Finite Group Databasebeta

← C3×#10596586 #13181386

Subgroups — C3×#10596586 #13181386

Order 1728.

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

normal subgroups only

This group has 847 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
1C2×C32×#41⋊C3C211yesmaximal, characteristic, complemented
2C3×#64267C211yesmaximal, characteristic, complemented
3C3×#64266C212yesmaximal, complemented
4C3×#64265C211yesmaximal, characteristic, complemented
5C3×#64264C212yesmaximal, complemented
6#10596586C311yesmaximal, characteristic, direct product factor
7C3×#5923—33nomaximal
8C3×#5822—31nomaximal
9C3×#3828—41nomaximal
10C32×#41⋊C3C2211yescharacteristic
11C32×#41⋊C3C2211yescharacteristic, complemented
12C2×C32×SL(2,𝔽3)C2211yescharacteristic, complemented
13C3×C3⋊GL(2,𝔽3)C2212yescomplemented
14C3×C3⋊#226C2212yescomplemented
15C2×C32×#41D311yescharacteristic, complemented
16C2×C3×#235—31no—
17C2×C3×#41⋊C3—21no—
18C2×C3×#41⋊C3—23no—
19C2×C3×#41⋊C3D313yescomplemented
20C2×C3×#41⋊C3C2×C311yescharacteristic, complemented
21#64267C2×C311yescharacteristic, complemented
22#64266C2×C312yescomplemented
23#64265C2×C311yescharacteristic, complemented
24#64264C2×C312yescomplemented
25C3×#986—33no—
26C3×#985—36no—
27C3×#984—33no—
28C3×#983—36no—
29C3×#951—31no—
30C3×#951—31no—
31C3×#950—32no—
32C3×#950—32no—
33C3×#949—31no—
34C3×#949—31no—
35C3×#942—32no—
36C3×#932—32no—
37C2×C3×#208—31no—
38C2×C33×C4—41no—
39C3×#638—42no—
40C3×C4⋊(C3⋊D3)—41no—
41C3×C4×C3⋊D3—42no—
42C3×#634—41no—
43C32×SL(2,𝔽3)C2311yescharacteristic
44#5923—33no—
45C3×#575—91no—
46#5822—31no—
47C32×#41—31no—
48C32×#41—31no—
49C32×#41C2×D311yescharacteristic
50C32×#41C2×D311yescharacteristic
51C2×C32×Q8C2×D311yescharacteristic
52C2×C32×D4—31no—
53C22×C32×C4—31no—
54#3828—41no—
55C2×C3×#67—61no—
56C3×#235—32no—
57C3×#235—32no—
58C3×#235—61no—
59C3×#235—61no—
60C3×#235—123no—
61C2×C3×D12—31no—
62C2×C3×C4×D3—61no—
63C2×C3×Dic6—31no—
64C3×#41⋊C3—21no—
65C3×#41⋊C3—21no—
66C3×#41⋊C3—23no—
67C3×#41⋊C3—23no—
68C3×#41⋊C3C2×D313yes—
69C3×#41⋊C3C2×D313yescomplemented
70C3×#41⋊C3C22×C311yescharacteristic
71C3×#41⋊C3C22×C311yescharacteristic, complemented
72C2×C3×SL(2,𝔽3)—21no—
73C2×C3×SL(2,𝔽3)—23no—
74C2×C3×SL(2,𝔽3)C2×D313yescomplemented
75C2×C3×SL(2,𝔽3)C22×C311yescharacteristic, complemented
76C3⋊GL(2,𝔽3)C22×C312yescomplemented
77C3⋊#226C22×C312yescomplemented
78C3×GL(2,𝔽3)—36no—
79C3×#226—36no—
80C3×#216—32no—
81C3×#216—32no—
82C3×#215—32no—
83C3×#215—32no—
84C3×#214—32no—
85C3×#214—32no—
86C3×#213—32no—
87C3×#213—32no—
88C3×#208—32no—
89C3×#208—31no—
90C3×#208—31no—
91C2×C3×C3⋊C8—32no—
92C22×C33—41no—
93C2×C3×C3⋊D3—42no—
94C33×C4—41no—
95C33×C4—41no—
96C3×C3⋊Dic3—42no—
97C2×C3×#41—21no—
98C2×C3×#41—91no—
99C2×C3×#41C3×D311yescharacteristic, complemented
100C2×C3×#41C3⋊D311yescharacteristic, complemented
101C2×#235—31no—
102C2×#41⋊C3—23no—
103C2×#41⋊C3C3×D313yescomplemented
104#986—33no—
105#985—36no—
106#984—33no—
107#983—36no—
108C3×#137—91no—
109C3×#137—91no—
110C3×#137—92no—
111C3×AGL(1,ℤ/8)—92no—
112C3×AGL(1,ℤ/8)—91no—
113C3×AGL(1,ℤ/8)—91no—
114C3×#135—92no—
115C3×#135—92no—
116C2×C3×M16—91no—
117#951—31no—
118#951—31no—
119#950—32no—
120#950—32no—
121#949—31no—
122#949—31no—
123#942—32no—
124#932—32no—
125C2×#208—31no—
126C23×C32—31no—
127C22×C3×D3—61no—
128C32×Q8—31no—
129C32×Q8C22×D311yescharacteristic
130C32×D4—31no—
131C32×D4—31no—
132C32×D4—61no—
133C2×C32×C4—31no—
134C2×C32×C4—31no—
