Finite Group Databasebeta

← C3×#10596307 #13179587

Subgroups — C3×#10596307 #13179587

Order 1728.

2212 conjugacy classes of subgroups.

The subgroup lattice has 374 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C33×H(5,𝔽2)C211yesmaximal, characteristic, complemented
2C3×#64367C211yesmaximal, characteristic, complemented
3C3×#64160C213yesmaximal, complemented
4C3×#64159C213yesmaximal, complemented
5C3×#64157C211yesmaximal, characteristic, complemented
6C2×C3×#3754C213yesmaximal, complemented
7C3×#64141C213yesmaximal, complemented
8#10596307C311yesmaximal, characteristic, direct product factor
9C3×#5836—34nomaximal
10C33×#41—21no—
11C33×#41C2213yes—
12C33×#41C2211yescharacteristic
13C2×C33×D4—23no—
14C2×C33×D4C2213yes—
15C3×Q8×C3⋊D3—21no—
16C3×#3830—23no—
17C3×#3828C2213yescomplemented
18C2×C3×#634C2213yescomplemented
19C3×#3756C2213yescomplemented
20C3×#3755C2211yescharacteristic, complemented
21C3×#3754C2216yescomplemented
22C3×#3754C2213yescomplemented
23C3×#3753C2213yescomplemented
24C3×#3748C2213yescomplemented
25C2×C3×C3⋊(C3⋊C8)C2213yescomplemented
26C32×H(5,𝔽2)—24no—
27C32×H(5,𝔽2)D314yescomplemented
28C32×H(5,𝔽2)C2×C311yescharacteristic, complemented
29#64367C2×C311yescharacteristic, complemented
30C3×#1010—34no—
31#64160C2×C313yescomplemented
32#64159C2×C313yescomplemented
33#64157C2×C311yescharacteristic, complemented
34C2×#3754C2×C313yescomplemented
35#64141C2×C313yescomplemented
36C3×#951—312no—
37C3×#950—312no—
38C3×#948—34no—
39C2×C3×#214—312no—
40C3×#932—312no—
41C23×C33—23no—
42C33×Q8D411yescharacteristic
43C33×Q8C2311yescharacteristic
44C33×D4—23no—
45C33×D4—23no—
46C33×D4D413yes—
47C33×D4C2313yes—
48C2×C33×C4—23no—
49C2×C33×C4C2313yes—
50C3×#638—23no—
51C2×C3×C3⋊Dic3—23no—
52C3×C4⋊(C3⋊D3)C2311yescharacteristic
53C3×C4×C3⋊D3—23no—
54C3×#634—23no—
55C3×#634C2313yes—
56C3×C3⋊(C3⋊C8)C2311yescharacteristic
57C3×C3⋊(C3⋊C8)C2313yes—
58C3×#577—91no—
59#5836—34no—
60C32×#41—28no—
61C32×#41—212no—
62C32×#41—24no—
63C32×#41—24no—
64C32×#41—21no—
65C32×#41C2×D3112yes—
66C32×#41C2×D314yes—
67C32×#41C22×C313yes—
68C32×#41C22×C311yescharacteristic
69C2×C32×D4—224no—
70C2×C32×D4—212no—
71C2×C32×D4—212no—
72C2×C32×D4—23no—
73C2×C32×D4C2×D3112yes—
74C2×C32×D4C22×C313yes—
75Q8×C3⋊D3—21no—
76#3830—23no—
77#3828C22×C313yescomplemented
78C2×#634C22×C313yescomplemented
79C3×D3×Q8—64no—
80C3×#237—612no—
81C3×#235—312no—
82C2×C3×Dic6—312no—
83#3756C22×C313yescomplemented
84#3755C22×C311yescharacteristic, complemented
85#3754C22×C316yescomplemented
86#3754C22×C313yescomplemented
87#3753C22×C313yescomplemented
88#3748C22×C313yescomplemented
89C2×C3⋊(C3⋊C8)C22×C313yescomplemented
90C3×#216—312no—
91C3×#215—34no—
92C3×#214—324no—
93C3×#214—312no—
94C3×#213—312no—
95C3×#208—312no—
96C2×C3×C3⋊C8—312no—
97C22×C33—23no—
98C22×C33—83no—
