Finite Group Databasebeta

← #11721321

Subgroups — #11721321

Order 864.

509 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 509 conjugacy classes of subgroups — too many to draw. Only its 68 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
1C23×(C3⋊D3)⋊C3C211yesmaximal, characteristic, complemented
2C2×#6235C211yesmaximal, characteristic, complemented
3C2×#6200C212yesmaximal, complemented
4#3757⋊C3C211yesmaximal, characteristic, complemented
5#88450C212yesmaximal, complemented
6#63981—31nomaximal
7#63976—31nomaximal
8C23×H(3,𝔽3)C2211yescharacteristic
9C22×#272C2211yescharacteristic, complemented
10C22×(C3⋊D3)⋊C3—22no—
11C22×(C3⋊D3)⋊C3—24no—
12C22×(C3⋊D3)⋊C3C2212yes—
13#6235—22no—
14C2×#1084C2211yescharacteristic, complemented
15C2×(C3⋊Dic3)⋊C3C2212yescomplemented
16#6200—24no—
17C23×C3⋊D3D311yescharacteristic, complemented
18C23×C3×D3—31no—
19C2×C3×#67—31no—
20C2×C3×#67—31no—
21C2×C3×#67—61no—
22C2×#626—32no—
23C2×#625—32no—
24#3757D311yescharacteristic, complemented
25C3×#217—31no—
26#3722—32no—
27(C2×D3)⋊Dic3—32no—
28C22×H(3,𝔽3)—22no—
29C22×H(3,𝔽3)D412yescomplemented
30C22×H(3,𝔽3)C2311yescharacteristic
31C2×#272—21no—
32C2×#272—22no—
33C2×(C3⋊D3)⋊C3—28no—
34C2×(C3⋊D3)⋊C3—22no—
35C2×(C3⋊D3)⋊C3—22no—
36C2×(C3⋊D3)⋊C3D414yescomplemented
37#1084—21no—
38(C3⋊Dic3)⋊C3—22no—
39#953—91no—
40#937—91no—
41#882—91no—
42C23×C32—21no—
43C23×C32C2×D311yescharacteristic
44C23×C32C2×D311yescharacteristic
45C22×C3⋊D3—22no—
46C22×C3⋊D3—24no—
47C22×C3⋊D3C2×D312yes—
48C22×C3×D3—32no—
49C22×C3×D3—31no—
50C22×C3×D3—31no—
51C22×C3×D3—61no—
52C22×C3×D3—64no—
53C22×C3×D3—62no—
54C2×C3⋊Dic3C2×D312yescomplemented
55C3×#67—64no—
56C3×#67—62no—
57C3×#67—62no—
58C2×C3×Dic3—32no—
59C2×C3×Dic3—31no—
60C2×C3×Dic3—31no—
61C2×C3×Dic3—61no—
62#626—64no—
63#625—64no—
64C2×H(3,𝔽3)—22no—
65C2×H(3,𝔽3)C2×D411yescharacteristic
66C2×H(3,𝔽3)C2×D412yes—
67#272—41no—
68(C3⋊D3)⋊C3—24no—
69C24×C3—91no—
70C23×D3—31no—
71C23×D3—91no—
72C2×C3×D4—91no—
73C2×#67—31no—
74C2×#67—31no—
75C2×#67—61no—
76C2×#67—92no—
77C2×#67—92no—
78C2×D12—92no—
79C3×C22⋊C4—91no—
80#217—31no—
81#217—91no—
82#217—92no—
83(C2×D3)⋊C4—92no—
84(C2×D3)⋊C4—92no—
85C22×C32—22no—
86C22×C32—22no—
87C22×C32—22no—
88C22×C32—21no—
89C22×C32—42no—
90C22×C32D1212yescomplemented
91C22×C32#6712yescomplemented
92C22×C32C22×D311yescharacteristic
93C22×C32C22×D311yescharacteristic
94C2×C3⋊D3—28no—
95C2×C3⋊D3—22no—
96C2×C3⋊D3—22no—
97C2×C3⋊D3#6714yescomplemented
98C2×C3×D3—34no—
99C2×C3×D3—68no—
100C2×C3×D3—62no—
101C2×C3×D3—62no—
102C2×C3×D3—62no—
103C2×C3×D3—61no—
104C2×C3×D3—62no—
105C2×C3×D3—62no—
106C2×C3×D3—61no—
107C2×C3×D3—122no—
108C3⋊Dic3—22no—
109C3×Dic3—62no—
110C3×Dic3—61no—
111C3×Dic3—62no—
112C3×Dic3—61no—
113#120—271noSylow
114H(3,𝔽3)#12011yescharacteristic, complemented, Sylow
115C23×C3—31no—
116C23×C3—31no—
117C23×C3—61no—
118C23×C3—92no—
119C23×C3—91no—
120C23×C3—182no—
121C23×C3—184no—
122C23×C3D3211yescharacteristic
123C22×D3—32no—
124C22×D3—31no—
125C22×D3—31no—
126C22×D3—61no—
127C22×D3—64no—
128C22×D3—62no—
129C22×D3—92no—
130C22×D3—184no—
131C22×D3—182no—
132C3×D4—182no—
133C2×C3×C4—92no—
134C2×C3×C4—91no—
