Finite Group Databasebeta

← #131×(C3⋊D3)⋊C3 #13153409

Subgroups — #131×(C3⋊D3)⋊C3 #13153409

Order 1728.

993 conjugacy classes of subgroups.

The subgroup lattice has 401 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#41×(C3⋊D3)⋊C3C211yesmaximal, characteristic
2H(3,𝔽3)×#131C211yesmaximal, characteristic, complemented
3#11721689C211yesmaximal, characteristic, complemented
4#64116⋊C3C213yesmaximal, complemented
5M16×(C3⋊D3)⋊C3C213yesmaximal, complemented
6#64110⋊C3C213yesmaximal, complemented
7C2×C8×(C3⋊D3)⋊C3C213yesmaximal, complemented
8C3⋊D3×#131C311yesmaximal, characteristic, complemented
9C3×D3×#131—31nomaximal
10#41×H(3,𝔽3)C2211yescharacteristic
11#88722C411yescharacteristic
12Q8×(C3⋊D3)⋊C3C411yescharacteristic
13#3830⋊C3C413yes—
14D4×(C3⋊D3)⋊C3C413yes—
15#3828⋊C3C2213yes—
16C2×C4×(C3⋊D3)⋊C3C2213yes—
17M16×H(3,𝔽3)C2213yescomplemented
18C2×C8×H(3,𝔽3)C2213yescomplemented
19#3748⋊C3C2213yescomplemented
20C2×(C3⋊(C3⋊C8))⋊C3C2213yescomplemented
21#3743⋊C3C2216yescomplemented
22C8×(C3⋊D3)⋊C3C2211yescharacteristic
23C8×(C3⋊D3)⋊C3C2213yes—
24C8×(C3⋊D3)⋊C3C2213yescomplemented
25C8×(C3⋊D3)⋊C3C2213yescomplemented
26#41×C3⋊D3C2×C311yescharacteristic
27C3×D3×#41—31no—
28C32×#131—21no—
29C32×#131D311yescharacteristic, complemented
30C32×#131C2×C311yescharacteristic, complemented
31#64157C2×C311yescharacteristic, complemented
32#64116C2×C313yescomplemented
33M16×C3⋊D3C2×C313yescomplemented
34#64110C2×C313yescomplemented
35C2×C8×C3⋊D3C2×C313yescomplemented
36C3×#948—31no—
37C3×#907—33no—
38C3×D3×M16—33no—
39C3×#901—33no—
40C2×C3×D3×C8—33no—
41C22×(C3⋊D3)⋊C3C2×C413yes—
42Q8×H(3,𝔽3)C2×C411yescharacteristic
43D4×H(3,𝔽3)C2×C413yes—
44C2×C4×H(3,𝔽3)C2313yes—
45#638⋊C3C2×C416yes—
46C2×(C3⋊Dic3)⋊C3C2×C413yes—
47C4⋊((C3⋊D3)⋊C3)C2×C413yes—
48C4×(C3⋊D3)⋊C3C2×C413yes—
49C4×(C3⋊D3)⋊C3C2×C413yes—
50C4×(C3⋊D3)⋊C3C2311yescharacteristic
51C4×(C3⋊D3)⋊C3C2313yes—
52#634⋊C3C2×C413yes—
53C8×H(3,𝔽3)C2311yescharacteristic
54C8×H(3,𝔽3)C2313yes—
55(C3⋊(C3⋊C8))⋊C3C2311yescharacteristic
56(C3⋊(C3⋊C8))⋊C3C2313yes—
57C2×C3×#131—91no—
58D3×#131—31no—
59D3×#131—91no—
60C32×#41—21no—
61C32×#41C2×D311yescharacteristic
62C32×#41C22×C311yescharacteristic
63#3832C3×C411yescharacteristic
64Q8×C3⋊D3C3×C411yescharacteristic
65#3830C3×C413yes—
66D4×C3⋊D3C3×C413yes—
67#3828C22×C313yes—
68C2×C4×C3⋊D3C22×C313yes—
69C3×#239—31no—
70C3×D3×Q8—31no—
71C3×#237—33no—
72C3×D3×D4—33no—
73C3×#235—33no—
74C2×C3×C4×D3—33no—
75C32×M16—23no—
76C32×M16C2×D313yescomplemented
77C32×M16C22×C313yescomplemented
78C2×C32×C8—23no—
79C2×C32×C8C2×D313yescomplemented
80C2×C32×C8C22×C313yescomplemented
81#3748C22×C313yescomplemented
82C2×C3⋊(C3⋊C8)C22×C313yescomplemented
83#3743C22×C316yescomplemented
84C8×C3⋊D3C22×C311yescharacteristic
85C8×C3⋊D3C22×C313yes—
86C8×C3⋊D3C22×C313yescomplemented
87C8×C3⋊D3C22×C313yescomplemented
88C3×#208—33no—
89C2×C3×C3⋊C8—33no—
90C3×ΓL(1,𝔽25)—36no—
91C3×D3×C8—31no—
92C3×D3×C8—33no—
93C3×D3×C8—33no—
94C3×D3×C8—33no—
95C22×H(3,𝔽3)C22×C413yes—
96C2×(C3⋊D3)⋊C3—26no—
97C2×(C3⋊D3)⋊C3C22×C413yes—
98C2×(C3⋊D3)⋊C3C22×C411yescharacteristic
