Finite Group Databasebeta

← C3×#10596589 #13181389

Subgroups — C3×#10596589 #13181389

Order 1728.

904 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 904 conjugacy classes of subgroups — too many to draw. Only its 106 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
1C32×2−1+4⋊C3C211yesmaximal, characteristic, complemented
2C3×#64267C211yesmaximal, characteristic, complemented
3C3×#64266C211yesmaximal, characteristic, complemented
4C3×#64264C212yesmaximal, complemented
5C2×C3×C3⋊GL(2,𝔽3)C212yesmaximal, complemented
6#10596589C311yesmaximal, characteristic, direct product factor
7C3×#5926—33nomaximal
8C3×#5837—31nomaximal
9C3×D4×C3⋊D3—41nomaximal
10C32×#41⋊C3C2211yescharacteristic, complemented
11C2×C32×SL(2,𝔽3)C2212yescomplemented
12C3×C3⋊GL(2,𝔽3)C2211yescharacteristic, complemented
13C3×C3⋊GL(2,𝔽3)C2212yescomplemented
14C3×C3⋊#226C2211yescharacteristic, complemented
15C32×2−1+4D311yescharacteristic, complemented
16C3×#1009—31no—
17C3×2−1+4⋊C3—21no—
18C3×2−1+4⋊C3—23no—
19C3×2−1+4⋊C3D313yescomplemented
20C3×2−1+4⋊C3C2×C311yescharacteristic, complemented
21#64267C2×C311yescharacteristic, complemented
22#64266C2×C311yescharacteristic, complemented
23#64264C2×C312yescomplemented
24C2×C3⋊GL(2,𝔽3)C2×C312yescomplemented
25C3×#986—33no—
26C3×#985—33no—
27C3×#983—36no—
28C2×C3×GL(2,𝔽3)—36no—
29C3×#950—32no—
30C3×#950—31no—
31C3×#949—32no—
32C3×#949—31no—
33C3×#948—31no—
34C3×#942—31no—
35C3×#942—32no—
36C2×C3×#215—31no—
37C2×C3×#215—32no—
38C22×C3×C3⋊D3—42no—
39C33×D4—41no—
40C3×#638—42no—
41C3×C4⋊(C3⋊D3)—41no—
42C3×C4×C3⋊D3—41no—
43C32×SL(2,𝔽3)C2311yescharacteristic
44#5926—33no—
45C3×#577—91no—
46#5837—31no—
47C32×#41—31no—
48C32×#41—32no—
49C32×#41C2×D311yescharacteristic, complemented
50C2×C32×Q8—31no—
51C2×C32×Q8C2×D312yescomplemented
52D4×C3⋊D3—41no—
53C3×#239—61no—
54C3×D3×D4—62no—
55C3×D3×D4—61no—
56C3×D3×D4—123no—
57C3×#235—32no—
58C3×#235—31no—
59C2×C3×D12—32no—
60C2×C3×D12—31no—
61C3×#41⋊C3—21no—
62C3×#41⋊C3—23no—
63C3×#41⋊C3C2×D313yescomplemented
64C3×#41⋊C3C22×C311yescharacteristic, complemented
65C2×C3×SL(2,𝔽3)—22no—
66C2×C3×SL(2,𝔽3)—26no—
67C2×C3×SL(2,𝔽3)C2×D316yescomplemented
68C2×C3×SL(2,𝔽3)C22×C312yescomplemented
69C3⋊GL(2,𝔽3)C22×C311yescharacteristic, complemented
70C3⋊GL(2,𝔽3)C22×C312yescomplemented
71C3⋊#226C22×C311yescharacteristic, complemented
72C3×GL(2,𝔽3)—36no—
73C3×GL(2,𝔽3)—33no—
74C3×#226—33no—
75C3×#216—32no—
76C3×#216—31no—
77C3×#215—32no—
78C3×#215—32no—
