Finite Group Databasebeta

← #13179741

Subgroups — #13179741

Order 1728.

3332 conjugacy classes of subgroups.

The subgroup lattice has 316 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C33×#121C211yesmaximal, characteristic, complemented
2#11723979C212yesmaximal, complemented
3#11723979C211yesmaximal, characteristic, complemented
4#11723976C212yesmaximal, complemented
5D4×C3⋊(C3⋊Dic3)C211yesmaximal, characteristic, complemented
6C4×#6312C211yesmaximal, characteristic, complemented
7#11723967C211yesmaximal, characteristic, complemented
8C4⋊C4×C3⋊(C3⋊D3)C211yesmaximal, characteristic, complemented
9#11723931C211yesmaximal, characteristic, complemented
10#11723922C212yesmaximal, complemented
11#11723920C212yesmaximal, complemented
12#10596085—313nomaximal
13C2×C33×D4C2211yescharacteristic
14C2×C33×D4C2212yes—
15C22×C33×C4C2211yescharacteristic
16C2×#6312C2212yes—
17C2×#6312C2211yescharacteristic
18C22×C3⋊(C3⋊Dic3)C2212yes—
19C2×C4×C3⋊(C3⋊D3)C2211yescharacteristic
20C2×C4×C3⋊(C3⋊D3)C2212yes—
21C33×C4⋊C4C2211yescharacteristic
22C33×C22⋊C4C2212yes—
23#88864C2212yes—
24#88864C2212yes—
25#88864C2212yes—
26#88864C2211yescharacteristic
27#88859C2211yescharacteristic
28C4⋊(C3⋊(C3⋊Dic3))C2212yes—
29C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
30C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
31#88857C2211yescharacteristic
32#88857C2212yes—
33#88857C2212yes—
34C4×C3⋊(C3⋊Dic3)C2212yes—
35C4×C3⋊(C3⋊Dic3)C2211yescharacteristic
36C32×#121D3113yescomplemented
37#64147—326no—
38#64147—313no—
39#64144—326no—
40D4×C3⋊Dic3—313no—
41C4×#638—313no—
42#64135—313no—
43C4⋊C4×C3⋊D3—313no—
44#64099—313no—
45#64090—326no—
46#64088—326no—
47C23×C33C2311yescharacteristic
48C23×C33C2312yes—
49C22×C3⋊(C3⋊D3)C2311yescharacteristic
50C33×D4—22no—
51C33×D4—42no—
52C2×C33×C4—22no—
53C2×C33×C4C2311yescharacteristic
54C2×C33×C4C2312yes—
55C2×C33×C4C2311yescharacteristic
56#6312—24no—
57#6312—22no—
58C2×C3⋊(C3⋊Dic3)—42no—
59C2×C3⋊(C3⋊Dic3)C2311yescharacteristic
60C2×C3⋊(C3⋊Dic3)C2312yes—
61C2×C3⋊(C3⋊Dic3)C2311yescharacteristic
62C2×C3⋊(C3⋊Dic3)C2312yes—
63C2×C3⋊(C3⋊Dic3)C2311yescharacteristic
64C4×C3⋊(C3⋊D3)—22no—
65C4×C3⋊(C3⋊D3)—42no—
66#5614—913no—
67C2×C32×D4C2×D3113yes—
68C2×C32×D4C2×D3126yes—
69C22×C32×C4C2×D3113yes—
70C2×#638—326no—
71C2×#638—313no—
72C22×C3⋊Dic3—326no—
73C2×C4×C3⋊D3—313no—
74C2×C4×C3⋊D3—326no—
75C32×C4⋊C4C2×D3113yes—
76C32×C22⋊C4C2×D3126yes—
77#3757—326no—
78#3757—326no—
79#3757—326no—
80#3757—313no—
81#3752—313no—
82C4⋊(C3⋊Dic3)—326no—
83C4⋊(C3⋊Dic3)—313no—
84C4⋊(C3⋊Dic3)—313no—
85#3750—313no—
86#3750—326no—
87#3750—326no—
88C4×C3⋊Dic3—326no—
89C4×C3⋊Dic3—313no—
90C22×C33—21no—
91C22×C33—22no—
92C22×C33—21no—
