Finite Group Databasebeta

← C32×#5835 #13179212

Subgroups — C32×#5835 #13179212

Order 1728.

2272 conjugacy classes of subgroups.

The subgroup lattice has 380 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C33×H(5,𝔽2)C211yesmaximal, characteristic, complemented
2C32×#1009C211yesmaximal, characteristic, complemented
3C32×#950C213yesmaximal, complemented
4C32×#949C213yesmaximal, complemented
5C32×#948C211yesmaximal, characteristic, complemented
6C32×#932C213yesmaximal, complemented
7C2×C32×#213C213yesmaximal, complemented
8C32×#576—31nomaximal
9C3×#5835C314yesmaximal, direct product factor
10C33×#41—21no—
11C33×#41C2211yescharacteristic
12C33×#41C2213yescomplemented
13C2×C33×D4—23no—
14C2×C33×D4C2213yescomplemented
15C32×#239—21no—
16C32×D3×D4—23no—
17C32×#235C2213yescomplemented
18C2×C32×D12C2213yescomplemented
19C32×#216C2211yescharacteristic, complemented
20C32×#215C2213yescomplemented
21C32×#214C2213yescomplemented
22C32×#213C2213yescomplemented
23C32×#213C2216yescomplemented
24C32×#208C2213yescomplemented
25C2×C32×C3⋊C8C2213yescomplemented
26C32×H(5,𝔽2)—24no—
27C32×H(5,𝔽2)—31no—
28C32×H(5,𝔽2)D311yescharacteristic, complemented
29C32×H(5,𝔽2)C2×C314yescomplemented
30C3×#1009C2×C314yescomplemented
31C32×AGL(1,ℤ/8)—33no—
32C32×AGL(1,ℤ/8)—33no—
33C32×#135—33no—
34C2×C32×D8—33no—
35C32×#131—31no—
36C3×#950C2×C3112yescomplemented
37C3×#949C2×C3112yescomplemented
38C3×#948C2×C314yescomplemented
39C3×#932C2×C3112yescomplemented
40C2×C3×#213C2×C3112yescomplemented
41C23×C33—23no—
42C22×C32×D3—23no—
43C33×Q8D411yescharacteristic
44C33×Q8C2311yescharacteristic
45C33×D4—23no—
46C33×D4—23no—
47C33×D4D413yes—
48C33×D4C2313yes—
49C2×C33×C4—23no—
50C2×C33×C4C2313yes—
51C32×#67—23no—
52C32×D12—23no—
53C32×D12C2313yes—
54C32×C4×D3—23no—
55C32×Dic6C2311yescharacteristic
56C32×C3⋊C8C2311yescharacteristic
57C32×C3⋊C8C2313yes—
58C3×#576—34no—
59#5835C3211yescharacteristic, direct product factor
60C32×#41—28no—
61C32×#41—24no—
62C32×#41—212no—
63C32×#41—21no—
64C32×#41—24no—
65C32×#41—33no—
66C32×#41—61no—
67C32×#41C2×D311yescharacteristic
68C32×#41C2×D313yescomplemented
69C32×#41C22×C314yes—
70C32×#41C22×C3112yescomplemented
71C2×C32×D4—224no—
72C2×C32×D4—212no—
73C2×C32×D4—23no—
74C2×C32×D4—212no—
75C2×C32×D4—33no—
76C2×C32×D4—63no—
77C2×C32×D4C2×D313yescomplemented
78C2×C32×D4C22×C3112yescomplemented
79C3×#239—24no—
80C3×D3×D4—212no—
81C3×#235C22×C3112yescomplemented
82C2×C3×D12C22×C3112yescomplemented
83C32×Q16—31no—
84C32×SD16—33no—
85C32×SD16—33no—
86C32×D8—33no—
87C32×D8—36no—
88C32×M16—33no—
89C2×C32×C8—33no—
90C3×#216C22×C314yescomplemented
91C3×#215C22×C3112yescomplemented
92C3×#214C22×C3112yescomplemented
93C3×#213C22×C3112yescomplemented
94C3×#213C22×C3124yescomplemented
95C3×#208C22×C3112yescomplemented
96C2×C3×C3⋊C8C22×C3112yescomplemented
97C22×C33—23no—
98C22×C33—83no—
99C22×C33C2×D413yes—
100C2×C32×D3—23no—
101C2×C32×D3—83no—
102C33×C4—21no—
103C33×C4C2×D413yes—
104C33×C4C2411yescharacteristic
