Finite Group Databasebeta

← C32×#5834 #13179211

Subgroups — C32×#5834 #13179211

Order 1728.

2528 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C33×#41C211yesmaximal, characteristic, complemented
2C2×C32×#239C211yesmaximal, characteristic, complemented
3C32×Q8×Dic3C211yesmaximal, characteristic, complemented
4C32×#940C213yesmaximal, complemented
5C32×#938C213yesmaximal, complemented
6C32×#930C213yesmaximal, complemented
7C32×C4×#67C213yesmaximal, complemented
8C32×#550—31nomaximal
9C3×#5834C314yesmaximal, direct product factor
10C33×#41—24no—
11C2×C33×Q8C2211yescharacteristic
12C2×C33×D4C2213yes—
13C22×C33×C4C2213yes—
14C2×C32×#67C2213yes—
15C2×C32×#67C2216yes—
16C32×#239—24no—
17C2×C32×D12C2213yes—
18C2×C32×C4×D3C2213yes—
19C32×#217C2213yes—
20C32×(C2×D3)⋊C4C2213yes—
21C32×C4⋊Dic3C2213yes—
22C32×Dic3⋊C4C2211yescharacteristic
23C32×C4×Dic3C2213yes—
24C2×C32×#41—24no—
25C2×C32×#41—31no—
26C2×C32×#41D311yescharacteristic, complemented
27C2×C32×#41C2×C314yescomplemented
28C2×C3×#239C2×C314yescomplemented
29C32×#127—33no—
30C32×#121—33no—
31C32×#121—33no—
32C32×C4×Q8—31no—
33C32×C4×D4—33no—
34C3×Q8×Dic3C2×C314yescomplemented
35C3×#940C2×C3112yescomplemented
36C3×#938C2×C3112yescomplemented
37C3×#930C2×C3112yescomplemented
38C3×C4×#67C2×C3112yescomplemented
39C23×C33C2313yes—
40C22×C32×D3C2313yes—
41C33×Q8D414yescomplemented
42C33×D4—26no—
43C2×C33×C4—26no—
44C2×C33×C4C2311yescharacteristic
45C2×C33×C4C2313yes—
46C32×#67—26no—
47C32×#67—46no—
48C2×C32×Dic3C2311yescharacteristic
49C2×C32×Dic3C2313yes—
50C2×C32×Dic3C2311yescharacteristic
51C32×D12—26no—
52C32×C4×D3—26no—
53C3×#550—34no—
54#5834C3211yescharacteristic, direct product factor
55C32×#41—232no—
56C32×#41—24no—
57C32×#41—216no—
58C32×#41—64no—
59C2×C32×Q8—24no—
60C2×C32×Q8C2×D311yescharacteristic
61C2×C32×Q8C22×C314yes—
62C2×C32×D4—212no—
63C2×C32×D4—33no—
64C2×C32×D4—36no—
65C2×C32×D4—33no—
66C2×C32×D4C2×D313yes—
67C2×C32×D4C22×C3112yes—
68C22×C32×C4—212no—
69C22×C32×C4—33no—
70C22×C32×C4C2×D313yes—
71C22×C32×C4C22×C3112yes—
72C2×C3×#67C22×C3112yes—
73C2×C3×#67C22×C3124yes—
74C3×#239—216no—
75C2×C3×D12C22×C3112yes—
76C2×C3×C4×D3C22×C3112yes—
77C32×C4⋊C4—31no—
78C32×C4⋊C4—33no—
79C32×C22⋊C4—33no—
80C32×C22⋊C4—33no—
81C32×C42—33no—
82C3×#217C22×C3112yes—
83C3×(C2×D3)⋊C4C22×C3112yes—
84C3×C4⋊Dic3C22×C3112yes—
85C3×Dic3⋊C4C22×C314yes—
86C3×C4×Dic3C22×C3112yes—
87C22×C33—23no—
88C22×C33—23no—
89C22×C33—23no—
90C22×C33C2411yescharacteristic
91C2×C32×D3—23no—
92C2×C32×D3—23no—
93C2×C32×D3—23no—
94C33×C4—21no—
95C33×C4C2×D416yes—
96C32×Dic3—21no—
97C32×Dic3—23no—
98C32×Dic3#4112yescomplemented
99C2×C3×#41—24no—
100C2×C3×#41—34no—
101C2×C3×#41C3×D314yescomplemented
102C2×C3×#41C2×C3211yescharacteristic, complemented
103C2×#239C2×C3211yescharacteristic, complemented
104C3×#127—312no—
105C3×#121—312no—
106C3×#121—312no—
107C3×C4×Q8—34no—
