Finite Group Databasebeta

← #10595992⋊C3 #13152515

Subgroups — #10595992⋊C3 #13152515

Order 1728.

1259 conjugacy classes of subgroups.

The subgroup lattice has 369 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4⋊((C3⋊D3)⋊C3)C211yesmaximal, characteristic
2H(3,𝔽3)×#117C211yesmaximal, characteristic, complemented
3#64135⋊C3C211yesmaximal, characteristic, complemented
4#64102⋊C3C214yesmaximal, complemented
5C22⋊C4×(C3⋊D3)⋊C3C214yesmaximal, complemented
6C4×C4⋊((C3⋊D3)⋊C3)C214yesmaximal, complemented
7#10595992C311yesmaximal, characteristic, complemented
8C3×#5521—31nomaximal
9C23×(C3⋊D3)⋊C3C2212yes—
10C22×C4×H(3,𝔽3)C2211yescharacteristic
11C2×C4⋊((C3⋊D3)⋊C3)C418yescomplemented
12C2×C4⋊((C3⋊D3)⋊C3)C2212yes—
13C2×C4⋊((C3⋊D3)⋊C3)C2212yes—
14C2×C4×(C3⋊D3)⋊C3C2218yescomplemented
15C4⋊C4×H(3,𝔽3)C2212yescomplemented
16C22⋊C4×H(3,𝔽3)C2212yes—
17C42×H(3,𝔽3)C2212yescomplemented
18#3757⋊C3C2212yes—
19#3752⋊C3C2218yescomplemented
20C4⋊((C3⋊Dic3)⋊C3)C2212yescomplemented
21C4×(C3⋊Dic3)⋊C3C2212yescomplemented
22C22×C4⋊(C3⋊D3)C2×C311yescharacteristic
23C22×C3×D12—31no—
24C32×#117—21no—
25C32×#117D311yescharacteristic, complemented
26C32×#117C2×C311yescharacteristic, complemented
27#64135C2×C311yescharacteristic, complemented
28#64102C2×C314yescomplemented
29C22⋊C4×C3⋊D3C2×C314yescomplemented
30C4×C4⋊(C3⋊D3)C2×C314yescomplemented
31C3×#926—31no—
32C3×#893—34no—
33C3×D3×C22⋊C4—34no—
34C3×C4×D12—34no—
35C23×H(3,𝔽3)C2311yescharacteristic
36C22×(C3⋊D3)⋊C3—44no—
37C22×(C3⋊D3)⋊C3C2×C418yescomplemented
38C22×(C3⋊D3)⋊C3C2314yes—
39C2×C4×H(3,𝔽3)C2×C414yescomplemented
40C2×C4×H(3,𝔽3)C2314yes—
41C2×C4×H(3,𝔽3)C2311yescharacteristic
42C2×C4×H(3,𝔽3)C2311yescharacteristic
43C2×(C3⋊Dic3)⋊C3C2314yes—
44C4⋊((C3⋊D3)⋊C3)C2×C418yescomplemented
45C4⋊((C3⋊D3)⋊C3)C2×C418yescomplemented
46C4×(C3⋊D3)⋊C3—48no—
47C3×#518—91no—
48#5521—31no—
49#5521—91no—
50C23×C3⋊D3C22×C312yes—
51C23×C3×D3—32no—
52C22×C32×C4—21no—
53C22×C32×C4C2×D311yescharacteristic
54C22×C32×C4C22×C311yescharacteristic
55C2×C4⋊(C3⋊D3)C3×C418yescomplemented
56C2×C4⋊(C3⋊D3)C22×C312yes—
57C2×C4⋊(C3⋊D3)C22×C312yes—
58C2×C4×C3⋊D3C22×C318yescomplemented
59C2×C3×D12—38no—
60C2×C3×D12—32no—
61C2×C3×D12—32no—
62C2×C3×C4×D3—38no—
63C32×C4⋊C4—22no—
64C32×C4⋊C4C2×D312yescomplemented
65C32×C4⋊C4C22×C312yescomplemented
66C32×C22⋊C4—22no—
67C32×C22⋊C4C2×D312yes—
68C32×C22⋊C4C22×C312yes—
69C32×C42—22no—
70C32×C42C2×D312yescomplemented
71C32×C42C22×C312yescomplemented
72#3757C22×C312yes—
73#3752C22×C318yescomplemented
74C4⋊(C3⋊Dic3)C22×C312yescomplemented
75C4×C3⋊Dic3C22×C312yescomplemented
76C3×#217—32no—
77C3×(C2×D3)⋊C4—38no—
78C3×C4⋊Dic3—32no—
79C3×C4×Dic3—32no—
80C22×H(3,𝔽3)—22no—
81C22×H(3,𝔽3)C22×C412yes—
82C22×H(3,𝔽3)C2411yescharacteristic
83C2×(C3⋊D3)⋊C3—28no—
84C2×(C3⋊D3)⋊C3—48no—
85C2×(C3⋊D3)⋊C3C22×C418yes—
86C4×H(3,𝔽3)—24no—
87C4×H(3,𝔽3)C22×C412yes—
88C4×H(3,𝔽3)C22×C412yes—
89(C3⋊Dic3)⋊C3—24no—
90C22×C3×D4—91no—
91C22×D12—31no—
92C22×D12—91no—
93C3×C4×D4—94no—