105C32×Dic3—21no—
106C3×H(5,𝔽2)—24no—
107C3×H(5,𝔽2)—34no—
108C3×H(5,𝔽2)C3×D314yescomplemented
109C3×H(5,𝔽2)C2×C3211yescharacteristic, complemented
110#1009C2×C3211yescharacteristic, complemented
111C3×AGL(1,ℤ/8)—312no—
112C3×AGL(1,ℤ/8)—312no—
113C3×#135—312no—
114C2×C3×D8—312no—
115C3×#131—34no—
116#950C2×C3213yescomplemented
117#949C2×C3213yescomplemented
118#948C2×C3211yescharacteristic, complemented
119#932C2×C3213yescomplemented
120C2×#213C2×C3213yescomplemented
121C23×C32—224no—
122C23×C32—23no—
123C23×C32—212no—
124C23×C32—63no—
125C22×C3×D3—212no—
126C32×Q8—24no—
127C32×Q8—24no—
128C32×Q8—31no—
129C32×Q8#6711yescharacteristic
130C32×Q8C3×D414yes—
131C32×Q8C22×D311yescharacteristic
132C32×Q8C23×C314yes—
133C32×D4—224no—
134C32×D4—224no—
135C32×D4—23no—
136C32×D4—212no—
137C32×D4—23no—
138C32×D4—212no—
139C32×D4—212no—
140C32×D4—212no—
141C32×D4—33no—
142C32×D4—63no—
143C32×D4—63no—
144C32×D4#6713yes—
145C32×D4C3×D4112yes—
146C32×D4C22×D313yes—
147C32×D4C23×C3112yes—
148C2×C32×C4—224no—
149C2×C32×C4—23no—
150C2×C32×C4—212no—
151C2×C32×C4—212no—
152C2×C32×C4—63no—
153C2×C32×C4C22×D313yes—
154C2×C32×C4C23×C3112yes—
155C3×#67—212no—
156C3×D12—212no—
157C3×D12C23×C3112yes—
158C3×C4×D3—212no—
159C3×Dic6C23×C314yes—
160C32×C8—31no—
161C32×C8—33no—
162C3×C3⋊C8C23×C314yes—
163C3×C3⋊C8C23×C3112yes—
164#576—31noSylow
165C2×C33—23no—
166C2×C33—43no—
167C2×C33C22×D411yescharacteristic
168C32×D3—43no—
169C3×#41—28no—
170C3×#41—24no—
171C3×#41—21no—
172C3×#41—24no—
173C3×#41—212no—
174C3×#41—312no—
175C3×#41—64no—
176C3×#41C2×C3×D314yes—
177C3×#41C2×C3×D3112yescomplemented
178C3×#41C22×C3211yescharacteristic
179C3×#41C22×C3213yescomplemented
180C2×C3×D4—224no—
181C2×C3×D4—212no—
182C2×C3×D4—23no—
183C2×C3×D4—212no—
184C2×C3×D4—312no—
185C2×C3×D4—612no—
186C2×C3×D4C2×C3×D3112yescomplemented
187C2×C3×D4C22×C3213yescomplemented
188#239—21no—
189D3×D4—23no—
190#235C22×C3213yescomplemented
191C2×D12C22×C3213yescomplemented
192C3×Q16—34no—
193C3×SD16—312no—
194C3×SD16—312no—
195C3×D8—312no—
196C3×D8—324no—
197C3×M16—312no—
198C2×C3×C8—312no—
199#216C22×C3211yescharacteristic, complemented
200#215C22×C3213yescomplemented
201#214C22×C3213yescomplemented
202#213C22×C3213yescomplemented
203#213C22×C3216yescomplemented
204#208C22×C3213yescomplemented
205C2×C3⋊C8C22×C3213yescomplemented
206C22×C32—224no—
207C22×C32—23no—
208C22×C32—212no—
209C22×C32—212no—
210C22×C32—63no—
211C22×C32—83no—
212C22×C32—812no—
213C22×C32—824no—
214C22×C32—243no—
215C22×C32C2×#6713yes—
216C22×C32C2×C3×D4112yes—
217C2×C3×D3—212no—
218C2×C3×D3—812no—
219C32×C4—28no—
220C32×C4—21no—
221C32×C4—24no—
222C32×C4—24no—
223C32×C4—212no—
224C32×C4—61no—
225C32×C4C2×#6713yes—
226C32×C4C2×C3×D4112yes—
227C32×C4C23×D311yescharacteristic
228C32×C4C24×C314yes—
229C3×Dic3—24no—
230H(5,𝔽2)—31no—
231H(5,𝔽2)C32×D311yescharacteristic, complemented
232AGL(1,ℤ/8)—33no—
233AGL(1,ℤ/8)—33no—
234#135—33no—
235C2×D8—33no—
236#131—31no—
237C33#57611yescharacteristic, complemented, Sylow