108C3×C4×D4—312no—
109Q8×Dic3C2×C3211yescharacteristic, complemented
110#940C2×C3213yescomplemented
111#938C2×C3213yescomplemented
112#930C2×C3213yescomplemented
113C4×#67C2×C3213yescomplemented
114C23×C32—212no—
115C23×C32—33no—
116C23×C32C22×D313yes—
117C23×C32C23×C3112yes—
118C22×C3×D3C23×C3112yes—
119C32×Q8—216no—
120C32×Q8#6714yescomplemented
121C32×Q8C3×D4116yescomplemented
122C32×D4—248no—
123C32×D4—26no—
124C32×D4—224no—
125C32×D4—66no—
126C32×D4—66no—
127C32×D4—126no—
128C2×C32×C4—248no—
129C2×C32×C4—26no—
130C2×C32×C4—224no—
131C2×C32×C4—24no—
132C2×C32×C4—212no—
133C2×C32×C4—31no—
134C2×C32×C4—33no—
135C2×C32×C4—31no—
136C2×C32×C4—66no—
137C2×C32×C4C22×D311yescharacteristic
138C2×C32×C4C22×D313yes—
139C2×C32×C4C23×C314yes—
140C2×C32×C4C23×C3112yes—
141C3×#67—224no—
142C3×#67—424no—
143C2×C3×Dic3C23×C3112yes—
144C2×C3×Dic3C23×C314yes—
145C2×C3×Dic3C23×C314yes—
146C3×D12—224no—
147C3×C4×D3—224no—
148#550—31noSylow
149C2×C33—43no—
150C2×C33C22×D411yescharacteristic
151C2×C33C2×#4111yescharacteristic
152C2×C33H(5,𝔽2)11yescharacteristic
153C32×D3—43no—
154C3×#41—232no—
155C3×#41—216no—
156C3×#41—24no—
157C3×#41—616no—
158C2×C3×Q8—24no—
159C2×C3×Q8C2×C3×D314yes—
160C2×C3×Q8C22×C3211yescharacteristic
161C2×C3×D4—212no—
162C2×C3×D4—312no—
163C2×C3×D4—324no—
164C2×C3×D4—312no—
165C2×C3×D4C2×C3×D3112yes—
166C2×C3×D4C22×C3213yes—
167C22×C3×C4—212no—
168C22×C3×C4—312no—
169C22×C3×C4C2×C3×D3112yes—
170C22×C3×C4C22×C3213yes—
171C2×#67C22×C3213yes—
172C2×#67C22×C3216yes—
173#239—24no—
174C2×D12C22×C3213yes—
175C2×C4×D3C22×C3213yes—
176C3×C4⋊C4—34no—
177C3×C4⋊C4—312no—
178C3×C22⋊C4—312no—
179C3×C22⋊C4—312no—
180C3×C42—312no—
181#217C22×C3213yes—
182(C2×D3)⋊C4C22×C3213yes—
183C4⋊Dic3C22×C3213yes—
184Dic3⋊C4C22×C3211yescharacteristic
185C4×Dic3C22×C3213yes—
186C22×C32—224no—
187C22×C32—23no—
188C22×C32—23no—
189C22×C32—23no—
190C22×C32—212no—
191C22×C32—24no—
192C22×C32—212no—
193C22×C32—212no—
194C22×C32—412no—
195C22×C32—412no—
196C22×C32—63no—
197C22×C32—63no—
198C22×C32—63no—
199C22×C32C23×D311yescharacteristic
200C22×C32C24×C314yes—
201C2×C3×D3—212no—
202C2×C3×D3—212no—
203C2×C3×D3—212no—
204C32×C4—28no—
205C32×C4—21no—
206C32×C4—24no—
207C32×C4—224no—
208C32×C4—32no—
209C32×C4—63no—
210C32×C4—61no—
211C32×C4C2×#6716yes—
212C32×C4C2×C3×D4124yes—
213C3×Dic3—24no—
214C3×Dic3—212no—
215C3×Dic3C3×#4118yescomplemented
216C2×#41—31no—
217C2×#41C32×D311yescharacteristic, complemented
218#127—33no—
219#121—33no—
220#121—33no—
221C4×Q8—31no—
222C4×D4—33no—
223C33#55011yescharacteristic, complemented, Sylow
224C23×C3—212no—
225C23×C3—312no—
226C23×C3C22×C3×D3112yes—
227C23×C3C23×C3213yes—
228C22×D3C23×C3213yes—
229C3×Q8—216no—
230C3×Q8C3×#67116yescomplemented
231C3×Q8C32×D414yescomplemented
232C3×D4—248no—
233C3×D4—224no—
234C3×D4—26no—
235C3×D4—624no—
236C3×D4—624no—
237C3×D4—1224no—
238C2×C3×C4—248no—
