Finite Group Databasebeta

← C3×#10596308 #13179588

Subgroups — C3×#10596308 #13179588

Order 1728.

2132 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C33×2−1+4C211yesmaximal, characteristic, complemented
2C3×#64366C211yesmaximal, characteristic, complemented
3C3×#64159C213yesmaximal, complemented
4C3×#64158C213yesmaximal, complemented
5C3×#64157C211yesmaximal, characteristic, complemented
6C3×#64151C213yesmaximal, complemented
7C2×C3×#3755C213yesmaximal, complemented
8#10596308C311yesmaximal, characteristic, direct product factor
9C3×#5837—34nomaximal
10C33×#41—23no—
11C33×#41C2211yescharacteristic
12C33×#41C2213yescomplemented
13C2×C33×Q8—21no—
14C2×C33×Q8C2213yescomplemented
15C3×#3832—21no—
16C3×D4×C3⋊D3—23no—
17C3×#3828C2213yes—
18C2×C3×C4⋊(C3⋊D3)C2213yes—
19C3×#3756C2213yescomplemented
20C3×#3755C2213yescomplemented
21C3×#3755C2216yescomplemented
22C3×#3754C2211yescharacteristic, complemented
23C3×#3753C2213yescomplemented
24C3×#3748C2213yescomplemented
25C2×C3×C3⋊(C3⋊C8)C2213yescomplemented
26C32×2−1+4—24no—
27C32×2−1+4D314yescomplemented
28C32×2−1+4C2×C311yescharacteristic, complemented
29#64366C2×C311yescharacteristic, complemented
30C3×#1009—34no—
31#64159C2×C313yescomplemented
32#64158C2×C313yescomplemented
33#64157C2×C311yescharacteristic, complemented
34#64151C2×C313yescomplemented
35C2×#3755C2×C313yescomplemented
36C3×#950—312no—
37C3×#949—312no—
38C3×#948—34no—
39C3×#942—312no—
40C2×C3×#215—312no—
41C22×C3×C3⋊D3—23no—
42C33×Q8—23no—
43C33×Q8D411yescharacteristic
44C33×Q8C2313yes—
45C33×D4—23no—
46C33×D4D413yes—
47C33×D4C2311yescharacteristic
48C2×C33×C4—23no—
49C2×C33×C4—23no—
50C2×C33×C4C2313yes—
51C3×#638—23no—
52C3×C4⋊(C3⋊D3)—23no—
53C3×C4⋊(C3⋊D3)C2313yes—
54C3×C4×C3⋊D3—23no—
55C3×#634C2311yescharacteristic
56C3×C3⋊(C3⋊C8)C2311yescharacteristic
57C3×C3⋊(C3⋊C8)C2313yes—
58C3×#577—91no—
59#5837—34no—
60C32×#41—224no—
61C32×#41—24no—
62C32×#41—212no—
63C32×#41—212no—
64C32×#41—23no—
65C32×#41C2×D314yes—
66C32×#41C2×D3112yescomplemented
67C32×#41C22×C311yescharacteristic
68C32×#41C22×C313yescomplemented
69C2×C32×Q8—28no—
70C2×C32×Q8—212no—
71C2×C32×Q8—24no—
72C2×C32×Q8—21no—
73C2×C32×Q8C2×D3112yescomplemented
74C2×C32×Q8C22×C313yescomplemented
75#3832—21no—
76D4×C3⋊D3—23no—
77#3828C22×C313yes—
78C2×C4⋊(C3⋊D3)C22×C313yes—
79C3×#239—64no—
80C3×D3×D4—612no—
81C3×#235—312no—
82C2×C3×D12—312no—
83#3756C22×C313yescomplemented
84#3755C22×C313yescomplemented
85#3755C22×C316yescomplemented
86#3754C22×C311yescharacteristic, complemented
87#3753C22×C313yescomplemented
88#3748C22×C313yescomplemented
89C2×C3⋊(C3⋊C8)C22×C313yescomplemented
90C3×#216—312no—
91C3×#215—312no—
92C3×#215—324no—
93C3×#214—34no—
94C3×#213—312no—
95C3×#208—312no—
96C2×C3×C3⋊C8—312no—
97C22×C33—21no—
98C22×C33C2×D413yes—