99C22×C33C2×D413yes—
100C2×C3×C3⋊D3—21no—
101C33×C4—21no—
102C33×C4C2×D413yes—
103C33×C4C2411yescharacteristic
104C3×C3⋊Dic3—23no—
105C3×2−1+4—91no—
106C3×H(5,𝔽2)—24no—
107C3×H(5,𝔽2)C3×D314yescomplemented
108C3×H(5,𝔽2)C3⋊D311yescharacteristic, complemented
109#1010—34no—
110C3×#137—93no—
111C3×AGL(1,ℤ/8)—93no—
112C3×#135—93no—
113C2×C3×SD16—93no—
114C3×#131—91no—
115#951—312no—
116#950—312no—
117#948—34no—
118C2×#214—312no—
119#932—312no—
120C23×C32—224no—
121C23×C32—212no—
122C23×C32—23no—
123C32×Q8—24no—
124C32×Q8—24no—
125C32×Q8#6714yes—
126C32×Q8C3×D411yescharacteristic
127C32×Q8C22×D314yes—
128C32×Q8C23×C311yescharacteristic
129C32×D4—224no—
130C32×D4—224no—
131C32×D4—212no—
132C32×D4—23no—
133C32×D4—212no—
134C32×D4—23no—
135C32×D4—212no—
136C32×D4—212no—
137C32×D4#67112yes—
138C32×D4C3×D413yes—
139C32×D4C22×D3112yes—
140C32×D4C23×C313yes—
141C2×C32×C4—224no—
142C2×C32×C4—212no—
143C2×C32×C4—23no—
144C2×C32×C4—212no—
145C2×C32×C4C22×D3112yes—
146C2×C32×C4C23×C313yes—
147#638—23no—
148C2×C3⋊Dic3—23no—
149C4⋊(C3⋊D3)C23×C311yescharacteristic
150C4×C3⋊D3—23no—
151#634—23no—
152#634C23×C313yes—
153C3×#67—612no—
154C2×C3×Dic3—612no—
155C3×D12—34no—
156C3×C4×D3—612no—
157C3×Dic6—312no—
158C3×Dic6—612no—
159C3⋊(C3⋊C8)C23×C311yescharacteristic
160C3⋊(C3⋊C8)C23×C313yes—
161C3×C3⋊C8—34no—
162C3×C3⋊C8—312no—
163#577—91noSylow
164C2×C33—23no—
165C2×C33—43no—
166C2×C33C22×D411yescharacteristic
167C3×C3⋊D3—41no—
168C3×#41—28no—
169C3×#41—24no—
170C3×#41—21no—
171C3×#41—212no—
172C3×#41—24no—
173C3×#41—93no—
174C3×#41—183no—
175C3×#41C2×C3×D3112yes—
176C3×#41C2×C3×D314yes—
177C3×#41C2×C3⋊D313yes—
178C3×#41C2×C3⋊D311yescharacteristic
179C2×C3×Q8—93no—
180C2×C3×Q8—181no—
181C2×C3×D4—224no—
182C2×C3×D4—212no—
183C2×C3×D4—23no—
184C2×C3×D4—212no—
185C2×C3×D4C2×C3×D3112yes—
186C2×C3×D4C2×C3⋊D313yes—
187D3×Q8—64no—
188#237—612no—
189#235—312no—
190C2×Dic6—312no—
191C3×Q16—93no—
192C3×SD16—91no—
193C3×SD16—93no—
194C3×SD16—96no—
195C3×D8—93no—
196C3×M16—93no—
197C2×C3×C8—93no—
198#216—312no—
199#215—34no—
200#214—324no—
201#214—312no—
202#213—312no—
203#208—312no—
204C2×C3⋊C8—312no—
205C22×C32—224no—
206C22×C32—212no—
207C22×C32—23no—
208C22×C32—212no—
209C22×C32—83no—
210C22×C32—812no—
211C22×C32—824no—
212C22×C32C2×#67112yes—
213C22×C32C2×C3×D413yes—
214C2×C3⋊D3—21no—
215C2×C3×D3—64no—
216C32×C4—28no—
217C32×C4—24no—
218C32×C4—21no—
219C32×C4—24no—
220C32×C4—212no—
221C32×C4C2×#67112yes—
222C32×C4C2×C3×D413yes—
223C32×C4C23×D314yes—
224C32×C4C24×C311yescharacteristic
225C3⋊Dic3—23no—
226C3×Dic3—612no—
2272−1+4—91no—
228H(5,𝔽2)C3×C3⋊D311yescharacteristic, complemented
229#137—93no—
230AGL(1,ℤ/8)—93no—
231#135—93no—
232C2×SD16—93no—
233#131—91no—
234C33#57711yescharacteristic, complemented, Sylow
235C23×C3—224no—
236C23×C3—212no—
237C23×C3—23no—
238C3×Q8—24no—
