Finite Group Databasebeta

← C3×#86776 #11909734

Subgroups — C3×#86776 #11909734

Order 1152.

2364 conjugacy classes of subgroups.

The subgroup lattice has 555 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C32×#550C211yesmaximal, characteristic, complemented
2C3×#5831C212yesmaximal, complemented
3C3×#5827C212yesmaximal, complemented
4C3×#5716C213yesmaximal, complemented
5C3×#5713C213yesmaximal, complemented
6C3×#5611C216yesmaximal, complemented
7C3×#5603C216yesmaximal, complemented
8C3×C4×D3×Q8C211yesmaximal, characteristic, complemented
9C3×Q8×Dic6C211yesmaximal, characteristic, complemented
10C3×#5557C213yesmaximal, complemented
11C3×#5543C213yesmaximal, complemented
12C3×#3522—31nomaximal
13#86776C311yesmaximal, characteristic, direct product factor
14C2×C32×#41C2212yes—
15C2×C3×D3×Q8C2211yescharacteristic
16C2×C3×#237C2216yes—
17C32×#127C2213yes—
18C32×#121C2216yes—
19C32×C4×Q8C2211yescharacteristic
20C32×C4×D4C2213yes—
21C3×Q8×Dic3C2212yes—
22C3×Q8×Dic3C2211yescharacteristic
23C3×#938C2213yes—
24C3×#938C2216yes—
25C3×#936C2213yes—
26C3×#936C2216yes—
27C3×#936C2216yes—
28C3×#935C22112yes—
29C3×D4×Dic3C2216yes—
30C3×D4×Dic3C2216yes—
31C3×D4×Dic3C2216yes—
32C3×C4×#67C2216yes—
33C3×#926C2216yes—
34C2×C3×C4⋊Dic3C2216yes—
35C3×#924C2216yes—
36C3×#897C2211yescharacteristic
37C3×D3×C4⋊C4C2213yes—
38C3×#890C2213yes—
39C3×#888C2216yes—
40C3×#887C2213yes—
41C3×#883C2216yes—
42C3×#879C2216yes—
43C3×#878C22112yes—
44C3×C42×D3C2213yes—
45C3×#869C2213yes—
46C3×C4×Dic6C2213yes—
47C3×C4×Dic6C2213yes—
48C3×#559—33no—
49C3×Q82—31no—
50C3×#554—33no—
51C3×#550—21no—
52C3×#550D311yescharacteristic, complemented
53C3×#550C2×C311yescharacteristic, complemented
54C3×#548—32no—
55C3×#547—36no—
56C3×#542—36no—
57C3×#532—33no—
58C3×#529—33no—
59C3×#520—32no—
60C2×C3×C4×Q8—31no—
61#5831C2×C312yescomplemented
62#5827C2×C312yescomplemented
63#5716C2×C313yescomplemented
64#5713C2×C313yescomplemented
65#5611C2×C316yescomplemented
66#5603C2×C316yescomplemented
67C4×D3×Q8C2×C311yescharacteristic, complemented
68Q8×Dic6C2×C311yescharacteristic, complemented
69#5557C2×C313yescomplemented
70#5543C2×C313yescomplemented
71C32×#41—28no—
72C2×C32×Q8C2311yescharacteristic
73C2×C32×D4C2316yes—
74C2×C32×D4C2313yes—
75C2×C32×D4C2316yes—
76C22×C32×C4C2316yes—
77C2×C3×#67C2316yes—
78C22×C3×Dic3C23112yes—
79C3×D3×Q8—24no—
80C3×#237—412no—
81C2×C3×C4×D3C2313yes—
82C2×C3×C4×D3C2313yes—
83C2×C3×C4×D3C2311yescharacteristic
84C2×C3×Dic6C2313yes—
85C2×C3×Dic6C2311yescharacteristic
86C2×C3×Dic6C2316yes—
87C32×C4⋊C4C2313yes—
88C32×C4⋊C4C2311yescharacteristic
89C32×C22⋊C4C2316yes—
90C32×C42C2313yes—
91C3×#217C23112yes—
92C3×#217C2316yes—
93C3×#217C2316yes—
94C3×#217C2316yes—