94C3×C4×D4—94no—
95C3×#117—31no—
96C3×#117—31no—
97C3×#117—61no—
98C3×#117—91no—
99C3×#117C3×D311yescharacteristic
100C2×C3×C22⋊C4—94no—
101#926—31no—
102#926—91no—
103#893—34no—
104#893—94no—
105D3×C22⋊C4—34no—
106D3×C22⋊C4—94no—
107C4×D12—34no—
108C4×D12—94no—
109C23×C32—21no—
110C23×C32C22×D311yescharacteristic
111C23×C32C23×C311yescharacteristic
112C22×C3⋊D3—44no—
113C22×C3⋊D3C2×C3×C418yescomplemented
114C22×C3⋊D3C23×C314yes—
115C22×C3×D3—34no—
116C22×C3×D3—38no—
117C22×C3×D3—124no—
118C2×C32×C4—24no—
119C2×C32×C4—21no—
120C2×C32×C4—21no—
121C2×C32×C4—24no—
122C2×C32×C4C4×D314yescomplemented
123C2×C32×C4C2×C3×C414yescomplemented
124C2×C32×C4C22×D314yes—
125C2×C32×C4C22×D311yescharacteristic
126C2×C32×C4C22×D311yescharacteristic
127C2×C32×C4C23×C314yes—
128C2×C32×C4C23×C311yescharacteristic
129C2×C32×C4C23×C311yescharacteristic
130C2×C3⋊Dic3C23×C314yes—
131C4⋊(C3⋊D3)C2×C3×C418yescomplemented
132C4⋊(C3⋊D3)C2×C3×C418yescomplemented
133C4×C3⋊D3—48no—
134C2×C3×Dic3—34no—
135C3×D12—38no—
136C3×D12—38no—
137C3×C4×D3—128no—
138#518—91noSylow
139C2×H(3,𝔽3)—22no—
140C2×H(3,𝔽3)C23×C411yescharacteristic
141C2×H(3,𝔽3)H(5,𝔽2)12yes—
142(C3⋊D3)⋊C3—28no—
143C24×C3—92no—
144C23×D3—32no—
145C23×D3—92no—
146C2×C3×D4—92no—
147C2×C3×D4—92no—
148C2×C3×D4—98no—
149C22×C3×C4—31no—
150C22×C3×C4—31no—
151C22×C3×C4—61no—
152C22×C3×C4—98no—
153C22×C3×C4C2×C3×D311yescharacteristic
154C2×D12—38no—
155C2×D12—32no—
156C2×D12—32no—
157C2×D12—92no—
158C2×D12—92no—
159C2×D12—98no—
160C2×C4×D3—38no—
161C2×C4×D3—98no—
162C3×C4⋊C4—32no—
163C3×C4⋊C4—32no—
164C3×C4⋊C4—62no—
165C3×C4⋊C4—92no—
166C3×C4⋊C4C2×C3×D312yes—
167C3×C22⋊C4—32no—
168C3×C22⋊C4—32no—
169C3×C22⋊C4—62no—
170C3×C22⋊C4—98no—
171C3×C22⋊C4—92no—
172C3×C22⋊C4C2×C3×D312yes—
173C3×C42—32no—
174C3×C42—32no—
175C3×C42—62no—
176C3×C42—92no—
177C3×C42C2×C3×D312yes—
178#217—32no—
179#217—92no—
180(C2×D3)⋊C4—38no—
181(C2×D3)⋊C4—98no—
182C4⋊Dic3—32no—
183C4⋊Dic3—92no—
184C4×Dic3—32no—
185C4×Dic3—92no—
186C22×C32—22no—
187C22×C32—22no—
188C22×C32—21no—
189C22×C32—22no—
190C22×C32—42no—
191C22×C32C2×C4×D312yes—
192C22×C32C22×C3×C412yes—
193C22×C32C23×D311yescharacteristic
194C22×C32C24×C311yescharacteristic
195C2×C3⋊D3—28no—
196C2×C3⋊D3—48no—
197C2×C3⋊D3C22×C3×C418yes—
198C2×C3×D3—38no—
199C2×C3×D3—68no—
200C2×C3×D3—128no—
201C32×C4—24no—
202C32×C4—24no—
203C32×C4—22no—
204C32×C4—22no—
205C32×C4—44no—
206C32×C4C2×C4×D312yes—
207C32×C4C2×C4×D312yes—
208C32×C4C22×C3×C412yes—
209C32×C4C22×C3×C412yes—
210C3⋊Dic3—24no—
211C3×Dic3—64no—
212C22×D4—91no—
213C4×D4—94no—
214C4×D4—94no—
215#117—91no—
216#117(C3⋊D3)⋊C311yescharacteristic, complemented
217C2×C22⋊C4—94no—
218H(3,𝔽3)#51811yescharacteristic, complemented, Sylow
219C23×C3—31no—
220C23×C3—31no—
221C23×C3—61no—
222C23×C3—94no—
223C23×C3—98no—
224C23×C3—364no—
225C23×C3C22×C3×D311yescharacteristic
226C22×D3—34no—
227C22×D3—38no—
228C22×D3—94no—
229C22×D3—98no—
230C22×D3—124no—
231C22×D3—364no—
232C3×D4—98no—
233C3×D4—98no—
234C2×C3×C4—34no—
