Finite Group Databasebeta

← C3×#10596306 #13179586

Subgroups — C3×#10596306 #13179586

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
2C3×#64366C211yesmaximal, characteristic, complemented
3C3×#64159C213yesmaximal, complemented
4C3×#64158C213yesmaximal, complemented
5C3×#64157C211yesmaximal, characteristic, complemented
6C3×#64141C213yesmaximal, complemented
7C2×C3×#3753C213yesmaximal, complemented
8#10596306C311yesmaximal, characteristic, direct product factor
9C3×#5835—34nomaximal
10C33×#41—21no—
11C33×#41C2211yescharacteristic
12C33×#41C2213yescomplemented
13C2×C33×D4—23no—
14C2×C33×D4C2213yescomplemented
15C3×#3832—21no—
16C3×D4×C3⋊D3—23no—
17C3×#3828C2213yescomplemented
18C2×C3×C4⋊(C3⋊D3)C2213yescomplemented
19C3×#3756C2211yescharacteristic, complemented
20C3×#3755C2213yescomplemented
21C3×#3754C2213yescomplemented
22C3×#3753C2213yescomplemented
23C3×#3753C2216yescomplemented
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#64366C2×C311yescharacteristic, complemented
30C3×#1009—34no—
31#64159C2×C313yescomplemented
32#64158C2×C313yescomplemented
33#64157C2×C311yescharacteristic, complemented
34#64141C2×C313yescomplemented
35C2×#3753C2×C313yescomplemented
36C3×#950—312no—
37C3×#949—312no—
38C3×#948—34no—
39C3×#932—312no—
40C2×C3×#213—312no—
41C23×C33—23no—
42C22×C3×C3⋊D3—23no—
43C33×Q8D411yescharacteristic
44C33×Q8C2311yescharacteristic
45C33×D4—23no—
46C33×D4—23no—
47C33×D4D413yes—
48C33×D4C2313yes—
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×#576—91no—
59#5835—34no—
60C32×#41—28no—
61C32×#41—24no—
62C32×#41—212no—
63C32×#41—24no—
64C32×#41—21no—
65C32×#41C2×D314yes—
66C32×#41C2×D3112yescomplemented
67C32×#41C22×C311yescharacteristic
68C32×#41C22×C313yescomplemented
69C2×C32×D4—224no—
70C2×C32×D4—212no—
71C2×C32×D4—212no—
72C2×C32×D4—23no—
73C2×C32×D4C2×D3112yescomplemented
74C2×C32×D4C22×C313yescomplemented
75#3832—21no—
76D4×C3⋊D3—23no—
77#3828C22×C313yescomplemented
78C2×C4⋊(C3⋊D3)C22×C313yescomplemented
79C3×#239—64no—
80C3×D3×D4—612no—
81C3×#235—312no—
82C2×C3×D12—312no—
83#3756C22×C311yescharacteristic, complemented
84#3755C22×C313yescomplemented
85#3754C22×C313yescomplemented
86#3753C22×C313yescomplemented
87#3753C22×C316yescomplemented
88#3748C22×C313yescomplemented
89C2×C3⋊(C3⋊C8)C22×C313yescomplemented
90C3×#216—34no—
91C3×#215—312no—
92C3×#214—312no—
93C3×#213—312no—
94C3×#213—324no—
95C3×#208—312no—
96C2×C3×C3⋊C8—312no—
97C22×C33—23no—
98C22×C33—83no—
99C22×C33C2×D413yes—
100C2×C3×C3⋊D3—23no—
101C2×C3×C3⋊D3—83no—
102C33×C4—21no—
103C33×C4C2×D413yes—
104C33×C4C2411yescharacteristic
105C3×C3⋊Dic3—21no—
106C3×H(5,𝔽2)—24no—
107C3×H(5,𝔽2)—91no—
108C3×H(5,𝔽2)C3×D314yescomplemented
109C3×H(5,𝔽2)C3⋊D311yescharacteristic, complemented
110#1009—34no—
111C3×AGL(1,ℤ/8)—93no—
112C3×AGL(1,ℤ/8)—93no—
113C3×#135—93no—
114C2×C3×D8—93no—
115C3×#131—91no—
116#950—312no—
117#949—312no—
118#948—34no—
119#932—312no—
120C2×#213—312no—
121C23×C32—224no—
122C23×C32—212no—
123C23×C32—23no—
124C22×C3⋊D3—23no—
125C22×C3×D3—612no—
126C32×Q8—24no—
127C32×Q8—24no—
128C32×Q8#6714yes—
129C32×Q8C3×D411yescharacteristic
130C32×Q8C22×D314yes—
131C32×Q8C23×C311yescharacteristic
