Finite Group Databasebeta

← C3×#10593528 #13166897

Subgroups — C3×#10593528 #13166897

Order 1728.

718 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 718 conjugacy classes of subgroups — too many to draw. Only its 62 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×#64330C211yesmaximal, characteristic, complemented
2C3×#64236C212yesmaximal, complemented
3C3×#63979C211yesmaximal, characteristic, complemented
4C3×#63788C211yesmaximal, characteristic, complemented
5C3×#63741C212yesmaximal, complemented
6#10593528C311yesmaximal, characteristic, direct product factor
7C2×C3×#638C2211yescharacteristic
8C2×C32×#67—21no—
9C2×C32×#67—21no—
10C3×#3811—21no—
11C3×D3×#67—22no—
12C2×C3×#626—21no—
13C2×C3×#625C2211yescharacteristic, complemented
14C3×#3805C2212yes—
15C3×#3804—22no—
16C3×#3792C2212yescomplemented
17C3×#3775—22no—
18C3×#3774—22no—
19C3×Dic32C2211yescharacteristic, complemented
20C3×#1004—61no—
21#64330C2×C311yescharacteristic, complemented
22#64236C2×C312yescomplemented
23C3×#940—61no—
24#63979C2×C311yescharacteristic, complemented
25#63788C2×C311yescharacteristic, complemented
26#63741C2×C312yescomplemented
27C23×C33D411yescharacteristic
28C22×C3×C3⋊D3D411yescharacteristic
29C22×C32×D3—21no—
30C22×C32×D3—21no—
31C2×C3×D32—21no—
32C3×#638—22no—
33C2×C3×C3⋊Dic3C2311yescharacteristic
34C32×#67—22no—
35C32×#67—22no—
36C32×#67—42no—
37C2×C32×Dic3—21no—
38C2×C32×Dic3—21no—
39C3×#627D412yescomplemented
40C3×#626—22no—
41C3×#626—42no—
42C3×#625—22no—
43C3×#625D412yescomplemented
44C3×#624—21no—
45C3×D3×Dic3—22no—
46C3×#622—22no—
47C3×#453—91nomaximal
48C2×C32×D4—61no—
49C2×C32×D4—61no—
50C2×#638C22×C311yescharacteristic
51C2×C3×#67—21no—
52C2×C3×#67—21no—
53C2×C3×#67—21no—
54C2×C3×#67—21no—
55C2×C3×#67—41no—
56C2×C3×#67—41no—
57C2×C3×#67—61no—
58C2×C3×#67—62no—
59C2×C3×#67—61no—
60C2×C3×#67—61no—
61C2×C3×#67—61no—
62C3×#237—62no—
63C3×#237—62no—
64C3×D3×D4—62no—
65C3×D3×D4—62no—
66C3×#235—62no—
67C2×C3×D12—61no—
68#3811—21no—
69D3×#67—22no—
70C2×#626—21no—
71C2×#625C22×C311yescharacteristic, complemented
72#3805C22×C312yes—
73#3804—22no—
74#3792C22×C312yescomplemented
75#3775—22no—
76#3774—22no—
77C3×C4×Dic3—61no—
78Dic32C22×C311yescharacteristic, complemented
79C22×C33—21no—
80C22×C33—41no—
81C22×C33C2×D411yescharacteristic
82C2×C3×C3⋊D3—21no—
83C2×C3×C3⋊D3—41no—
84C2×C32×D3—22no—
85C2×C32×D3—41no—
86C2×C32×D3—41no—
87C2×C32×D3—41no—
88C2×C32×D3—81no—
89C3×D32—81no—
90C3×C3⋊Dic3C2×D412yes—
91C32×Dic3—22no—
92C32×Dic3—22no—
93C3×H(5,𝔽2)—91no—
94#1004—61no—
95C3×AGL(1,ℤ/8)—92no—
96C3×#127—91no—
97#940—61no—
98C3×#104—92no—
99C3×#100—91no—
100C23×C32—21no—
101C23×C32—21no—
102C23×C32—41no—
103C23×C32—41no—
