Finite Group Databasebeta

← #13179684

Subgroups — #13179684

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×C4×D4C211yesmaximal, characteristic, complemented
2#11723979C211yesmaximal, characteristic, complemented
3#11723979C211yesmaximal, characteristic, complemented
4#11723970C212yesmaximal, complemented
5C4×#6312C212yesmaximal, complemented
6#11723933C211yesmaximal, characteristic, complemented
7C4⋊C4×C3⋊(C3⋊D3)C211yesmaximal, characteristic, complemented
8#11723925C212yesmaximal, complemented
9#11723920C212yesmaximal, complemented
10C4×C4⋊(C3⋊(C3⋊D3))C211yesmaximal, characteristic, complemented
11#11723911C211yesmaximal, characteristic, complemented
12#10596029—313nomaximal
13C2×C33×D4C2211yescharacteristic
14C22×C33×C4C2212yes—
15C2×#6312C2212yes—
16C2×#6312C2212yes—
17C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic
18C2×C4×C3⋊(C3⋊D3)C2211yescharacteristic
19C2×C4×C3⋊(C3⋊D3)C2211yescharacteristic
20C2×C4×C3⋊(C3⋊D3)C2212yes—
21C33×C4⋊C4C2211yescharacteristic
22C33×C22⋊C4C2212yes—
23C33×C42C2211yescharacteristic
24#88864C2212yes—
25#88864C2212yes—
26#88859C2211yescharacteristic
27#88859C2211yescharacteristic
28#88859C2212yes—
29C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
30C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
31C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
32#88857C2212yes—
33#88857C2212yes—
34#88857C2212yes—
35C4×C3⋊(C3⋊Dic3)C2212yes—
36C32×C4×D4D3113yescomplemented
37#64147—313no—
38#64147—313no—
39#64138—326no—
40C4×#638—326no—
41#64101—313no—
42C4⋊C4×C3⋊D3—313no—
43#64093—326no—
44#64088—326no—
45C4×C4⋊(C3⋊D3)—313no—
46#64079—313no—
47C23×C33C2312yes—
48C22×C3⋊(C3⋊D3)C2311yescharacteristic
49C22×C3⋊(C3⋊D3)C2311yescharacteristic
50C33×D4—22no—
51C2×C33×C4—42no—
52C2×C33×C4C2312yes—
53C2×C33×C4C2311yescharacteristic
54C2×C33×C4C2311yescharacteristic
55C2×C33×C4C2311yescharacteristic
56#6312—24no—
57#6312—42no—
58C2×C3⋊(C3⋊Dic3)C2311yescharacteristic
59C2×C3⋊(C3⋊Dic3)C2312yes—
60C2×C3⋊(C3⋊Dic3)C2312yes—
61C2×C3⋊(C3⋊Dic3)C2311yescharacteristic
62C4⋊(C3⋊(C3⋊D3))—22no—
63C4×C3⋊(C3⋊D3)—22no—
64C4×C3⋊(C3⋊D3)—22no—
65C4×C3⋊(C3⋊D3)—42no—
66#5558—913no—
67C2×C32×D4C2×D3113yes—
68C22×C32×C4C2×D3126yes—
69C2×#638—326no—
70C2×#638—326no—
71C2×C4⋊(C3⋊D3)—313no—
72C2×C4×C3⋊D3—313no—
73C2×C4×C3⋊D3—313no—
74C2×C4×C3⋊D3—326no—
75C32×C4⋊C4C2×D3113yes—
76C32×C22⋊C4C2×D3126yes—
77C32×C42C2×D3113yes—
78#3757—326no—
79#3757—326no—
80#3752—313no—
81#3752—313no—
82#3752—326no—
83C4⋊(C3⋊Dic3)—313no—
84C4⋊(C3⋊Dic3)—313no—
85C4⋊(C3⋊Dic3)—313no—
86#3750—326no—
87#3750—326no—
88#3750—326no—
89C4×C3⋊Dic3—326no—
90C22×C33—22no—
91C22×C33—22no—
92C22×C33—22no—