79C3×#215—32no—
80C3×#215—31no—
81C3×#215—32no—
82C3×#214—31no—
83C3×#213—31no—
84C3×#213—32no—
85C3×#208—32no—
86C3×#208—31no—
87C2×C3×C3⋊C8—32no—
88C2×C3×C3⋊C8—31no—
89C22×C33—42no—
90C2×C3×C3⋊D3—41no—
91C2×C3×C3⋊D3—42no—
92C2×C3×C3⋊D3—84no—
93C33×C4—41no—
94C3×C3⋊Dic3—41no—
95C3×2−1+4—21no—
96C3×2−1+4C3×D311yescharacteristic, complemented
97C3×2−1+4C3⋊D311yescharacteristic, complemented
98C3×H(5,𝔽2)—91no—
99#1009—31no—
1002−1+4⋊C3—23no—
1012−1+4⋊C3C3×D313yescomplemented
102#986—33no—
103#985—33no—
104#983—36no—
105C2×GL(2,𝔽3)—36no—
106C3×#137—91no—
107C3×#137—92no—
108C3×AGL(1,ℤ/8)—92no—
109C3×AGL(1,ℤ/8)—91no—
110C3×#135—92no—
111C3×#135—91no—
112C2×C3×SD16—91no—
113C2×C3×SD16—92no—
114C3×#131—91no—
115#950—32no—
116#950—31no—
117#949—32no—
118#949—31no—
119#948—31no—
120#942—31no—
121#942—32no—
122C2×#215—31no—
123C2×#215—32no—
124C22×C3⋊D3—42no—
125C22×C3×D3—61no—
126C22×C3×D3—62no—
127C22×C3×D3—126no—
128C32×Q8—32no—
129C32×Q8—31no—
130C32×Q8C22×D311yescharacteristic
131C32×D4—31no—
132C32×D4—32no—
133C32×D4—41no—
134C32×D4—43no—
135C32×D4—83no—
136C32×D4—81no—
137C32×D4S411yescharacteristic
138C2×C32×C4—32no—
139C2×C32×C4—31no—
140C2×C32×C4—61no—
141#638—42no—
142C4⋊(C3⋊D3)—41no—
143C4×C3⋊D3—41no—
144C3×#67—61no—
145C3×#67—62no—
146C3×#67—126no—
147C3×D12—32no—
148C3×D12—31no—
149C3×D12—62no—
150C3×D12—61no—
151C3×D12—123no—
152C3×C4×D3—62no—
153C3×C4×D3—61no—
154C3×C4×D3—123no—
155C3×Dic6—31no—
156C3×SL(2,𝔽3)—21no—
157C3×SL(2,𝔽3)—23no—
158C3×SL(2,𝔽3)C22×D313yes—
159C3×SL(2,𝔽3)C23×C311yescharacteristic
160C3×C3⋊C8—31no—
161C3×C3⋊C8—31no—
162C3×C3⋊C8—32no—
163#577—91noSylow
164C2×C33—41no—
165C2×C33—82no—
166C3×C3⋊D3—42no—
167C3×C3⋊D3—82no—
168C3×#41—21no—
169C3×#41—31no—
170C3×#41—31no—
171C3×#41—32no—
172C3×#41—32no—
173C3×#41—61no—
174C3×#41—62no—
175C3×#41—92no—
176C3×#41—91no—
177C3×#41—181no—
178C3×#41C2×C3×D311yescharacteristic, complemented
179C3×#41C2×C3⋊D311yescharacteristic, complemented
180C2×C3×Q8—22no—
181C2×C3×Q8—31no—
182C2×C3×Q8—31no—
183C2×C3×Q8—61no—
184C2×C3×Q8C2×C3×D312yescomplemented
185C2×C3×Q8C2×C3⋊D312yescomplemented
186C2×C3×D4—92no—
187C2×C3×D4—91no—
188C2×C3×D4—182no—
189C2×C3×D4—181no—
190#239—61no—
191D3×D4—62no—
192D3×D4—61no—
193D3×D4—123no—
194#235—32no—
195#235—31no—
196C2×D12—32no—
197C2×D12—31no—
198#41⋊C3—23no—
199#41⋊C3C2×C3×D313yescomplemented
200C2×SL(2,𝔽3)—26no—
201C2×SL(2,𝔽3)C2×C3×D316yescomplemented
202GL(2,𝔽3)—33no—
203GL(2,𝔽3)—36no—
204#226—33no—