135C2×C32×C4—31no—
136C2×C32×C4—31no—
137C2×C32×C4—31no—
138C2×C32×C4—41no—
139C2×C32×C4—43no—
140C2×C32×C4—83no—
141C2×C32×C4—81no—
142C2×C32×C4S411yescharacteristic
143#638—42no—
144C4⋊(C3⋊D3)—41no—
145C4×C3⋊D3—42no—
146#634—41no—
147C3×#67—62no—
148C3×#67—62no—
149C3×#67—126no—
150C2×C3×Dic3—61no—
151C3×D12—32no—
152C3×D12—61no—
153C3×D12—123no—
154C3×C4×D3—62no—
155C3×C4×D3—62no—
156C3×C4×D3—126no—
157C3×Dic6—32no—
158C3×Dic6—61no—
159C3×Dic6—123no—
160C3×SL(2,𝔽3)—21no—
161C3×SL(2,𝔽3)—23no—
162C3×SL(2,𝔽3)C22×D313yes—
163C3×SL(2,𝔽3)C23×C311yescharacteristic
164C3×C3⋊C8—32no—
165C3×C3⋊C8—32no—
166#575—91noSylow
167C2×C33—41no—
168C2×C33—81no—
169C3×C3⋊D3—82no—
170C3×#41—21no—
171C3×#41—21no—
172C3×#41—31no—
173C3×#41—31no—
174C3×#41—31no—
175C3×#41—31no—
176C3×#41—61no—
177C3×#41—61no—
178C3×#41—92no—
179C3×#41—92no—
180C3×#41—181no—
181C3×#41—181no—
182C3×#41C2×C3×D311yescharacteristic
183C3×#41C2×C3×D311yescharacteristic
184C3×#41C2×C3⋊D311yescharacteristic
185C3×#41C2×C3⋊D311yescharacteristic
186C2×C3×Q8—21no—
187C2×C3×Q8—91no—
188C2×C3×Q8C2×C3×D311yescharacteristic
189C2×C3×Q8C2×C3⋊D311yescharacteristic
190C2×C3×D4—31no—
191C2×C3×D4—31no—
192C2×C3×D4—61no—
193C2×C3×D4—91no—
194C2×C3×D4—181no—
195C22×C3×C4—31no—
196C22×C3×C4—31no—
197C22×C3×C4—61no—
198C22×C3×C4—181no—
199C2×#67—61no—
200#235—32no—
201#235—32no—
202#235—61no—
203#235—61no—
204#235—123no—
205C2×D12—31no—
206C2×C4×D3—61no—
207C2×Dic6—31no—
208#41⋊C3—23no—
209#41⋊C3—23no—
210#41⋊C3C2×C3×D313yes—
211#41⋊C3C2×C3×D313yescomplemented
212C2×SL(2,𝔽3)—23no—
213C2×SL(2,𝔽3)C2×C3×D313yescomplemented
214GL(2,𝔽3)—36no—
215#226—36no—
216C3×Q16—92no—
217C3×Q16—92no—
218C3×SD16—92no—
219C3×SD16—92no—
220C3×SD16—92no—
221C3×SD16—92no—
222C3×D8—92no—
223C3×D8—92no—
224C3×M16—92no—
225C3×M16—91no—
226C3×M16—91no—
227C2×C3×C8—92no—
228#216—32no—
229#216—32no—
230#215—32no—
231#215—32no—
232#214—32no—
233#214—32no—
234#213—32no—
235#213—32no—
236#208—32no—
237#208—31no—
238#208—31no—
239C2×C3⋊C8—32no—
240C22×C32—31no—
241C22×C32—31no—
242C22×C32—41no—
243C22×C32—43no—
244C22×C32—83no—
245C22×C32—81no—
246C22×C32—121no—
247C22×C32C2×S411yescharacteristic
248C2×C3⋊D3—42no—
249C2×C3×D3—62no—
250C2×C3×D3—126no—
251C2×C3×D3—122no—
252C32×C4—31no—
253C32×C4—31no—
254C32×C4—41no—
255C32×C4—41no—
256C32×C4—43no—
257C32×C4—43no—
258C32×C4—83no—
259C32×C4—83no—
260C32×C4—81no—
261C32×C4—81no—
262C32×C4C2×S411yescharacteristic
263C32×C4C2×S411yescharacteristic
264C3⋊Dic3—42no—
265C3×Dic3—62no—
266C3×Dic3—126no—
267C2×#41—91no—
268C2×#41C3×C3⋊D311yescharacteristic, complemented
269#137—91no—
270#137—91no—
271#137—92no—
272AGL(1,ℤ/8)—92no—
273AGL(1,ℤ/8)—91no—
274AGL(1,ℤ/8)—91no—
275#135—92no—
276#135—92no—
277C2×M16—91no—
278C33—41noSylow
279C23×C3—31no—
280C23×C3—31no—
281C23×C3—61no—
282C23×C3—181no—
283C22×D3—61no—
284C3×Q8—21no—
285C3×Q8—31no—
286C3×Q8—31no—
287C3×Q8—61no—
288C3×Q8—92no—
289C3×Q8—181no—
290C3×Q8C22×C3×D311yescharacteristic
291C3×Q8C22×C3⋊D311yescharacteristic
292C3×D4—31no—
293C3×D4—31no—
294C3×D4—31no—
295C3×D4—31no—
296C3×D4—61no—
297C3×D4—61no—
298C3×D4—61no—
299C3×D4—61no—
300C3×D4—62no—
301C3×D4—92no—
302C3×D4—182no—
303C3×D4—182no—
304C3×D4—181no—
305C2×C3×C4—21no—
306C2×C3×C4—31no—
307C2×C3×C4—31no—
308C2×C3×C4—31no—
309C2×C3×C4—31no—
310C2×C3×C4—31no—
311C2×C3×C4—31no—
312C2×C3×C4—31no—
313C2×C3×C4—31no—
314C2×C3×C4—31no—
315C2×C3×C4—31no—
316C2×C3×C4—43no—
