Finite Group Databasebeta

← C32×#5576 #13178953

Subgroups — C32×#5576 #13178953

Order 1728.

2020 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C32×#239C211yesmaximal, characteristic, complemented
2C2×C32×D3×Q8C211yesmaximal, characteristic, complemented
3C33×C4×Q8C211yesmaximal, characteristic, complemented
4C32×#897C213yesmaximal, complemented
5C32×#895C213yesmaximal, complemented
6C32×#875C213yesmaximal, complemented
7C32×C4×D12C213yesmaximal, complemented
8C32×#548—31nomaximal
9C3×#5576C314yesmaximal, direct product factor
10C2×C33×Q8C2211yescharacteristic
11C32×#239—24no—
12C32×D3×Q8—24no—
13C2×C32×D12C2213yes—
14C2×C32×D12C2213yes—
15C2×C32×C4×D3C2213yes—
16C2×C32×C4×D3C2213yes—
17C2×C32×Dic6C2213yes—
18C33×C4⋊C4C2213yes—
19C33×C42C2213yes—
20C32×(C2×D3)⋊C4C2211yescharacteristic
21C32×(C2×D3)⋊C4C2213yes—
22C32×(C2×D3)⋊C4C2216yes—
23C32×C4⋊Dic3C2213yes—
24C2×C32×#41—31no—
25C22×C32×Q8—31no—
26C2×C3×#239C2×C314yescomplemented
27C2×C3×D3×Q8C2×C314yescomplemented
28C32×#124—33no—
29C32×C22⋊Q8—33no—
30C32×#121—33no—
31C32×C4×Q8—24no—
32C32×C4×Q8D311yescharacteristic, complemented
33C32×C4×Q8C2×C314yescomplemented
34C32×C4×D4—33no—
35C3×#897C2×C3112yescomplemented
36C3×#895C2×C3112yescomplemented
37C3×#875C2×C3112yescomplemented
38C3×C4×D12C2×C3112yescomplemented
39C22×C32×D3C2313yes—
40C22×C32×D3C2311yescharacteristic
41C33×Q8D414yescomplemented
42C2×C33×C4C2311yescharacteristic
43C2×C33×C4C2313yes—
44C2×C33×C4C2313yes—
45C2×C32×Dic3C2313yes—
46C2×C32×Dic3C2311yescharacteristic
47C32×D12—26no—
48C32×D12—26no—
49C32×C4×D3—26no—
50C32×C4×D3—26no—
51C32×Dic6—26no—
52C3×#548—34no—
53#5576C3211yescharacteristic, direct product factor
54C32×#41—64no—
55C2×C32×Q8—24no—
56C2×C32×Q8—33no—
57C2×C32×Q8—64no—
58C2×C32×Q8C2×D311yescharacteristic
59C2×C32×Q8C22×C314yes—
60C2×C32×D4—33no—
61C2×C32×D4—33no—
62C22×C32×C4—33no—
63C22×C32×C4—33no—
64C3×#239—216no—
65C3×D3×Q8—216no—
66C2×C3×D12C22×C3112yes—
67C2×C3×D12C22×C3112yes—
68C2×C3×C4×D3C22×C3112yes—
69C2×C3×C4×D3C22×C3112yes—
70C2×C3×Dic6C22×C3112yes—
71C32×C4⋊C4—212no—
72C32×C4⋊C4—33no—
73C32×C4⋊C4C2×D313yes—
74C32×C4⋊C4C22×C3112yes—
75C32×C22⋊C4—31no—
76C32×C22⋊C4—33no—
77C32×C22⋊C4—36no—
78C32×C42—212no—
79C32×C42C2×D313yes—
80C32×C42C22×C3112yes—
81C3×(C2×D3)⋊C4C22×C3124yes—
82C3×(C2×D3)⋊C4C22×C314yes—
83C3×(C2×D3)⋊C4C22×C3112yes—
84C3×C4⋊Dic3C22×C3112yes—
85C22×C33C2411yescharacteristic
86C2×C32×D3—21no—
87C2×C32×D3—21no—
88C2×C32×D3—23no—
89C2×C32×D3—23no—
90C2×C32×D3—23no—
91C2×C32×D3#4112yescomplemented
92C33×C4—23no—
93C33×C4—21no—
94C33×C4C2×D416yes—
95C32×Dic3—23no—
96C32×Dic3—21no—
97C2×C3×#41—34no—
98C22×C3×Q8—34no—
99C2×#239C2×C3211yescharacteristic, complemented
100C2×D3×Q8C2×C3211yescharacteristic, complemented
101C3×#124—312no—