99C2×C3×C3⋊D3—23no—
100C2×C3×C3⋊D3—83no—
101C33×C4—23no—
102C33×C4C2×D413yes—
103C33×C4C2411yescharacteristic
104C3×C3⋊Dic3—21no—
105C3×2−1+4—24no—
106C3×2−1+4C3×D314yescomplemented
107C3×2−1+4C3⋊D311yescharacteristic, complemented
108C3×H(5,𝔽2)—91no—
109#1009—34no—
110C3×#137—93no—
111C3×AGL(1,ℤ/8)—93no—
112C3×#135—93no—
113C2×C3×SD16—93no—
114C3×#131—91no—
115#950—312no—
116#949—312no—
117#948—34no—
118#942—312no—
119C2×#215—312no—
120C22×C3⋊D3—23no—
121C22×C3×D3—612no—
122C32×Q8—224no—
123C32×Q8—212no—
124C32×Q8—23no—
125C32×Q8—24no—
126C32×Q8—212no—
127C32×Q8#6714yes—
128C32×Q8C3×D411yescharacteristic
129C32×Q8C22×D3112yes—
130C32×Q8C23×C313yes—
131C32×D4—224no—
132C32×D4—212no—
133C32×D4—23no—
134C32×D4—212no—
135C32×D4—24no—
136C32×D4#67112yes—
137C32×D4C3×D413yes—
138C32×D4C22×D314yes—
139C32×D4C23×C311yescharacteristic
140C2×C32×C4—224no—
141C2×C32×C4—224no—
142C2×C32×C4—212no—
143C2×C32×C4—23no—
144C2×C32×C4—212no—
145C2×C32×C4—23no—
146C2×C32×C4—212no—
147C2×C32×C4C22×D3112yes—
148C2×C32×C4C23×C313yes—
149#638—23no—
150C4⋊(C3⋊D3)—23no—
151C4⋊(C3⋊D3)C23×C313yes—
152C4×C3⋊D3—23no—
153#634C23×C311yescharacteristic
154C3×#67—612no—
155C3×D12—312no—
156C3×D12—612no—
157C3×C4×D3—612no—
158C3×Dic6—34no—
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—41no—
166C2×C33C22×D411yescharacteristic
167C3×C3⋊D3—43no—
168C3×#41—224no—
169C3×#41—212no—
170C3×#41—23no—
171C3×#41—24no—
172C3×#41—212no—
173C3×#41—93no—
174C3×#41—181no—
175C3×#41C2×C3×D314yes—
176C3×#41C2×C3×D3112yescomplemented
177C3×#41C2×C3⋊D311yescharacteristic
178C3×#41C2×C3⋊D313yescomplemented
179C2×C3×Q8—28no—
180C2×C3×Q8—24no—
181C2×C3×Q8—21no—
182C2×C3×Q8—212no—
183C2×C3×Q8C2×C3×D3112yescomplemented
184C2×C3×Q8C2×C3⋊D313yescomplemented
185C2×C3×D4—93no—
186C2×C3×D4—183no—
187#239—64no—
188D3×D4—612no—
189#235—312no—
190C2×D12—312no—
191C3×Q16—93no—
192C3×SD16—93no—
193C3×SD16—96no—
194C3×SD16—91no—
195C3×D8—93no—
196C3×M16—93no—
197C2×C3×C8—93no—
198#216—312no—
199#215—312no—
200#215—324no—
201#214—34no—
202#213—312no—
203#208—312no—
204C2×C3⋊C8—312no—
205C22×C32—28no—
206C22×C32—24no—
207C22×C32—21no—
208C22×C32—212no—
209C22×C32C2×#67112yes—
210C22×C32C2×C3×D413yes—
211C2×C3⋊D3—23no—
212C2×C3⋊D3—83no—
213C2×C3×D3—612no—
214C2×C3×D3—2412no—
215C32×C4—224no—
216C32×C4—212no—
217C32×C4—23no—
218C32×C4—24no—
219C32×C4—212no—
220C32×C4C2×#67112yes—
221C32×C4C2×C3×D413yes—
222C32×C4C23×D314yes—
223C32×C4C24×C311yescharacteristic
224C3⋊Dic3—21no—
225C3×Dic3—64no—
2262−1+4C3×C3⋊D311yescharacteristic, complemented
227H(5,𝔽2)—91no—
228#137—93no—
229AGL(1,ℤ/8)—93no—
230#135—93no—
231C2×SD16—93no—
232#131—91no—
233C33#57711yescharacteristic, complemented, Sylow
234C23×C3—183no—
235C22×D3—612no—
236C3×Q8—224no—
237C3×Q8—212no—
238C3×Q8—23no—
239C3×Q8—24no—