135#67—62no—
136#67—62no—
137#67—64no—
138#67—184no—
139#67—184no—
140C2×Dic3—31no—
141C2×Dic3—31no—
142C2×Dic3—32no—
143C2×Dic3—61no—
144C2×Dic3—92no—
145D12—184no—
146C2×C32—22no—
147C2×C32—22no—
148C2×C32—21no—
149C2×C32—22no—
150C2×C32—42no—
151C2×C32C2×D1211yescharacteristic
152C2×C32D3×D412yes—
153C2×C32C2×#6711yescharacteristic
154C2×C32C2×#6712yes—
155C3⋊D3—24no—
156C3×D3—64no—
157C3×D3—122no—
158C3×D3—121no—
159C3×D3—121no—
160C24—91no—
161C2×D4—91no—
162C2×D4—272no—
163C22⋊C4—91no—
164C22⋊C4—272no—
165C22×C3—22no—
166C22×C3—32no—
167C22×C3—32no—
168C22×C3—31no—
169C22×C3—31no—
170C22×C3—62no—
171C22×C3—62no—
172C22×C3—62no—
173C22×C3—61no—
174C22×C3—94no—
175C22×C3—122no—
176C22×C3—182no—
177C22×C3—182no—
178C22×C3—188no—
179C22×C3—181no—
180C22×C3—182no—
181C22×C3#62612yes—
182C22×C3C2×D3211yescharacteristic
183C2×D3—34no—
184C2×D3—68no—
185C2×D3—62no—
186C2×D3—62no—
187C2×D3—62no—
188C2×D3—61no—
189C2×D3—61no—
190C2×D3—62no—
191C2×D3—62no—
192C2×D3—94no—
193C2×D3—122no—
194C2×D3—182no—
195C2×D3—182no—
196C2×D3—188no—
197C3×C4—182no—
198C3×C4—181no—
199Dic3—61no—
200Dic3—61no—
201Dic3—62no—
202Dic3—62no—
203Dic3—182no—
204C32—21no—
205C32#88211yescharacteristic, complemented
206C32#95311yescharacteristic, complemented
207C23—92no—
208C23—91no—
209C23—184no—
210C23—182no—
211C23((C3⋊D3)⋊C3)⋊C211yescharacteristic
212D4—182no—
213D4—544no—
214C2×C4—92no—
215C2×C4—91no—
216C2×C3—22no—
217C2×C3—31no—
218C2×C3—31no—
219C2×C3—32no—
220C2×C3—32no—
221C2×C3—62no—
222C2×C3—62no—
223C2×C3—61no—
224C2×C3—62no—
225C2×C3—122no—
226C2×C3—184no—
227C2×C3—361no—
228C2×C3C2×#62611yescharacteristic
229C2×C3D3×#6712yes—
230D3—64no—
231D3—121no—
232D3—121no—
233D3—122no—
234D3—184no—
235C22—22no—
236C22—94no—
237C22—188no—
238C22—182no—
239C22—182no—
240C22—181no—
241C22—182no—
242C22#620012yescomplemented
243C22C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
244C4—181no—
245C4—182no—
246C3—31no—
247C3—31no—
248C3—61no—
249C3#6397711yescharacteristic
250C2—22no—
251C2—184no—
252C2—361no—
253C2C2×#620011yescharacteristic
254C2#8867612yes—
subgroupG/H
centerC22C2×((C3⋊D3)⋊C3)⋊C2
commutator, cosocleC22×H(3,𝔽3)C23
Frattini, socleC22×C3C2×D32
FittingC23×H(3,𝔽3)C22
hypercenterC23((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#120
LCSG ▷ C22×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C23

Complements

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

NH ≅ G/Ntype
1. C23×(C3⋊D3)⋊C3C2semidirect
2. C2×#6235C2semidirect
3. C2×#6200C2semidirect
4. #3757⋊C3C2semidirect
5. #88450C2semidirect
9. C22×#272C22semidirect
14. C2×#1084C22semidirect
15. C2×(C3⋊Dic3)⋊C3C22semidirect
17. C23×C3⋊D3D3semidirect
24. #3757D3semidirect
29. C22×H(3,𝔽3)D4semidirect
36. C2×(C3⋊D3)⋊C3D4semidirect
54. C2×C3⋊Dic3C2×D3semidirect
90. C22×C32D12semidirect
91. C22×C32#67semidirect
97. C2×C3⋊D3#67semidirect
114. H(3,𝔽3)#120semidirect
205. C32#882semidirect
206. C32#953semidirect
242. C22#6200semidirect

Maximal subgroups

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

subgroupindex
C23×(C3⋊D3)⋊C32
C2×#62352
C2×#62002
#3757⋊C32
#884502
#639813
#639763