Finite Group Databasebeta

← #13179705

Subgroups — #13179705

Order 1728.

3276 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C33×C4×Q8C211yesmaximal, characteristic, complemented
2#11723988C211yesmaximal, characteristic, complemented
3#11723987C211yesmaximal, characteristic, complemented
4#11723939C212yesmaximal, complemented
5#11723936C211yesmaximal, characteristic
6#11723935C212yesmaximal, complemented
7#11723931C211yesmaximal, characteristic, complemented
8C4×C4⋊(C3⋊(C3⋊D3))C212yesmaximal, complemented
9C4×C4⋊(C3⋊(C3⋊D3))C211yesmaximal, characteristic, complemented
10#11723913C212yesmaximal, complemented
11C42×C3⋊(C3⋊D3)C211yesmaximal, characteristic, complemented
12#10596051—313nomaximal
13C2×C33×Q8C2211yescharacteristic
14C2×C4⋊(C3⋊(C3⋊D3))C2212yes—
15C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic
16C2×C4×C3⋊(C3⋊D3)C2211yescharacteristic
17C2×C4×C3⋊(C3⋊D3)C2211yescharacteristic
18C2×C4×C3⋊(C3⋊D3)C2211yescharacteristic
19C2×C4×C3⋊(C3⋊D3)C2212yes—
20C33×C4⋊C4C2211yescharacteristic
21C33×C4⋊C4C2212yes—
22C33×C42C2211yescharacteristic
23C33×C42C2212yes—
24#88859C2212yes—
25#88859C2212yes—
26#88859C2212yes—
27#88859C2212yes—
28#88859C2212yes—
29C4⋊(C3⋊(C3⋊Dic3))C2212yes—
30C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
31#88857C2212yes—
32#88857C2212yes—
33C4×C3⋊(C3⋊Dic3)C2211yescharacteristic
34C4×C3⋊(C3⋊Dic3)C2211yescharacteristic
35C4×C3⋊(C3⋊Dic3)C2211yescharacteristic
36C32×C4×Q8D3113yescomplemented
37#64156—313no—
38#64155—313no—
39#64107—326no—
40#64104—313no—
41#64103—326no—
42#64099—313no—
43C4×C4⋊(C3⋊D3)—326no—
44C4×C4⋊(C3⋊D3)—313no—
45#64081—326no—
46C42×C3⋊D3—313no—
47C22×C3⋊(C3⋊D3)C2312yes—
48C22×C3⋊(C3⋊D3)C2311yescharacteristic
49C33×Q8—22no—
50C2×C33×C4C2311yescharacteristic
51C2×C33×C4C2312yes—
52C2×C33×C4C2312yes—
53C2×C33×C4C2311yescharacteristic
54C2×C33×C4C2311yescharacteristic
55C2×C3⋊(C3⋊Dic3)C2311yescharacteristic
56C2×C3⋊(C3⋊Dic3)C2311yescharacteristic
57C2×C3⋊(C3⋊Dic3)C2311yescharacteristic
58C2×C3⋊(C3⋊Dic3)C2312yes—
59C4⋊(C3⋊(C3⋊D3))—24no—
60C4⋊(C3⋊(C3⋊D3))—22no—
61C4×C3⋊(C3⋊D3)—22no—
62C4×C3⋊(C3⋊D3)—22no—
63C4×C3⋊(C3⋊D3)—22no—
64C4×C3⋊(C3⋊D3)—24no—
65#5580—913no—
66C2×C32×Q8C2×D3113yes—
67C2×C4⋊(C3⋊D3)—326no—
68C2×C4⋊(C3⋊D3)—313no—
69C2×C4×C3⋊D3—313no—
70C2×C4×C3⋊D3—313no—
71C2×C4×C3⋊D3—313no—
72C2×C4×C3⋊D3—326no—
73C32×C4⋊C4C2×D3113yes—
74C32×C4⋊C4C2×D3126yes—
75C32×C42C2×D3113yes—
76C32×C42C2×D3126yes—
77#3752—326no—
78#3752—326no—
79#3752—326no—
80#3752—326no—
81#3752—326no—
82C4⋊(C3⋊Dic3)—326no—
83C4⋊(C3⋊Dic3)—313no—
84#3750—326no—
85#3750—326no—
86C4×C3⋊Dic3—313no—
87C4×C3⋊Dic3—313no—
88C4×C3⋊Dic3—313no—
