Finite Group Databasebeta

← C3×#86768 #11904228

Subgroups — C3×#86768 #11904228

Order 1152.

2560 conjugacy classes of subgroups.

The subgroup lattice has 567 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C32×#550C211yesmaximal, characteristic, complemented
2C3×#5834C212yesmaximal, complemented
3C3×Dic3×#41C212yesmaximal, complemented
4C3×#5718C213yesmaximal, complemented
5C3×#5716C213yesmaximal, complemented
6C3×#5616C216yesmaximal, complemented
7C3×#5596C216yesmaximal, complemented
8C3×C4×#239C211yesmaximal, characteristic, complemented
9C3×Q8×Dic6C211yesmaximal, characteristic, complemented
10C3×#5563C213yesmaximal, complemented
11C3×C4×#237C213yesmaximal, complemented
12C3×#3483—31nomaximal
13#86768C311yesmaximal, characteristic, direct product factor
14C2×C32×#41C2212yes—
15C2×C3×#239C2211yescharacteristic
16C2×C3×#237C2213yes—
17C2×C3×#237C2216yes—
18C32×#127C2213yes—
19C32×#121C2216yes—
20C32×C4×Q8C2211yescharacteristic
21C32×C4×D4C2213yes—
22C3×Q8×Dic3C2212yes—
23C3×Q8×Dic3C2211yescharacteristic
24C3×#940C2216yes—
25C3×#939C22112yes—
26C3×#938C2216yes—
27C3×#938C2216yes—
28C3×#936C2216yes—
29C3×D4×Dic3C2213yes—
30C3×D4×Dic3C2216yes—
31C3×D4×Dic3C2216yes—
32C3×#930C2216yes—
33C3×C4×#67C2216yes—
34C3×#926C2216yes—
35C2×C3×C4×Dic3C2216yes—
36C3×#893C2213yes—
37C3×#892C2213yes—
38C3×#892C2213yes—
39C3×#888C2216yes—
40C3×#887C2211yescharacteristic
41C3×#885C22112yes—
42C3×#67⋊C4C2216yes—
43C3×#877C2216yes—
44C3×C4×D12C2213yes—
45C3×C42×D3C2213yes—
46C3×#869C2213yes—
47C3×C4×Dic6C2213yes—
48C3×Q82—31no—
49C3×#555—36no—
50C3×#555—33no—
51C3×#550—21no—
52C3×#550—32no—
53C3×#550—33no—
54C3×#550D311yescharacteristic, complemented
55C3×#550C2×C311yescharacteristic, complemented
56C3×#532—33no—
57C3×#532—36no—
58C3×C4×#41—32no—
59C3×C4×#41—31no—
60C3×C4×#41—33no—
61#5834C2×C312yescomplemented
62Dic3×#41C2×C312yescomplemented
63#5718C2×C313yescomplemented
64#5716C2×C313yescomplemented
65#5616C2×C316yescomplemented
66#5596C2×C316yescomplemented
67C4×#239C2×C311yescharacteristic, complemented
68Q8×Dic6C2×C311yescharacteristic, complemented
69#5563C2×C313yescomplemented
70C4×#237C2×C313yescomplemented
71C32×#41—28no—
72C2×C32×Q8C2311yescharacteristic
73C2×C32×D4C2316yes—
74C2×C32×D4C2313yes—
75C2×C32×D4C2316yes—
76C22×C32×C4C2316yes—
77C2×C3×#67C2316yes—
78C2×C3×#67C23112yes—
79C22×C3×Dic3C2316yes—
80C22×C3×Dic3C2316yes—
81C3×#239—24no—
82C3×#237—212no—
83C3×#237—412no—
84C2×C3×D12C2313yes—
85C2×C3×C4×D3C2313yes—
86C2×C3×C4×D3C2313yes—
87C2×C3×C4×D3C2313yes—
88C2×C3×Dic6C2316yes—
89C2×C3×Dic6C2311yescharacteristic
90C32×C4⋊C4C2311yescharacteristic
91C32×C4⋊C4C2313yes—
92C32×C22⋊C4C2316yes—
93C32×C42C2313yes—
94C3×#217C2316yes—
95C3×#217C23112yes—
96C3×(C2×D3)⋊C4C2316yes—
97C3×(C2×D3)⋊C4C2316yes—
98C3×C4⋊Dic3C2316yes—
99C3×C4⋊Dic3C2313yes—
100C3×C4⋊Dic3C2313yes—
101C3×Dic3⋊C4C2312yes—
102C3×Dic3⋊C4C2316yes—
103C3×C4×Dic3C2313yes—
104C3×C4×Dic3C2316yes—
105C3×C4×Dic3C2312yes—
106C3×C4×Dic3C2311yescharacteristic
107C3×C4×Dic3C2316yes—
108C3×C4×Dic3C2313yes—
109#3483—31noSylow
110C2×C3×#41—22no—
111C2×C3×#41—31no—
112C2×C3×#41—33no—