104C23×C32—61no—
105C23×C32—61no—
106C23×C32C3×D411yescharacteristic
107C22×C3⋊D3C3×D411yescharacteristic
108C22×C3×D3—21no—
109C22×C3×D3—21no—
110C22×C3×D3—21no—
111C22×C3×D3—21no—
112C22×C3×D3—41no—
113C22×C3×D3—41no—
114C22×C3×D3—61no—
115C22×C3×D3—61no—
116C22×C3×D3—62no—
117C2×D32—21no—
118C32×D4—62no—
119C32×D4—62no—
120C32×D4—122no—
121C2×C32×C4—61no—
122C2×C32×C4—61no—
123#638—22no—
124C2×C3⋊Dic3C23×C311yescharacteristic
125C3×#67—22no—
126C3×#67—22no—
127C3×#67—22no—
128C3×#67—22no—
129C3×#67—42no—
130C3×#67—42no—
131C3×#67—42no—
132C3×#67—42no—
133C3×#67—44no—
134C3×#67—62no—
135C3×#67—62no—
136C3×#67—62no—
137C3×#67—62no—
138C3×#67—122no—
139C3×#67—122no—
140C3×#67—122no—
141C2×C3×Dic3—21no—
142C2×C3×Dic3—21no—
143C2×C3×Dic3—21no—
144C2×C3×Dic3—21no—
145C2×C3×Dic3—41no—
146C2×C3×Dic3—41no—
147C2×C3×Dic3—61no—
148C2×C3×Dic3—61no—
149C2×C3×Dic3—62no—
150C3×D12—62no—
151C3×D12—122no—
152C3×C4×D3—62no—
153C3×C4×D3—62no—
154C3×Dic6—62no—
155#627C3×D412yescomplemented
156#626—22no—
157#626—42no—
158#625—22no—
159#625C3×D412yescomplemented
160#624—21no—
161D3×Dic3—22no—
162#622—22no—
163#453—91noSylow
164C2×C33—21no—
165C2×C33—41no—
166C2×C33#12011yescharacteristic
167C3×C3⋊D3—41no—
168C32×D3—42no—
169C32×D3—81no—
170C3×#41—92no—
171C3×#41—182no—
172C2×C3×D4—61no—
173C2×C3×D4—61no—
174C2×C3×D4—61no—
175C2×C3×D4—61no—
176C2×C3×D4—91no—
177C2×C3×D4—91no—
178C2×C3×D4—121no—
179C2×C3×D4—121no—
180C2×C3×D4—181no—
181C2×C3×D4—181no—
182C2×C3×D4—182no—
183C2×#67—21no—
184C2×#67—21no—
185C2×#67—61no—
186C2×#67—61no—
187C2×#67—61no—
188C2×#67—61no—
189C2×#67—62no—
190#237—62no—
191#237—62no—
192D3×D4—62no—
193D3×D4—62no—
194#235—62no—
195C2×D12—61no—
196C3×SD16—182no—
197C3×D8—182no—
198C3×M16—92no—
199C3×C42—91no—
200C4×Dic3—61no—
201C22×C32—21no—
202C22×C32—21no—
203C22×C32—21no—
204C22×C32—21no—
205C22×C32—21no—
206C22×C32—41no—
207C22×C32—41no—
208C22×C32—41no—
209C22×C32—41no—
210C22×C32—41no—
211C22×C32—41no—
212C22×C32—41no—
213C22×C32—41no—
214C22×C32—42no—
215C22×C32—62no—
216C22×C32—81no—
217C22×C32—81no—
218C22×C32—81no—
219C22×C32—82no—
220C22×C32—121no—
221C22×C32—121no—
222C22×C32—121no—
223C22×C32—241no—
224C22×C32C2×C3×D411yescharacteristic
225C2×C3⋊D3—21no—
226C2×C3⋊D3—41no—
227C2×C3×D3—22no—
228C2×C3×D3—22no—
229C2×C3×D3—42no—
230C2×C3×D3—41no—
231C2×C3×D3—41no—
232C2×C3×D3—41no—
233C2×C3×D3—41no—
234C2×C3×D3—41no—
235C2×C3×D3—42no—
236C2×C3×D3—41no—
237C2×C3×D3—62no—
238C2×C3×D3—81no—
239C2×C3×D3—81no—
240C2×C3×D3—81no—
241C2×C3×D3—81no—
242C2×C3×D3—82no—
243C2×C3×D3—121no—
244C2×C3×D3—121no—
245C2×C3×D3—121no—
246C2×C3×D3—241no—
247C2×C3×D3—242no—
248D32—81no—
249C32×C4—62no—
250C32×C4—62no—
