Finite Group Databasebeta

← #10596168⋊C3 #13153187

Subgroups — #10596168⋊C3 #13153187

Order 1728.

896 conjugacy classes of subgroups.

The subgroup lattice has 371 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1H(3,𝔽3)×#125C211yesmaximal, characteristic, complemented
2#64104⋊C3C212yesmaximal, complemented
3#64104⋊C3C214yesmaximal, complemented
4#64103⋊C3C214yesmaximal, complemented
5#64102⋊C3C212yesmaximal, complemented
6#11721385C211yesmaximal, characteristic, complemented
7#64081⋊C3C211yesmaximal, characteristic, complemented
8#10596168C311yesmaximal, characteristic, complemented
9C3×#5697—31nomaximal
10C2×C4⋊((C3⋊D3)⋊C3)C2212yes—
11C2×C4⋊((C3⋊D3)⋊C3)C2214yes—
12C2×C4⋊((C3⋊D3)⋊C3)C2214yes—
13C2×C4×(C3⋊D3)⋊C3C2211yescharacteristic
14C2×C4×(C3⋊D3)⋊C3C2214yes—
15C4⋊C4×H(3,𝔽3)C2212yes—
16C4⋊C4×H(3,𝔽3)C2214yes—
17C42×H(3,𝔽3)C2211yescharacteristic
18#3752⋊C3C2212yes—
19#3752⋊C3C2214yes—
20#3752⋊C3C2214yes—
21#3750⋊C3C2212yes—
22C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
23C32×#125—21no—
24C32×#125D311yescharacteristic, complemented
25C32×#125C2×C311yescharacteristic, complemented
26#64104C2×C314yescomplemented
27#64104C2×C312yescomplemented
28#64103C2×C314yescomplemented
29#64102C2×C312yescomplemented
30#64083C2×C311yescharacteristic, complemented
31#64081C2×C311yescharacteristic, complemented
32C3×#895—34no—
33C3×#895—32no—
34C3×#894—34no—
35C3×#893—32no—
36C3×#874—31no—
37C3×#872—31no—
38C22×(C3⋊D3)⋊C3C2311yescharacteristic
39C22×(C3⋊D3)⋊C3C2314yes—
40C2×C4×H(3,𝔽3)C2314yes—
41C2×C4×H(3,𝔽3)C2312yes—
42C2×C4×H(3,𝔽3)C2311yescharacteristic
43C2×(C3⋊Dic3)⋊C3C2312yes—
44C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
45C4⋊((C3⋊D3)⋊C3)—24no—
46C4⋊((C3⋊D3)⋊C3)—44no—
47C4⋊((C3⋊D3)⋊C3)—44no—
48C4×(C3⋊D3)⋊C3—22no—
49C4×(C3⋊D3)⋊C3—44no—
50C3×#540—91no—
51#5697—31no—
52#5697—91no—
53C2×C4⋊(C3⋊D3)C22×C312yes—
54C2×C4⋊(C3⋊D3)C22×C314yes—
55C2×C4⋊(C3⋊D3)C22×C314yes—
56C2×C4×C3⋊D3C22×C311yescharacteristic
57C2×C4×C3⋊D3C22×C314yes—
58C2×C3×D12—32no—
59C2×C3×D12—34no—
60C2×C3×D12—34no—
61C2×C3×C4×D3—31no—
62C2×C3×C4×D3—34no—
63C32×C4⋊C4—22no—
64C32×C4⋊C4—24no—
65C32×C4⋊C4C2×D312yes—
66C32×C4⋊C4C2×D314yes—
67C32×C4⋊C4C22×C312yes—
68C32×C4⋊C4C22×C314yes—
69C32×C42—21no—
70C32×C42C2×D311yescharacteristic