317C2×C3×C4—61no—
318C2×C3×C4—61no—
319C2×C3×C4—61no—
320C2×C3×C4—61no—
321C2×C3×C4—61no—
322C2×C3×C4—83no—
323C2×C3×C4—182no—
324C2×C3×C4—181no—
325C2×C3×C4—182no—
326C2×C3×C4C3×S411yescharacteristic
327C2×C3×C4C3⋊S411yescharacteristic
328#67—62no—
329#67—62no—
330#67—126no—
331C2×Dic3—61no—
332D12—32no—
333D12—61no—
334D12—123no—
335C4×D3—62no—
336C4×D3—62no—
337C4×D3—126no—
338Dic6—32no—
339Dic6—61no—
340Dic6—123no—
341SL(2,𝔽3)—23no—
342SL(2,𝔽3)C22×C3×D313yes—
343C3×C8—92no—
344C3×C8—92no—
345C3⋊C8—32no—
346C3⋊C8—32no—
347C2×C32—21no—
348C2×C32—41no—
349C2×C32—43no—
350C2×C32—61no—
351C2×C32—61no—
352C2×C32—83no—
353C2×C32—81no—
354C2×C32—83no—
355C2×C32—86no—
356C2×C32—82no—
357C2×C32—81no—
358C2×C32C22×S411yescharacteristic
359C3⋊D3—82no—
360C3×D3—122no—
361C3×D3—246no—
362#41—31no—
363#41—31no—
364#41—92no—
365#41—92no—
366#41—181no—
367#41—181no—
368#41C2×C3×C3⋊D311yescharacteristic
369#41C2×C3×C3⋊D311yescharacteristic
370C2×Q8—91no—
371C2×Q8C2×C3×C3⋊D311yescharacteristic
372C2×D4—31no—
373C2×D4—91no—
374C2×D4—181no—
375C22×C4—31no—
376C22×C4—181no—
377Q16—92no—
378Q16—92no—
379SD16—92no—
380SD16—92no—
381SD16—92no—
382SD16—92no—
383D8—92no—
384D8—92no—
385M16—92no—
386M16—91no—
387M16—91no—
388C2×C8—92no—
389C22×C3—21no—
390C22×C3—31no—
391C22×C3—31no—
392C22×C3—31no—
393C22×C3—31no—
394C22×C3—43no—
395C22×C3—61no—
396C22×C3—61no—
397C22×C3—83no—
398C22×C3—121no—
399C22×C3—121no—
400C22×C3—122no—
401C22×C3—182no—
402C22×C3—362no—
403C22×C3C2×C3×S411yescharacteristic
404C22×C3C2×C3⋊S411yescharacteristic
405C2×D3—62no—
406C2×D3—126no—
407C2×D3—122no—
408C3×C4—21no—
409C3×C4—21no—
410C3×C4—31no—
411C3×C4—31no—
412C3×C4—31no—
413C3×C4—31no—
414C3×C4—43no—
415C3×C4—43no—
416C3×C4—61no—
417C3×C4—61no—
418C3×C4—83no—
419C3×C4—83no—
420C3×C4—182no—
421C3×C4C2×C3×S411yescharacteristic
422C3×C4C2×C3×S411yescharacteristic
423C3×C4C2×C3⋊S411yescharacteristic
424C3×C4C2×C3⋊S411yescharacteristic
425Dic3—62no—
426Dic3—126no—
427C32—41no—
428C32—43no—
429C32—83no—
430C32—81no—
431C32#592311yescharacteristic, complemented
432C23—31no—
433C23—181no—
434Q8—31no—
435Q8—92no—
436Q8—181no—
437Q8C22×C3×C3⋊D311yescharacteristic
438D4—31no—
439D4—31no—
440D4—61no—
441D4—92no—
442D4—182no—
443D4—182no—
444D4—181no—
445C2×C4—31no—
446C2×C4—31no—
447C2×C4—31no—
448C2×C4—31no—
449C2×C4—31no—
450C2×C4—182no—
451C2×C4—181no—
452C2×C4—182no—
453C2×C4C3×C3⋊S411yescharacteristic
454C8—92no—
455C8—92no—
456C2×C3—22no—
457C2×C3—21no—
458C2×C3—21no—
459C2×C3—21no—
460C2×C3—43no—
461C2×C3—61no—
462C2×C3—61no—
463C2×C3—61no—
464C2×C3—61no—
465C2×C3—83no—
466C2×C3—86no—
467C2×C3—83no—
468C2×C3—121no—
469C2×C3—121no—
470C2×C3—362no—
471C2×C3C22×C3×S411yescharacteristic
472C2×C3C22×C3⋊S411yescharacteristic
473D3—122no—
474D3—246no—
475C22—31no—
476C22—31no—
477C22—121no—
478C22—182no—
479C22—362no—
480C22C2×C3×C3⋊S411yescharacteristic
481C4—31no—
482C4—31no—
483C4—182no—
484C4C2×C3×C3⋊S411yescharacteristic
485C4C2×C3×C3⋊S411yescharacteristic
486C3—21no—
487C3—43no—
488C3—83no—
489C3C3×#592311yescharacteristic, complemented
490C3#1059658611yescharacteristic, direct product factor
491C2—21no—
492C2—61no—
493C2—61no—
494C2—362no—
495C2C22×C3×C3⋊S411yescharacteristic
subgroupG/H
centerC3×C4C2×C3⋊S4
commutator, cosocle, hypocenterC3×SL(2,𝔽3)C23×C3
FrattiniC2C22×C3×C3⋊S4
FittingC2×C32×#41D3
socleC2×C32C22×S4
hypercenterC2×C3×C4C3⋊S4
LCSG ▷ C3×SL(2,𝔽3)
UCSC1 ◁ C3×C4 ◁ C2×C3×C4

