Finite Group Databasebeta

← #11723520

Subgroups — #11723520

Order 864.

700 conjugacy classes of subgroups; only the 112 normal subgroups are shown.

normal subgroups only

This group has 700 conjugacy classes of subgroups — too many to draw. Only its 112 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⋊D3C211yesmaximal, characteristic, complemented
2C2×#6292C211yesmaximal, characteristic, complemented
3C2×#6291C211yesmaximal, characteristic, complemented
4C2×#6290C211yesmaximal, characteristic, complemented
5#88859C211yesmaximal, characteristic, complemented
6C32×(C2×D3)⋊C4C211yesmaximal, characteristic, complemented
7#88807C211yesmaximal, characteristic, complemented
8#64093—31nomaximal
9#63906—34nomaximal
10C22×C3⋊(C3⋊D3)C2211yescharacteristic
11C22×C3×C3⋊D3C2211yescharacteristic
12C22×C32×D3C2211yescharacteristic
13C2×C33×C4C2211yescharacteristic
14C2×C3⋊(C3⋊Dic3)C2211yescharacteristic
15C2×C3×C3⋊Dic3C2211yescharacteristic
16C3×C4×C3⋊D3—22no—
17C2×C32×Dic3C2211yescharacteristic
18#6292—22no—
19#6291—22no—
20#6290—22no—
21C2×#638—31no—
22C2×#638—31no—
23C2×C4⋊(C3⋊D3)—31no—
24C2×C4×C3⋊D3D311yescharacteristic, complemented
25C2×C3×C4×D3—34no—
26C2×#626—34no—
27C2×#626—34no—
28C2×#625—34no—
29C32×C22⋊C4—31no—
30#3752—31no—
31#3752—34no—
32#3752—64no—
33#3750—31no—
34C3×(C2×D3)⋊C4D314yescomplemented
35Dic3⋊Dic3—34no—
36C22×C33C2311yescharacteristic
37C2×C3⋊(C3⋊D3)—21no—
38C2×C3⋊(C3⋊D3)—21no—
39C2×C3⋊(C3⋊D3)—21no—
40C2×C3×C3⋊D3—21no—
41C2×C3×C3⋊D3—21no—
42C2×C3×C3⋊D3D412yescomplemented
43C2×C32×D3—21no—
44C2×C32×D3—21no—
45C2×C32×D3—21no—
46C33×C4—21no—
47C3⋊(C3⋊Dic3)—21no—
48C3×C3⋊Dic3D412yescomplemented
49C32×Dic3—21no—
50#930—91no—
51#884—94no—
52C23×C32—31no—
53C22×C3⋊D3—31no—
54C22×C3⋊D3—34no—
55C22×C3⋊D3—64no—
56C22×C3⋊D3C2×D311yescharacteristic
57C22×C3×D3—34no—
58C22×C3×D3C2×D314yes—
59C2×C32×C4—24no—
60C2×C32×C4—31no—
61C2×C32×C4C2×D314yes—
62C2×C32×C4C2×D311yescharacteristic
63#638—62no—
64#638—62no—
65C2×C3⋊Dic3—31no—
66C2×C3⋊Dic3—34no—
67C2×C3⋊Dic3—64no—
68C2×C3⋊Dic3C2×D311yescharacteristic
69C4⋊(C3⋊D3)—62no—
70C4×C3⋊D3—22no—
71C2×C3×Dic3—34no—
72C2×C3×Dic3C2×D314yes—
73C3×C4×D3—68no—
74#626—68no—
75#626—68no—
76#625—68no—
77C2×C33C2×D411yescharacteristic
78C2×C33C2×D411yescharacteristic
79C2×C33#4111yescharacteristic
80C3⋊(C3⋊D3)—41no—
81C3×C3⋊D3—22no—
82C32×D3—41no—
83C22×C3×C4—91no—
84C2×#67—94no—
85C2×#67—94no—
86C2×#67—91no—
87C2×#67—91no—
88C2×D12—94no—
89C2×D12—91no—
90C2×C4×D3—34no—
91C3×C22⋊C4—34no—
92(C2×D3)⋊C4—94no—
93(C2×D3)⋊C4—91no—
94(C2×D3)⋊C4—184no—
95(C2×D3)⋊C4C3⋊D311yescharacteristic, complemented
96C4⋊Dic3—91no—
97Dic3⋊C4—94no—
98C22×C32—24no—
99C22×C32—61no—
100C22×C32—61no—
101C22×C32—61no—
102C22×C32C22×D314yes—
103C22×C32C22×D311yescharacteristic
104C2×C3⋊D3—21no—
105C2×C3⋊D3—21no—
106C2×C3⋊D3—68no—
107C2×C3⋊D3—61no—
108C2×C3⋊D3—64no—
109C2×C3⋊D3—61no—
110C2×C3⋊D3—61no—
111C2×C3⋊D3—64no—
112C2×C3⋊D3—64no—
113C2×C3⋊D3—124no—
114C2×C3⋊D3—124no—
115C2×C3⋊D3D1212yescomplemented
116C2×C3×D3—24no—
117C2×C3×D3—24no—
118C2×C3×D3—24no—
119C2×C3×D3—38no—
120C2×C3×D3—64no—
121C2×C3×D3—64no—
122C32×C4—21no—
123C32×C4—24no—
124C32×C4—44no—
125C32×C4—61no—
126C3⋊Dic3—61no—
127C3⋊Dic3—64no—
128C3⋊Dic3—124no—
129C3⋊Dic3#6712yescomplemented
130C3×Dic3—24no—
131C3×Dic3—38no—
132#121—271noSylow
133C33#12111yescharacteristic, complemented, Sylow