71C32×C42C22×C311yescharacteristic
72#3752C22×C312yes—
73#3752C22×C314yes—
74#3752C22×C314yes—
75#3750C22×C312yes—
76C4×C3⋊Dic3C22×C311yescharacteristic
77C3×(C2×D3)⋊C4—34no—
78C3×(C2×D3)⋊C4—34no—
79C3×(C2×D3)⋊C4—32no—
80C3×Dic3⋊C4—32no—
81C3×C4×Dic3—31no—
82C22×H(3,𝔽3)C2411yescharacteristic
83C2×(C3⋊D3)⋊C3—21no—
84C2×(C3⋊D3)⋊C3—24no—
85C2×(C3⋊D3)⋊C3—24no—
86C2×(C3⋊D3)⋊C3—22no—
87C2×(C3⋊D3)⋊C3—24no—
88C4×H(3,𝔽3)—24no—
89C4×H(3,𝔽3)—22no—
90C4×H(3,𝔽3)#4112yes—
91(C3⋊Dic3)⋊C3—21no—
92(C3⋊Dic3)⋊C3—22no—
93C3×#127—91no—
94C3×#125—31no—
95C3×#125—31no—
96C3×#125—61no—
97C3×#125C3×D311yescharacteristic
98C3×#123—94no—
99C3×#121—94no—
100C3×#121—92no—
101C3×C4×D4—92no—
102C3×#117—91no—
103#895—34no—
104#895—32no—
105#895—94no—
106#895—92no—
107#894—34no—
108#894—94no—
109#893—32no—
110#893—92no—
111#874—31no—
112#874—91no—
113#872—31no—
114#872—91no—
115C22×C3⋊D3C23×C311yescharacteristic
116C22×C3⋊D3C23×C314yes—
117C22×C3×D3—31no—
118C22×C3×D3—34no—
119C2×C32×C4—21no—
120C2×C32×C4—22no—
121C2×C32×C4—24no—
122C2×C32×C4C22×D312yes—
123C2×C32×C4C22×D314yes—
124C2×C32×C4C22×D311yescharacteristic
125C2×C32×C4C23×C312yes—
126C2×C32×C4C23×C314yes—
127C2×C32×C4C23×C311yescharacteristic
128C2×C3⋊Dic3C23×C312yes—
129C2×C3⋊Dic3C23×C311yescharacteristic
130C4⋊(C3⋊D3)—24no—
131C4⋊(C3⋊D3)—44no—
132C4⋊(C3⋊D3)—44no—
133C4×C3⋊D3—22no—
134C4×C3⋊D3—44no—
135C2×C3×Dic3—32no—
136C2×C3×Dic3—31no—
137C3×D12—64no—
138C3×D12—124no—
139C3×D12—124no—
140C3×C4×D3—62no—
141C3×C4×D3—124no—
142#540—91noSylow
143C2×H(3,𝔽3)C2×#4111yescharacteristic
144C2×H(3,𝔽3)H(5,𝔽2)12yes—
145(C3⋊D3)⋊C3—41no—
146(C3⋊D3)⋊C3—44no—
147C2×C3×D4—92no—
148C2×C3×D4—94no—
149C2×C3×D4—94no—
150C22×C3×C4—91no—
151C22×C3×C4—94no—
152C2×D12—32no—
153C2×D12—34no—
154C2×D12—34no—
155C2×D12—92no—
156C2×D12—94no—
157C2×D12—94no—
158C2×C4×D3—31no—
159C2×C4×D3—34no—
160C2×C4×D3—91no—
161C2×C4×D3—94no—
162C3×C4⋊C4—32no—
163C3×C4⋊C4—34no—
164C3×C4⋊C4—32no—
165C3×C4⋊C4—34no—
166C3×C4⋊C4—62no—
167C3×C4⋊C4—64no—
168C3×C4⋊C4—92no—
169C3×C4⋊C4C2×C3×D312yes—
170C3×C4⋊C4C2×C3×D314yes—
171C3×C22⋊C4—92no—
172C3×C22⋊C4—94no—
173C3×C22⋊C4—94no—
174C3×C42—31no—
175C3×C42—31no—
176C3×C42—61no—
177C3×C42—91no—
178C3×C42C2×C3×D311yescharacteristic
179(C2×D3)⋊C4—32no—
180(C2×D3)⋊C4—34no—
