Finite Group Databasebeta

← C3×#10596459 #13180718

Subgroups — C3×#10596459 #13180718

Order 1728.

709 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 709 conjugacy classes of subgroups — too many to draw. Only its 74 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#1010⋊C3C211yesmaximal, characteristic, complemented
2C3×#64267C211yesmaximal, characteristic, complemented
3C32×#985C211yesmaximal, characteristic, complemented
4C3×D3×GL(2,𝔽3)C212yesmaximal, complemented
5C3×#64202C212yesmaximal, complemented
6C3×#5926—31nomaximal
7#10596459C311yesmaximal, characteristic, direct product factor
8C3×#5770—31nomaximal
9C32×#41⋊C3C2211yescharacteristic, complemented
10C3×D3×D12—41nomaximal
11C3×D3×SL(2,𝔽3)C2212yescomplemented
12C3×C3⋊GL(2,𝔽3)C2212yescomplemented
13C32×GL(2,𝔽3)C2211yescharacteristic, complemented
14C32×#226C2211yescharacteristic, complemented
15C3×#1010D311yescharacteristic, complemented
16C3×#1009—31no—
17C3×2−1+4⋊C3—31no—
18#1010⋊C3—21no—
19#1010⋊C3C2×C311yescharacteristic, complemented
20#64267C2×C311yescharacteristic, complemented
21C3×#986—31no—
22C3×#986—61no—
23C3×#985—21no—
24C3×#985D311yescharacteristic, complemented
25C3×#985C2×C311yescharacteristic, complemented
26C3×#983—32no—
27C2×C3×GL(2,𝔽3)—32no—
28D3×GL(2,𝔽3)C2×C312yescomplemented
29#64202C2×C312yescomplemented
30C32×#135—31no—
31C3×#951—31no—
32C3×#949—31no—
33C3×#918—32no—
34C3×#916—32no—
35C3×D3×SD16—32no—
36C3×#911—32no—
37C2×C3×#204—31no—
38C3×#901—31no—
39C2×C3×D32—42no—
40C3×C4⋊(C3⋊D3)—41no—
41C32×D12—41no—
42C32×C4×D3—41no—
43C32×SL(2,𝔽3)C2311yescharacteristic
44C3×#626—42no—
45#5926—31no—
46C3×#577—91no—
47#5770—31no—
48C32×#41—31no—
49C32×#41C2×D311yescharacteristic, complemented
50C3×#239—32no—
51C3×D3×Q8C2×D312yescomplemented
52C3×#237—32no—
53C3×D3×D4—32no—
54C3×D3×D4—62no—
55C3×D3×D4—121no—
56C3×#235—31no—
57C3×#235—61no—
58C2×C3×D12—31no—
59C2×C3×D12—61no—
60C2×C3×Dic6—31no—
61C3×#41⋊C3—21no—
62C3×#41⋊C3—21no—
63C3×#41⋊C3—21no—
64C3×#41⋊C3—41no—
65C3×#41⋊C3C2×D311yescharacteristic, complemented
66C3×#41⋊C3C22×C311yescharacteristic, complemented
67C2×C3×SL(2,𝔽3)—32no—
68D3×D12—41no—
69D3×SL(2,𝔽3)—22no—
70D3×SL(2,𝔽3)C22×C312yescomplemented
71C3⋊GL(2,𝔽3)C22×C312yescomplemented
72C3×GL(2,𝔽3)—21no—
73C3×GL(2,𝔽3)—32no—
74C3×GL(2,𝔽3)—62no—
75C3×GL(2,𝔽3)C2×D311yescharacteristic, complemented
76C3×GL(2,𝔽3)C22×C311yescharacteristic, complemented
77C3×#226—21no—
78C3×#226C2×D311yescharacteristic, complemented
79C3×#226C22×C311yescharacteristic, complemented
80C32×Q16—31no—
81C32×SD16—31no—
82C32×SD16—31no—
83C32×D8—31no—