102C3×C22⋊Q8—312no—
103C3×#121—312no—
104C3×C4×Q8—24no—
105C3×C4×Q8C3×D314yescomplemented
106C3×C4×Q8C2×C3211yescharacteristic, complemented
107C3×C4×D4—312no—
108#897C2×C3213yescomplemented
109#895C2×C3213yescomplemented
110#875C2×C3213yescomplemented
111C4×D12C2×C3213yescomplemented
112C23×C32—31no—
113C23×C32—33no—
114C22×C3×D3C23×C3112yes—
115C22×C3×D3C23×C314yes—
116C32×Q8—216no—
117C32×Q8—66no—
118C32×Q8D1214yescomplemented
119C32×Q8C3×D4116yescomplemented
120C32×D4—66no—
121C32×D4—66no—
122C2×C32×C4—24no—
123C2×C32×C4—212no—
124C2×C32×C4—212no—
125C2×C32×C4—33no—
126C2×C32×C4—31no—
127C2×C32×C4—66no—
128C2×C32×C4—66no—
129C2×C32×C4C22×D311yescharacteristic
130C2×C32×C4C22×D313yes—
131C2×C32×C4C22×D313yes—
132C2×C32×C4C23×C3112yes—
133C2×C32×C4C23×C314yes—
134C2×C32×C4C23×C3112yes—
135C2×C3×Dic3C23×C314yes—
136C2×C3×Dic3C23×C3112yes—
137C3×D12—224no—
138C3×D12—224no—
139C3×C4×D3—224no—
140C3×C4×D3—224no—
141C3×Dic6—224no—
142#548—31noSylow
143C2×C33C22×D411yescharacteristic
144C2×C33C2×#4111yescharacteristic
145C2×C332−1+411yescharacteristic
146C32×D3—22no—
147C32×D3—43no—
148C3×#41—616no—
149C2×C3×Q8—24no—
150C2×C3×Q8—312no—
151C2×C3×Q8—616no—
152C2×C3×Q8C2×C3×D314yes—
153C2×C3×Q8C22×C3211yescharacteristic
154C2×C3×D4—312no—
155C2×C3×D4—312no—
156C22×C3×C4—312no—
157C22×C3×C4—312no—
158#239—24no—
159D3×Q8—24no—
160C2×D12C22×C3213yes—
161C2×D12C22×C3213yes—
162C2×C4×D3C22×C3213yes—
163C2×C4×D3C22×C3213yes—
164C2×Dic6C22×C3213yes—
165C3×C4⋊C4—212no—
166C3×C4⋊C4—312no—
167C3×C4⋊C4C2×C3×D3112yes—
168C3×C4⋊C4C22×C3213yes—
169C3×C22⋊C4—34no—
170C3×C22⋊C4—312no—
171C3×C22⋊C4—324no—
172C3×C42—212no—
173C3×C42C2×C3×D3112yes—
174C3×C42C22×C3213yes—
175(C2×D3)⋊C4C22×C3211yescharacteristic
176(C2×D3)⋊C4C22×C3213yes—
177(C2×D3)⋊C4C22×C3216yes—
178C4⋊Dic3C22×C3213yes—
179C22×C32—24no—
180C22×C32—32no—
181C22×C32—61no—
182C22×C32—61no—
183C22×C32—63no—
184C22×C32—63no—
185C22×C32—63no—
186C22×C32C23×D311yescharacteristic
187C22×C32C24×C314yes—
188C2×C3×D3—24no—
189C2×C3×D3—24no—
190C2×C3×D3—212no—
191C2×C3×D3—212no—
192C2×C3×D3—212no—
193C2×C3×D3C3×#4118yescomplemented
194C32×C4—28no—
195C32×C4—21no—
196C32×C4—23no—
197C32×C4—24no—
198C32×C4—224no—
199C32×C4—212no—
200C32×C4—412no—
201C32×C4—61no—
202C32×C4—63no—
203C32×C4C2×D1216yes—
204C32×C4C2×C3×D4124yes—
205C3×Dic3—212no—
206C3×Dic3—24no—
207C2×#41—31no—
208C22×Q8—31no—
209#124—33no—
210C22⋊Q8—33no—
211#121—33no—
212C4×Q8C32×D311yescharacteristic, complemented
213C4×D4—33no—
214C33#54811yescharacteristic, complemented, Sylow
215C23×C3—34no—
216C23×C3—312no—
217C22×D3C23×C3213yes—
218C22×D3C23×C3211yescharacteristic
219C3×Q8—216no—
220C3×Q8—624no—
221C3×Q8C3×D12116yescomplemented
222C3×Q8C32×D414yescomplemented
223C3×D4—624no—
224C3×D4—624no—