99C4×H(3,𝔽3)C22×C413yes—
100C4×H(3,𝔽3)C2411yescharacteristic
101(C3⋊Dic3)⋊C3C22×C413yes—
102(C3⋊Dic3)⋊C3C22×C411yescharacteristic
103C2×C3×#41—91no—
104D3×#41—31no—
105D3×#41—91no—
106C3×#131—31no—
107C3×#131—31no—
108C3×#131—61no—
109C3×#131—93no—
110C3×#131—93no—
111C3×#131—91no—
112C3×#131C3×D311yescharacteristic
113C2×C3×M16—93no—
114C22×C3×C8—93no—
115#948—31no—
116#948—91no—
117#907—33no—
118#907—93no—
119D3×M16—33no—
120D3×M16—93no—
121#901—33no—
122#901—93no—
123C2×D3×C8—33no—
124C2×D3×C8—93no—
125C22×C3⋊D3C2×C3×C413yes—
126C22×C3×D3—33no—
127C32×Q8—21no—
128C32×Q8C4×D311yescharacteristic
129C32×Q8C2×C3×C411yescharacteristic
130C32×D4—23no—
131C32×D4C4×D313yes—
132C32×D4C2×C3×C413yes—
133C2×C32×C4—23no—
134C2×C32×C4C22×D313yes—
135C2×C32×C4C23×C313yes—
136#638C2×C3×C416yes—
137C2×C3⋊Dic3C2×C3×C413yes—
138C4⋊(C3⋊D3)C2×C3×C413yes—
139C4×C3⋊D3C2×C3×C413yes—
140C4×C3⋊D3C2×C3×C413yes—
141C4×C3⋊D3C23×C311yescharacteristic
142C4×C3⋊D3C23×C313yes—
143#634C2×C3×C413yes—
144C3×#67—36no—
145C2×C3×Dic3—33no—
146C3×D12—33no—
147C3×C4×D3—33no—
148C3×C4×D3—33no—
149C3×C4×D3—31no—
150C3×C4×D3—33no—
151C3×Dic6—33no—
152C32×C8—21no—
153C32×C8—23no—
154C32×C8C22×D311yescharacteristic
155C32×C8C22×D313yes—
156C32×C8C23×C311yescharacteristic
157C32×C8C23×C313yes—
158C3⋊(C3⋊C8)C23×C311yescharacteristic
159C3⋊(C3⋊C8)C23×C313yes—
160C3×C3⋊C8—31no—
161C3×C3⋊C8—33no—
162C2×#131—91noSylow
163C2×H(3,𝔽3)—23no—
164C2×H(3,𝔽3)C23×C411yescharacteristic
165(C3⋊D3)⋊C3—23no—
166(C3⋊D3)⋊C3#13112yesdirect product factor
167C3×#41—31no—
168C3×#41—31no—
169C3×#41—61no—
170C3×#41—93no—
171C3×#41—93no—
172C3×#41—91no—
173C3×#41C2×C3×D311yescharacteristic
174C2×C3×Q8—91no—
175C2×C3×D4—93no—
176C22×C3×C4—93no—
177#239—31no—
178#239—91no—
179D3×Q8—31no—
180D3×Q8—91no—
181#237—33no—
182#237—93no—
183D3×D4—33no—
184D3×D4—93no—
185#235—33no—
186#235—93no—
187C2×C4×D3—33no—
188C2×C4×D3—93no—
189C3×M16—33no—
190C3×M16—33no—
191C3×M16—63no—
192C3×M16—96no—
193C3×M16—93no—
194C3×M16C2×C3×D313yes—
195C2×C3×C8—33no—
196C2×C3×C8—33no—
197C2×C3×C8—63no—
198C2×C3×C8—91no—
199C2×C3×C8—93no—
200C2×C3×C8—93no—
201C2×C3×C8—93no—
202C2×C3×C8—93no—
203C2×C3×C8C2×C3×D313yes—
204#208—33no—
205#208—93no—
206C2×C3⋊C8—33no—
207C2×C3⋊C8—93no—
208ΓL(1,𝔽25)—36no—
209ΓL(1,𝔽25)—96no—
210D3×C8—31no—
211D3×C8—33no—
212D3×C8—33no—
213D3×C8—33no—
214D3×C8—91no—
215D3×C8—93no—
216D3×C8—93no—
217D3×C8—93no—
218C22×C32—23no—
219C22×C32C2×C4×D313yes—
220C22×C32C22×C3×C413yes—
221C2×C3⋊D3—26no—
222C2×C3⋊D3C22×C3×C413yes—
223C2×C3⋊D3C22×C3×C411yescharacteristic
224C2×C3×D3—31no—
225C2×C3×D3—33no—
226C2×C3×D3—66no—
227C32×C4—21no—
228C32×C4—23no—
229C32×C4C2×C4×D313yes—
230C32×C4C22×C3×C413yes—
231C32×C4C23×D311yescharacteristic
232C32×C4C24×C311yescharacteristic
233C3⋊Dic3C22×C3×C413yes—
234C3⋊Dic3C22×C3×C411yescharacteristic
235C3×Dic3—33no—
236C3×Dic3—31no—
237C2×#41—91no—
238#131—91no—
239#131—93no—
240#131—93no—
241#131(C3⋊D3)⋊C311yescharacteristic, direct product factor
242C2×M16—93no—
243C22×C8—93no—
244H(3,𝔽3)C2×#13111yescharacteristic, complemented, Sylow
245C23×C3—93no—