235C2×C3×C4—34no—
236C2×C3×C4—31no—
237C2×C3×C4—31no—
238C2×C3×C4—31no—
239C2×C3×C4—31no—
240C2×C3×C4—34no—
241C2×C3×C4—34no—
242C2×C3×C4—64no—
243C2×C3×C4—61no—
244C2×C3×C4—61no—
245C2×C3×C4—64no—
246C2×C3×C4—94no—
247C2×C3×C4—368no—
248C2×C3×C4C3×C4×D314yes—
249C2×C3×C4C22×C3×D314yes—
250C2×C3×C4C22×C3×D311yescharacteristic
251C2×C3×C4C22×C3×D311yescharacteristic
252C2×Dic3—34no—
253C2×Dic3—94no—
254D12—38no—
255D12—38no—
256D12—98no—
257D12—98no—
258C4×D3—128no—
259C4×D3—368no—
260C2×C32—22no—
261C2×C32—22no—
262C2×C32—21no—
263C2×C32—22no—
264C2×C32—42no—
265C2×C32C22×C4×D311yescharacteristic
266C2×C32#100912yes—
267C2×C32C23×C3×C411yescharacteristic
268C2×C32C3×H(5,𝔽2)12yes—
269C3⋊D3—28no—
270C3×D3—68no—
271C24—92no—
272C2×D4—98no—
273C2×D4—92no—
274C2×D4—92no—
275C22×C4—98no—
276C22×C4C2×(C3⋊D3)⋊C311yescharacteristic
277C4⋊C4—92no—
278C4⋊C4C2×(C3⋊D3)⋊C312yescomplemented
279C22⋊C4—92no—
280C22⋊C4—98no—
281C22⋊C4C2×(C3⋊D3)⋊C312yes—
282C42—92no—
283C42C2×(C3⋊D3)⋊C312yescomplemented
284C22×C3—22no—
285C22×C3—31no—
286C22×C3—31no—
287C22×C3—32no—
288C22×C3—32no—
289C22×C3—62no—
290C22×C3—62no—
291C22×C3—61no—
292C22×C3—62no—
293C22×C3—98no—
294C22×C3—122no—
295C22×C3—188no—
296C22×C3—368no—
297C22×C3C2×C3×C4×D312yes—
298C22×C3C23×C3×D311yescharacteristic
299C2×D3—38no—
300C2×D3—68no—
301C2×D3—98no—
302C2×D3—128no—
303C2×D3—188no—
304C2×D3—368no—
305C3×C4—24no—
306C3×C4—32no—
307C3×C4—32no—
308C3×C4—32no—
309C3×C4—32no—
310C3×C4—64no—
311C3×C4—64no—
312C3×C4—62no—
313C3×C4—62no—
314C3×C4—124no—
315C3×C4—184no—
316C3×C4C2×C3×C4×D312yes—
317C3×C4C2×C3×C4×D312yes—
318Dic3—64no—
319Dic3—184no—
320C32—21no—
321C32#552111yescharacteristic, complemented
322C32C3×#51811yescharacteristic, complemented
323C23—94no—
324C23—98no—
325C23—364no—
326C23C22×(C3⋊D3)⋊C311yescharacteristic
327D4—98no—
328D4—98no—
329C2×C4—94no—
330C2×C4—368no—
331C2×C4C4×(C3⋊D3)⋊C314yescomplemented
332C2×C4C22×(C3⋊D3)⋊C314yes—
333C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
334C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
335C2×C3—22no—
336C2×C3—31no—
337C2×C3—31no—
338C2×C3—32no—
339C2×C3—32no—
340C2×C3—62no—
341C2×C3—62no—
342C2×C3—61no—
343C2×C3—62no—
344C2×C3—122no—
345C2×C3—188no—
346C2×C3C22×C3×C4×D311yescharacteristic
347C2×C3C3×#100912yes—
348D3—68no—
349D3—188no—
350C22—22no—
351C22—98no—
352C22—188no—
353C22—368no—
354C22C2×C4×(C3⋊D3)⋊C312yes—
355C22C23×(C3⋊D3)⋊C311yescharacteristic
356C4—24no—
357C4—184no—
358C4C2×C4×(C3⋊D3)⋊C312yes—
359C4C2×C4×(C3⋊D3)⋊C312yes—
360C3—31no—
361C3—31no—
362C3—61no—
363C3C3×#552111yescharacteristic
364C2—22no—
365C2—188no—
366C2C22×C4×(C3⋊D3)⋊C311yescharacteristic
367C2#1172263812yes—
subgroupG/H
centerC22C23×(C3⋊D3)⋊C3
commutatorC2×C32C23×C3×C4
Frattini, socleC22×C3C23×C3×D3
FittingH(3,𝔽3)×#117C2
prime radicalC22×C4⋊((C3⋊D3)⋊C3)C2
cosocleC22×C32C24×C3
hypercenter#117(C3⋊D3)⋊C3
hypocenterC32C3×#518
LCSG ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ #117