225C2×C3×C4—24no—
226C2×C3×C4—212no—
227C2×C3×C4—212no—
228C2×C3×C4—312no—
229C2×C3×C4—34no—
230C2×C3×C4—624no—
231C2×C3×C4—624no—
232C2×C3×C4C22×C3×D314yes—
233C2×C3×C4C22×C3×D3112yes—
234C2×C3×C4C22×C3×D3112yes—
235C2×C3×C4C23×C3213yes—
236C2×C3×C4C23×C3211yescharacteristic
237C2×C3×C4C23×C3213yes—
238C2×Dic3C23×C3211yescharacteristic
239C2×Dic3C23×C3213yes—
240D12—26no—
241D12—26no—
242C4×D3—26no—
243C4×D3—26no—
244Dic6—26no—
245C2×C32—24no—
246C2×C32—24no—
247C2×C32—24no—
248C2×C32—62no—
249C2×C32—123no—
250C2×C32C22×D1211yescharacteristic
251C2×C32#100711yescharacteristic
252C2×C32D3×#4111yescharacteristic
253C2×C32C22×C3×D414yes—
254C2×C32C2×C3×#4114yes—
255C2×C32C3×2−1+414yes—
256C3×D3—28no—
257C3×D3—412no—
258#41—64no—
259C2×Q8—33no—
260C2×Q8—64no—
261C2×Q8C2×C32×D311yescharacteristic
262C2×D4—33no—
263C2×D4—33no—
264C22×C4—33no—
265C22×C4—33no—
266C4⋊C4—33no—
267C4⋊C4C2×C32×D313yes—
268C22⋊C4—31no—
269C22⋊C4—33no—
270C22⋊C4—36no—
271C42C2×C32×D313yes—
272C22×C3—24no—
273C22×C3—38no—
274C22×C3—64no—
275C22×C3—64no—
276C22×C3—612no—
277C22×C3—612no—
278C22×C3—612no—
279C22×C3C23×C3×D314yes—
280C22×C3C24×C3211yescharacteristic
281C2×D3—21no—
282C2×D3—21no—
283C2×D3—23no—
284C2×D3—23no—
285C2×D3—23no—
286C2×D3C32×#4112yescomplemented
287C3×C4—28no—
288C3×C4—24no—
289C3×C4—212no—
290C3×C4—23no—
291C3×C4—21no—
292C3×C4—224no—
293C3×C4—412no—
294C3×C4—64no—
295C3×C4—612no—
296C3×C4C2×C3×D12124yes—
297C3×C4C2×C32×D416yes—
298Dic3—23no—
299Dic3—21no—
300C32—24no—
301C32#557611yescharacteristic, direct product factor
302C32C3×#54814yescomplemented
303C23—31no—
304C23—33no—
305Q8—66no—
306Q8C32×D1214yescomplemented
307D4—66no—
308D4—66no—
309C2×C4—33no—
310C2×C4—31no—
311C2×C4—66no—
312C2×C4—66no—
313C2×C4C22×C32×D311yescharacteristic
314C2×C4C22×C32×D313yes—
315C2×C4C22×C32×D313yes—
316C2×C3—24no—
317C2×C3—24no—
318C2×C3—24no—
319C2×C3—68no—
320C2×C3—1212no—
321C2×C3C22×C3×D1214yes—
322C2×C3C3×#100714yes—
323C2×C3C3×D3×#4114yes—
324C2×C3C22×C32×D411yescharacteristic
325C2×C3C2×C32×#4111yescharacteristic
326C2×C3C32×2−1+411yescharacteristic
327D3—22no—
328D3—43no—
329C22—32no—
330C22—61no—
331C22—61no—
332C22—63no—
333C22—63no—
334C22—63no—
335C22C23×C32×D311yescharacteristic
336C4—21no—
337C4—23no—
338C4—61no—
339C4—63no—
340C4C2×C32×D1216yes—
341C3—24no—
342C3C3×#557614yesdirect product factor
343C3C32×#54811yescharacteristic, complemented
344C2—62no—
345C2—123no—
346C2C22×C32×D1211yescharacteristic
347C2C32×#100711yescharacteristic
348C2C32×D3×#4111yescharacteristic
subgroupG/H
centerC22×C32C23×D3
commutator, cosocleC22×C3C24×C32
FrattiniC22C23×C32×D3
FittingC33×C4×Q8C2
socleC22×C33C24
hypercenterC32×C4×Q8D3
hypocenterC3C32×#548
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C22×C32 ◁ C32×C4×Q8

