Finite Group Databasebeta

← #11723488

Subgroups — #11723488

Order 864.

572 conjugacy classes of subgroups; only the 108 normal subgroups are shown.

normal subgroups only

This group has 572 conjugacy classes of subgroups — too many to draw. Only its 108 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×C4⋊(C3⋊D3)C211yesmaximal, characteristic
2C2×Dic3×C3⋊D3C211yesmaximal, characteristic, complemented
3#88857C211yesmaximal, characteristic, complemented
4C32×Dic3⋊C4C211yesmaximal, characteristic, complemented
5#88803C211yesmaximal, characteristic, complemented
6#88803C211yesmaximal, characteristic, complemented
7#88803C211yesmaximal, characteristic, complemented
8#64103—31nomaximal
9#63884—34nomaximal
10C22×C3×C3⋊D3C2211yescharacteristic
11C22×C3×C3⋊D3C2211yescharacteristic
12C2×C33×C4C2211yescharacteristic
13C2×C3⋊(C3⋊Dic3)C2211yescharacteristic
14C2×C3⋊(C3⋊Dic3)C2211yescharacteristic
15C3×C4⋊(C3⋊D3)—22no—
16C2×C32×Dic3C2211yescharacteristic
17C2×C32×Dic3C2211yescharacteristic
18Dic3×C3⋊D3—22no—
19C2×C4⋊(C3⋊D3)D311yescharacteristic
20C2×C4×C3⋊D3—31no—
21C2×C3×D12—34no—
22C2×D3×Dic3—34no—
23C32×C4⋊C4—31no—
24#3752—31no—
25#3752—31no—
26#3752—31no—
27#3750—31no—
28#3750—34no—
29#3750—64no—
30C3×Dic3⋊C4D314yescomplemented
31(C2×D3)⋊Dic3—34no—
32(C2×D3)⋊Dic3—34no—
33(C2×D3)⋊Dic3—34no—
34C22×C33C2311yescharacteristic
35C2×C3×C3⋊D3—21no—
36C2×C3×C3⋊D3—21no—
37C2×C3×C3⋊D3—21no—
38C2×C3×C3⋊D3—21no—
39C2×C3×C3⋊D3—21no—
40C2×C3×C3⋊D3D412yes—
41C33×C4—21no—
42C3⋊(C3⋊Dic3)—21no—
43C3⋊(C3⋊Dic3)—21no—
44C32×Dic3—21no—
45C32×Dic3—21no—
46#935—91no—
47#894—94no—
48C22×C3⋊D3C2×D311yescharacteristic
49C22×C3⋊D3C2×D311yescharacteristic
50C22×C3×D3—34no—
51C22×C3×D3—34no—
52C2×C32×C4—24no—
53C2×C32×C4—31no—
54C2×C32×C4—31no—
55C2×C32×C4C2×D314yes—
56C2×C32×C4C2×D311yescharacteristic
57C2×C3⋊Dic3—31no—
58C2×C3⋊Dic3—31no—
59C2×C3⋊Dic3—34no—
60C2×C3⋊Dic3—34no—
61C2×C3⋊Dic3—64no—
62C2×C3⋊Dic3—64no—
63C4⋊(C3⋊D3)—22no—
64C4×C3⋊D3—62no—
65C2×C3×Dic3C2×D314yes—
66C2×C3×Dic3C2×D314yes—
67C3×D12—68no—
68D3×Dic3—68no—
69C2×C33C2×D411yescharacteristic
70C2×C33#4111yescharacteristic
71C2×C33#4111yescharacteristic
72C3×C3⋊D3—22no—
73C3×C3⋊D3—41no—
74C2×C3×D4—91no—
75C22×Dic3—91no—
76C2×D12—34no—
77C2×C4×D3—94no—
78C3×C4⋊C4—34no—
79#217—91no—
80#217—91no—
81#217—91no—
82(C2×D3)⋊C4—94no—
83(C2×D3)⋊C4—94no—
84(C2×D3)⋊C4—94no—
85Dic3⋊C4—94no—
86Dic3⋊C4—91no—
87Dic3⋊C4—184no—
88Dic3⋊C4C3⋊D311yescharacteristic, complemented
89C22×C32—24no—
90C22×C32C22×D314yes—
91C22×C32C22×D311yescharacteristic
92C2×C3⋊D3—21no—
93C2×C3⋊D3—21no—
94C2×C3⋊D3—21no—
95C2×C3⋊D3—21no—
96C2×C3⋊D3—21no—
97C2×C3⋊D3#6712yes—
98C2×C3×D3—38no—
99C2×C3×D3—64no—
100C2×C3×D3—64no—
101C2×C3×D3—64no—
102C2×C3×D3—64no—
103C2×C3×D3—64no—
104C32×C4—21no—
105C32×C4—24no—
106C32×C4—44no—
107C32×C4—61no—
108C32×C4—61no—
109C3⋊Dic3—68no—
110C3⋊Dic3—61no—
111C3⋊Dic3—64no—
112C3⋊Dic3—61no—
113C3⋊Dic3—64no—
114C3⋊Dic3—124no—
115C3×Dic3—24no—
116C3×Dic3—24no—