205C3×Q16—92no—
206C3×Q16—91no—
207C3×SD16—91no—
208C3×SD16—92no—
209C3×SD16—92no—
210C3×SD16—92no—
211C3×SD16—91no—
212C3×SD16—92no—
213C3×D8—92no—
214C3×D8—91no—
215C3×M16—92no—
216C3×M16—91no—
217C2×C3×C8—92no—
218C2×C3×C8—91no—
219#216—32no—
220#216—31no—
221#215—32no—
222#215—32no—
223#215—32no—
224#215—31no—
225#215—32no—
226#214—31no—
227#213—31no—
228#213—32no—
229#208—32no—
230#208—31no—
231C2×C3⋊C8—32no—
232C2×C3⋊C8—31no—
233C22×C32—31no—
234C22×C32—42no—
235C22×C32—46no—
236C22×C32—86no—
237C22×C32—82no—
238C22×C32C2×S412yes—
239C2×C3⋊D3—41no—
240C2×C3⋊D3—42no—
241C2×C3⋊D3—84no—
242C2×C3×D3—61no—
243C2×C3×D3—62no—
244C2×C3×D3—123no—
245C2×C3×D3—126no—
246C2×C3×D3—242no—
247C2×C3×D3—241no—
248C2×C3×D3—2412no—
249C32×C4—31no—
250C32×C4—32no—
251C32×C4—41no—
252C32×C4—43no—
253C32×C4—83no—
254C32×C4—81no—
255C32×C4C2×S411yescharacteristic
256C3⋊Dic3—41no—
257C3×Dic3—61no—
258C3×Dic3—123no—
2592−1+4C3×C3⋊D311yescharacteristic, complemented
260H(5,𝔽2)—91no—
261#137—91no—
262#137—92no—
263AGL(1,ℤ/8)—92no—
264AGL(1,ℤ/8)—91no—
265#135—92no—
266#135—91no—
267C2×SD16—91no—
268C2×SD16—92no—
269#131—91no—
270C33—41noSylow
271C23×C3—181no—
272C23×C3—182no—
273C22×D3—61no—
274C22×D3—62no—
275C22×D3—126no—
276C3×Q8—21no—
277C3×Q8—32no—
278C3×Q8—32no—
279C3×Q8—31no—
280C3×Q8—31no—
281C3×Q8—61no—
282C3×Q8—62no—
283C3×Q8—91no—
284C3×Q8C22×C3×D311yescharacteristic
285C3×Q8C22×C3⋊D311yescharacteristic
286C3×D4—21no—
287C3×D4—31no—
288C3×D4—31no—
289C3×D4—32no—
290C3×D4—32no—
291C3×D4—43no—
292C3×D4—62no—
293C3×D4—61no—
294C3×D4—83no—
295C3×D4—92no—
296C3×D4—91no—
297C3×D4—182no—
298C3×D4—181no—
299C3×D4—182no—
300C3×D4—181no—
301C3×D4C3×S411yescharacteristic
302C3×D4C3⋊S411yescharacteristic
303C2×C3×C4—32no—
304C2×C3×C4—32no—
305C2×C3×C4—31no—
306C2×C3×C4—31no—
307C2×C3×C4—61no—
308C2×C3×C4—62no—
309C2×C3×C4—61no—
310C2×C3×C4—61no—
311C2×C3×C4—62no—
312C2×C3×C4—182no—
313C2×C3×C4—181no—
314#67—61no—
315#67—62no—
316#67—126no—
317D12—32no—
318D12—31no—
319D12—62no—
320D12—61no—
321D12—123no—
322C4×D3—62no—
323C4×D3—61no—
324C4×D3—123no—
325Dic6—31no—
326SL(2,𝔽3)—23no—
327SL(2,𝔽3)C22×C3×D313yes—
328C3×C8—91no—
329C3×C8—91no—
330C3×C8—92no—
331C3⋊C8—31no—
332C3⋊C8—31no—
333C3⋊C8—32no—
334C2×C32—22no—
335C2×C32—41no—
336C2×C32—43no—
337C2×C32—61no—
338C2×C32—83no—
339C2×C32—82no—
340C2×C32—86no—
341C2×C32—81no—
342C2×C32—162no—
343C2×C32—166no—
344C2×C32C22×S411yescharacteristic
345C3⋊D3—42no—
346C3⋊D3—82no—