89C22×C33C2411yescharacteristic
90C2×C3⋊(C3⋊D3)—21no—
91C2×C3⋊(C3⋊D3)—21no—
92C2×C3⋊(C3⋊D3)—22no—
93C2×C3⋊(C3⋊D3)—22no—
94C2×C3⋊(C3⋊D3)—22no—
95C2×C3⋊(C3⋊D3)#4112yes—
96C33×C4—21no—
97C33×C4—22no—
98C33×C4—22no—
99C33×C4#4112yes—
100C33×C4#4112yes—
101C3⋊(C3⋊Dic3)—21no—
102C3⋊(C3⋊Dic3)—21no—
103C3⋊(C3⋊Dic3)—22no—
104C3⋊(C3⋊Dic3)—21no—
105C3×C4×Q8C3⋊D3113yescomplemented
106#947—913no—
107#946—913no—
108#898—926no—
109#895—913no—
110#894—926no—
111#890—913no—
112C4×D12—926no—
113C4×D12—913no—
114#872—926no—
115C42×D3—913no—
116C22×C3⋊D3—313no—
117C22×C3⋊D3—326no—
118C32×Q8—226no—
119C2×C32×C4C22×D3113yes—
120C2×C32×C4C22×D3126yes—
121C2×C32×C4C22×D3126yes—
122C2×C32×C4C22×D3113yes—
123C2×C32×C4C22×D3113yes—
124C2×C3⋊Dic3—313no—
125C2×C3⋊Dic3—313no—
126C2×C3⋊Dic3—313no—
127C2×C3⋊Dic3—326no—
128C4⋊(C3⋊D3)—652no—
129C4⋊(C3⋊D3)—626no—
130C4×C3⋊D3—626no—
131C4×C3⋊D3—626no—
132C4×C3⋊D3—626no—
133C4×C3⋊D3—652no—
134#529—271noSylow
135C2×C33C2×#4111yescharacteristic
136C2×C33C2×#4111yescharacteristic
137C2×C33C2×#4111yescharacteristic
138C3⋊(C3⋊D3)—22no—
139C3⋊(C3⋊D3)—42no—
140C2×C3×Q8C2×C3⋊D3113yes—
141C2×D12—926no—
142C2×D12—913no—
143C2×C4×D3—913no—
144C2×C4×D3—913no—
145C2×C4×D3—913no—
146C2×C4×D3—926no—
147C3×C4⋊C4C2×C3⋊D3113yes—
148C3×C4⋊C4C2×C3⋊D3126yes—
149C3×C42C2×C3⋊D3113yes—
150C3×C42C2×C3⋊D3126yes—
151(C2×D3)⋊C4—926no—
152(C2×D3)⋊C4—926no—
153(C2×D3)⋊C4—926no—
154(C2×D3)⋊C4—926no—
155(C2×D3)⋊C4—926no—
156C4⋊Dic3—926no—
157C4⋊Dic3—913no—
158Dic3⋊C4—926no—
159Dic3⋊C4—926no—
160C4×Dic3—913no—
161C4×Dic3—913no—
162C4×Dic3—913no—
163C22×C32C23×D3113yes—
164C2×C3⋊D3—326no—
165C2×C3⋊D3—613no—
166C2×C3⋊D3—613no—
167C2×C3⋊D3—626no—
168C2×C3⋊D3—626no—
169C2×C3⋊D3—626no—
170C32×C4—226no—
171C32×C4—213no—
172C32×C4—226no—
173C32×C4#235126yes—
174C32×C4#239126yes—
175C3⋊Dic3—613no—
176C3⋊Dic3—613no—
177C3⋊Dic3—626no—
178C3⋊Dic3—613no—
179#126—272no—
180#125—271no—
181#124—271no—
182#123—272no—
183C22⋊Q8—271no—
184#121—271no—
185C4×Q8C3⋊(C3⋊D3)11yescharacteristic, complemented
186C4×D4—271no—
187C4×D4—272no—
188#117—272no—
189C2×C42—271no—
190C33#52911yescharacteristic, complemented, Sylow
191C22×D3—913no—
192C22×D3—926no—
193C3×Q8—226no—
194C2×C3×C4C22×C3⋊D3113yes—
195C2×C3×C4C22×C3⋊D3126yes—
196C2×C3×C4C22×C3⋊D3126yes—
197C2×C3×C4C22×C3⋊D3113yes—
198C2×C3×C4C22×C3⋊D3113yes—
199C2×Dic3—913no—
200C2×Dic3—913no—
201C2×Dic3—913no—
202C2×Dic3—926no—
203D12—1852no—
204D12—1826no—
205C4×D3—1826no—
206C4×D3—1826no—
207C4×D3—1826no—
208C4×D3—1852no—
209C2×C32C2×#235113yes—
210C2×C32C2×#239113yes—
211C2×C32D3×#41113yes—
212C3⋊D3—626no—
213C3⋊D3—1226no—
214C2×Q8C2×C3⋊(C3⋊D3)11yescharacteristic
215C2×D4—272no—