251C3⋊Dic3C2×C3×D412yes—
252C3×Dic3—22no—
253C3×Dic3—22no—
254C3×Dic3—22no—
255C3×Dic3—22no—
256C3×Dic3—42no—
257C3×Dic3—42no—
258C3×Dic3—62no—
259C3×Dic3—62no—
260H(5,𝔽2)—91no—
261AGL(1,ℤ/8)—92no—
262#127—91no—
263#104—92no—
264#100—91no—
265C33#45311yescharacteristic, complemented, Sylow
266C23×C3—21no—
267C23×C3—21no—
268C23×C3—41no—
269C23×C3—41no—
270C23×C3—61no—
271C23×C3—61no—
272C23×C3—61no—
273C23×C3—61no—
274C23×C3—91no—
275C23×C3—121no—
276C23×C3—121no—
277C23×C3—181no—
278C23×C3GO+(4,𝔽2)11yescharacteristic
279C22×D3—21no—
280C22×D3—21no—
281C22×D3—61no—
282C22×D3—61no—
283C22×D3—62no—
284C3×Q8—92no—
285C3×D4—62no—
286C3×D4—62no—
287C3×D4—62no—
288C3×D4—62no—
289C3×D4—92no—
290C3×D4—122no—
291C3×D4—122no—
292C3×D4—122no—
293C3×D4—122no—
294C3×D4—124no—
295C3×D4—182no—
296C3×D4—182no—
297C3×D4—182no—
298C3×D4—362no—
299C2×C3×C4—61no—
300C2×C3×C4—61no—
301C2×C3×C4—61no—
302C2×C3×C4—61no—
303C2×C3×C4—91no—
304C2×C3×C4—121no—
305C2×C3×C4—121no—
306C2×C3×C4—181no—
307C2×C3×C4—182no—
308#67—22no—
309#67—22no—
310#67—42no—
311#67—62no—
312#67—62no—
313#67—62no—
314#67—62no—
315#67—122no—
316#67—122no—
317#67—122no—
318C2×Dic3—21no—
319C2×Dic3—21no—
320C2×Dic3—61no—
321C2×Dic3—61no—
322C2×Dic3—62no—
323D12—62no—
324D12—122no—
325C4×D3—62no—
326C4×D3—62no—
327Dic6—62no—
328C3×C8—182no—
329C2×C32—21no—
330C2×C32—21no—
331C2×C32—21no—
332C2×C32—21no—
333C2×C32—21no—
334C2×C32—41no—
335C2×C32—41no—
336C2×C32—41no—
337C2×C32—41no—
338C2×C32—41no—
339C2×C32—41no—
340C2×C32—41no—
341C2×C32—42no—
342C2×C32—41no—
343C2×C32—81no—
344C2×C32—81no—
345C2×C32—81no—
346C2×C32—82no—
347C2×C32—122no—
348C2×C32—241no—
349C2×C32C3×#12011yescharacteristic
350C3⋊D3—41no—
351C3×D3—42no—
352C3×D3—42no—
353C3×D3—82no—
354C3×D3—81no—
355C3×D3—82no—
356C3×D3—81no—
357C3×D3—122no—
358C3×D3—241no—
359#41—92no—
360#41—182no—
361C2×D4—61no—
362C2×D4—61no—
363C2×D4—91no—
364C2×D4—91no—
365C2×D4—181no—
366C2×D4—181no—
367C2×D4—182no—
368SD16—182no—
369D8—182no—
370M16—92no—
371C42—91no—
372C22×C3—21no—
373C22×C3—21no—
374C22×C3—21no—
375C22×C3—21no—
376C22×C3—21no—
377C22×C3—41no—
378C22×C3—41no—
379C22×C3—41no—
380C22×C3—41no—
381C22×C3—41no—
382C22×C3—41no—
383C22×C3—42no—
384C22×C3—41no—
385C22×C3—41no—
386C22×C3—62no—
387C22×C3—62no—
388C22×C3—81no—
389C22×C3—81no—
390C22×C3—81no—
391C22×C3—82no—
392C22×C3—122no—
393C22×C3—121no—
394C22×C3—121no—
395C22×C3—121no—
396C22×C3—121no—
397C22×C3—121no—
398C22×C3—122no—
399C22×C3—121no—
400C22×C3—181no—
401C22×C3—241no—
402C22×C3—241no—
403C22×C3—241no—
404C22×C3—241no—
405C22×C3—242no—
406C22×C3—361no—
407C22×C3—721no—
408C22×C3C2×GO+(4,𝔽2)11yescharacteristic
409C2×D3—22no—
410C2×D3—41no—