239C2×C3×C4—224no—
240C2×C3×C4—26no—
241C2×C3×C4—24no—
242C2×C3×C4—212no—
243C2×C3×C4—34no—
244C2×C3×C4—312no—
245C2×C3×C4—34no—
246C2×C3×C4—624no—
247C2×C3×C4C22×C3×D314yes—
248C2×C3×C4C22×C3×D3112yes—
249C2×C3×C4C23×C3211yescharacteristic
250C2×C3×C4C23×C3213yes—
251#67—26no—
252#67—46no—
253C2×Dic3C23×C3213yes—
254C2×Dic3C23×C3211yescharacteristic
255C2×Dic3C23×C3211yescharacteristic
256D12—26no—
257C4×D3—26no—
258C2×C32—24no—
259C2×C32—24no—
260C2×C32—24no—
261C2×C32—424no—
262C2×C32—43no—
263C2×C32—412no—
264C2×C32—123no—
265C2×C32D3×#4111yescharacteristic
266C2×C32#100911yescharacteristic
267C2×C32C22×#6711yescharacteristic
268C2×C32C22×C3×D414yes—
269C2×C32C2×C3×#4114yes—
270C2×C32C3×H(5,𝔽2)14yes—
271C3×D3—412no—
272#41—24no—
273#41—64no—
274C2×Q8C2×C32×D311yescharacteristic
275C2×D4—33no—
276C2×D4—36no—
277C2×D4—33no—
278C2×D4C2×C32×D313yes—
279C22×C4—33no—
280C22×C4C2×C32×D313yes—
281C4⋊C4—31no—
282C4⋊C4—33no—
283C22⋊C4—33no—
284C22⋊C4—33no—
285C42—33no—
286C22×C3—224no—
287C22×C3—212no—
288C22×C3—212no—
289C22×C3—212no—
290C22×C3—23no—
291C22×C3—23no—
292C22×C3—23no—
293C22×C3—24no—
294C22×C3—412no—
295C22×C3—412no—
296C22×C3—612no—
297C22×C3—612no—
298C22×C3—612no—
299C22×C3C23×C3×D314yes—
300C22×C3C24×C3211yescharacteristic
301C2×D3—23no—
302C2×D3—23no—
303C2×D3—23no—
304C3×C4—28no—
305C3×C4—24no—
306C3×C4—21no—
307C3×C4—224no—
308C3×C4—38no—
309C3×C4—612no—
310C3×C4—64no—
311C3×C4C2×C3×#67124yes—
312C3×C4C2×C32×D416yes—
313Dic3—21no—
314Dic3—23no—
315Dic3C32×#4112yescomplemented
316C32—24no—
317C32#583411yescharacteristic, direct product factor
318C32C3×#55014yescomplemented
319C23—33no—
320C23C22×C32×D313yes—
321Q8C32×#6714yescomplemented
322D4—26no—
323D4—66no—
324D4—66no—
325D4—126no—
326C2×C4—26no—
327C2×C4—31no—
328C2×C4—33no—
329C2×C4—31no—
330C2×C4—66no—
331C2×C4C22×C32×D311yescharacteristic
332C2×C4C22×C32×D313yes—
333C2×C3—24no—
334C2×C3—24no—
335C2×C3—24no—
336C2×C3—424no—
337C2×C3—412no—
338C2×C3—43no—
339C2×C3—1212no—
340C2×C3C3×D3×#4114yes—
341C2×C3C3×#100914yes—
342C2×C3C22×C3×#6714yes—
343C2×C3C22×C32×D411yescharacteristic
344C2×C3C2×C32×#4111yescharacteristic
345C2×C3C32×H(5,𝔽2)11yescharacteristic
346D3—43no—
347C22—23no—
348C22—23no—
349C22—23no—
350C22—63no—
351C22—63no—
352C22—63no—
353C22C23×C32×D311yescharacteristic
354C4—21no—
355C4—32no—
356C4—63no—
357C4—61no—
358C4C2×C32×#6716yes—
359C3—24no—
360C3C3×#583414yesdirect product factor
361C3C32×#55011yescharacteristic, complemented
362C2—43no—
363C2—123no—
364C2C32×D3×#4111yescharacteristic
365C2C32×#100911yescharacteristic
366C2C22×C32×#6711yescharacteristic
subgroupG/H
centerC22×C32C23×D3
commutator, cosocleC22×C3C24×C32
FrattiniC22C23×C32×D3
FittingC2×C33×#41C2
socleC22×C33C24
hypercenterC2×C32×#41D3
hypocenterC3C32×#550
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C22×C32 ◁ C2×C32×#41