238C23×C3—224no—
239C23×C3—212no—
240C23×C3—23no—
241C23×C3—612no—
242C22×D3—23no—
243C3×Q8—24no—
244C3×Q8—24no—
245C3×Q8—34no—
246C3×Q8C3×#6714yes—
247C3×Q8C32×D411yescharacteristic
248C3×Q8C22×C3×D314yes—
249C3×Q8C23×C3211yescharacteristic
250C3×D4—224no—
251C3×D4—224no—
252C3×D4—212no—
253C3×D4—23no—
254C3×D4—212no—
255C3×D4—23no—
256C3×D4—212no—
257C3×D4—212no—
258C3×D4—312no—
259C3×D4—612no—
260C3×D4—612no—
261C3×D4C3×#67112yes—
262C3×D4C32×D413yes—
263C3×D4C22×C3×D3112yes—
264C3×D4C23×C3213yes—
265C2×C3×C4—224no—
266C2×C3×C4—212no—
267C2×C3×C4—23no—
268C2×C3×C4—212no—
269C2×C3×C4—612no—
270C2×C3×C4C22×C3×D3112yes—
271C2×C3×C4C23×C3213yes—
272#67—23no—
273D12—23no—
274D12C23×C3213yes—
275C4×D3—23no—
276Dic6C23×C3211yescharacteristic
277C3×C8—34no—
278C3×C8—312no—
279C3⋊C8C23×C3211yescharacteristic
280C3⋊C8C23×C3213yes—
281C2×C32—23no—
282C2×C32—24no—
283C2×C32—212no—
284C2×C32—412no—
285C2×C32—424no—
286C2×C32—412no—
287C2×C32—43no—
288C2×C32—123no—
289C2×C32C22×#6711yescharacteristic
290C2×C32C22×C3×D414yes—
291C3×D3—412no—
292#41—21no—
293#41—33no—
294#41—61no—
295#41C2×C32×D311yescharacteristic
296#41C2×C32×D313yescomplemented
297C2×D4—23no—
298C2×D4—33no—
299C2×D4—63no—
300C2×D4C2×C32×D313yescomplemented
301Q16—31no—
302SD16—33no—
303SD16—33no—
304D8—33no—
305D8—36no—
306M16—33no—
307C2×C8—33no—
308C22×C3—224no—
309C22×C3—212no—
310C22×C3—23no—
311C22×C3—212no—
312C22×C3—612no—
313C22×C3—83no—
314C22×C3—812no—
315C22×C3—824no—
316C22×C3—2412no—
317C22×C3C2×C3×#67112yes—
318C22×C3C2×C32×D413yes—
319C2×D3—23no—
320C2×D3—83no—
321C3×C4—28no—
322C3×C4—24no—
323C3×C4—21no—
324C3×C4—24no—
325C3×C4—212no—
326C3×C4—64no—
327C3×C4C2×C3×#67112yes—
328C3×C4C2×C32×D413yes—
329C3×C4C23×C3×D314yes—
330C3×C4C24×C3211yescharacteristic
331Dic3—21no—
332C32—24no—
333C32#583511yescharacteristic, direct product factor
334C32C3×#57614yescomplemented
335C23—23no—
336C23—63no—
337Q8—31no—
338Q8C32×#6711yescharacteristic
339Q8C22×C32×D311yescharacteristic
340D4—23no—
341D4—23no—
342D4—33no—
343D4—63no—
344D4—63no—
345D4C32×#6713yes—
346D4C22×C32×D313yes—
347C2×C4—23no—
348C2×C4—63no—
349C2×C4C22×C32×D313yes—
350C8—31no—
351C8—33no—
352C2×C3—23no—
353C2×C3—212no—
354C2×C3—24no—
355C2×C3—412no—
356C2×C3—424no—
357C2×C3—43no—
358C2×C3—412no—
359C2×C3—1212no—
360C2×C3C22×C3×#6714yes—
361C2×C3C22×C32×D411yescharacteristic
362D3—43no—
363C22—23no—
364C22—63no—
365C22—83no—
366C22—243no—
367C22C2×C32×#6713yes—
368C4—21no—
369C4—61no—
370C4C2×C32×#6713yes—
371C4C23×C32×D311yescharacteristic
372C3—24no—
373C3C3×#583514yesdirect product factor
374C3C32×#57611yescharacteristic, complemented
375C2—23no—
376C2—43no—
377C2—123no—
378C2C22×C32×#6711yescharacteristic
subgroupG/H
centerC2×C32C22×#67
commutator, cosocleC3×C4C24×C32
FrattiniC4C23×C32×D3
FittingC33×H(5,𝔽2)C2
socleC2×C33C22×D4
hypercenterC32×H(5,𝔽2)D3
hypocenterC3C32×#576
LCSG ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C32 ◁ C32×#41 ◁ C32×H(5,𝔽2)