240C3×Q8—212no—
241C3×Q8—91no—
242C3×Q8C3×#6714yes—
243C3×Q8#63811yescharacteristic
244C3×Q8C22×C3×D3112yes—
245C3×Q8C22×C3⋊D313yes—
246C3×D4—224no—
247C3×D4—212no—
248C3×D4—23no—
249C3×D4—212no—
250C3×D4—24no—
251C3×D4—93no—
252C3×D4—183no—
253C3×D4—183no—
254C3×D4C3×#67112yes—
255C3×D4#63813yes—
256C3×D4C22×C3×D314yes—
257C3×D4C22×C3⋊D311yescharacteristic
258C2×C3×C4—224no—
259C2×C3×C4—224no—
260C2×C3×C4—212no—
261C2×C3×C4—23no—
262C2×C3×C4—212no—
263C2×C3×C4—23no—
264C2×C3×C4—212no—
265C2×C3×C4—183no—
266C2×C3×C4C22×C3×D3112yes—
267C2×C3×C4C22×C3⋊D313yes—
268#67—612no—
269D12—312no—
270D12—612no—
271C4×D3—612no—
272Dic6—34no—
273C3×C8—91no—
274C3×C8—93no—
275C3⋊C8—34no—
276C3⋊C8—312no—
277C2×C32—23no—
278C2×C32—24no—
279C2×C32—212no—
280C2×C32—412no—
281C2×C32—48no—
282C2×C32—41no—
283C2×C32—44no—
284C2×C32C22×#6714yes—
285C2×C32C22×C3×D411yescharacteristic
286C3⋊D3—43no—
287C3×D3—1212no—
288#41—23no—
289#41—93no—
290#41—181no—
291#41C2×C3×C3⋊D311yescharacteristic
292#41C2×C3×C3⋊D313yescomplemented
293C2×Q8—21no—
294C2×Q8C2×C3×C3⋊D313yescomplemented
295C2×D4—93no—
296C2×D4—183no—
297Q16—93no—
298SD16—93no—
299SD16—96no—
300SD16—91no—
301D8—93no—
302M16—93no—
303C2×C8—93no—
304C22×C3—28no—
305C22×C3—24no—
306C22×C3—21no—
307C22×C3—212no—
308C22×C3—183no—
309C22×C3—723no—
310C22×C3C2×C3×#67112yes—
311C22×C3C2×#63813yes—
312C2×D3—612no—
313C2×D3—2412no—
314C3×C4—224no—
315C3×C4—212no—
316C3×C4—23no—
317C3×C4—24no—
318C3×C4—212no—
319C3×C4—181no—
320C3×C4C2×C3×#67112yes—
321C3×C4C2×#63813yes—
322C3×C4C23×C3×D314yes—
323C3×C4C23×C3⋊D311yescharacteristic
324Dic3—64no—
325C32—24no—
326C32#583714yescomplemented
327C32C3×#57711yescharacteristic, complemented
328C23—183no—
329Q8—23no—
330Q8—91no—
331Q8C3×#63811yescharacteristic
332Q8C22×C3×C3⋊D313yes—
333D4—23no—
334D4—93no—
335D4—183no—
336D4—183no—
337D4C3×#63813yes—
338D4C22×C3×C3⋊D311yescharacteristic
339C2×C4—23no—
340C2×C4—23no—
341C2×C4—183no—
342C2×C4C22×C3×C3⋊D313yes—
343C8—91no—
344C8—93no—
345C2×C3—212no—
346C2×C3—23no—
347C2×C3—24no—
348C2×C3—412no—
349C2×C3—48no—
350C2×C3—44no—
351C2×C3—41no—
352C2×C3—363no—
353C2×C3C22×C3×#6714yes—
354C2×C3C22×#63811yescharacteristic
355D3—1212no—
356C22—21no—
357C22—183no—
358C22—723no—
359C22C2×C3×#63813yes—
360C4—23no—
361C4—181no—
362C4C2×C3×#63813yes—
363C4C23×C3×C3⋊D311yescharacteristic
364C3—24no—
365C3C3×#583714yescomplemented
366C3#1059630811yescharacteristic, direct product factor
367C2—23no—
368C2—41no—
369C2—363no—
370C2C22×C3×#63811yescharacteristic
subgroupG/H
centerC2×C3C22×#638
commutator, cosocleC32×C4C24×C3
FrattiniC4C23×C3×C3⋊D3
FittingC33×2−1+4C2
socleC2×C33C22×D4
hypercenterC3×2−1+4C3⋊D3
hypocenterC32C3×#577
LCSG ▷ C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2×C3 ◁ C3×#41 ◁ C3×2−1+4