Complements

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

NH ≅ G/Ntype
1. C2×C32×#239C2semidirect
2. C2×C32×D3×Q8C2semidirect
3. C33×C4×Q8C2semidirect
4. C32×#897C2semidirect
5. C32×#895C2semidirect
6. C32×#875C2semidirect
7. C32×C4×D12C2semidirect
9. C3×#5576C3direct (N × H)
26. C2×C3×#239C2×C3semidirect
27. C2×C3×D3×Q8C2×C3semidirect
32. C32×C4×Q8D3semidirect
33. C32×C4×Q8C2×C3semidirect
35. C3×#897C2×C3semidirect
36. C3×#895C2×C3semidirect
37. C3×#875C2×C3semidirect
38. C3×C4×D12C2×C3semidirect
41. C33×Q8D4semidirect
53. #5576C32direct (N × H)
91. C2×C32×D3#41semidirect
99. C2×#239C2×C32semidirect
100. C2×D3×Q8C2×C32semidirect
105. C3×C4×Q8C3×D3semidirect
106. C3×C4×Q8C2×C32semidirect
108. #897C2×C32semidirect
109. #895C2×C32semidirect
110. #875C2×C32semidirect
111. C4×D12C2×C32semidirect
118. C32×Q8D12semidirect
119. C32×Q8C3×D4semidirect
193. C2×C3×D3C3×#41semidirect
212. C4×Q8C32×D3semidirect
214. C33#548semidirect
221. C3×Q8C3×D12semidirect
222. C3×Q8C32×D4semidirect
286. C2×D3C32×#41semidirect
301. C32#5576direct (N × H)
302. C32C3×#548semidirect
306. Q8C32×D12semidirect
342. C3C3×#5576direct (N × H)
343. C3C32×#548semidirect

Maximal subgroups

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

subgroupindex
C2×C32×#2392
C2×C32×D3×Q82
C33×C4×Q82
C32×#8972
C32×#8952
C32×#8752
C32×C4×D122
C32×#5483
C3×#55763