Finite Group Databasebeta

← C32×#5750 #13179127

Subgroups — C32×#5750 #13179127

Order 1728.

1796 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C32×D3×#41C211yesmaximal, characteristic
2C33×#131C211yesmaximal, characteristic, complemented
3C32×#948C211yesmaximal, characteristic, complemented
4C32×#907C213yesmaximal, complemented
5C32×D3×M16C213yesmaximal, complemented
6C32×#901C213yesmaximal, complemented
7C2×C32×ΓL(1,𝔽25)C213yesmaximal, complemented
8C32×#568—31nomaximal
9C3×#5750C314yesmaximal, direct product factor
10C33×#41C2211yescharacteristic
11C32×#239C411yescharacteristic
12C32×D3×Q8C411yescharacteristic
13C32×#237C413yes—
14C32×D3×D4C413yes—
15C32×#235C2213yes—
16C2×C32×C4×D3C2213yes—
17C33×M16C2213yescomplemented
18C2×C33×C8C2213yescomplemented
19C32×#208C2213yescomplemented
20C2×C32×C3⋊C8C2213yescomplemented
21C32×ΓL(1,𝔽25)C2211yescharacteristic
22C32×ΓL(1,𝔽25)C2213yes—
23C32×ΓL(1,𝔽25)C2213yescomplemented
24C32×ΓL(1,𝔽25)C2213yescomplemented
25C32×D3×C8C2216yescomplemented
26C2×C32×#41—31no—
27C3×D3×#41C2×C314yes—
28C32×#131—24no—
29C32×#131—31no—
30C32×#131—33no—
31C32×#131—33no—
32C32×#131D311yescharacteristic, complemented
33C32×#131C2×C314yescomplemented
34C2×C32×M16—33no—
35C2×C32×M16—33no—
36C3×#948C2×C314yescomplemented
37C3×#907C2×C3112yescomplemented
38C3×D3×M16C2×C3112yescomplemented
39C3×#901C2×C3112yescomplemented
40C2×C3×ΓL(1,𝔽25)C2×C3112yescomplemented
41C22×C32×D3C2×C413yes—
42C33×Q8C2×C411yescharacteristic
43C33×D4C2×C413yes—
44C2×C33×C4C2313yes—
45C32×#67C2×C416yes—
46C2×C32×Dic3C2×C413yes—
47C32×D12C2×C413yes—
48C32×C4×D3C2×C413yes—
49C32×C4×D3C2×C413yes—
50C32×C4×D3C2311yescharacteristic
51C32×C4×D3C2313yes—
52C32×Dic6C2×C413yes—
53C33×C8C2311yescharacteristic
54C33×C8C2313yes—
55C32×C3⋊C8C2313yes—
56C32×C3⋊C8C2311yescharacteristic
57C3×#568—34no—
58#5750C3211yescharacteristic, direct product factor
59C32×#41—24no—
60C32×#41—31no—
61C32×#41—33no—
62C32×#41—33no—
63C32×#41C2×D311yescharacteristic
64C32×#41C22×C314yes—
65C2×C32×Q8—31no—
66C2×C32×D4—33no—
67C22×C32×C4—33no—
68C3×#239C3×C414yes—
69C3×D3×Q8C3×C414yes—
70C3×#237C3×C4112yes—
71C3×D3×D4C3×C4112yes—
72C3×#235C22×C3112yes—
73C2×C3×C4×D3C22×C3112yes—
74C32×M16—212no—
75C32×M16—33no—
76C32×M16—33no—
77C32×M16—31no—
78C32×M16—33no—
79C32×M16—33no—
80C32×M16C2×D313yescomplemented
81C32×M16C22×C3112yescomplemented
82C2×C32×C8—212no—
83C2×C32×C8—33no—
84C2×C32×C8—36no—
85C2×C32×C8C2×D313yescomplemented
86C2×C32×C8C22×C3112yescomplemented
87C3×#208C22×C3112yescomplemented
88C2×C3×C3⋊C8C22×C3112yescomplemented
89C3×ΓL(1,𝔽25)C22×C314yes—
90C3×ΓL(1,𝔽25)C22×C3112yes—
91C3×ΓL(1,𝔽25)C22×C3112yescomplemented
92C3×ΓL(1,𝔽25)C22×C3112yescomplemented
93C3×D3×C8C22×C3124yescomplemented
94C22×C33C22×C413yes—
95C2×C32×D3—43no—
96C2×C32×D3C22×C411yescharacteristic
97C2×C32×D3C22×C413yes—
98C33×C4C22×C413yes—
99C33×C4C2411yescharacteristic