Complements

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

NH ≅ G/Ntype
1. C33×2−1+4C2semidirect
2. C3×#64366C2semidirect
3. C3×#64159C2semidirect
4. C3×#64158C2semidirect
5. C3×#64157C2semidirect
6. C3×#64151C2semidirect
7. C2×C3×#3755C2semidirect
8. #10596308C3direct (N × H)
12. C33×#41C22semidirect
14. C2×C33×Q8C22semidirect
19. C3×#3756C22semidirect
20. C3×#3755C22semidirect
21. C3×#3755C22semidirect
22. C3×#3754C22semidirect
23. C3×#3753C22semidirect
24. C3×#3748C22semidirect
25. C2×C3×C3⋊(C3⋊C8)C22semidirect
27. C32×2−1+4D3semidirect
28. C32×2−1+4C2×C3semidirect
29. #64366C2×C3semidirect
31. #64159C2×C3semidirect
32. #64158C2×C3semidirect
33. #64157C2×C3semidirect
34. #64151C2×C3semidirect
35. C2×#3755C2×C3semidirect
66. C32×#41C2×D3semidirect
68. C32×#41C22×C3semidirect
73. C2×C32×Q8C2×D3semidirect
74. C2×C32×Q8C22×C3semidirect
83. #3756C22×C3semidirect
84. #3755C22×C3semidirect
85. #3755C22×C3semidirect
86. #3754C22×C3semidirect
87. #3753C22×C3semidirect
88. #3748C22×C3semidirect
89. C2×C3⋊(C3⋊C8)C22×C3semidirect
106. C3×2−1+4C3×D3semidirect
107. C3×2−1+4C3⋊D3semidirect
176. C3×#41C2×C3×D3semidirect
178. C3×#41C2×C3⋊D3semidirect
183. C2×C3×Q8C2×C3×D3semidirect
184. C2×C3×Q8C2×C3⋊D3semidirect
226. 2−1+4C3×C3⋊D3semidirect
233. C33#577semidirect
292. #41C2×C3×C3⋊D3semidirect
294. C2×Q8C2×C3×C3⋊D3semidirect
326. C32#5837semidirect
327. C32C3×#577semidirect
365. C3C3×#5837semidirect
366. C3#10596308direct (N × H)

Maximal subgroups

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

subgroupindex
C33×2−1+42
C3×#643662
C3×#641592
C3×#641582
C3×#641572
C3×#641512
C2×C3×#37552
#105963083
C3×#58373