181(C2×D3)⋊C4—34no—
182(C2×D3)⋊C4—92no—
183(C2×D3)⋊C4—94no—
184(C2×D3)⋊C4—94no—
185Dic3⋊C4—32no—
186Dic3⋊C4—92no—
187C4×Dic3—31no—
188C4×Dic3—91no—
189C22×C32—21no—
190C22×C32C23×D311yescharacteristic
191C22×C32C24×C311yescharacteristic
192C2×C3⋊D3—21no—
193C2×C3⋊D3—24no—
194C2×C3⋊D3—24no—
195C2×C3⋊D3—22no—
196C2×C3⋊D3—24no—
197C2×C3×D3—61no—
198C2×C3×D3—64no—
199C2×C3×D3—64no—
200C2×C3×D3—62no—
201C2×C3×D3—64no—
202C32×C4—24no—
203C32×C4—22no—
204C32×C4—24no—
205C32×C4—22no—
206C32×C4—22no—
207C32×C4—44no—
208C32×C4—42no—
209C32×C4#23912yes—
210C32×C4C3×#4112yes—
211C3⋊Dic3—21no—
212C3⋊Dic3—22no—
213C3×Dic3—61no—
214C3×Dic3—62no—
215#127—91no—
216#125(C3⋊D3)⋊C311yescharacteristic, complemented
217#123—94no—
218#121—94no—
219#121—92no—
220C4×D4—92no—
221#117—91no—
222H(3,𝔽3)#54011yescharacteristic, complemented, Sylow
223C23×C3—91no—
224C23×C3—94no—
225C22×D3—31no—
226C22×D3—34no—
227C22×D3—91no—
228C22×D3—94no—
229C3×D4—184no—
230C3×D4—364no—
231C3×D4—364no—
232C2×C3×C4—31no—
233C2×C3×C4—31no—
234C2×C3×C4—32no—
235C2×C3×C4—34no—
236C2×C3×C4—32no—
237C2×C3×C4—34no—
238C2×C3×C4—61no—
239C2×C3×C4—62no—
240C2×C3×C4—64no—
241C2×C3×C4—92no—
242C2×C3×C4—91no—
243C2×C3×C4—182no—
244C2×C3×C4—364no—
245C2×C3×C4C22×C3×D312yes—
246C2×C3×C4C22×C3×D314yes—
247C2×C3×C4C22×C3×D311yescharacteristic
248C2×Dic3—32no—
249C2×Dic3—31no—
250C2×Dic3—92no—
251C2×Dic3—91no—
252D12—64no—
253D12—124no—
254D12—124no—
255D12—184no—
256D12—364no—
257D12—364no—
258C4×D3—62no—
259C4×D3—124no—
260C4×D3—182no—
261C4×D3—364no—
262C2×C32—21no—
263C2×C32—22no—
264C2×C32C2×#23911yescharacteristic
265C2×C32#100912yes—
266C2×C32C2×C3×#4111yescharacteristic
267C2×C32C3×H(5,𝔽2)12yes—
268C3⋊D3—41no—
269C3⋊D3—44no—
270C3×D3—121no—
271C3×D3—124no—
272C2×D4—92no—
273C2×D4—94no—
274C2×D4—94no—
275C22×C4—91no—
276C22×C4—94no—
277C4⋊C4—92no—
278C4⋊C4C2×(C3⋊D3)⋊C312yes—
279C4⋊C4C2×(C3⋊D3)⋊C314yes—
280C22⋊C4—92no—
281C22⋊C4—94no—
282C22⋊C4—94no—
283C42—91no—
284C42C2×(C3⋊D3)⋊C311yescharacteristic
285C22×C3—31no—
286C22×C3—31no—
287C22×C3—61no—
288C22×C3—181no—
289C22×C3—184no—
290C22×C3—184no—
291C22×C3—182no—
292C22×C3—184no—
293C22×C3C23×C3×D311yescharacteristic
294C2×D3—61no—
295C2×D3—64no—
296C2×D3—64no—
297C2×D3—62no—
298C2×D3—64no—
299C2×D3—181no—
300C2×D3—184no—