93C22×C33—22no—
94C22×C33—22no—
95C22×C33—21no—
96C22×C33C2411yescharacteristic
97C2×C3⋊(C3⋊D3)—21no—
98C2×C3⋊(C3⋊D3)—21no—
99C2×C3⋊(C3⋊D3)#4112yes—
100C33×C4—21no—
101C33×C4—22no—
102C33×C4#4112yes—
103C3⋊(C3⋊Dic3)—21no—
104C3⋊(C3⋊Dic3)—22no—
105C3⋊(C3⋊Dic3)—21no—
106C3⋊(C3⋊Dic3)—22no—
107C3⋊(C3⋊Dic3)—21no—
108C3×#121C3⋊D3113yescomplemented
109#938—926no—
110#938—913no—
111#935—926no—
112D4×Dic3—913no—
113C4×#67—913no—
114#926—913no—
115D3×C4⋊C4—913no—
116#890—913no—
117#67⋊C4—926no—
118#879—926no—
119C23×C32C22×D3113yes—
120C23×C32C22×D3126yes—
121C22×C3⋊D3—313no—
122C32×D4—226no—
123C32×D4—426no—
124C2×C32×C4—226no—
125C2×C32×C4C22×D3113yes—
126C2×C32×C4C22×D3126yes—
127C2×C32×C4C22×D3113yes—
128#638—652no—
129#638—626no—
130C2×C3⋊Dic3—313no—
131C2×C3⋊Dic3—326no—
132C2×C3⋊Dic3—313no—
133C2×C3⋊Dic3—326no—
134C2×C3⋊Dic3—313no—
135C2×C3⋊Dic3—1226no—
136C4×C3⋊D3—626no—
137C4×C3⋊D3—1226no—
138#553—271noSylow
139C2×C33—41no—
140C2×C33—42no—
141C2×C33C2×#4111yescharacteristic
142C2×C33C2×#4111yescharacteristic
143C2×C33H(5,𝔽2)11yescharacteristic
144C3⋊(C3⋊D3)—22no—
145C2×C3×D4C2×C3⋊D3113yes—
146C2×C3×D4C2×C3⋊D3126yes—
147C22×C3×C4C2×C3⋊D3113yes—
148C2×#67—926no—
149C2×#67—913no—
150C22×Dic3—926no—
151C2×C4×D3—913no—
152C2×C4×D3—926no—
153C3×C4⋊C4C2×C3⋊D3113yes—
154C3×C22⋊C4C2×C3⋊D3126yes—
155#217—926no—
156#217—926no—
157#217—926no—
158#217—913no—
159(C2×D3)⋊C4—913no—
160C4⋊Dic3—926no—
161C4⋊Dic3—913no—
162C4⋊Dic3—913no—
163Dic3⋊C4—913no—
164Dic3⋊C4—926no—
165Dic3⋊C4—926no—
166C4×Dic3—926no—
167C4×Dic3—913no—
168C22×C32—213no—
169C22×C32—226no—
170C22×C32—226no—
171C22×C32—213no—
172C22×C32—213no—
173C22×C32—226no—
174C22×C32C23×D3113yes—
175C2×C3⋊D3—326no—
176C2×C3⋊D3—613no—
177C2×C3⋊D3—613no—
178C32×C4—213no—
179C32×C4—226no—
180C32×C4#237126yes—
181C3⋊Dic3—613no—
182C3⋊Dic3—626no—
183C3⋊Dic3—613no—
184C3⋊Dic3—626no—
185C3⋊Dic3—613no—
186#126—272no—
187#125—271no—
188#123—272no—
189#121—272no—
190#121—271no—
191#121C3⋊(C3⋊D3)11yescharacteristic, complemented
192C4×D4—271no—
193C4×D4—271no—
194C4×D4—272no—
195#117—271no—
196C2×C4⋊C4—271no—
197C33#55311yescharacteristic, complemented, Sylow
198C23×C3C22×C3⋊D3113yes—
199C23×C3C22×C3⋊D3126yes—
200C22×D3—913no—
201C3×D4—226no—
202C3×D4—426no—
203C2×C3×C4—226no—
204C2×C3×C4C22×C3⋊D3113yes—
205C2×C3×C4C22×C3⋊D3126yes—
206C2×C3×C4C22×C3⋊D3113yes—
207#67—1852no—
208#67—1826no—
209C2×Dic3—913no—
210C2×Dic3—926no—
211C2×Dic3—913no—
212C2×Dic3—926no—
213C2×Dic3—913no—
214C2×Dic3—3626no—
215C4×D3—1826no—
216C4×D3—3626no—
217C2×C32—413no—
218C2×C32—426no—
219C2×C32C2×#237113yes—
220C2×C32#1004113yes—
221C2×C32D3×#41113yes—
222C3⋊D3—626no—
223C2×D4—271no—
224C2×D4—272no—