93C22×C33C2411yescharacteristic
94C2×C3⋊(C3⋊D3)—21no—
95C2×C3⋊(C3⋊D3)—21no—
96C2×C3⋊(C3⋊D3)—21no—
97C2×C3⋊(C3⋊D3)—21no—
98C2×C3⋊(C3⋊D3)—21no—
99C2×C3⋊(C3⋊D3)#4112yes—
100C33×C4—21no—
101C33×C4—22no—
102C33×C4—21no—
103C33×C4#4112yes—
104C3⋊(C3⋊Dic3)—21no—
105C3⋊(C3⋊Dic3)—21no—
106C3⋊(C3⋊Dic3)—22no—
107C3⋊(C3⋊Dic3)—22no—
108C3×C4×D4C3⋊D3113yescomplemented
109#938—913no—
110#938—913no—
111#929—926no—
112C4×#67—926no—
113#892—913no—
114D3×C4⋊C4—913no—
115#884—926no—
116#879—926no—
117C4×D12—913no—
118#870—913no—
119C23×C32C22×D3126yes—
120C22×C3⋊D3—313no—
121C22×C3⋊D3—313no—
122C32×D4—226no—
123C2×C32×C4—426no—
124C2×C32×C4C22×D3126yes—
125C2×C32×C4C22×D3113yes—
126C2×C32×C4C22×D3113yes—
127C2×C32×C4C22×D3113yes—
128#638—652no—
129#638—1226no—
130C2×C3⋊Dic3—313no—
131C2×C3⋊Dic3—326no—
132C2×C3⋊Dic3—326no—
133C2×C3⋊Dic3—313no—
134C4⋊(C3⋊D3)—626no—
135C4×C3⋊D3—626no—
136C4×C3⋊D3—626no—
137C4×C3⋊D3—1226no—
138#553—271noSylow
139C2×C33—42no—
140C2×C33C2×#4111yescharacteristic
141C2×C33C2×#4111yescharacteristic
142C2×C33H(5,𝔽2)11yescharacteristic
143C3⋊(C3⋊D3)—22no—
144C3⋊(C3⋊D3)—41no—
145C2×C3×D4C2×C3⋊D3113yes—
146C22×C3×C4C2×C3⋊D3126yes—
147C2×#67—926no—
148C2×#67—926no—
149C2×D12—913no—
150C2×C4×D3—913no—
151C2×C4×D3—913no—
152C2×C4×D3—926no—
153C3×C4⋊C4C2×C3⋊D3113yes—
154C3×C22⋊C4C2×C3⋊D3126yes—
155C3×C42C2×C3⋊D3113yes—
156#217—926no—
157#217—926no—
158(C2×D3)⋊C4—913no—
159(C2×D3)⋊C4—913no—
160(C2×D3)⋊C4—926no—
161C4⋊Dic3—913no—
162C4⋊Dic3—913no—
163C4⋊Dic3—913no—
164Dic3⋊C4—926no—
165Dic3⋊C4—926no—
166Dic3⋊C4—926no—
167C4×Dic3—926no—
168C22×C32—226no—
169C22×C32—226no—
170C22×C32—226no—
171C22×C32C23×D3113yes—
172C2×C3⋊D3—326no—
173C2×C3⋊D3—613no—
174C2×C3⋊D3—613no—
175C2×C3⋊D3—613no—
176C2×C3⋊D3—613no—
177C2×C3⋊D3—613no—
178C32×C4—226no—
179C32×C4—213no—
180C32×C4—213no—
181C32×C4#235126yes—
182C3⋊Dic3—613no—
183C3⋊Dic3—613no—
184C3⋊Dic3—626no—
185C3⋊Dic3—626no—
186#126—272no—
187#125—271no—
188#123—272no—
189#121—271no—
190#121—271no—
191#121—272no—
192C4×D4—272no—
193C4×D4—271no—
194C4×D4C3⋊(C3⋊D3)11yescharacteristic, complemented
195#117—271no—
196C2×C4⋊C4—271no—
197C33#55311yescharacteristic, complemented, Sylow
198C23×C3C22×C3⋊D3126yes—
199C22×D3—913no—
200C22×D3—913no—
201C3×D4—226no—
202C2×C3×C4—426no—
203C2×C3×C4C22×C3⋊D3126yes—
204C2×C3×C4C22×C3⋊D3113yes—
205C2×C3×C4C22×C3⋊D3113yes—
206C2×C3×C4C22×C3⋊D3113yes—
207#67—1852no—
208#67—3626no—
209C2×Dic3—913no—
210C2×Dic3—926no—
211C2×Dic3—926no—
212C2×Dic3—913no—
213D12—1826no—
214C4×D3—1826no—
215C4×D3—1826no—
216C4×D3—3626no—
217C2×C32—426no—
218C2×C32C2×#235113yes—
219C2×C32#1004113yes—
220C2×C32D3×#41113yes—
221C3⋊D3—626no—