239C3×Q8—24no—
240C3×Q8—93no—
241C3×Q8—183no—
242C3×Q8C3×#6714yes—
243C3×Q8#63811yescharacteristic
244C3×Q8C22×C3×D314yes—
245C3×Q8C22×C3⋊D311yescharacteristic
246C3×D4—224no—
247C3×D4—224no—
248C3×D4—212no—
249C3×D4—23no—
250C3×D4—212no—
251C3×D4—23no—
252C3×D4—212no—
253C3×D4—212no—
254C3×D4—91no—
255C3×D4—183no—
256C3×D4C3×#67112yes—
257C3×D4#63813yes—
258C3×D4C22×C3×D3112yes—
259C3×D4C22×C3⋊D313yes—
260C2×C3×C4—224no—
261C2×C3×C4—212no—
262C2×C3×C4—23no—
263C2×C3×C4—212no—
264C2×C3×C4—183no—
265C2×C3×C4—183no—
266C2×C3×C4C22×C3×D3112yes—
267C2×C3×C4C22×C3⋊D313yes—
268#67—612no—
269C2×Dic3—612no—
270D12—34no—
271C4×D3—612no—
272Dic6—312no—
273Dic6—612no—
274C3×C8—91no—
275C3×C8—93no—
276C3⋊C8—34no—
277C3⋊C8—312no—
278C2×C32—23no—
279C2×C32—24no—
280C2×C32—212no—
281C2×C32—412no—
282C2×C32—424no—
283C2×C32—43no—
284C2×C32—412no—
285C2×C32C22×#6714yes—
286C2×C32C22×C3×D411yescharacteristic
287C3⋊D3—41no—
288C3×D3—124no—
289#41—21no—
290#41—93no—
291#41—183no—
292#41C2×C3×C3⋊D313yes—
293#41C2×C3×C3⋊D311yescharacteristic
294C2×Q8—93no—
295C2×Q8—181no—
296C2×D4—23no—
297C2×D4C2×C3×C3⋊D313yes—
298Q16—93no—
299SD16—91no—
300SD16—93no—
301SD16—96no—
302D8—93no—
303M16—93no—
304C2×C8—93no—
305C22×C3—224no—
306C22×C3—212no—
307C22×C3—23no—
308C22×C3—212no—
309C22×C3—812no—
310C22×C3—83no—
311C22×C3—824no—
312C22×C3—181no—
313C22×C3C2×C3×#67112yes—
314C22×C3C2×#63813yes—
315C2×D3—64no—
316C3×C4—28no—
317C3×C4—24no—
318C3×C4—21no—
319C3×C4—24no—
320C3×C4—212no—
321C3×C4—183no—
322C3×C4C2×C3×#67112yes—
323C3×C4C2×#63813yes—
324C3×C4C23×C3×D314yes—
325C3×C4C23×C3⋊D311yescharacteristic
326Dic3—612no—
327C32—24no—
328C32#583614yescomplemented
329C32C3×#57711yescharacteristic, complemented
330C23—23no—
331Q8—93no—
332Q8—183no—
333Q8C3×#63811yescharacteristic
334Q8C22×C3×C3⋊D311yescharacteristic
335D4—23no—
336D4—23no—
337D4—91no—
338D4—183no—
339D4C3×#63813yes—
340D4C22×C3×C3⋊D313yes—
341C2×C4—23no—
342C2×C4—183no—
343C2×C4—183no—
344C2×C4C22×C3×C3⋊D313yes—
345C8—91no—
346C8—93no—
347C2×C3—212no—
348C2×C3—23no—
349C2×C3—24no—
350C2×C3—412no—
351C2×C3—424no—
352C2×C3—412no—
353C2×C3—43no—
354C2×C3—361no—
355C2×C3C22×C3×#6714yes—
356C2×C3C22×#63811yescharacteristic
357D3—124no—
358C22—23no—
359C22—83no—
360C22—181no—
361C22C2×C3×#63813yes—
362C4—21no—
363C4—183no—
364C4C2×C3×#63813yes—
365C4C23×C3×C3⋊D311yescharacteristic
366C3—24no—
367C3C3×#583614yescomplemented
368C3#1059630711yescharacteristic, direct product factor
369C2—23no—
370C2—43no—
371C2—361no—
372C2C22×C3×#63811yescharacteristic
subgroupG/H
centerC2×C3C22×#638
commutator, cosocleC32×C4C24×C3
FrattiniC4C23×C3×C3⋊D3
FittingC33×H(5,𝔽2)C2
socleC2×C33C22×D4
hypercenterC3×H(5,𝔽2)C3⋊D3
hypocenterC32C3×#577
LCSG ▷ C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2×C3 ◁ C3×#41 ◁ C3×H(5,𝔽2)