225C2×D4C2×C3⋊(C3⋊D3)11yescharacteristic
226C2×D4C2×C3⋊(C3⋊D3)12yes—
227C22×C4—272no—
228C22×C4—271no—
229C22×C4—272no—
230C22×C4C2×C3⋊(C3⋊D3)11yescharacteristic
231C4⋊C4—271no—
232C4⋊C4—272no—
233C4⋊C4—271no—
234C4⋊C4—272no—
235C4⋊C4—271no—
236C4⋊C4—272no—
237C4⋊C4C2×C3⋊(C3⋊D3)11yescharacteristic
238C22⋊C4—272no—
239C22⋊C4—272no—
240C22⋊C4—271no—
241C22⋊C4—272no—
242C22⋊C4—271no—
243C22⋊C4C2×C3⋊(C3⋊D3)12yes—
244C42—272no—
245C42—271no—
246C22×C3—213no—
247C22×C3—226no—
248C22×C3—226no—
249C22×C3—213no—
250C22×C3—213no—
251C22×C3—226no—
252C22×C3C23×C3⋊D3113yes—
253C2×D3—926no—
254C2×D3—1813no—
255C2×D3—1813no—
256C3×C4—213no—
257C3×C4—226no—
258C3×C4#3830126yes—
259Dic3—1813no—
260Dic3—1826no—
261Dic3—1813no—
262Dic3—1826no—
263Dic3—1813no—
264C32#5614113yescomplemented
265C23—271no—
266C23C22×C3⋊(C3⋊D3)11yescharacteristic
267C23C22×C3⋊(C3⋊D3)12yes—
268D4—22no—
269D4—42no—
270D4—542no—
271D4—544no—
272C2×C4—22no—
273C2×C4—272no—
274C2×C4—271no—
275C2×C4—272no—
276C2×C4—271no—
277C2×C4—271no—
278C2×C4—542no—
279C2×C4—1082no—
280C2×C4—1082no—
281C2×C4C22×C3⋊(C3⋊D3)11yescharacteristic
282C2×C4C22×C3⋊(C3⋊D3)12yes—
283C2×C4C22×C3⋊(C3⋊D3)11yescharacteristic
284C2×C3—413no—
285C2×C3—426no—
286C2×C3C2×#3830113yes—
287C2×C3#64361113yes—
288C2×C3#41×C3⋊D3113yes—
289D3—1826no—
290C22—21no—
291C22—22no—
292C22—22no—
293C22—21no—
294C22—21no—
295C22—22no—
296C22—272no—
297C22—541no—
298C22—541no—
299C22C23×C3⋊(C3⋊D3)11yescharacteristic
300C4—21no—
301C4—22no—
302C4—541no—
303C4—542no—
304C4—541no—
305C4—542no—
306C4—541no—
307C4#8907812yes—
308C3#10596085113yescomplemented
309C2—41no—
310C2—42no—
311C2—542no—
312C2C2×#8907811yescharacteristic
313C2#1172467911yescharacteristic
314C2#41×C3⋊(C3⋊D3)11yescharacteristic
subgroupG/H
center, FrattiniC22C23×C3⋊(C3⋊D3)
commutator, socle, cosocleC22×C33C24
FittingC33×#121C2
hypercenter#121C3⋊(C3⋊D3)
hypocenterC33#553
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ #121

Complements

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

NH ≅ G/Ntype
1. C33×#121C2semidirect
2. #11723979C2semidirect
3. #11723979C2semidirect
4. #11723976C2semidirect
5. D4×C3⋊(C3⋊Dic3)C2semidirect
6. C4×#6312C2semidirect
7. #11723967C2semidirect
8. C4⋊C4×C3⋊(C3⋊D3)C2semidirect
9. #11723931C2semidirect
10. #11723922C2semidirect
11. #11723920C2semidirect
36. C32×#121D3semidirect
108. C3×#121C3⋊D3semidirect
191. #121C3⋊(C3⋊D3)semidirect
197. C33#553semidirect
264. C32#5614semidirect
308. C3#10596085semidirect

Maximal subgroups

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

subgroupindex
C33×#1212
#117239792
#117239762
D4×C3⋊(C3⋊Dic3)2
C4×#63122
#117239672
C4⋊C4×C3⋊(C3⋊D3)2
#117239312
#117239222
#117239202
#105960853