84C2×C32×C8—31no—
85C3×#216—32no—
86C3×#215—32no—
87C3×#214—32no—
88C3×#213—32no—
89C3×#208—31no—
90C3×#204—32no—
91C3×#204—32no—
92C3×ΓL(1,𝔽25)—32no—
93C3×D3×C8—32no—
94C2×C3×C3⋊D3—42no—
95C2×C32×D3—41no—
96C2×C32×D3—42no—
97C3×D32—84no—
98C33×C4—41no—
99C32×Dic3—41no—
100C3×2−1+4—31no—
101C3×H(5,𝔽2)—91no—
102#1010C3×D311yescharacteristic, complemented
103#1009—31no—
1042−1+4⋊C3—31no—
1052−1+4⋊C3—61no—
106#986—31no—
107#986—61no—
108#985C3×D311yescharacteristic, complemented
109#983—32no—
110C2×GL(2,𝔽3)—32no—
111C3×#137—92no—
112C3×#137—91no—
113C3×AGL(1,ℤ/8)—92no—
114C3×AGL(1,ℤ/8)—91no—
115C3×#135—31no—
116C3×#135—31no—
117C3×#135—61no—
118C3×#135—92no—
119C2×C3×SD16—91no—
120C2×C3×SD16—92no—
121C3×#131—91no—
122#951—31no—
123#949—31no—
124#918—32no—
125#916—32no—
126D3×SD16—32no—
127#911—32no—
128C2×#204—31no—
129#901—31no—
130C22×C3×D3—61no—
131C22×C3×D3—62no—
132C22×C3×D3—122no—
133C2×D32—42no—
134C32×Q8—31no—
135C32×Q8C22×D311yescharacteristic
136C32×D4—31no—
137C32×D4—31no—
138C32×D4—61no—
139C32×D4—121no—
140C2×C32×C4—31no—
141C2×C32×C4—61no—
142C4⋊(C3⋊D3)—41no—
143C3×#67—32no—
144C3×#67—62no—
145C3×#67—122no—
146C2×C3×Dic3—61no—
147C3×D12—32no—
148C3×D12—41no—
149C3×D12—62no—
150C3×D12—61no—
151C3×D12—81no—
152C3×D12—121no—
153C3×D12—241no—
154C3×D12S411yescharacteristic
155C3×C4×D3—32no—
156C3×C4×D3—41no—
157C3×C4×D3—41no—
158C3×C4×D3—62no—
159C3×C4×D3—81no—
160C3×Dic6—31no—
161C3×Dic6—32no—
162C3×SL(2,𝔽3)—21no—
163C3×SL(2,𝔽3)—21no—
164C3×SL(2,𝔽3)—21no—
165C3×SL(2,𝔽3)—41no—
166C3×SL(2,𝔽3)C22×D311yescharacteristic
167C3×SL(2,𝔽3)C23×C311yescharacteristic
168#626—42no—
169C32×C8—31no—
170C32×C8—31no—
171C3×C3⋊C8—32no—
172#577—91noSylow
173C2×C33—41no—
174C3×C3⋊D3—82no—
175C32×D3—42no—
176C32×D3—82no—
177C3×#41—21no—
178C3×#41—31no—
179C3×#41—31no—
180C3×#41—61no—
181C3×#41—91no—
182C3×#41—92no—
183C3×#41—92no—
184C3×#41—181no—
185C3×#41D3211yescharacteristic, complemented
186C3×#41C2×C3×D311yescharacteristic, complemented
187C2×C3×Q8—32no—
188C2×C3×Q8—91no—
189C2×C3×D4—91no—
190C2×C3×D4—92no—
191C2×C3×D4—181no—
192C2×C3×D4—182no—
193#239—32no—
194D3×Q8C2×C3×D312yescomplemented
195#237—32no—
196D3×D4—32no—
197D3×D4—62no—
198D3×D4—121no—
199#235—31no—
200#235—61no—
201C2×D12—31no—
202C2×D12—61no—
203C2×Dic6—31no—
204#41⋊C3—21no—
205#41⋊C3—21no—
206#41⋊C3—41no—
207#41⋊C3C2×C3×D311yescharacteristic, complemented
208C2×SL(2,𝔽3)—32no—
209C2×SL(2,𝔽3)—62no—
210GL(2,𝔽3)—32no—
211GL(2,𝔽3)—62no—
212GL(2,𝔽3)C2×C3×D311yescharacteristic, complemented
213#226C2×C3×D311yescharacteristic, complemented
214C3×Q16—31no—
