Finite Group Databasebeta

← #11723493

Subgroups — #11723493

Order 864.

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

normal subgroups only

This group has 604 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×#6292C211yesmaximal, characteristic
2C2×Dic3×C3⋊D3C211yesmaximal, characteristic, complemented
3#88859C211yesmaximal, characteristic, complemented
4C3×#3752C211yesmaximal, characteristic, complemented
5C32×C4⋊Dic3C211yesmaximal, characteristic, complemented
6#88807C211yesmaximal, characteristic, complemented
7#88804C211yesmaximal, characteristic, complemented
8#64103—31nomaximal
9#63890—34nomaximal
10C22×C3⋊(C3⋊D3)C2211yescharacteristic
11C22×C3×C3⋊D3C2211yescharacteristic
12C2×C33×C4C2211yescharacteristic
13C2×C3⋊(C3⋊Dic3)C2211yescharacteristic
14C2×C3×C3⋊Dic3C2211yescharacteristic
15C2×C32×Dic3C2211yescharacteristic
16C2×C32×Dic3C2211yescharacteristic
17#6292—22no—
18Dic3×C3⋊D3—22no—
19C2×C4⋊(C3⋊D3)—31no—
20C2×C4×C3⋊D3—31no—
21C2×#626—34no—
22C2×D3×Dic3—34no—
23C32×C4⋊C4—31no—
24#3752—31no—
25#3752—34no—
26#3752—31no—
27#3752—64no—
28#3752D311yescharacteristic, complemented
29#3750—31no—
30C3×(C2×D3)⋊C4—34no—
31C3×C4⋊Dic3D314yescomplemented
32Dic3⋊Dic3—34no—
33#3722—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#894—94no—
47#886—91no—
48C22×C3⋊D3—31no—
49C22×C3⋊D3—34no—
50C22×C3⋊D3—64no—
51C22×C3⋊D3C2×D311yescharacteristic
52C22×C3×D3—34no—
53C2×C32×C4—24no—
54C2×C32×C4—31no—
55C2×C32×C4—31no—
56C2×C32×C4C2×D314yes—
57C2×C32×C4C2×D311yescharacteristic
58C2×C3⋊Dic3—31no—
59C2×C3⋊Dic3—34no—
60C2×C3⋊Dic3—64no—
61C2×C3⋊Dic3C2×D311yescharacteristic
62C4⋊(C3⋊D3)—62no—
63C4×C3⋊D3—62no—
64C2×C3×Dic3—34no—
65C2×C3×Dic3C2×D314yes—
66C2×C3×Dic3C2×D314yes—
67#626—68no—
68D3×Dic3—68no—
69C2×C33C2×D411yescharacteristic
70C2×C33#4111yescharacteristic
71C2×C33#4111yescharacteristic
72C3⋊(C3⋊D3)—41no—
73C3×C3⋊D3—22no—
74C2×#67—91no—
75C22×Dic3—91no—
76C2×D12—94no—
77C2×C4×D3—94no—
78C3×C4⋊C4—34no—
79C3×C22⋊C4—91no—
80(C2×D3)⋊C4—34no—
81(C2×D3)⋊C4—94no—
82(C2×D3)⋊C4—94no—
83(C2×D3)⋊C4—91no—
84(C2×D3)⋊C4—91no—
85(C2×D3)⋊C4—184no—
86C4⋊Dic3—91no—
87C4⋊Dic3C3⋊D311yescharacteristic, complemented
88Dic3⋊C4—94no—
89C22×C32—24no—
90C22×C32C22×D314yes—
91C22×C32C22×D311yescharacteristic
92C2×C3⋊D3—21no—
93C2×C3⋊D3—21no—
94C2×C3⋊D3—68no—
95C2×C3⋊D3—61no—
96C2×C3⋊D3—61no—
97C2×C3⋊D3—61no—
98C2×C3⋊D3—64no—
99C2×C3⋊D3—64no—
100C2×C3⋊D3—64no—
101C2×C3⋊D3—124no—
102C2×C3⋊D3—124no—
103C2×C3⋊D3D1212yes—
104C2×C3×D3—38no—
105C2×C3×D3—64no—
106C2×C3×D3—64no—
107C32×C4—21no—
108C32×C4—24no—
109C32×C4—44no—
110C32×C4—61no—
111C32×C4—61no—
112C3⋊Dic3—21no—
113C3⋊Dic3—61no—
114C3⋊Dic3—64no—
115C3⋊Dic3—124no—
116C3×Dic3—24no—
117C3×Dic3—24no—
118C3×Dic3—64no—
119#123—271noSylow
120C33#12311yescharacteristic, complemented, Sylow
121C23×C3—91no—
122C22×D3—34no—
123C22×D3—94no—
124C22×D3—91no—