347C3×D3—126no—
348C3×D3—121no—
349C3×D3—122no—
350C3×D3—246no—
351#41—31no—
352#41—32no—
353#41—92no—
354#41—91no—
355#41—181no—
356#41C2×C3×C3⋊D311yescharacteristic, complemented
357C2×Q8—31no—
358C2×Q8C2×C3×C3⋊D312yescomplemented
359C2×D4—92no—
360C2×D4—91no—
361C2×D4—182no—
362C2×D4—181no—
363Q16—92no—
364Q16—91no—
365SD16—91no—
366SD16—92no—
367SD16—92no—
368SD16—92no—
369SD16—91no—
370SD16—92no—
371D8—92no—
372D8—91no—
373M16—92no—
374M16—91no—
375C2×C8—92no—
376C2×C8—91no—
377C22×C3—22no—
378C22×C3—31no—
379C22×C3—31no—
380C22×C3—46no—
381C22×C3—61no—
382C22×C3—86no—
383C22×C3—181no—
384C22×C3—182no—
385C22×C3—722no—
386C22×C3—721no—
387C22×C3C2×C3×S412yes—
388C22×C3C2×C3⋊S412yes—
389C2×D3—61no—
390C2×D3—62no—
391C2×D3—123no—
392C2×D3—126no—
393C2×D3—242no—
394C2×D3—241no—
395C2×D3—2412no—
396C3×C4—21no—
397C3×C4—31no—
398C3×C4—32no—
399C3×C4—31no—
400C3×C4—32no—
401C3×C4—43no—
402C3×C4—61no—
403C3×C4—62no—
404C3×C4—83no—
405C3×C4—181no—
406C3×C4C2×C3×S411yescharacteristic
407C3×C4C2×C3⋊S411yescharacteristic
408Dic3—61no—
409Dic3—123no—
410C32—41no—
411C32—43no—
412C32—83no—
413C32—81no—
414C32#592611yescharacteristic, complemented
415C23—181no—
416C23—182no—
417Q8—32no—
418Q8—31no—
419Q8—91no—
420Q8C22×C3×C3⋊D311yescharacteristic
421D4—31no—
422D4—32no—
423D4—92no—
424D4—91no—
425D4—182no—
426D4—181no—
427D4—182no—
428D4—181no—
429D4C3×C3⋊S411yescharacteristic
430C2×C4—32no—
431C2×C4—31no—
432C2×C4—61no—
433C2×C4—182no—
434C2×C4—181no—
435C8—91no—
436C8—91no—
437C8—92no—
438C2×C3—22no—
439C2×C3—22no—
440C2×C3—21no—
441C2×C3—42no—
442C2×C3—43no—
443C2×C3—61no—
444C2×C3—61no—
445C2×C3—86no—
446C2×C3—83no—
447C2×C3—121no—
448C2×C3—166no—
449C2×C3—361no—
450C2×C3—362no—
451C2×C3C22×C3×S411yescharacteristic
452C2×C3C22×C3⋊S411yescharacteristic
453D3—126no—
454D3—121no—
455D3—122no—
456D3—246no—
457C22—31no—
458C22—181no—
459C22—182no—
460C22—722no—
461C22—721no—
462C22C2×C3×C3⋊S412yes—
463C4—31no—
464C4—32no—
465C4—181no—
466C4C2×C3×C3⋊S411yescharacteristic
467C3—21no—
468C3—43no—
469C3—83no—
470C3C3×#592611yescharacteristic, complemented
471C3#1059658911yescharacteristic, direct product factor
472C2—22no—
473C2—61no—
474C2—361no—
475C2—362no—
476C2C22×C3×C3⋊S411yescharacteristic
subgroupG/H
centerC2×C3C22×C3⋊S4
commutator, cosocle, hypocenterC3×SL(2,𝔽3)C23×C3
FrattiniC2C22×C3×C3⋊S4
FittingC32×2−1+4D3
socleC2×C32C22×S4
hypercenterC3×D4C3⋊S4
LCSG ▷ C3×SL(2,𝔽3)
UCSC1 ◁ C2×C3 ◁ C3×D4

