Finite Group Databasebeta

← #11721297

Subgroups — #11721297

Order 864.

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

normal subgroups only

This group has 244 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
1#88552C211yesmaximal, characteristic
2#3757⋊C3C211yesmaximal, characteristic
3#3757⋊C3C211yesmaximal, characteristic
4#88449C211yesmaximal, characteristic, complemented
5#88448C211yesmaximal, characteristic, complemented
6#88448C211yesmaximal, characteristic, complemented
7#88447C211yesmaximal, characteristic, complemented
8#63956—31nomaximal
9#63956—31nomaximal
10C23×H(3,𝔽3)C2211yescharacteristic
11C2×#1084C2211yescharacteristic
12C2×#1084C2211yescharacteristic
13C2×(C3⋊Dic3)⋊C3C2211yescharacteristic
14C2×(C3⋊Dic3)⋊C3C2211yescharacteristic
15C2×(C3⋊Dic3)⋊C3C2211yescharacteristic
16C2×(C3⋊Dic3)⋊C3C2211yescharacteristic
17#3757D311yescharacteristic
18#3757D311yescharacteristic
19C3×#217—31no—
20C3×#217—31no—
21C3×#217—31no—
22C3×#217—31no—
23C3×#217—61no—
24#3724—31no—
25#3724—31no—
26Dic3⋊Dic3—31no—
27Dic3⋊Dic3—31no—
28Dic3⋊Dic3—31no—
29Dic3⋊Dic3—31no—
30Dic32—31no—
31Dic32—31no—
32C22×H(3,𝔽3)—21no—
33C22×H(3,𝔽3)—21no—
34C22×H(3,𝔽3)—21no—
35C22×H(3,𝔽3)C2311yescharacteristic
36#1084—21no—
37#1084—21no—
38(C3⋊Dic3)⋊C3—21no—
39(C3⋊Dic3)⋊C3—21no—
40(C3⋊Dic3)⋊C3—21no—
41(C3⋊Dic3)⋊C3—21no—
42#879—91no—
43#879—91no—
44#879—91no—
45C23×C32—21no—
46C23×C32C2×D311yescharacteristic
47C23×C32C2×D311yescharacteristic
48C2×C3⋊Dic3C2×D311yescharacteristic
49C2×C3⋊Dic3C2×D311yescharacteristic
50C2×C3⋊Dic3C2×D311yescharacteristic
51C2×C3⋊Dic3C2×D311yescharacteristic
52C2×C3×Dic3—31no—
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)—41no—
63C2×H(3,𝔽3)#4111yescharacteristic
64C2×H(3,𝔽3)#4111yescharacteristic
65C2×H(3,𝔽3)#4111yescharacteristic
66C3×C22⋊C4—91no—
67C3×C22⋊C4—91no—
68C3×C22⋊C4—91no—
69#217—31no—
70#217—31no—
71#217—31no—
72#217—31no—
73#217—61no—
74#217—91no—
75#217—91no—
76C4⋊Dic3—91no—
77C4⋊Dic3—91no—
78C4⋊Dic3—91no—
79Dic3⋊C4—91no—
80Dic3⋊C4—91no—
81Dic3⋊C4—91no—
82Dic3⋊C4—91no—
83Dic3⋊C4—91no—
84Dic3⋊C4—91no—
85C4×Dic3—91no—
86C4×Dic3—91no—
87C4×Dic3—91no—
88C22×C32—22no—
89C22×C32—21no—
90C22×C32—21no—
91C22×C32—21no—
92C22×C32—21no—
93C22×C32—21no—
94C22×C32—21no—
95C22×C32—21no—
96C22×C32—41no—
97C22×C32—41no—
98C22×C32C22×D311yescharacteristic
99C22×C32C22×D311yescharacteristic
100C3⋊Dic3—21no—
101C3⋊Dic3—21no—
102C3⋊Dic3—21no—
103C3⋊Dic3—21no—
104C3×Dic3—61no—
105C3×Dic3—61no—
106C3×Dic3—61no—
107C3×Dic3—61no—
108C3×Dic3—61no—
109C3×Dic3—61no—
110C3×Dic3—61no—
111C3×Dic3—61no—
112C3×Dic3—121no—
113C3×Dic3—121no—
114#126—271noSylow
115H(3,𝔽3)#12611yescharacteristic, complemented, Sylow
116C23×C3—31no—
117C23×C3—31no—
118C23×C3—61no—
119C23×C3D3211yescharacteristic
120C2×C3×C4—91no—