100C32×Dic3C22×C413yes—
101C32×Dic3C22×C411yescharacteristic
102C2×C3×#41—34no—
103D3×#41C2×C3211yescharacteristic
104C3×#131—24no—
105C3×#131—34no—
106C3×#131—312no—
107C3×#131—312no—
108C3×#131C3×D314yescomplemented
109C3×#131C2×C3211yescharacteristic, complemented
110C2×C3×M16—312no—
111C2×C3×M16—312no—
112#948C2×C3211yescharacteristic, complemented
113#907C2×C3213yescomplemented
114D3×M16C2×C3213yescomplemented
115#901C2×C3213yescomplemented
116C2×ΓL(1,𝔽25)C2×C3213yescomplemented
117C23×C32—33no—
118C22×C3×D3C2×C3×C4112yes—
119C32×Q8—24no—
120C32×Q8—33no—
121C32×Q8C4×D311yescharacteristic
122C32×Q8C2×C3×C414yes—
123C32×D4—212no—
124C32×D4—36no—
125C32×D4—33no—
126C32×D4C4×D313yes—
127C32×D4C2×C3×C4112yes—
128C2×C32×C4—212no—
129C2×C32×C4—33no—
130C2×C32×C4—33no—
131C2×C32×C4—33no—
132C2×C32×C4—33no—
133C2×C32×C4—31no—
134C2×C32×C4C22×D313yes—
135C2×C32×C4C23×C3112yes—
136C3×#67C2×C3×C4124yes—
137C2×C3×Dic3C2×C3×C4112yes—
138C3×D12C2×C3×C4112yes—
139C3×C4×D3C2×C3×C4112yes—
140C3×C4×D3C2×C3×C4112yes—
141C3×C4×D3C23×C3112yes—
142C3×C4×D3C23×C314yes—
143C3×Dic6C2×C3×C4112yes—
144C32×C8—24no—
145C32×C8—212no—
146C32×C8—33no—
147C32×C8—31no—
148C32×C8C22×D313yes—
149C32×C8C22×D311yescharacteristic
150C32×C8C23×C3112yes—
151C32×C8C23×C314yes—
152C3×C3⋊C8C23×C3112yes—
153C3×C3⋊C8C23×C314yes—
154#568—31noSylow
155C2×C33—23no—
156C2×C33C23×C411yescharacteristic
157C32×D3—21no—
158C32×D3—23no—
159C3×#41—24no—
160C3×#41—34no—
161C3×#41—312no—
162C3×#41—312no—
163C3×#41C2×C3×D314yes—
164C3×#41C22×C3211yescharacteristic
165C2×C3×Q8—34no—
166C2×C3×D4—312no—
167C22×C3×C4—312no—
168#239C32×C411yescharacteristic
169D3×Q8C32×C411yescharacteristic
170#237C32×C413yes—
171D3×D4C32×C413yes—
172#235C22×C3213yes—
173C2×C4×D3C22×C3213yes—
174C3×M16—212no—
175C3×M16—312no—
176C3×M16—312no—
177C3×M16—34no—
178C3×M16—312no—
179C3×M16—312no—
180C3×M16C2×C3×D3112yescomplemented
181C3×M16C22×C3213yescomplemented
182C2×C3×C8—212no—
183C2×C3×C8—312no—
184C2×C3×C8—324no—
185C2×C3×C8C2×C3×D3112yescomplemented
186C2×C3×C8C22×C3213yescomplemented
187#208C22×C3213yescomplemented
188C2×C3⋊C8C22×C3213yescomplemented
189ΓL(1,𝔽25)C22×C3211yescharacteristic
190ΓL(1,𝔽25)C22×C3213yes—
191ΓL(1,𝔽25)C22×C3213yescomplemented
192ΓL(1,𝔽25)C22×C3213yescomplemented
193D3×C8C22×C3216yescomplemented
194C22×C32—212no—
195C22×C32—31no—
196C22×C32—33no—
197C22×C32—123no—
198C22×C32C2×C4×D313yes—
199C22×C32C22×C3×C4112yes—
200C2×C3×D3—412no—
201C2×C3×D3C22×C3×C414yes—
202C2×C3×D3C22×C3×C4112yes—
203C32×C4—24no—
204C32×C4—212no—
205C32×C4—31no—
206C32×C4—33no—
207C32×C4C2×C4×D313yes—
208C32×C4C22×C3×C4112yes—
209C32×C4C23×D311yescharacteristic
210C32×C4C24×C314yes—
211C3×Dic3C22×C3×C4112yes—
212C3×Dic3C22×C3×C414yes—
213C2×#41—31no—
214#131—31no—
215#131—33no—
216#131—33no—
217#131C32×D311yescharacteristic, complemented
218C2×M16—33no—
219C2×M16—33no—
220C33#56811yescharacteristic, complemented, Sylow
