
52C2×C3⋊Dic3C2×D311yescharacteristic
53C2×C3⋊Dic3C2×D311yescharacteristic
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—31no—
61C2×C3×Dic3—61no—
62C2×C3×Dic3—61no—
63#624—34no—
64D3×Dic3—34no—
65C2×H(3,𝔽3)C22×C411yescharacteristic
66C2×H(3,𝔽3)#4111yescharacteristic
67C2×H(3,𝔽3)#4111yescharacteristic
68(C3⋊D3)⋊C3—22no—
69C22×Dic3—91no—
70C2×C4×D3—91no—
71C2×C4×D3—91no—
72C3×C4⋊C4—91no—
73C3×C22⋊C4—91no—
74C3×C42—91no—
75#217—91no—
76(C2×D3)⋊C4—31no—
77(C2×D3)⋊C4—91no—
78(C2×D3)⋊C4—91no—
79(C2×D3)⋊C4—181no—
80C4⋊Dic3—91no—
81Dic3⋊C4—31no—
82Dic3⋊C4—91no—
83Dic3⋊C4—91no—
84Dic3⋊C4—181no—
85C4×Dic3—31no—
86C4×Dic3—31no—
87C4×Dic3—61no—
88C4×Dic3—91no—
89C4×Dic3—91no—
90C4×Dic3—91no—
91C22×C32—21no—
92C22×C32C22×D311yescharacteristic
93C22×C32C22×D311yescharacteristic
94C2×C3⋊D3—21no—
95C2×C3⋊D3—21no—
96C2×C3⋊D3C4×D312yes—
97C2×C3×D3—32no—
98C2×C3×D3—61no—
99C2×C3×D3—61no—
100C32×C4—22no—
101C32×C4—21no—
102C32×C4—21no—
103C3⋊Dic3—21no—
104C3⋊Dic3—21no—
105C3⋊Dic3C4×D312yes—
106C3×Dic3—32no—
107C3×Dic3—32no—
108C3×Dic3—32no—
109C3×Dic3—62no—
110C3×Dic3—62no—
111C3×Dic3—61no—
112C3×Dic3—61no—
113C3×Dic3—61no—
114C3×Dic3—61no—
115#117—271noSylow
116H(3,𝔽3)#11711yescharacteristic, complemented, Sylow
117C23×C3—91no—
118C22×D3—31no—
119C22×D3—181no—
120C2×C3×C4—61no—
121C2×C3×C4—61no—
122C2×C3×C4—91no—
123C2×C3×C4—91no—
124C2×C3×C4—91no—
125C2×C3×C4—91no—
126C2×C3×C4—91no—
127C2×C3×C4—121no—
128C2×C3×C4D3211yescharacteristic
129C2×Dic3—31no—
130C2×Dic3—31no—
131C2×Dic3—31no—
132C2×Dic3—31no—
133C2×Dic3—31no—
134C2×Dic3—31no—
135C2×Dic3—31no—
136C2×Dic3—61no—
137C2×Dic3—61no—
138C2×Dic3—94no—
139C2×Dic3—181no—
140C2×Dic3—181no—
141C2×Dic3—181no—
142C4×D3—94no—
143C4×D3—94no—
144C2×C32—21no—
145C2×C32—21no—
146C2×C32—21no—
147C2×C32C2×C4×D311yescharacteristic
148C2×C32C2×C4×D311yescharacteristic
149C2×C32#23711yescharacteristic
150C2×C32#23711yescharacteristic
151C2×C32#23711yescharacteristic
152C2×C32#23911yescharacteristic
153C3⋊D3—22no—
154C3×D3—62no—
155C22×C4—271no—
156C4⋊C4—91no—
157C4⋊C4—271no—
158C22⋊C4—91no—
159C22⋊C4—271no—
160C42—91no—
161C42—271no—
162C22×C3—61no—
163C22×C3—61no—
164C22×C3—92no—
165C22×C3—121no—
166C22×C3—181no—
167C22×C3—181no—
168C22×C3C2×D3211yescharacteristic
169C2×D3—32no—
170C2×D3—61no—
171C2×D3—61no—
172C2×D3—182no—
173C2×D3—361no—
174C2×D3—361no—
175C3×C4—21no—
176C3×C4—92no—
177C3×C4—92no—
178C3×C4—121no—
179C3×C4—121no—
180C3×C4—122no—
181C3×C4—181no—
182C3×C4—181no—
183C3×C4—181no—
184Dic3—32no—
185Dic3—32no—
186Dic3—32no—
187Dic3—62no—
188Dic3—61no—
189Dic3—61no—
190Dic3—62no—
191Dic3—61no—
192Dic3—61no—
193Dic3—182no—
194Dic3—361no—
195Dic3—361no—
196C32—21no—
197C32#87711yescharacteristic, complemented
198C32#89211yescharacteristic, complemented
199C23—91no—
200C2×C4—91no—
201C2×C4—91no—
202C2×C4—91no—
203C2×C4—91no—
204C2×C4—91no—
205C2×C4—274no—
206C2×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
207C2×C3—61no—
208C2×C3—61no—
209C2×C3—61no—
210C2×C3—61no—
211C2×C3—61no—
212C2×C3—61no—
213C2×C3—121no—
214C2×C3—121no—
215C2×C3—121no—
216C2×C3—182no—
217C2×C3#379611yescharacteristic
218C2×C3C4×D3211yescharacteristic
219C2×C3#380511yescharacteristic
220D3—62no—
221D3—362no—
222C22—92no—
223C22—181no—
224C22—181no—
225C22C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
226C4—21no—
227C4—92no—
228C4—92no—
229C4—181no—
230C4—181no—
231C4—181no—
232C3—61no—
233C3—61no—
234C3—121no—
235C3#6387811yescharacteristic
236C2—182no—
237C2#8864911yescharacteristic
238C2C4×((C3⋊D3)⋊C3)⋊C211yescharacteristic
239C2#8867211yescharacteristic
subgroupG/H
centerC22C2×((C3⋊D3)⋊C3)⋊C2
commutatorC2×H(3,𝔽3)C22×C4
Frattini, socleC22×C3C2×D32
FittingC2×C4×H(3,𝔽3)C22
prime radicalC22×(C3⋊D3)⋊C3C22
cosocleC22×H(3,𝔽3)C23
hypercenterC2×C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#117
LCSG ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
2. C4×#1084C2semidirect
4. #3750⋊C3C2semidirect
6. #88448C2semidirect
7. #88447C2semidirect
17. ((C3⋊D3)⋊C3)⋊C4C4semidirect
21. #3750D3semidirect
116. H(3,𝔽3)#117semidirect
197. C32#877semidirect
198. C32#892semidirect

Maximal subgroups

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

subgroupindex
C2×((C3⋊D3)⋊C3)⋊C42
C4×#10842
#3752⋊C32
#3750⋊C32
#884502
#884482
#884472
#638773
#638363