246C22×D3—33no—
247C22×D3—93no—
248C3×Q8—31no—
249C3×Q8—31no—
250C3×Q8—61no—
251C3×Q8—93no—
252C3×Q8C3×C4×D311yescharacteristic
253C3×D4—33no—
254C3×D4—33no—
255C3×D4—63no—
256C3×D4—93no—
257C3×D4—96no—
258C3×D4C3×C4×D313yes—
259C2×C3×C4—33no—
260C2×C3×C4—33no—
261C2×C3×C4—63no—
262C2×C3×C4—91no—
263C2×C3×C4—93no—
264C2×C3×C4—93no—
265C2×C3×C4—93no—
266C2×C3×C4—93no—
267C2×C3×C4C22×C3×D313yes—
268#67—36no—
269#67—96no—
270C2×Dic3—33no—
271C2×Dic3—93no—
272D12—33no—
273D12—93no—
274C4×D3—31no—
275C4×D3—33no—
276C4×D3—33no—
277C4×D3—33no—
278C4×D3—93no—
279C4×D3—93no—
280C4×D3—91no—
281C4×D3—93no—
282Dic6—33no—
283Dic6—93no—
284C3×C8—33no—
285C3×C8—33no—
286C3×C8—31no—
287C3×C8—31no—
288C3×C8—63no—
289C3×C8—61no—
290C3×C8—91no—
291C3×C8—93no—
292C3×C8C22×C3×D313yes—
293C3×C8C22×C3×D311yescharacteristic
294C3⋊C8—31no—
295C3⋊C8—33no—
296C3⋊C8—91no—
297C3⋊C8—93no—
298C2×C32—23no—
299C2×C32—23no—
300C2×C32—21no—
301C2×C32—43no—
302C2×C32C22×C4×D311yescharacteristic
303C2×C32C23×C3×C411yescharacteristic
304C3⋊D3—23no—
305C3⋊D3C3×#13112yescomplemented
306C3×D3—32no—
307C3×D3—63no—
308#41—91no—
309#41—93no—
310#41—93no—
311#41C2×(C3⋊D3)⋊C311yescharacteristic
312C2×Q8—91no—
313C2×D4—93no—
314C22×C4—93no—
315M16—93no—
316M16—96no—
317M16C2×(C3⋊D3)⋊C313yescomplemented
318C2×C8—93no—
319C2×C8—93no—
320C2×C8—93no—
321C2×C8—91no—
322C2×C8—93no—
323C2×C8C2×(C3⋊D3)⋊C313yescomplemented
324C22×C3—33no—
325C22×C3—33no—
326C22×C3—63no—
327C22×C3—91no—
328C22×C3—93no—
329C22×C3—186no—
330C22×C3C2×C3×C4×D313yes—
331C2×D3—31no—
332C2×D3—33no—
333C2×D3—66no—
334C2×D3—91no—
335C2×D3—93no—
336C2×D3—186no—
337C3×C4—31no—
338C3×C4—31no—
339C3×C4—33no—
340C3×C4—33no—
341C3×C4—61no—
342C3×C4—63no—
343C3×C4—91no—
344C3×C4—93no—
345C3×C4C2×C3×C4×D313yes—
346C3×C4C23×C3×D311yescharacteristic
347Dic3—33no—
348Dic3—31no—
349Dic3—93no—
350Dic3—91no—
351C32—21no—
352C32D3×#13111yescharacteristic, complemented
353C32C2×C3×#13111yescharacteristic, complemented
354C23—93no—
355Q8—93no—
356Q8C4×(C3⋊D3)⋊C311yescharacteristic
357D4—96no—
358D4—93no—
359D4C4×(C3⋊D3)⋊C313yes—
360C2×C4—93no—
361C2×C4—91no—
362C2×C4—93no—
363C2×C4—93no—
364C2×C4—93no—
365C2×C4C22×(C3⋊D3)⋊C313yes—
366C8—91no—
367C8—93no—
368C8C22×(C3⋊D3)⋊C313yes—
369C8C22×(C3⋊D3)⋊C311yescharacteristic
370C2×C3—23no—
371C2×C3—31no—
372C2×C3—31no—
373C2×C3—63no—
374C2×C3—63no—
375C2×C3—61no—
376C2×C3—92no—
377C2×C3—123no—
378C2×C3—183no—
379C2×C3C22×C3×C4×D311yescharacteristic
380D3—32no—
381D3—63no—
382D3—92no—
383D3—183no—
384C22—91no—
385C22—93no—
386C22—186no—
387C22C2×C4×(C3⋊D3)⋊C313yes—
388C4—91no—
389C4—93no—
390C4C2×C4×(C3⋊D3)⋊C313yes—
391C4C23×(C3⋊D3)⋊C311yescharacteristic
392C3—31no—
393C3—31no—
394C3—61no—
395C3C3×D3×#13111yescharacteristic
396C2—23no—
397C2—92no—
398C2—183no—
399C2C22×C4×(C3⋊D3)⋊C311yescharacteristic
subgroupG/H
centerC8C22×(C3⋊D3)⋊C3
commutatorC2×C32C23×C3×C4
FrattiniC3×C4C23×C3×D3
FittingH(3,𝔽3)×#131C2
socleC2×C3C22×C3×C4×D3
prime radical#41×(C3⋊D3)⋊C3C2
cosocleC32×C4C24×C3
hypercenter#131(C3⋊D3)⋊C3
hypocenterC32C2×C3×#131
LCSG ▷ C2×C32 ▷ C32
UCSC1 ◁ C8 ◁ #131