Complements

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

NH ≅ G/Ntype
1. C2×C33×#41C2semidirect
2. C2×C32×#239C2semidirect
3. C32×Q8×Dic3C2semidirect
4. C32×#940C2semidirect
5. C32×#938C2semidirect
6. C32×#930C2semidirect
7. C32×C4×#67C2semidirect
9. C3×#5834C3direct (N × H)
26. C2×C32×#41D3semidirect
27. C2×C32×#41C2×C3semidirect
28. C2×C3×#239C2×C3semidirect
34. C3×Q8×Dic3C2×C3semidirect
35. C3×#940C2×C3semidirect
36. C3×#938C2×C3semidirect
37. C3×#930C2×C3semidirect
38. C3×C4×#67C2×C3semidirect
41. C33×Q8D4semidirect
54. #5834C32direct (N × H)
98. C32×Dic3#41semidirect
101. C2×C3×#41C3×D3semidirect
102. C2×C3×#41C2×C32semidirect
103. C2×#239C2×C32semidirect
109. Q8×Dic3C2×C32semidirect
110. #940C2×C32semidirect
111. #938C2×C32semidirect
112. #930C2×C32semidirect
113. C4×#67C2×C32semidirect
120. C32×Q8#67semidirect
121. C32×Q8C3×D4semidirect
215. C3×Dic3C3×#41semidirect
217. C2×#41C32×D3semidirect
223. C33#550semidirect
230. C3×Q8C3×#67semidirect
231. C3×Q8C32×D4semidirect
315. Dic3C32×#41semidirect
317. C32#5834direct (N × H)
318. C32C3×#550semidirect
321. Q8C32×#67semidirect
360. C3C3×#5834direct (N × H)
361. C3C32×#550semidirect

Maximal subgroups

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

subgroupindex
C2×C33×#412
C2×C32×#2392
C32×Q8×Dic32
C32×#9402
C32×#9382
C32×#9302
C32×C4×#672
C32×#5503
C3×#58343