215C3×Q16—31no—
216C3×Q16—61no—
217C3×Q16—92no—
218C3×SD16—31no—
219C3×SD16—31no—
220C3×SD16—31no—
221C3×SD16—31no—
222C3×SD16—61no—
223C3×SD16—61no—
224C3×SD16—92no—
225C3×SD16—92no—
226C3×SD16—92no—
227C3×SD16—92no—
228C3×D8—31no—
229C3×D8—31no—
230C3×D8—61no—
231C3×D8—92no—
232C3×M16—92no—
233C3×M16—91no—
234C2×C3×C8—31no—
235C2×C3×C8—31no—
236C2×C3×C8—61no—
237C2×C3×C8—92no—
238#216—32no—
239#215—32no—
240#214—32no—
241#213—32no—
242#208—31no—
243#204—32no—
244#204—32no—
245ΓL(1,𝔽25)—32no—
246D3×C8—32no—
247C22×C32—31no—
248C22×C32—61no—
249C22×C32—122no—
250C2×C3⋊D3—42no—
251C2×C3×D3—41no—
252C2×C3×D3—41no—
253C2×C3×D3—42no—
254C2×C3×D3—62no—
255C2×C3×D3—81no—
256C2×C3×D3—82no—
257C2×C3×D3—122no—
258C2×C3×D3—242no—
259C2×C3×D3—242no—
260C2×C3×D3—244no—
261C2×C3×D3—241no—
262C2×C3×D3C2×S412yes—
263D32—84no—
264C32×C4—31no—
265C32×C4—41no—
266C32×C4—41no—
267C32×C4—61no—
268C32×C4—81no—
269C32×C4—81no—
270C32×C4—81no—
271C32×C4—161no—
272C32×C4C2×S411yescharacteristic
273C3×Dic3—32no—
274C3×Dic3—41no—
275C3×Dic3—41no—
276C3×Dic3—81no—
2772−1+4—31no—
278H(5,𝔽2)—91no—
279#137—92no—
280#137—91no—
281AGL(1,ℤ/8)—92no—
282AGL(1,ℤ/8)—91no—
283#135—31no—
284#135—92no—
285C2×SD16—91no—
286C2×SD16—92no—
287#131—91no—
288C33—41noSylow
289C23×C3—182no—
290C23×C3—181no—
291C22×D3—61no—
292C22×D3—62no—
293C22×D3—122no—
294C3×Q8—21no—
295C3×Q8—31no—
296C3×Q8—31no—
297C3×Q8—61no—
298C3×Q8—92no—
299C3×Q8—91no—
300C3×Q8C2×D3211yescharacteristic
301C3×Q8C22×C3×D311yescharacteristic
302C3×D4—31no—
303C3×D4—31no—
304C3×D4—31no—
305C3×D4—31no—
306C3×D4—31no—
307C3×D4—61no—
308C3×D4—61no—
309C3×D4—61no—
310C3×D4—61no—
311C3×D4—92no—
312C3×D4—92no—
313C3×D4—121no—
314C3×D4—121no—
315C3×D4—182no—
316C3×D4—182no—
317C3×D4—181no—
318C3×D4—241no—
319C2×C3×C4—31no—
320C2×C3×C4—31no—
321C2×C3×C4—61no—
322C2×C3×C4—61no—
323C2×C3×C4—61no—
324C2×C3×C4—92no—
325C2×C3×C4—121no—
326C2×C3×C4—182no—
327C2×C3×C4—181no—
328#67—32no—
329#67—62no—
330#67—122no—
331C2×Dic3—61no—
332D12—32no—
333D12—62no—
334D12—61no—
335D12—121no—
336D12—241no—
337D12C3×S411yescharacteristic
338C4×D3—32no—
339C4×D3—41no—
340C4×D3—62no—
341Dic6—31no—
342Dic6—32no—
343SL(2,𝔽3)—21no—
344SL(2,𝔽3)—21no—
345SL(2,𝔽3)—41no—
346SL(2,𝔽3)C22×C3×D311yescharacteristic
347C3×C8—31no—
348C3×C8—31no—
349C3×C8—31no—
350C3×C8—31no—
351C3×C8—61no—
352C3×C8—61no—
353C3×C8—92no—
354C3⋊C8—32no—
355C2×C32—41no—
356C2×C32—41no—
357C2×C32—61no—
358C2×C32—81no—
359C2×C32—81no—
360C2×C32—81no—
361C2×C32—121no—
362C2×C32—161no—
363C2×C32—242no—
364C2×C32C22×S411yescharacteristic
365C3⋊D3—82no—