221C23×C3—312no—
222C22×D3C2×C32×C413yes—
223C3×Q8—24no—
224C3×Q8—312no—
225C3×Q8C3×C4×D314yes—
226C3×Q8C2×C32×C411yescharacteristic
227C3×D4—212no—
228C3×D4—324no—
229C3×D4—312no—
230C3×D4C3×C4×D3112yes—
231C3×D4C2×C32×C413yes—
232C2×C3×C4—212no—
233C2×C3×C4—312no—
234C2×C3×C4—312no—
235C2×C3×C4—34no—
236C2×C3×C4—312no—
237C2×C3×C4—312no—
238C2×C3×C4C22×C3×D3112yes—
239C2×C3×C4C23×C3213yes—
240#67C2×C32×C416yes—
241C2×Dic3C2×C32×C413yes—
242D12C2×C32×C413yes—
243C4×D3C2×C32×C413yes—
244C4×D3C2×C32×C413yes—
245C4×D3C23×C3213yes—
246C4×D3C23×C3211yescharacteristic
247Dic6C2×C32×C413yes—
248C3×C8—24no—
249C3×C8—212no—
250C3×C8—312no—
251C3×C8—34no—
252C3×C8C22×C3×D3112yes—
253C3×C8C22×C3×D314yes—
254C3×C8C23×C3213yes—
255C3×C8C23×C3211yescharacteristic
256C3⋊C8C23×C3213yes—
257C3⋊C8C23×C3211yescharacteristic
258C2×C32—23no—
259C2×C32—24no—
260C2×C32—212no—
261C2×C32—412no—
262C2×C32—63no—
263C2×C32—61no—
264C2×C32C22×C4×D311yescharacteristic
265C2×C32C23×C3×C414yes—
266C3×D3—24no—
267C3×D3—212no—
268#41—31no—
269#41—33no—
270#41—33no—
271#41C2×C32×D311yescharacteristic
272C2×Q8—31no—
273C2×D4—33no—
274C22×C4—33no—
275M16—33no—
276M16—33no—
277M16—31no—
278M16—33no—
279M16—33no—
280M16C2×C32×D313yescomplemented
281C2×C8—33no—
282C2×C8—36no—
283C2×C8C2×C32×D313yescomplemented
284C22×C3—212no—
285C22×C3—34no—
286C22×C3—312no—
287C22×C3—1212no—
288C22×C3C2×C3×C4×D3112yes—
289C22×C3C22×C32×C413yes—
290C2×D3—43no—
291C2×D3C22×C32×C411yescharacteristic
292C2×D3C22×C32×C413yes—
293C3×C4—24no—
294C3×C4—212no—
295C3×C4—34no—
296C3×C4—312no—
297C3×C4C2×C3×C4×D3112yes—
298C3×C4C22×C32×C413yes—
299C3×C4C23×C3×D314yes—
300C3×C4C24×C3211yescharacteristic
301Dic3C22×C32×C413yes—
302Dic3C22×C32×C411yescharacteristic
303C32—24no—
304C32#575011yescharacteristic, direct product factor
305C32C3×#56814yescomplemented
306C23—33no—
307Q8—33no—
308Q8C32×C4×D311yescharacteristic
309D4—36no—
310D4—33no—
311D4C32×C4×D313yes—
312C2×C4—33no—
313C2×C4—33no—
314C2×C4—31no—
315C2×C4—33no—
316C2×C4—33no—
317C2×C4C22×C32×D313yes—
318C8—31no—
319C8—33no—
320C8C22×C32×D313yes—
321C8C22×C32×D311yescharacteristic
322C2×C3—212no—
323C2×C3—23no—
324C2×C3—24no—
325C2×C3—412no—
326C2×C3—64no—
327C2×C3—612no—
328C2×C3C22×C3×C4×D314yes—
329C2×C3C23×C32×C411yescharacteristic
330D3—21no—
331D3—23no—
332C22—31no—
333C22—33no—
334C22—123no—
335C22C2×C32×C4×D313yes—
336C4—31no—
337C4—33no—
338C4C2×C32×C4×D313yes—
339C4C23×C32×D311yescharacteristic
340C3—24no—
341C3C3×#575014yesdirect product factor
342C3C32×#56811yescharacteristic, complemented
343C2—23no—
344C2—61no—
345C2—63no—
346C2C22×C32×C4×D311yescharacteristic
subgroupG/H
centerC32×C4C23×D3
commutatorC2×C3C23×C32×C4
FrattiniC4C23×C32×D3
FittingC33×#131C2
socleC2×C33C23×C4
prime radicalC32×D3×#41C2
cosocleC3×C4C24×C32
hypercenterC32×#131D3
hypocenterC3C32×#568
LCSG ▷ C2×C3 ▷ C3
UCSC1 ◁ C32×C4 ◁ C32×#131