Complements

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

NH ≅ G/Ntype
1. C33×H(5,𝔽2)C2semidirect
2. C32×#1009C2semidirect
3. C32×#950C2semidirect
4. C32×#949C2semidirect
5. C32×#948C2semidirect
6. C32×#932C2semidirect
7. C2×C32×#213C2semidirect
9. C3×#5835C3direct (N × H)
12. C33×#41C22semidirect
14. C2×C33×D4C22semidirect
17. C32×#235C22semidirect
18. C2×C32×D12C22semidirect
19. C32×#216C22semidirect
20. C32×#215C22semidirect
21. C32×#214C22semidirect
22. C32×#213C22semidirect
23. C32×#213C22semidirect
24. C32×#208C22semidirect
25. C2×C32×C3⋊C8C22semidirect
28. C32×H(5,𝔽2)D3semidirect
29. C32×H(5,𝔽2)C2×C3semidirect
30. C3×#1009C2×C3semidirect
36. C3×#950C2×C3semidirect
37. C3×#949C2×C3semidirect
38. C3×#948C2×C3semidirect
39. C3×#932C2×C3semidirect
40. C2×C3×#213C2×C3semidirect
59. #5835C32direct (N × H)
68. C32×#41C2×D3semidirect
70. C32×#41C22×C3semidirect
77. C2×C32×D4C2×D3semidirect
78. C2×C32×D4C22×C3semidirect
81. C3×#235C22×C3semidirect
82. C2×C3×D12C22×C3semidirect
90. C3×#216C22×C3semidirect
91. C3×#215C22×C3semidirect
92. C3×#214C22×C3semidirect
93. C3×#213C22×C3semidirect
94. C3×#213C22×C3semidirect
95. C3×#208C22×C3semidirect
96. C2×C3×C3⋊C8C22×C3semidirect
108. C3×H(5,𝔽2)C3×D3semidirect
109. C3×H(5,𝔽2)C2×C32semidirect
110. #1009C2×C32semidirect
116. #950C2×C32semidirect
117. #949C2×C32semidirect
118. #948C2×C32semidirect
119. #932C2×C32semidirect
120. C2×#213C2×C32semidirect
177. C3×#41C2×C3×D3semidirect
179. C3×#41C22×C32semidirect
186. C2×C3×D4C2×C3×D3semidirect
187. C2×C3×D4C22×C32semidirect
190. #235C22×C32semidirect
191. C2×D12C22×C32semidirect
199. #216C22×C32semidirect
200. #215C22×C32semidirect
201. #214C22×C32semidirect
202. #213C22×C32semidirect
203. #213C22×C32semidirect
204. #208C22×C32semidirect
205. C2×C3⋊C8C22×C32semidirect
231. H(5,𝔽2)C32×D3semidirect
237. C33#576semidirect
296. #41C2×C32×D3semidirect
300. C2×D4C2×C32×D3semidirect
333. C32#5835direct (N × H)
334. C32C3×#576semidirect
373. C3C3×#5835direct (N × H)
374. C3C32×#576semidirect

Maximal subgroups

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

subgroupindex
C33×H(5,𝔽2)2
C32×#10092
C32×#9502
C32×#9492
C32×#9482
C32×#9322
C2×C32×#2132
C32×#5763
C3×#58353