366C3×D3—22no—
367C3×D3—42no—
368C3×D3—42no—
369C3×D3—82no—
370C3×D3—82no—
371C3×D3—122no—
372C3×D3—162no—
373C3×D3—242no—
374C3×D3—482no—
375#41—31no—
376#41—91no—
377#41—92no—
378#41—92no—
379#41—181no—
380#41C3×D3211yescharacteristic, complemented
381C2×Q8—32no—
382C2×Q8—91no—
383C2×D4—91no—
384C2×D4—92no—
385C2×D4—181no—
386C2×D4—182no—
387Q16—31no—
388Q16—92no—
389SD16—31no—
390SD16—31no—
391SD16—92no—
392SD16—92no—
393SD16—92no—
394SD16—92no—
395D8—31no—
396D8—92no—
397M16—92no—
398M16—91no—
399C2×C8—31no—
400C2×C8—92no—
401C22×C3—31no—
402C22×C3—32no—
403C22×C3—31no—
404C22×C3—61no—
405C22×C3—61no—
406C22×C3—61no—
407C22×C3—122no—
408C22×C3—121no—
409C22×C3—182no—
410C22×C3—242no—
411C22×C3—722no—
412C22×C3—721no—
413C2×D3—41no—
414C2×D3—62no—
415C2×D3—122no—
416C2×D3—242no—
417C2×D3—242no—
418C2×D3—244no—
419C2×D3—241no—
420C2×D3C2×C3×S412yes—
421C3×C4—21no—
422C3×C4—31no—
423C3×C4—31no—
424C3×C4—41no—
425C3×C4—61no—
426C3×C4—61no—
427C3×C4—61no—
428C3×C4—81no—
429C3×C4—81no—
430C3×C4—92no—
431C3×C4—121no—
432C3×C4—161no—
433C3×C4D3×S411yescharacteristic
434C3×C4C2×C3×S411yescharacteristic
435Dic3—32no—
436Dic3—41no—
437C32—41no—
438C32—41no—
439C32—81no—
440C32—81no—
441C32—81no—
442C32—161no—
443C32#592611yescharacteristic, complemented
444C23—182no—
445C23—181no—
446Q8—31no—
447Q8—92no—
448Q8—91no—
449Q8C2×C3×D3211yescharacteristic
450D4—31no—
451D4—31no—
452D4—31no—
453D4—61no—
454D4—92no—
455D4—92no—
456D4—182no—
457D4—182no—
458D4—181no—
459C2×C4—31no—
460C2×C4—61no—
461C2×C4—92no—
462C2×C4—182no—
463C2×C4—181no—
464C8—31no—
465C8—31no—
466C8—92no—
467C2×C3—21no—
468C2×C3—41no—
469C2×C3—62no—
470C2×C3—61no—
471C2×C3—61no—
472C2×C3—81no—
473C2×C3—81no—
474C2×C3—121no—
475C2×C3—121no—
476C2×C3—121no—
477C2×C3—161no—
478C2×C3—242no—
479C2×C3—241no—
480C2×C3—362no—
481C2×C3—482no—
482C2×C3C2×D3×S411yescharacteristic
483C2×C3C22×C3×S411yescharacteristic
484D3—22no—
485D3—42no—
486D3—122no—
487D3—242no—
488D3—482no—
489C22—31no—
490C22—32no—
491C22—61no—
492C22—182no—
493C22—722no—
494C22—721no—
495C4—31no—
496C4—61no—
497C4—92no—
498C4C3×D3×S411yescharacteristic
499C3—21no—
500C3—41no—
501C3—81no—
502C3—81no—
503C3—161no—
504C3#1059645911yescharacteristic, direct product factor
505C3C3×#592611yescharacteristic, complemented
506C2—62no—
507C2—61no—
508C2—121no—
509C2—362no—
510C2C2×C3×D3×S411yescharacteristic
subgroupG/H
centerC2×C3C2×D3×S4
commutator, cosocle, hypocenterC3×SL(2,𝔽3)C23×C3
FrattiniC2C2×C3×D3×S4
FittingC32×#41C2×D3
socleC2×C32C22×S4
hypercenterC3×C4D3×S4
LCSG ▷ C3×SL(2,𝔽3)
UCSC1 ◁ C2×C3 ◁ C3×C4