113C2×C3×#41—36no—
114C2×C3×#41C2×D312yes—
115C2×C3×#41C22×C312yes—
116C2×#239C22×C311yescharacteristic
117C2×#237C22×C313yes—
118C2×#237C22×C316yes—
119C3×C4⋊Q8—36no—
120C3×C4⋊Q8—33no—
121C3×#127—23no—
122C3×#127—36no—
123C3×#127C2×D313yes—
124C3×#127C22×C313yes—
125C3×#124—36no—
126C3×#124—312no—
127C3×#121—26no—
128C3×#121—36no—
129C3×#121—36no—
130C3×#121—312no—
131C3×#121—36no—
132C3×#121C2×D316yes—
133C3×#121C22×C316yes—
134C3×C4×Q8—21no—
135C3×C4×Q8—32no—
136C3×C4×Q8—31no—
137C3×C4×Q8—31no—
138C3×C4×Q8—33no—
139C3×C4×Q8C2×D311yescharacteristic
140C3×C4×Q8C22×C311yescharacteristic
141C3×C4×D4—23no—
142C3×C4×D4—36no—
143C3×C4×D4—33no—
144C3×C4×D4—36no—
145C3×C4×D4—36no—
146C3×C4×D4—33no—
147C3×C4×D4—36no—
148C3×C4×D4—33no—
149C3×C4×D4C2×D313yes—
150C3×C4×D4C22×C313yes—
151C3×#117—36no—
152C3×#117—33no—
153C3×#117—33no—
154C3×#117—36no—
155C2×C3×C42—36no—
156C2×C3×C42—33no—
157Q8×Dic3C22×C312yes—
158Q8×Dic3C22×C311yescharacteristic
159#940C22×C316yes—
160#939C22×C3112yes—
161#938C22×C316yes—
162#938C22×C316yes—
163#936C22×C316yes—
164D4×Dic3C22×C313yes—
165D4×Dic3C22×C316yes—
166D4×Dic3C22×C316yes—
167#930C22×C316yes—
168C4×#67C22×C316yes—
169#926C22×C316yes—
170C2×C4×Dic3C22×C316yes—
171#893C22×C313yes—
172#892C22×C313yes—
173#892C22×C313yes—
174#888C22×C316yes—
175#887C22×C311yescharacteristic
176#885C22×C3112yes—
177#67⋊C4C22×C316yes—
178#877C22×C316yes—
179C4×D12C22×C313yes—
180C42×D3C22×C313yes—
181#869C22×C313yes—
182C4×Dic6C22×C313yes—
183C23×C32C2416yes—
184C22×C3×D3C2413yes—
185C32×Q8#4114yescomplemented
186C32×D4—26no—
187C32×D4—212no—
188C32×D4—46no—
189C2×C32×C4—212no—
190C2×C32×C4C2412yes—
191C2×C32×C4C2413yes—
192C2×C32×C4C2413yes—
193C2×C32×C4C2411yescharacteristic
194C3×#67—212no—
195C3×#67—412no—
196C2×C3×Dic3—212no—
197C2×C3×Dic3—212no—
198C2×C3×Dic3C2411yescharacteristic
199C2×C3×Dic3C2413yes—
200C2×C3×Dic3C2416yes—
201C2×C3×Dic3C2412yes—
202C2×C3×Dic3C2411yescharacteristic
203C3×D12—26no—
204C3×C4×D3—26no—
205C3×C4×D3—26no—
206C3×C4×D3—26no—
207C3×Dic6—46no—
208C3×Dic6#4114yescomplemented
209Q82—31no—
210#555—36no—
211#555—33no—
212#550—32no—
213#550—33no—
214#550C3×D311yescharacteristic, complemented
215#532—33no—
216#532—36no—
217C4×#41—32no—
218C4×#41—31no—
219C4×#41—33no—
220C3×#41—28no—
221C3×#41—28no—
222C3×#41—48no—
223C3×#41—612no—
224C3×#41—64no—
225C3×#41—1212no—
226C2×C3×Q8—21no—
227C2×C3×Q8—31no—
228C2×C3×Q8—36no—
229C2×C3×Q8C22×D311yescharacteristic
230C2×C3×Q8C23×C311yescharacteristic
231C2×C3×D4—23no—
232C2×C3×D4—26no—
233C2×C3×D4—26no—
234C2×C3×D4—36no—
235C2×C3×D4—312no—
236C2×C3×D4—33no—
237C2×C3×D4C22×D316yes—
238C2×C3×D4C22×D313yes—
239C2×C3×D4C22×D316yes—
240C2×C3×D4C23×C316yes—
241C2×C3×D4C23×C313yes—
242C2×C3×D4C23×C316yes—
243C22×C3×C4—26no—
244C22×C3×C4—36no—
245C22×C3×C4—36no—
246C22×C3×C4—33no—
247C22×C3×C4—33no—
248C22×C3×C4—33no—
249C22×C3×C4C22×D316yes—
250C22×C3×C4C23×C316yes—
251C2×#67C23×C316yes—
252C2×#67C23×C3112yes—
253C22×Dic3C23×C316yes—
254C22×Dic3C23×C316yes—
255#239—24no—
256#237—212no—
257#237—412no—
258C2×D12C23×C313yes—
259C2×C4×D3C23×C313yes—
260C2×C4×D3C23×C313yes—