Complements

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

NH ≅ G/Ntype
1. C32×2−1+4⋊C3C2semidirect
2. C3×#64267C2semidirect
3. C3×#64266C2semidirect
4. C3×#64264C2semidirect
5. C2×C3×C3⋊GL(2,𝔽3)C2semidirect
6. #10596589C3direct (N × H)
10. C32×#41⋊C3C22semidirect
11. C2×C32×SL(2,𝔽3)C22semidirect
12. C3×C3⋊GL(2,𝔽3)C22semidirect
13. C3×C3⋊GL(2,𝔽3)C22semidirect
14. C3×C3⋊#226C22semidirect
15. C32×2−1+4D3semidirect
19. C3×2−1+4⋊C3D3semidirect
20. C3×2−1+4⋊C3C2×C3semidirect
21. #64267C2×C3semidirect
22. #64266C2×C3semidirect
23. #64264C2×C3semidirect
24. C2×C3⋊GL(2,𝔽3)C2×C3semidirect
49. C32×#41C2×D3semidirect
51. C2×C32×Q8C2×D3semidirect
63. C3×#41⋊C3C2×D3semidirect
64. C3×#41⋊C3C22×C3semidirect
67. C2×C3×SL(2,𝔽3)C2×D3semidirect
68. C2×C3×SL(2,𝔽3)C22×C3semidirect
69. C3⋊GL(2,𝔽3)C22×C3semidirect
70. C3⋊GL(2,𝔽3)C22×C3semidirect
71. C3⋊#226C22×C3semidirect
96. C3×2−1+4C3×D3semidirect
97. C3×2−1+4C3⋊D3semidirect
101. 2−1+4⋊C3C3×D3semidirect
178. C3×#41C2×C3×D3semidirect
179. C3×#41C2×C3⋊D3semidirect
184. C2×C3×Q8C2×C3×D3semidirect
185. C2×C3×Q8C2×C3⋊D3semidirect
199. #41⋊C3C2×C3×D3semidirect
201. C2×SL(2,𝔽3)C2×C3×D3semidirect
259. 2−1+4C3×C3⋊D3semidirect
356. #41C2×C3×C3⋊D3semidirect
358. C2×Q8C2×C3×C3⋊D3semidirect
414. C32#5926semidirect
470. C3C3×#5926semidirect
471. C3#10596589direct (N × H)

Maximal subgroups

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

subgroupindex
C32×2−1+4⋊C32
C3×#642672
C3×#642662
C3×#642642
C2×C3×C3⋊GL(2,𝔽3)2
#105965893
C3×#59263
C3×#58373
C3×D4×C3⋊D34