216C2×D4—271no—
217C22×C4—271no—
218C22×C4—271no—
219C22×C4—271no—
220C22×C4—272no—
221C4⋊C4—272no—
222C4⋊C4—272no—
223C4⋊C4—271no—
224C4⋊C4—272no—
225C4⋊C4C2×C3⋊(C3⋊D3)11yescharacteristic
226C4⋊C4C2×C3⋊(C3⋊D3)12yes—
227C22⋊C4—272no—
228C22⋊C4—272no—
229C22⋊C4—272no—
230C22⋊C4—272no—
231C22⋊C4—272no—
232C42—271no—
233C42—271no—
234C42—271no—
235C42C2×C3⋊(C3⋊D3)11yescharacteristic
236C42C2×C3⋊(C3⋊D3)12yes—
237C22×C3C23×C3⋊D3113yes—
238C2×D3—926no—
239C2×D3—1813no—
240C2×D3—1813no—
241C2×D3—1826no—
242C2×D3—1826no—
243C2×D3—1826no—
244C3×C4—226no—
245C3×C4—213no—
246C3×C4—226no—
247C3×C4#3828126yes—
248C3×C4#3832126yes—
249Dic3—1813no—
250Dic3—1813no—
251Dic3—1826no—
252Dic3—1813no—
253C32#5580113yescomplemented
254C23—271no—
255C23—272no—
256Q8—22no—
257D4—544no—
258D4—542no—
259C2×C4—271no—
260C2×C4—271no—
261C2×C4—271no—
262C2×C4—272no—
263C2×C4—542no—
264C2×C4—542no—
265C2×C4—542no—
266C2×C4—544no—
267C2×C4C22×C3⋊(C3⋊D3)11yescharacteristic
268C2×C4C22×C3⋊(C3⋊D3)12yes—
269C2×C4C22×C3⋊(C3⋊D3)12yes—
270C2×C4C22×C3⋊(C3⋊D3)11yescharacteristic
271C2×C4C22×C3⋊(C3⋊D3)11yescharacteristic
272C2×C3C2×#3828113yes—
273C2×C3C2×#3832113yes—
274C2×C3#41×C3⋊D3113yes—
275D3—1826no—
276D3—3626no—
277C22—272no—
278C22—541no—
279C22—541no—
280C22—542no—
281C22—542no—
282C22—542no—
283C22C23×C3⋊(C3⋊D3)11yescharacteristic
284C4—22no—
285C4—21no—
286C4—22no—
287C4—541no—
288C4—541no—
289C4—542no—
290C4—541no—
291C4#8907612yes—
292C4#8908012yes—
293C3#10596051113yescomplemented
294C2—542no—
295C2—1082no—
296C2C2×#8907611yescharacteristic
297C2C2×#8908011yescharacteristic
298C2#41×C3⋊(C3⋊D3)11yescharacteristic
subgroupG/H
centerC2×C4C22×C3⋊(C3⋊D3)
commutator, socle, cosocleC22×C33C24
FrattiniC22C23×C3⋊(C3⋊D3)
FittingC33×C4×Q8C2
prime radical#11723936C2
hypercenterC4×Q8C3⋊(C3⋊D3)
hypocenterC33#529
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C2×C4 ◁ C4×Q8

Complements

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

NH ≅ G/Ntype
1. C33×C4×Q8C2semidirect
2. #11723988C2semidirect
3. #11723987C2semidirect
4. #11723939C2semidirect
6. #11723935C2semidirect
7. #11723931C2semidirect
8. C4×C4⋊(C3⋊(C3⋊D3))C2semidirect
9. C4×C4⋊(C3⋊(C3⋊D3))C2semidirect
10. #11723913C2semidirect
11. C42×C3⋊(C3⋊D3)C2semidirect
36. C32×C4×Q8D3semidirect
105. C3×C4×Q8C3⋊D3semidirect
185. C4×Q8C3⋊(C3⋊D3)semidirect
190. C33#529semidirect
253. C32#5580semidirect
293. C3#10596051semidirect

Maximal subgroups

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

subgroupindex
C33×C4×Q82
#117239882
#117239872
#117239392
#117239362
#117239352
#117239312
C4×C4⋊(C3⋊(C3⋊D3))2
#117239132
C42×C3⋊(C3⋊D3)2
#105960513