261C2×C4×D3C23×C313yes—
262C2×Dic6C23×C316yes—
263C2×Dic6C23×C311yescharacteristic
264C3×C4⋊C4—21no—
265C3×C4⋊C4—23no—
266C3×C4⋊C4—32no—
267C3×C4⋊C4—36no—
268C3×C4⋊C4—33no—
269C3×C4⋊C4—36no—
270C3×C4⋊C4—33no—
271C3×C4⋊C4C22×D311yescharacteristic
272C3×C4⋊C4C22×D313yes—
273C3×C4⋊C4C23×C311yescharacteristic
274C3×C4⋊C4C23×C313yes—
275C3×C22⋊C4—26no—
276C3×C22⋊C4—36no—
277C3×C22⋊C4—312no—
278C3×C22⋊C4—36no—
279C3×C22⋊C4—36no—
280C3×C22⋊C4C22×D316yes—
281C3×C22⋊C4C23×C316yes—
282C3×C42—23no—
283C3×C42—32no—
284C3×C42—33no—
285C3×C42—31no—
286C3×C42—33no—
287C3×C42—36no—
288C3×C42—36no—
289C3×C42C22×D313yes—
290C3×C42C23×C313yes—
291#217C23×C316yes—
292#217C23×C3112yes—
293(C2×D3)⋊C4C23×C316yes—
294(C2×D3)⋊C4C23×C316yes—
295C4⋊Dic3C23×C313yes—
296C4⋊Dic3C23×C316yes—
297C4⋊Dic3C23×C313yes—
298Dic3⋊C4C23×C316yes—
299Dic3⋊C4C23×C312yes—
300C4×Dic3C23×C316yes—
301C4×Dic3C23×C311yescharacteristic
302C4×Dic3C23×C313yes—
303C4×Dic3C23×C313yes—
304C4×Dic3C23×C316yes—
305C4×Dic3C23×C312yes—
306C22×C32—26no—
307C22×C32—26no—
308C22×C32—26no—
309C22×C32C2511yescharacteristic
310C2×C3×D3—23no—
311C2×C3×D3—23no—
312C2×C3×D3—23no—
313C32×C4—22no—
314C32×C4—23no—
315C32×C4C2×#4116yes—
316C32×C4C2×#4112yes—
317C3×Dic3—21no—
318C3×Dic3—23no—
319C3×Dic3—26no—
320C3×Dic3C2×#4114yes—
321C3×Dic3H(5,𝔽2)12yes—
322C2×#41—31no—
323C2×#41—33no—
324C2×#41—36no—
325C2×#41C2×C3×D312yes—
326C4⋊Q8—36no—
327C4⋊Q8—33no—
328#127—36no—
329#127C2×C3×D313yes—
330#124—36no—
331#124—312no—
332#121—36no—
333#121—36no—
334#121—312no—
335#121—36no—
336#121C2×C3×D316yes—
337C4×Q8—31no—
338C4×Q8—32no—
339C4×Q8—31no—
340C4×Q8—33no—
341C4×Q8C2×C3×D311yescharacteristic
342C4×D4—36no—
343C4×D4—33no—
344C4×D4—36no—
345C4×D4—36no—
346C4×D4—33no—
347C4×D4—36no—
348C4×D4—33no—
349C4×D4C2×C3×D313yes—
350#117—36no—
351#117—33no—
352#117—33no—
353#117—36no—
354C2×C42—36no—
355C2×C42—33no—
356C23×C3—26no—
357C23×C3—33no—
358C23×C3C23×D316yes—
359C23×C3C24×C316yes—
360C22×D3C24×C313yes—
361C3×Q8—24no—
362C3×Q8—34no—
363C3×Q8—126no—
364C3×Q8#23714yescomplemented
365C3×Q8C3×#4114yescomplemented
366C3×D4—212no—
367C3×D4—26no—
368C3×D4—26no—
369C3×D4—212no—
370C3×D4—46no—
371C3×D4—412no—
372C3×D4—46no—
373C3×D4—46no—
374C3×D4—612no—
375C3×D4—66no—
376C3×D4—86no—
377C3×D4—1212no—
378C2×C3×C4—212no—
379C2×C3×C4—212no—
380C2×C3×C4—21no—
381C2×C3×C4—23no—
382C2×C3×C4—23no—
383C2×C3×C4—22no—
384C2×C3×C4—31no—
385C2×C3×C4—32no—
386C2×C3×C4—31no—
387C2×C3×C4—33no—
388C2×C3×C4—36no—
389C2×C3×C4—412no—
390C2×C3×C4—66no—
391C2×C3×C4—612no—
392C2×C3×C4—612no—
393C2×C3×C4—66no—
394C2×C3×C4—66no—
395C2×C3×C4C23×D312yes—
396C2×C3×C4C23×D313yes—
397C2×C3×C4C23×D313yes—
398C2×C3×C4C23×D311yescharacteristic
399C2×C3×C4C24×C312yes—
400C2×C3×C4C24×C313yes—
401C2×C3×C4C24×C313yes—
402C2×C3×C4C24×C311yescharacteristic
403#67—212no—
404#67—412no—
405C2×Dic3—212no—
406C2×Dic3—212no—
407C2×Dic3C24×C311yescharacteristic
408C2×Dic3C24×C313yes—
409C2×Dic3C24×C316yes—
410C2×Dic3C24×C312yes—
411C2×Dic3C24×C311yescharacteristic
412D12—26no—
413C4×D3—26no—
414C4×D3—26no—
415C4×D3—26no—
416Dic6—46no—
417Dic6C3×#4114yescomplemented