Complements

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

NH ≅ G/Ntype
1. C3×#1010⋊C3C2semidirect
2. C3×#64267C2semidirect
3. C32×#985C2semidirect
4. C3×D3×GL(2,𝔽3)C2semidirect
5. C3×#64202C2semidirect
7. #10596459C3direct (N × H)
9. C32×#41⋊C3C22semidirect
11. C3×D3×SL(2,𝔽3)C22semidirect
12. C3×C3⋊GL(2,𝔽3)C22semidirect
13. C32×GL(2,𝔽3)C22semidirect
14. C32×#226C22semidirect
15. C3×#1010D3semidirect
19. #1010⋊C3C2×C3semidirect
20. #64267C2×C3semidirect
24. C3×#985D3semidirect
25. C3×#985C2×C3semidirect
28. D3×GL(2,𝔽3)C2×C3semidirect
29. #64202C2×C3semidirect
49. C32×#41C2×D3semidirect
51. C3×D3×Q8C2×D3semidirect
65. C3×#41⋊C3C2×D3semidirect
66. C3×#41⋊C3C22×C3semidirect
70. D3×SL(2,𝔽3)C22×C3semidirect
71. C3⋊GL(2,𝔽3)C22×C3semidirect
75. C3×GL(2,𝔽3)C2×D3semidirect
76. C3×GL(2,𝔽3)C22×C3semidirect
78. C3×#226C2×D3semidirect
79. C3×#226C22×C3semidirect
102. #1010C3×D3semidirect
108. #985C3×D3semidirect
185. C3×#41D32semidirect
186. C3×#41C2×C3×D3semidirect
194. D3×Q8C2×C3×D3semidirect
207. #41⋊C3C2×C3×D3semidirect
212. GL(2,𝔽3)C2×C3×D3semidirect
213. #226C2×C3×D3semidirect
380. #41C3×D32semidirect
443. C32#5926semidirect
504. C3#10596459direct (N × H)
505. C3C3×#5926semidirect

Maximal subgroups

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

subgroupindex
C3×#1010⋊C32
C3×#642672
C32×#9852
C3×D3×GL(2,𝔽3)2
C3×#642022
C3×#59263
#105964593
C3×#57703
C3×D3×D124