125C22×D3—184no—
126C2×C3×C4—24no—
127C2×C3×C4—34no—
128C2×C3×C4—34no—
129C2×C3×C4—91no—
130C2×C3×C4D3214yes—
131C2×C3×C4C2×C3⋊D311yescharacteristic
132#67—182no—
133C2×Dic3—34no—
134C2×Dic3—94no—
135C2×Dic3—91no—
136C2×Dic3—184no—
137C2×Dic3—182no—
138C2×Dic3C2×C3⋊D311yescharacteristic
139C2×Dic3C2×C3⋊D311yescharacteristic
140D12—188no—
141C4×D3—188no—
142C2×C32—24no—
143C2×C32—24no—
144C2×C32—24no—
145C2×C32C2×D1211yescharacteristic
146C2×C32#23514yes—
147C2×C32D3×D414yes—
148C2×C32#23711yescharacteristic
149C2×C32#23711yescharacteristic
150C2×C32#23914yes—
151C3⋊D3—22no—
152C3⋊D3—128no—
153C3⋊D3—121no—
154C3⋊D3—124no—
155C3×D3—68no—
156C2×D4—271no—
157C22×C4—271no—
158C4⋊C4—31no—
159C4⋊C4—271no—
160C22⋊C4—91no—
161C22⋊C4—271no—
162C22⋊C4—271no—
163C22×C3—24no—
164C22×C3—92no—
165C22×C3—181no—
166C22×C3—181no—
167C22×C3C2×D3214yes—
168C22×C3C22×C3⋊D311yescharacteristic
169C2×D3—38no—
170C2×D3—64no—
171C2×D3—64no—
172C2×D3—188no—
173C2×D3—184no—
174C2×D3—184no—
175C2×D3—184no—
176C2×D3—181no—
177C2×D3—181no—
178C2×D3—181no—
179C2×D3—364no—
180C2×D3—364no—
181C3×C4—24no—
182C3×C4—21no—
183C3×C4—44no—
184C3×C4—64no—
185C3×C4—64no—
186C3×C4—181no—
187Dic3—21no—
188Dic3—21no—
189Dic3—64no—
190Dic3—184no—
191Dic3—181no—
192Dic3—364no—
193C32—24no—
194C32#88611yescharacteristic, complemented
195C32#89414yescomplemented
196C23—91no—
197C23—271no—
198D4—542no—
199C2×C4—31no—
200C2×C4—31no—
201C2×C4—91no—
202C2×C4—271no—
203C2×C4—542no—
204C2×C4D3×C3⋊D311yescharacteristic
205C2×C3—24no—
206C2×C3—24no—
207C2×C3—24no—
208C2×C3—182no—
209C2×C3#379614yes—
210C2×C3D3×D1214yes—
211C2×C3#380414yes—
212C2×C3#382811yescharacteristic
213C2×C3D4×C3⋊D311yescharacteristic
214C2×C3#383211yescharacteristic
215D3—68no—
216D3—368no—
217D3—361no—
218D3—364no—
219C22—92no—
220C22—181no—
221C22—181no—
222C22—541no—
223C22—541no—
224C22—541no—
225C22C2×D3×C3⋊D311yescharacteristic
226C4—21no—
227C4—61no—
228C4—61no—
229C4—181no—
230C4—541no—
231C3—24no—
232C3#6389014yescomplemented
233C3#6410311yescharacteristic, complemented
234C2—182no—
235C2—1081no—
236C2#8901711yescharacteristic
237C2C3⋊D3×D1211yescharacteristic
238C2#8903111yescharacteristic
subgroupG/H
center, FrattiniC22C2×D3×C3⋊D3
commutator, socle, cosocleC22×C33C23
FittingC2×C33×C4C22
prime radicalC2×#6292C2
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. #88859C2semidirect
4. C3×#3752C2semidirect
5. C32×C4⋊Dic3C2semidirect
6. #88807C2semidirect
7. #88804C2semidirect
28. #3752D3semidirect
31. C3×C4⋊Dic3D3semidirect
87. C4⋊Dic3C3⋊D3semidirect
120. C33#123semidirect
194. C32#886semidirect
195. C32#894semidirect
232. C3#63890semidirect
233. C3#64103semidirect

Maximal subgroups

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

subgroupindex
C2×#62922
C2×Dic3×C3⋊D32
#888592
C3×#37522
C32×C4⋊Dic32
#888072
#888042
#641033
#638903