134C23×C3—34no—
135C23×C3—91no—
136C22×D3—34no—
137C22×D3—94no—
138C22×D3—91no—
139C22×D3—184no—
140C22×D3C2×C3⋊D311yescharacteristic
141C2×C3×C4—24no—
142C2×C3×C4—34no—
143C2×C3×C4—91no—
144C2×C3×C4—182no—
145C2×C3×C4D3214yes—
146C2×C3×C4C2×C3⋊D311yescharacteristic
147#67—182no—
148#67—188no—
149#67—182no—
150#67—188no—
151C2×Dic3—34no—
152C2×Dic3—94no—
153C2×Dic3—91no—
154C2×Dic3—184no—
155C2×Dic3C2×C3⋊D311yescharacteristic
156D12—188no—
157D12—182no—
158C4×D3—68no—
159C2×C32—24no—
160C2×C32—24no—
161C2×C32—24no—
162C2×C32—121no—
163C2×C32C2×D1211yescharacteristic
164C2×C32#23514yes—
165C2×C32#23511yescharacteristic
166C2×C32D3×D414yes—
167C2×C32D3×D414yes—
168C2×C32C2×#6711yescharacteristic
169C3⋊D3—22no—
170C3⋊D3—128no—
171C3⋊D3—124no—
172C3⋊D3—121no—
173C3×D3—44no—
174C3×D3—68no—
175C2×D4—271no—
176C2×D4—271no—
177C2×D4—271no—
178C22×C4—91no—
179C4⋊C4—271no—
180C22⋊C4—31no—
181C22⋊C4—271no—
182C22×C3—24no—
183C22×C3—64no—
184C22×C3—64no—
185C22×C3—64no—
186C22×C3—92no—
187C22×C3—181no—
188C22×C3—181no—
189C22×C3C2×D3214yes—
190C22×C3C22×C3⋊D311yescharacteristic
191C2×D3—21no—
192C2×D3—21no—
193C2×D3—21no—
194C2×D3—38no—
195C2×D3—64no—
196C2×D3—64no—
197C2×D3—188no—
198C2×D3—184no—
199C2×D3—184no—
200C2×D3—184no—
201C2×D3—181no—
202C2×D3—181no—
203C2×D3—181no—
204C2×D3—364no—
205C2×D3—364no—
206C3×C4—24no—
207C3×C4—21no—
208C3×C4—44no—
209C3×C4—64no—
210C3×C4—92no—
211Dic3—21no—
212Dic3—38no—
213Dic3—184no—
214Dic3—181no—
215Dic3—364no—
216C32—24no—
217C32#88414yescomplemented
218C32#93011yescharacteristic, complemented
219C23—31no—
220C23—91no—
221C23—271no—
222D4—542no—
223D4—542no—
224D4—542no—
225C2×C4—31no—
226C2×C4—91no—
227C2×C4—182no—
228C2×C4—271no—
229C2×C4D3×C3⋊D311yescharacteristic
230C2×C3—24no—
231C2×C3—24no—
232C2×C3—24no—
233C2×C3—124no—
234C2×C3—182no—
235C2×C3#379814yes—
236C2×C3D3×D1214yes—
237C2×C3D3×#6714yes—
238C2×C3#382811yescharacteristic
239C2×C3D4×C3⋊D311yescharacteristic
240C2×C3D4×C3⋊D311yescharacteristic
241D3—41no—
242D3—68no—
243D3—368no—
244D3—361no—
245D3—364no—
246C22—61no—
247C22—61no—
248C22—61no—
249C22—92no—
250C22—181no—
251C22—181no—
252C22—541no—
253C22—541no—
254C22—541no—
255C22C2×D3×C3⋊D311yescharacteristic
256C4—21no—
257C4—61no—
258C4—92no—
259C4—541no—
260C3—24no—
261C3#6390614yescomplemented
262C3#6409311yescharacteristic, complemented
263C2—121no—
264C2—182no—
265C2—1081no—
266C2#8902011yescharacteristic
267C2C3⋊D3×D1211yescharacteristic
268C2C3⋊D3×#6711yescharacteristic
subgroupG/H
center, FrattiniC22C2×D3×C3⋊D3
commutator, socle, cosocleC22×C33C23
FittingC2×C33×C4C22
hypercenterC2×C4D3×C3⋊D3
hypocenterC33#121
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
1. C2×C3×C4×C3⋊D3C2semidirect
2. C2×#6292C2semidirect
3. C2×#6291C2semidirect
4. C2×#6290C2semidirect
5. #88859C2semidirect
6. C32×(C2×D3)⋊C4C2semidirect
7. #88807C2semidirect
24. C2×C4×C3⋊D3D3semidirect
34. C3×(C2×D3)⋊C4D3semidirect
42. C2×C3×C3⋊D3D4semidirect
48. C3×C3⋊Dic3D4semidirect
95. (C2×D3)⋊C4C3⋊D3semidirect
115. C2×C3⋊D3D12semidirect
129. C3⋊Dic3#67semidirect
133. C33#121semidirect
217. C32#884semidirect
218. C32#930semidirect
261. C3#63906semidirect
262. C3#64093semidirect

Maximal subgroups

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

subgroupindex
C2×C3×C4×C3⋊D32
C2×#62922
C2×#62912
C2×#62902
#888592
C32×(C2×D3)⋊C42
#888072
#640933
#639063