Complements

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

NH ≅ G/Ntype
1. C33×H(5,𝔽2)C2semidirect
2. C3×#64367C2semidirect
3. C3×#64160C2semidirect
4. C3×#64159C2semidirect
5. C3×#64157C2semidirect
6. C2×C3×#3754C2semidirect
7. C3×#64141C2semidirect
8. #10596307C3direct (N × H)
17. C3×#3828C22semidirect
18. C2×C3×#634C22semidirect
19. C3×#3756C22semidirect
20. C3×#3755C22semidirect
21. C3×#3754C22semidirect
22. C3×#3754C22semidirect
23. C3×#3753C22semidirect
24. C3×#3748C22semidirect
25. C2×C3×C3⋊(C3⋊C8)C22semidirect
27. C32×H(5,𝔽2)D3semidirect
28. C32×H(5,𝔽2)C2×C3semidirect
29. #64367C2×C3semidirect
31. #64160C2×C3semidirect
32. #64159C2×C3semidirect
33. #64157C2×C3semidirect
34. C2×#3754C2×C3semidirect
35. #64141C2×C3semidirect
77. #3828C22×C3semidirect
78. C2×#634C22×C3semidirect
83. #3756C22×C3semidirect
84. #3755C22×C3semidirect
85. #3754C22×C3semidirect
86. #3754C22×C3semidirect
87. #3753C22×C3semidirect
88. #3748C22×C3semidirect
89. C2×C3⋊(C3⋊C8)C22×C3semidirect
107. C3×H(5,𝔽2)C3×D3semidirect
108. C3×H(5,𝔽2)C3⋊D3semidirect
228. H(5,𝔽2)C3×C3⋊D3semidirect
234. C33#577semidirect
328. C32#5836semidirect
329. C32C3×#577semidirect
367. C3C3×#5836semidirect
368. C3#10596307direct (N × H)

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
C3×#643672
C3×#641602
C3×#641592
C3×#641572
C2×C3×#37542
C3×#641412
#105963073
C3×#58363