132C32×D4—224no—
133C32×D4—224no—
134C32×D4—212no—
135C32×D4—23no—
136C32×D4—212no—
137C32×D4—23no—
138C32×D4—212no—
139C32×D4—212no—
140C32×D4#67112yes—
141C32×D4C3×D413yes—
142C32×D4C22×D3112yes—
143C32×D4C23×C313yes—
144C2×C32×C4—224no—
145C2×C32×C4—212no—
146C2×C32×C4—23no—
147C2×C32×C4—212no—
148C2×C32×C4C22×D3112yes—
149C2×C32×C4C23×C313yes—
150#638—23no—
151C4⋊(C3⋊D3)—23no—
152C4⋊(C3⋊D3)C23×C313yes—
153C4×C3⋊D3—23no—
154#634C23×C311yescharacteristic
155C3×#67—612no—
156C3×D12—312no—
157C3×D12—612no—
158C3×C4×D3—612no—
159C3×Dic6—34no—
160C3⋊(C3⋊C8)C23×C311yescharacteristic
161C3⋊(C3⋊C8)C23×C313yes—
162C3×C3⋊C8—34no—
163C3×C3⋊C8—312no—
164#576—91noSylow
165C2×C33—23no—
166C2×C33—43no—
167C2×C33C22×D411yescharacteristic
168C3×C3⋊D3—43no—
169C3×#41—28no—
170C3×#41—24no—
171C3×#41—21no—
172C3×#41—24no—
173C3×#41—212no—
174C3×#41—93no—
175C3×#41—181no—
176C3×#41C2×C3×D314yes—
177C3×#41C2×C3×D3112yescomplemented
178C3×#41C2×C3⋊D311yescharacteristic
179C3×#41C2×C3⋊D313yescomplemented
180C2×C3×D4—224no—
181C2×C3×D4—212no—
182C2×C3×D4—23no—
183C2×C3×D4—212no—
184C2×C3×D4—93no—
185C2×C3×D4—183no—
186C2×C3×D4C2×C3×D3112yescomplemented
187C2×C3×D4C2×C3⋊D313yescomplemented
188#239—64no—
189D3×D4—612no—
190#235—312no—
191C2×D12—312no—
192C3×Q16—91no—
193C3×SD16—93no—
194C3×SD16—93no—
195C3×D8—93no—
196C3×D8—96no—
197C3×M16—93no—
198C2×C3×C8—93no—
199#216—34no—
200#215—312no—
201#214—312no—
202#213—312no—
203#213—324no—
204#208—312no—
205C2×C3⋊C8—312no—
206C22×C32—224no—
207C22×C32—212no—
208C22×C32—23no—
209C22×C32—212no—
210C22×C32—83no—
211C22×C32—812no—
212C22×C32—824no—
213C22×C32C2×#67112yes—
214C22×C32C2×C3×D413yes—
215C2×C3⋊D3—23no—
216C2×C3⋊D3—83no—
217C2×C3×D3—612no—
218C2×C3×D3—2412no—
219C32×C4—28no—
220C32×C4—24no—
221C32×C4—21no—
222C32×C4—24no—
223C32×C4—212no—
224C32×C4C2×#67112yes—
225C32×C4C2×C3×D413yes—
226C32×C4C23×D314yes—
227C32×C4C24×C311yescharacteristic
228C3⋊Dic3—21no—
229C3×Dic3—64no—
230H(5,𝔽2)—91no—
231H(5,𝔽2)C3×C3⋊D311yescharacteristic, complemented
232AGL(1,ℤ/8)—93no—
233AGL(1,ℤ/8)—93no—
234#135—93no—
235C2×D8—93no—
236#131—91no—
237C33#57611yescharacteristic, complemented, Sylow
238C23×C3—224no—
239C23×C3—212no—
240C23×C3—23no—
241C23×C3—183no—
242C22×D3—612no—
243C3×Q8—24no—
244C3×Q8—24no—
245C3×Q8—91no—
246C3×Q8C3×#6714yes—
247C3×Q8#63811yescharacteristic
248C3×Q8C22×C3×D314yes—
249C3×Q8C22×C3⋊D311yescharacteristic
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—93no—
259C3×D4—183no—
260C3×D4—183no—
261C3×D4C3×#67112yes—
262C3×D4#63813yes—
263C3×D4C22×C3×D3112yes—
264C3×D4C22×C3⋊D313yes—
265C2×C3×C4—224no—
266C2×C3×C4—212no—
267C2×C3×C4—23no—
268C2×C3×C4—212no—
269C2×C3×C4—183no—
270C2×C3×C4C22×C3×D3112yes—
271C2×C3×C4C22×C3⋊D313yes—
272#67—612no—
273D12—312no—
274D12—612no—
275C4×D3—612no—
276Dic6—34no—
277C3×C8—91no—
278C3×C8—93no—
279C3⋊C8—34no—
280C3⋊C8—312no—
281C2×C32—23no—
282C2×C32—24no—
283C2×C32—212no—
284C2×C32—412no—
285C2×C32—424no—
286C2×C32—43no—
287C2×C32—412no—
288C2×C32C22×#6714yes—
289C2×C32C22×C3×D411yescharacteristic
290C3⋊D3—43no—
291C3×D3—1212no—
292#41—21no—
293#41—93no—
294#41—181no—
295#41C2×C3×C3⋊D311yescharacteristic
296#41C2×C3×C3⋊D313yescomplemented