418C2×C32—46no—
419C2×C32C22×#4111yescharacteristic
420C2×C32C22×#4111yescharacteristic
421C2×C32C2×H(5,𝔽2)11yescharacteristic
422C3×D3—43no—
423#41—28no—
424#41—612no—
425#41—64no—
426#41—1212no—
427C2×Q8—31no—
428C2×Q8—36no—
429C2×Q8C22×C3×D311yescharacteristic
430C2×D4—36no—
431C2×D4—312no—
432C2×D4—33no—
433C2×D4C22×C3×D316yes—
434C2×D4C22×C3×D313yes—
435C2×D4C22×C3×D316yes—
436C22×C4—36no—
437C22×C4—36no—
438C22×C4—33no—
439C22×C4—33no—
440C22×C4—33no—
441C22×C4C22×C3×D316yes—
442C4⋊C4—32no—
443C4⋊C4—33no—
444C4⋊C4—36no—
445C4⋊C4—36no—
446C4⋊C4—33no—
447C4⋊C4C22×C3×D311yescharacteristic
448C4⋊C4C22×C3×D313yes—
449C22⋊C4—312no—
450C22⋊C4—36no—
451C22⋊C4—36no—
452C22⋊C4—36no—
453C22⋊C4C22×C3×D316yes—
454C42—32no—
455C42—33no—
456C42—31no—
457C42—33no—
458C42—36no—
459C42—36no—
460C42C22×C3×D313yes—
461C22×C3—26no—
462C22×C3—26no—
463C22×C3—26no—
464C22×C3—26no—
465C22×C3—26no—
466C22×C3—26no—
467C22×C3—21no—
468C22×C3—46no—
469C22×C3—46no—
470C22×C3—46no—
471C22×C3—63no—
472C22×C3—63no—
473C22×C3—63no—
474C22×C3C24×D311yescharacteristic
475C22×C3C25×C311yescharacteristic
476C2×D3—23no—
477C2×D3—23no—
478C2×D3—23no—
479C3×C4—22no—
480C3×C4—23no—
481C3×C4—22no—
482C3×C4—23no—
483C3×C4—26no—
484C3×C4—22no—
485C3×C4—34no—
486C3×C4—32no—
487C3×C4—42no—
488C3×C4—43no—
489C3×C4—61no—
490C3×C4—63no—
491C3×C4—66no—
492C3×C4C2×#23716yes—
493C3×C4D3×#4112yes—
494C3×C4C2×C3×#4116yes—
495C3×C4C2×C3×#4112yes—
496Dic3—21no—
497Dic3—23no—
498Dic3—26no—
499Dic3C2×C3×#4114yes—
500Dic3C3×H(5,𝔽2)12yes—
501C32#348311yescharacteristic, complemented, Sylow
502C23—33no—
503C23C23×C3×D316yes—
504Q8—34no—
505Q8—126no—
506Q8C3×#23714yescomplemented
507D4—212no—
508D4—26no—
509D4—46no—
510D4—612no—
511D4—66no—
512D4—1212no—
513C2×C4—212no—
514C2×C4—31no—
515C2×C4—32no—
516C2×C4—31no—
517C2×C4—33no—
518C2×C4—36no—
519C2×C4—66no—
520C2×C4—612no—
521C2×C4—612no—
522C2×C4—66no—
523C2×C4—66no—
524C2×C4C23×C3×D312yes—
525C2×C4C23×C3×D313yes—
526C2×C4C23×C3×D313yes—
527C2×C4C23×C3×D311yescharacteristic
528C2×C3—21no—
529C2×C3—21no—
530C2×C3—21no—
531C2×C3—46no—
532C2×C3—46no—
533C2×C3—86no—
534C2×C3—123no—
535C2×C3C22×#23711yescharacteristic
536C2×C3C2×D3×#4111yescharacteristic
537C2×C3D3×H(5,𝔽2)11yescharacteristic
538C2×C3C22×C3×#4111yescharacteristic
539C2×C3C22×C3×#4111yescharacteristic
540C2×C3C2×C3×H(5,𝔽2)11yescharacteristic
541D3—43no—
542C22—26no—
543C22—26no—
544C22—26no—
545C22—63no—
546C22—63no—
547C22—63no—
548C22C24×C3×D311yescharacteristic
549C4—22no—
550C4—23no—
551C4—34no—
552C4—32no—
553C4—61no—
554C4—63no—
555C4—66no—
556C4C2×C3×#23716yes—
557C4C3×D3×#4112yes—
558C3—21no—
559C3#8676811yescharacteristic, direct product factor
560C3C3×#348311yescharacteristic, complemented
561C2—46no—
562C2—123no—
563C2C22×C3×#23711yescharacteristic
564C2C2×C3×D3×#4111yescharacteristic
565C2C3×D3×H(5,𝔽2)11yescharacteristic
subgroupG/H
centerC22×C3C24×D3
commutator, cosocleC22×C3C25×C3
FrattiniC22C24×C3×D3
FittingC32×#550C2
socleC22×C32C25
hypercenterC3×#550D3
hypocenterC3C3×#3483
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C22×C3 ◁ C3×#550