Complements

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

NH ≅ G/Ntype
2. C33×#131C2semidirect
3. C32×#948C2semidirect
4. C32×#907C2semidirect
5. C32×D3×M16C2semidirect
6. C32×#901C2semidirect
7. C2×C32×ΓL(1,𝔽25)C2semidirect
9. C3×#5750C3direct (N × H)
17. C33×M16C22semidirect
18. C2×C33×C8C22semidirect
19. C32×#208C22semidirect
20. C2×C32×C3⋊C8C22semidirect
23. C32×ΓL(1,𝔽25)C22semidirect
24. C32×ΓL(1,𝔽25)C22semidirect
25. C32×D3×C8C22semidirect
32. C32×#131D3semidirect
33. C32×#131C2×C3semidirect
36. C3×#948C2×C3semidirect
37. C3×#907C2×C3semidirect
38. C3×D3×M16C2×C3semidirect
39. C3×#901C2×C3semidirect
40. C2×C3×ΓL(1,𝔽25)C2×C3semidirect
58. #5750C32direct (N × H)
80. C32×M16C2×D3semidirect
81. C32×M16C22×C3semidirect
85. C2×C32×C8C2×D3semidirect
86. C2×C32×C8C22×C3semidirect
87. C3×#208C22×C3semidirect
88. C2×C3×C3⋊C8C22×C3semidirect
91. C3×ΓL(1,𝔽25)C22×C3semidirect
92. C3×ΓL(1,𝔽25)C22×C3semidirect
93. C3×D3×C8C22×C3semidirect
108. C3×#131C3×D3semidirect
109. C3×#131C2×C32semidirect
112. #948C2×C32semidirect
113. #907C2×C32semidirect
114. D3×M16C2×C32semidirect
115. #901C2×C32semidirect
116. C2×ΓL(1,𝔽25)C2×C32semidirect
180. C3×M16C2×C3×D3semidirect
181. C3×M16C22×C32semidirect
185. C2×C3×C8C2×C3×D3semidirect
186. C2×C3×C8C22×C32semidirect
187. #208C22×C32semidirect
188. C2×C3⋊C8C22×C32semidirect
191. ΓL(1,𝔽25)C22×C32semidirect
192. ΓL(1,𝔽25)C22×C32semidirect
193. D3×C8C22×C32semidirect
217. #131C32×D3semidirect
220. C33#568semidirect
280. M16C2×C32×D3semidirect
283. C2×C8C2×C32×D3semidirect
304. C32#5750direct (N × H)
305. C32C3×#568semidirect
341. C3C3×#5750direct (N × H)
342. C3C32×#568semidirect

Maximal subgroups

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

subgroupindex
C32×D3×#412
C33×#1312
C32×#9482
C32×#9072
C32×D3×M162
C32×#9012
C2×C32×ΓL(1,𝔽25)2
C32×#5683
C3×#57503