Complements

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

NH ≅ G/Ntype
2. H(3,𝔽3)×#117C2semidirect
3. #64135⋊C3C2semidirect
4. #64102⋊C3C2semidirect
5. C22⋊C4×(C3⋊D3)⋊C3C2semidirect
6. C4×C4⋊((C3⋊D3)⋊C3)C2semidirect
7. #10595992C3semidirect
11. C2×C4⋊((C3⋊D3)⋊C3)C4semidirect
14. C2×C4×(C3⋊D3)⋊C3C22semidirect
15. C4⋊C4×H(3,𝔽3)C22semidirect
17. C42×H(3,𝔽3)C22semidirect
19. #3752⋊C3C22semidirect
20. C4⋊((C3⋊Dic3)⋊C3)C22semidirect
21. C4×(C3⋊Dic3)⋊C3C22semidirect
25. C32×#117D3semidirect
26. C32×#117C2×C3semidirect
27. #64135C2×C3semidirect
28. #64102C2×C3semidirect
29. C22⋊C4×C3⋊D3C2×C3semidirect
30. C4×C4⋊(C3⋊D3)C2×C3semidirect
37. C22×(C3⋊D3)⋊C3C2×C4semidirect
39. C2×C4×H(3,𝔽3)C2×C4semidirect
44. C4⋊((C3⋊D3)⋊C3)C2×C4semidirect
45. C4⋊((C3⋊D3)⋊C3)C2×C4semidirect
55. C2×C4⋊(C3⋊D3)C3×C4semidirect
58. C2×C4×C3⋊D3C22×C3semidirect
64. C32×C4⋊C4C2×D3semidirect
65. C32×C4⋊C4C22×C3semidirect
70. C32×C42C2×D3semidirect
71. C32×C42C22×C3semidirect
73. #3752C22×C3semidirect
74. C4⋊(C3⋊Dic3)C22×C3semidirect
75. C4×C3⋊Dic3C22×C3semidirect
113. C22×C3⋊D3C2×C3×C4semidirect
122. C2×C32×C4C4×D3semidirect
123. C2×C32×C4C2×C3×C4semidirect
131. C4⋊(C3⋊D3)C2×C3×C4semidirect
132. C4⋊(C3⋊D3)C2×C3×C4semidirect
216. #117(C3⋊D3)⋊C3semidirect
218. H(3,𝔽3)#518semidirect
278. C4⋊C4C2×(C3⋊D3)⋊C3semidirect
283. C42C2×(C3⋊D3)⋊C3semidirect
321. C32#5521semidirect
322. C32C3×#518semidirect
331. C2×C4C4×(C3⋊D3)⋊C3semidirect

Maximal subgroups

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

subgroupindex
C22×C4⋊((C3⋊D3)⋊C3)2
H(3,𝔽3)×#1172
#64135⋊C32
#64102⋊C32
C22⋊C4×(C3⋊D3)⋊C32
C4×C4⋊((C3⋊D3)⋊C3)2
#105959923
C3×#55213