Complements

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

NH ≅ G/Ntype
1. C2×C32×#41⋊C3C2semidirect
2. C3×#64267C2semidirect
3. C3×#64266C2semidirect
4. C3×#64265C2semidirect
5. C3×#64264C2semidirect
6. #10596586C3direct (N × H)
11. C32×#41⋊C3C22semidirect
12. C2×C32×SL(2,𝔽3)C22semidirect
13. C3×C3⋊GL(2,𝔽3)C22semidirect
14. C3×C3⋊#226C22semidirect
15. C2×C32×#41D3semidirect
19. C2×C3×#41⋊C3D3semidirect
20. C2×C3×#41⋊C3C2×C3semidirect
21. #64267C2×C3semidirect
22. #64266C2×C3semidirect
23. #64265C2×C3semidirect
24. #64264C2×C3semidirect
69. C3×#41⋊C3C2×D3semidirect
71. C3×#41⋊C3C22×C3semidirect
74. C2×C3×SL(2,𝔽3)C2×D3semidirect
75. C2×C3×SL(2,𝔽3)C22×C3semidirect
76. C3⋊GL(2,𝔽3)C22×C3semidirect
77. C3⋊#226C22×C3semidirect
99. C2×C3×#41C3×D3semidirect
100. C2×C3×#41C3⋊D3semidirect
103. C2×#41⋊C3C3×D3semidirect
211. #41⋊C3C2×C3×D3semidirect
213. C2×SL(2,𝔽3)C2×C3×D3semidirect
268. C2×#41C3×C3⋊D3semidirect
431. C32#5923semidirect
489. C3C3×#5923semidirect
490. C3#10596586direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C32×#41⋊C32
C3×#642672
C3×#642662
C3×#642652
C3×#642642
#105965863
C3×#59233
C3×#58223
C3×#38284