Complements

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

NH ≅ G/Ntype
1. C32×#550C2semidirect
2. C3×#5834C2semidirect
3. C3×Dic3×#41C2semidirect
4. C3×#5718C2semidirect
5. C3×#5716C2semidirect
6. C3×#5616C2semidirect
7. C3×#5596C2semidirect
8. C3×C4×#239C2semidirect
9. C3×Q8×Dic6C2semidirect
10. C3×#5563C2semidirect
11. C3×C4×#237C2semidirect
13. #86768C3direct (N × H)
54. C3×#550D3semidirect
55. C3×#550C2×C3semidirect
61. #5834C2×C3semidirect
62. Dic3×#41C2×C3semidirect
63. #5718C2×C3semidirect
64. #5716C2×C3semidirect
65. #5616C2×C3semidirect
66. #5596C2×C3semidirect
67. C4×#239C2×C3semidirect
68. Q8×Dic6C2×C3semidirect
69. #5563C2×C3semidirect
70. C4×#237C2×C3semidirect
185. C32×Q8#41semidirect
208. C3×Dic6#41semidirect
214. #550C3×D3semidirect
364. C3×Q8#237semidirect
365. C3×Q8C3×#41semidirect
417. Dic6C3×#41semidirect
501. C32#3483semidirect
506. Q8C3×#237semidirect
559. C3#86768direct (N × H)
560. C3C3×#3483semidirect

Maximal subgroups

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

subgroupindex
C32×#5502
C3×#58342
C3×Dic3×#412
C3×#57182
C3×#57162
C3×#56162
C3×#55962
C3×C4×#2392
C3×Q8×Dic62
C3×#55632
C3×C4×#2372
C3×#34833
#867683