Complements

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

NH ≅ G/Ntype
2. H(3,𝔽3)×#131C2semidirect
3. #11721689C2semidirect
4. #64116⋊C3C2semidirect
5. M16×(C3⋊D3)⋊C3C2semidirect
6. #64110⋊C3C2semidirect
7. C2×C8×(C3⋊D3)⋊C3C2semidirect
8. C3⋊D3×#131C3semidirect
17. M16×H(3,𝔽3)C22semidirect
18. C2×C8×H(3,𝔽3)C22semidirect
19. #3748⋊C3C22semidirect
20. C2×(C3⋊(C3⋊C8))⋊C3C22semidirect
21. #3743⋊C3C22semidirect
24. C8×(C3⋊D3)⋊C3C22semidirect
25. C8×(C3⋊D3)⋊C3C22semidirect
29. C32×#131D3semidirect
30. C32×#131C2×C3semidirect
31. #64157C2×C3semidirect
32. #64116C2×C3semidirect
33. M16×C3⋊D3C2×C3semidirect
34. #64110C2×C3semidirect
35. C2×C8×C3⋊D3C2×C3semidirect
76. C32×M16C2×D3semidirect
77. C32×M16C22×C3semidirect
79. C2×C32×C8C2×D3semidirect
80. C2×C32×C8C22×C3semidirect
81. #3748C22×C3semidirect
82. C2×C3⋊(C3⋊C8)C22×C3semidirect
83. #3743C22×C3semidirect
86. C8×C3⋊D3C22×C3semidirect
87. C8×C3⋊D3C22×C3semidirect
166. (C3⋊D3)⋊C3#131direct (N × H)
241. #131(C3⋊D3)⋊C3direct (N × H)
244. H(3,𝔽3)C2×#131semidirect
305. C3⋊D3C3×#131semidirect
317. M16C2×(C3⋊D3)⋊C3semidirect
323. C2×C8C2×(C3⋊D3)⋊C3semidirect
352. C32D3×#131semidirect
353. C32C2×C3×#131semidirect

Maximal subgroups

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

subgroupindex
#41×(C3⋊D3)⋊C32
H(3,𝔽3)×#1312
#117216892
#64116⋊C32
M16×(C3⋊D3)⋊C32
#64110⋊C32
C2×C8×(C3⋊D3)⋊C32
C3⋊D3×#1313
C3×D3×#1313