121C2×C3×C4—91no—
122C2×C3×C4—91no—
123C2×C3×C4—91no—
124C2×C3×C4—91no—
125C2×C3×C4—91no—
126C2×Dic3—31no—
127C2×Dic3—31no—
128C2×Dic3—31no—
129C2×Dic3—31no—
130C2×Dic3—31no—
131C2×Dic3—31no—
132C2×Dic3—31no—
133C2×Dic3—31no—
134C2×Dic3—61no—
135C2×Dic3—61no—
136C2×Dic3—91no—
137C2×Dic3—91no—
138C2×Dic3—91no—
139C2×Dic3—91no—
140C2×C32—21no—
141C2×C32—21no—
142C2×C32—21no—
143C2×C32—42no—
144C2×C32—41no—
145C2×C32—41no—
146C2×C32#23511yescharacteristic
147C2×C32#23511yescharacteristic
148C2×C32#23711yescharacteristic
149C2×C32#23711yescharacteristic
150C2×C32#23711yescharacteristic
151C2×C32#23711yescharacteristic
152C4⋊C4—271no—
153C4⋊C4—271no—
154C4⋊C4—271no—
155C22⋊C4—91no—
156C22⋊C4—91no—
157C22⋊C4—91no—
158C42—271no—
159C22×C3—21no—
160C22×C3—21no—
161C22×C3—21no—
162C22×C3—31no—
163C22×C3—31no—
164C22×C3—61no—
165C22×C3—61no—
166C22×C3—61no—
167C22×C3—61no—
168C22×C3—61no—
169C22×C3—61no—
170C22×C3—62no—
171C22×C3—61no—
172C22×C3—121no—
173C22×C3—121no—
174C22×C3C2×D3211yescharacteristic
175C3×C4—181no—
176C3×C4—181no—
177C3×C4—181no—
178C3×C4—181no—
179C3×C4—181no—
180C3×C4—181no—
181Dic3—61no—
182Dic3—61no—
183Dic3—61no—
184Dic3—61no—
185Dic3—61no—
186Dic3—61no—
187Dic3—61no—
188Dic3—61no—
189Dic3—121no—
190Dic3—121no—
191Dic3—181no—
192Dic3—181no—
193Dic3—181no—
194Dic3—181no—
195C32—21no—
196C32#87911yescharacteristic, complemented
197C32#87911yescharacteristic, complemented
198C23((C3⋊D3)⋊C3)⋊C211yescharacteristic
199C2×C4—91no—
200C2×C4—91no—
201C2×C4—91no—
202C2×C4—91no—
203C2×C4—91no—
204C2×C4—91no—
205C2×C3—31no—
206C2×C3—31no—
207C2×C3—31no—
208C2×C3—31no—
209C2×C3—31no—
210C2×C3—31no—
211C2×C3—41no—
212C2×C3—61no—
213C2×C3—61no—
214C2×C3—61no—
215C2×C3—121no—
216C2×C3—121no—
217C2×C3—122no—
218C2×C3#380411yescharacteristic
219C2×C3#380411yescharacteristic
220C2×C3#380511yescharacteristic
221C22—21no—
222C22—21no—
223C22—21no—
224C22C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
225C4—181no—
226C4—181no—
227C4—181no—
228C4—181no—
229C4—181no—
230C4—181no—
231C3—31no—
232C3—31no—
233C3—61no—
234C3#6395611yescharacteristic
235C2—41no—
236C2#8867111yescharacteristic
237C2#8867111yescharacteristic
238C2#8867211yescharacteristic
subgroupG/H
centerC22C2×((C3⋊D3)⋊C3)⋊C2
commutator, cosocleC22×H(3,𝔽3)C23
Frattini, socleC22×C3C2×D32
Fitting, prime radicalC23×H(3,𝔽3)C22
hypercenterC23((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#126
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
4. #88449C2semidirect
5. #88448C2semidirect
6. #88448C2semidirect
7. #88447C2semidirect
115. H(3,𝔽3)#126semidirect
196. C32#879semidirect
197. C32#879semidirect

Maximal subgroups

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

subgroupindex
#885522
#3757⋊C32
#884492
#884482
#884472
#639563