297C2×D4—23no—
298C2×D4—93no—
299C2×D4—183no—
300C2×D4C2×C3×C3⋊D313yescomplemented
301Q16—91no—
302SD16—93no—
303SD16—93no—
304D8—93no—
305D8—96no—
306M16—93no—
307C2×C8—93no—
308C22×C3—224no—
309C22×C3—212no—
310C22×C3—23no—
311C22×C3—212no—
312C22×C3—812no—
313C22×C3—83no—
314C22×C3—824no—
315C22×C3—183no—
316C22×C3—723no—
317C22×C3C2×C3×#67112yes—
318C22×C3C2×#63813yes—
319C2×D3—612no—
320C2×D3—2412no—
321C3×C4—28no—
322C3×C4—24no—
323C3×C4—21no—
324C3×C4—24no—
325C3×C4—212no—
326C3×C4—181no—
327C3×C4C2×C3×#67112yes—
328C3×C4C2×#63813yes—
329C3×C4C23×C3×D314yes—
330C3×C4C23×C3⋊D311yescharacteristic
331Dic3—64no—
332C32—24no—
333C32#583514yescomplemented
334C32C3×#57611yescharacteristic, complemented
335C23—23no—
336C23—183no—
337Q8—91no—
338Q8C3×#63811yescharacteristic
339Q8C22×C3×C3⋊D311yescharacteristic
340D4—23no—
341D4—23no—
342D4—93no—
343D4—183no—
344D4—183no—
345D4C3×#63813yes—
346D4C22×C3×C3⋊D313yes—
347C2×C4—23no—
348C2×C4—183no—
349C2×C4C22×C3×C3⋊D313yes—
350C8—91no—
351C8—93no—
352C2×C3—212no—
353C2×C3—23no—
354C2×C3—24no—
355C2×C3—412no—
356C2×C3—424no—
357C2×C3—412no—
358C2×C3—43no—
359C2×C3—363no—
360C2×C3C22×C3×#6714yes—
361C2×C3C22×#63811yescharacteristic
362D3—1212no—
363C22—23no—
364C22—83no—
365C22—183no—
366C22—723no—
367C22C2×C3×#63813yes—
368C4—21no—
369C4—181no—
370C4C2×C3×#63813yes—
371C4C23×C3×C3⋊D311yescharacteristic
372C3—24no—
373C3C3×#583514yescomplemented
374C3#1059630611yescharacteristic, direct product factor
375C2—23no—
376C2—43no—
377C2—363no—
378C2C22×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×#576
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×#64366C2semidirect
3. C3×#64159C2semidirect
4. C3×#64158C2semidirect
5. C3×#64157C2semidirect
6. C3×#64141C2semidirect
7. C2×C3×#3753C2semidirect
8. #10596306C3direct (N × H)
12. C33×#41C22semidirect
14. C2×C33×D4C22semidirect
17. C3×#3828C22semidirect
18. C2×C3×C4⋊(C3⋊D3)C22semidirect
19. C3×#3756C22semidirect
20. C3×#3755C22semidirect
21. C3×#3754C22semidirect
22. C3×#3753C22semidirect
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. #64366C2×C3semidirect
31. #64159C2×C3semidirect
32. #64158C2×C3semidirect
33. #64157C2×C3semidirect
34. #64141C2×C3semidirect
35. C2×#3753C2×C3semidirect
66. C32×#41C2×D3semidirect
68. C32×#41C22×C3semidirect
73. C2×C32×D4C2×D3semidirect
74. C2×C32×D4C22×C3semidirect
77. #3828C22×C3semidirect
78. C2×C4⋊(C3⋊D3)C22×C3semidirect
83. #3756C22×C3semidirect
84. #3755C22×C3semidirect
85. #3754C22×C3semidirect
86. #3753C22×C3semidirect
87. #3753C22×C3semidirect
88. #3748C22×C3semidirect
89. C2×C3⋊(C3⋊C8)C22×C3semidirect
108. C3×H(5,𝔽2)C3×D3semidirect
109. C3×H(5,𝔽2)C3⋊D3semidirect
177. C3×#41C2×C3×D3semidirect
179. C3×#41C2×C3⋊D3semidirect
186. C2×C3×D4C2×C3×D3semidirect
187. C2×C3×D4C2×C3⋊D3semidirect
231. H(5,𝔽2)C3×C3⋊D3semidirect
237. C33#576semidirect
296. #41C2×C3×C3⋊D3semidirect
300. C2×D4C2×C3×C3⋊D3semidirect
333. C32#5835semidirect
334. C32C3×#576semidirect
373. C3C3×#5835semidirect
374. C3#10596306direct (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×#643662
C3×#641592
C3×#641582
C3×#641572
C3×#641412
C2×C3×#37532
#105963063
C3×#58353