301C2×D3—184no—
302C2×D3—182no—
303C2×D3—184no—
304C3×C4—24no—
305C3×C4—22no—
306C3×C4—32no—
307C3×C4—32no—
308C3×C4—64no—
309C3×C4—62no—
310C3×C4—64no—
311C3×C4—62no—
312C3×C4—62no—
313C3×C4—124no—
314C3×C4—122no—
315C3×C4—181no—
316C3×C4—182no—
317C3×C4C3×#23912yes—
318Dic3—61no—
319Dic3—62no—
320Dic3—181no—
321Dic3—182no—
322C32—21no—
323C32#569711yescharacteristic, complemented
324C32C3×#54011yescharacteristic, complemented
325C23—91no—
326C23—94no—
327D4—184no—
328D4—364no—
329D4—364no—
330C2×C4—92no—
331C2×C4—91no—
332C2×C4—182no—
333C2×C4—364no—
334C2×C4C22×(C3⋊D3)⋊C312yes—
335C2×C4C22×(C3⋊D3)⋊C314yes—
336C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
337C2×C3—31no—
338C2×C3—31no—
339C2×C3—32no—
340C2×C3—32no—
341C2×C3—61no—
342C2×C3—62no—
343C2×C3—361no—
344C2×C3—364no—
345C2×C3C2×C3×#23911yescharacteristic
346C2×C3C3×#100912yes—
347D3—121no—
348D3—124no—
349D3—361no—
350D3—364no—
351C22—181no—
352C22—184no—
353C22—184no—
354C22—182no—
355C22—184no—
356C22C23×(C3⋊D3)⋊C311yescharacteristic
357C4—24no—
358C4—22no—
359C4—181no—
360C4—182no—
361C4#8872212yes—
362C3—31no—
363C3—31no—
364C3—61no—
365C3C3×#569711yescharacteristic
366C2—361no—
367C2—364no—
368C2C2×#8872211yescharacteristic
369C2#1172263812yes—
subgroupG/H
centerC22C23×(C3⋊D3)⋊C3
commutator, cosocleC22×C32C24×C3
Frattini, socleC22×C3C23×C3×D3
FittingH(3,𝔽3)×#125C2
hypercenter#125(C3⋊D3)⋊C3
hypocenterC32C3×#540
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ #125

Complements

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

NH ≅ G/Ntype
1. H(3,𝔽3)×#125C2semidirect
2. #64104⋊C3C2semidirect
3. #64104⋊C3C2semidirect
4. #64103⋊C3C2semidirect
5. #64102⋊C3C2semidirect
6. #11721385C2semidirect
7. #64081⋊C3C2semidirect
8. #10596168C3semidirect
24. C32×#125D3semidirect
25. C32×#125C2×C3semidirect
26. #64104C2×C3semidirect
27. #64104C2×C3semidirect
28. #64103C2×C3semidirect
29. #64102C2×C3semidirect
30. #64083C2×C3semidirect
31. #64081C2×C3semidirect
216. #125(C3⋊D3)⋊C3semidirect
222. H(3,𝔽3)#540semidirect
323. C32#5697semidirect
324. C32C3×#540semidirect

Maximal subgroups

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

subgroupindex
H(3,𝔽3)×#1252
#64104⋊C32
#64103⋊C32
#64102⋊C32
#117213852
#64081⋊C32
#105961683
C3×#56973