117#123—271noSylow
118C33#12311yescharacteristic, complemented, Sylow
119C23×C3—91no—
120C23×C3—91no—
121C22×D3—34no—
122C22×D3—34no—
123C3×D4—182no—
124C2×C3×C4—24no—
125C2×C3×C4—34no—
126C2×C3×C4—34no—
127C2×C3×C4D3214yes—
128C2×C3×C4C2×C3⋊D311yescharacteristic
129C2×Dic3—94no—
130C2×Dic3—94no—
131C2×Dic3—91no—
132C2×Dic3—91no—
133C2×Dic3—184no—
134C2×Dic3—184no—
135C2×Dic3—182no—
136C2×Dic3C2×C3⋊D311yescharacteristic
137C2×Dic3C2×C3⋊D311yescharacteristic
138D12—68no—
139C4×D3—188no—
140C2×C32—24no—
141C2×C32—24no—
142C2×C32—24no—
143C2×C32#23514yes—
144C2×C32D3×D414yes—
145C2×C32#23711yescharacteristic
146C2×C32#23711yescharacteristic
147C2×C32#23914yes—
148C2×C32C2×#6711yescharacteristic
149C3⋊D3—22no—
150C3⋊D3—41no—
151C3×D3—68no—
152C3×D3—124no—
153C2×D4—91no—
154C22×C4—271no—
155C4⋊C4—31no—
156C4⋊C4—271no—
157C22⋊C4—271no—
158C22⋊C4—271no—
159C22⋊C4—271no—
160C22×C3—24no—
161C22×C3—92no—
162C22×C3—181no—
163C22×C3—181no—
164C22×C3—181no—
165C22×C3—181no—
166C22×C3—181no—
167C22×C3C2×D3214yes—
168C22×C3C22×C3⋊D311yescharacteristic
169C2×D3—38no—
170C2×D3—64no—
171C2×D3—64no—
172C2×D3—64no—
173C2×D3—64no—
174C2×D3—64no—
175C3×C4—24no—
176C3×C4—21no—
177C3×C4—44no—
178C3×C4—64no—
179C3×C4—64no—
180Dic3—21no—
181Dic3—21no—
182Dic3—188no—
183Dic3—184no—
184Dic3—184no—
185Dic3—181no—
186Dic3—181no—
187Dic3—364no—
188C32—24no—
189C32#89414yescomplemented
190C32#93511yescharacteristic, complemented
191C23—91no—
192C23—91no—
193D4—182no—
194C2×C4—31no—
195C2×C4—31no—
196C2×C4—271no—
197C2×C4—271no—
198C2×C4—542no—
199C2×C4D3×C3⋊D311yescharacteristic
200C2×C3—24no—
201C2×C3—24no—
202C2×C3—24no—
203C2×C3—182no—
204C2×C3—361no—
205C2×C3#379514yes—
206C2×C3#379614yes—
207C2×C3D3×#6714yes—
208C2×C3#382811yescharacteristic
209C2×C3D4×C3⋊D311yescharacteristic
210C2×C3#383211yescharacteristic
211D3—68no—
212D3—124no—
213C22—92no—
214C22—181no—
215C22—181no—
216C22—181no—
217C22—181no—
218C22—181no—
219C22C2×D3×C3⋊D311yescharacteristic
220C4—21no—
221C4—61no—
222C4—61no—
223C4—541no—
224C4—541no—
225C3—24no—
226C3#6388414yescomplemented
227C3#6410311yescharacteristic, complemented
228C2—182no—
229C2—361no—
230C2#8901711yescharacteristic
231C2#8902111yescharacteristic
232C2C3⋊D3×#6711yescharacteristic
subgroupG/H
center, FrattiniC22C2×D3×C3⋊D3
commutator, socle, cosocleC22×C33C23
FittingC2×C33×C4C22
prime radicalC2×C3×C4⋊(C3⋊D3)C2
hypercenterC2×C4D3×C3⋊D3
hypocenterC33#123
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
2. C2×Dic3×C3⋊D3C2semidirect
3. #88857C2semidirect
4. C32×Dic3⋊C4C2semidirect
5. #88803C2semidirect
6. #88803C2semidirect
7. #88803C2semidirect
30. C3×Dic3⋊C4D3semidirect
88. Dic3⋊C4C3⋊D3semidirect
118. C33#123semidirect
189. C32#894semidirect
190. C32#935semidirect
226. C3#63884semidirect
227. C3#64103semidirect

Maximal subgroups

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

subgroupindex
C2×C3×C4⋊(C3⋊D3)2
C2×Dic3×C3⋊D32
#888572
C32×Dic3⋊C42
#888032
#641033
#638843