222C3⋊D3—1213no—
223C2×D4—272no—
224C2×D4—272no—
225C2×D4—271no—
226C2×D4C2×C3⋊(C3⋊D3)11yescharacteristic
227C22×C4—271no—
228C22×C4—271no—
229C22×C4—272no—
230C22×C4C2×C3⋊(C3⋊D3)12yes—
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—272no—
241C22⋊C4—271no—
242C22⋊C4—271no—
243C22⋊C4C2×C3⋊(C3⋊D3)12yes—
244C42—272no—
245C42C2×C3⋊(C3⋊D3)11yescharacteristic
246C22×C3—226no—
247C22×C3—226no—
248C22×C3—226no—
249C22×C3C23×C3⋊D3113yes—
250C2×D3—926no—
251C2×D3—1813no—
252C2×D3—1813no—
253C2×D3—1813no—
254C2×D3—1813no—
255C2×D3—1813no—
256C3×C4—226no—
257C3×C4—213no—
258C3×C4—213no—
259C3×C4#3828126yes—
260Dic3—1813no—
261Dic3—1813no—
262Dic3—1826no—
263Dic3—1826no—
264C32#5558113yescomplemented
265C23—271no—
266C23—271no—
267C23C22×C3⋊(C3⋊D3)12yes—
268D4—22no—
269D4—542no—
270D4—544no—
271D4—1082no—
272C2×C4—42no—
273C2×C4—271no—
274C2×C4—272no—
275C2×C4—271no—
276C2×C4—272no—
277C2×C4—542no—
278C2×C4—542no—
279C2×C4—1082no—
280C2×C4C22×C3⋊(C3⋊D3)12yes—
281C2×C4C22×C3⋊(C3⋊D3)11yescharacteristic
282C2×C4C22×C3⋊(C3⋊D3)11yescharacteristic
283C2×C4C22×C3⋊(C3⋊D3)11yescharacteristic
284C2×C3—426no—
285C2×C3C2×#3828113yes—
286C2×C3#64361113yes—
287C2×C3#41×C3⋊D3113yes—
288D3—1826no—
289D3—3613no—
290C22—22no—
291C22—22no—
292C22—22no—
293C22—272no—
294C22—541no—
295C22—541no—
296C22—541no—
297C22—541no—
298C22—541no—
299C22C23×C3⋊(C3⋊D3)11yescharacteristic
300C4—21no—
301C4—22no—
302C4—21no—
303C4—541no—
304C4—541no—
305C4—542no—
306C4—542no—
307C4#8907612yes—
308C3#10596029113yescomplemented
309C2—42no—
310C2—542no—
311C2—1081no—
312C2C2×#8907611yescharacteristic
313C2#1172467911yescharacteristic
314C2#41×C3⋊(C3⋊D3)11yescharacteristic
subgroupG/H
center, FrattiniC22C23×C3⋊(C3⋊D3)
commutator, socle, cosocleC22×C33C24
FittingC33×C4×D4C2
hypercenterC4×D4C3⋊(C3⋊D3)
hypocenterC33#553
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C4×D4

Complements

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

NH ≅ G/Ntype
1. C33×C4×D4C2semidirect
2. #11723979C2semidirect
3. #11723979C2semidirect
4. #11723970C2semidirect
5. C4×#6312C2semidirect
6. #11723933C2semidirect
7. C4⋊C4×C3⋊(C3⋊D3)C2semidirect
8. #11723925C2semidirect
9. #11723920C2semidirect
10. C4×C4⋊(C3⋊(C3⋊D3))C2semidirect
11. #11723911C2semidirect
36. C32×C4×D4D3semidirect
108. C3×C4×D4C3⋊D3semidirect
194. C4×D4C3⋊(C3⋊D3)semidirect
197. C33#553semidirect
264. C32#5558semidirect
308. C3#10596029semidirect

Maximal subgroups

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

subgroupindex
C33×C4×D42
#117239792
#117239702
C4×#63122
#117239332
C4⋊C4×C3⋊(C3⋊D3)2
#117239252
#117239202
C4×C4⋊(C3⋊(C3⋊D3))2
#117239112
#105960293