411C2×D3—41no—
412C2×D3—41no—
413C2×D3—62no—
414C2×D3—81no—
415C2×D3—121no—
416C2×D3—121no—
417C2×D3—121no—
418C2×D3—241no—
419C2×D3—242no—
420C3×C4—62no—
421C3×C4—62no—
422C3×C4—62no—
423C3×C4—62no—
424C3×C4—92no—
425C3×C4—122no—
426C3×C4—122no—
427Dic3—22no—
428Dic3—22no—
429Dic3—62no—
430Dic3—62no—
431C32—21no—
432C32—21no—
433C32—41no—
434C32—41no—
435C32C3×#45311yescharacteristic, complemented
436C23—61no—
437C23—61no—
438C23—91no—
439C23—181no—
440C23C3×GO+(4,𝔽2)11yescharacteristic
441Q8—92no—
442D4—62no—
443D4—62no—
444D4—92no—
445D4—122no—
446D4—182no—
447D4—182no—
448D4—182no—
449D4—362no—
450C2×C4—61no—
451C2×C4—61no—
452C2×C4—91no—
453C2×C4—181no—
454C2×C4—182no—
455C8—182no—
456C2×C3—21no—
457C2×C3—21no—
458C2×C3—21no—
459C2×C3—21no—
460C2×C3—21no—
461C2×C3—41no—
462C2×C3—41no—
463C2×C3—41no—
464C2×C3—41no—
465C2×C3—41no—
466C2×C3—42no—
467C2×C3—41no—
468C2×C3—41no—
469C2×C3—41no—
470C2×C3—81no—
471C2×C3—81no—
472C2×C3—81no—
473C2×C3—82no—
474C2×C3—122no—
475C2×C3—122no—
476C2×C3—242no—
477C2×C3—241no—
478C2×C3—242no—
479C2×C3—241no—
480C2×C3—361no—
481C2×C3#6424111yescharacteristic
482D3—42no—
483D3—81no—
484D3—122no—
485D3—241no—
486C22—21no—
487C22—41no—
488C22—62no—
489C22—121no—
490C22—121no—
491C22—121no—
492C22—181no—
493C22—241no—
494C22—361no—
495C22—721no—
496C22C2×C3×GO+(4,𝔽2)11yescharacteristic
497C4—62no—
498C4—62no—
499C4—92no—
500C3—21no—
501C3—21no—
502C3—41no—
503C3—41no—
504C3#1059352811yescharacteristic, direct product factor
505C2—21no—
506C2—41no—
507C2—122no—
508C2—241no—
509C2—361no—
510C2C3×#6424111yescharacteristic
subgroupG/H
centerC2×C3#64241
commutator, cosocleC2×C3⋊Dic3C23×C3
FrattiniC22C2×C3×GO+(4,𝔽2)
FittingC23×C33D4
socleC2×C33#120
hypercenterC23×C3GO+(4,𝔽2)
hypocenterC32C3×#453
LCSG ▷ C2×C3⋊Dic3 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2×C3 ◁ C22×C3 ◁ C23×C3

Complements

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

NH ≅ G/Ntype
1. C3×#64330C2semidirect
2. C3×#64236C2semidirect
3. C3×#63979C2semidirect
4. C3×#63788C2semidirect
5. C3×#63741C2semidirect
6. #10593528C3direct (N × H)
13. C2×C3×#625C22semidirect
16. C3×#3792C22semidirect
19. C3×Dic32C22semidirect
21. #64330C2×C3semidirect
22. #64236C2×C3semidirect
24. #63979C2×C3semidirect
25. #63788C2×C3semidirect
26. #63741C2×C3semidirect
39. C3×#627D4semidirect
43. C3×#625D4semidirect
71. C2×#625C22×C3semidirect
74. #3792C22×C3semidirect
78. Dic32C22×C3semidirect
155. #627C3×D4semidirect
159. #625C3×D4semidirect
265. C33#453semidirect
435. C32C3×#453semidirect
504. C3#10593528direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#643302
C3×#642362
C3×#639792
C3×#637882
C3×#637412
#105935283
C3×#4539