95C3×(C2×D3)⋊C4C2312yes—
96C3×C4⋊Dic3C2316yes—
97C3×C4⋊Dic3C2312yes—
98C3×C4⋊Dic3C2316yes—
99C3×C4⋊Dic3C2313yes—
100C3×C4⋊Dic3C2313yes—
101C3×Dic3⋊C4C2316yes—
102C3×Dic3⋊C4C2316yes—
103C3×Dic3⋊C4C2316yes—
104C3×C4×Dic3C2313yes—
105C3×C4×Dic3C2313yes—
106C3×C4×Dic3C2316yes—
107C3×C4×Dic3C2313yes—
108#3522—31noSylow
109C2×C3×#41—22no—
110C2×C3×#41—36no—
111C2×C3×#41C2×D312yes—
112C2×C3×#41C22×C312yes—
113C22×C3×Q8—31no—
114C2×D3×Q8C22×C311yescharacteristic
115C2×#237C22×C316yes—
116C3×C4⋊Q8—36no—
117C3×C4⋊Q8—33no—
118C3×#127—23no—
119C3×#127C2×D313yes—
120C3×#127C22×C313yes—
121C3×#126—36no—
122C3×#125—33no—
123C3×#124—33no—
124C3×#124—36no—
125C3×#124—36no—
126C3×#123—312no—
127C3×#123—36no—
128C3×C22⋊Q8—36no—
129C3×C22⋊Q8—31no—
130C3×C22⋊Q8—312no—
131C3×#121—26no—
132C3×#121—33no—
133C3×#121—36no—
134C3×#121C2×D316yes—
135C3×#121C22×C316yes—
136C3×C4×Q8—21no—
137C3×C4×Q8—32no—
138C3×C4×Q8—31no—
139C3×C4×Q8—33no—
140C3×C4×Q8—33no—
141C3×C4×Q8—33no—
142C3×C4×Q8C2×D311yescharacteristic
143C3×C4×Q8C22×C311yescharacteristic
144C3×C4×D4—23no—
145C3×C4×D4—36no—
146C3×C4×D4—36no—
147C3×C4×D4—36no—
148C3×C4×D4—36no—
149C3×C4×D4C2×D313yes—
150C3×C4×D4C22×C313yes—
151C3×#117—36no—
152C2×C3×C4⋊C4—36no—
153C2×C3×C4⋊C4—33no—
154C2×C3×C42—33no—
155Q8×Dic3C22×C312yes—
156Q8×Dic3C22×C311yescharacteristic
157#938C22×C313yes—
158#938C22×C316yes—
159#936C22×C313yes—
160#936C22×C316yes—
161#936C22×C316yes—
162#935C22×C3112yes—
163D4×Dic3C22×C316yes—
164D4×Dic3C22×C316yes—
165D4×Dic3C22×C316yes—
166C4×#67C22×C316yes—
167#926C22×C316yes—
168C2×C4⋊Dic3C22×C316yes—
169#924C22×C316yes—
170#897C22×C311yescharacteristic
171D3×C4⋊C4C22×C313yes—
172#890C22×C313yes—
173#888C22×C316yes—
174#887C22×C313yes—
175#883C22×C316yes—
176#879C22×C316yes—
177#878C22×C3112yes—
178C42×D3C22×C313yes—
179#869C22×C313yes—
180C4×Dic6C22×C313yes—
181C4×Dic6C22×C313yes—
182C23×C32C2416yes—
183C22×C3×D3C2411yescharacteristic
184C32×Q8#4114yescomplemented
185C32×D4—212no—
186C32×D4—26no—
187C32×D4—46no—
188C2×C32×C4—212no—
189C2×C32×C4C2413yes—
190C2×C32×C4C2412yes—
191C2×C32×C4C2413yes—
192C2×C32×C4C2411yescharacteristic
193C3×#67—212no—
194C2×C3×Dic3—412no—
195C2×C3×Dic3C2411yescharacteristic
196C2×C3×Dic3C2412yes—
197C2×C3×Dic3C2413yes—
198C2×C3×Dic3C2413yes—
199C2×C3×Dic3C2416yes—
200C3×C4×D3—26no—
201C3×C4×D3—26no—
202C3×C4×D3—22no—
203C3×Dic6—26no—
204C3×Dic6—22no—
205C3×Dic6—46no—
206#559—33no—
207Q82—31no—
208#554—33no—
209#550C3×D311yescharacteristic, complemented
210#548—32no—
211#547—36no—
212#542—36no—
213#532—33no—
214#529—33no—
215#520—32no—
216C2×C4×Q8—31no—
217C3×#41—28no—
218C3×#41—28no—
219C3×#41—48no—
220C3×#41—1212no—
221C2×C3×Q8—21no—
222C2×C3×Q8—31no—
223C2×C3×Q8—33no—
224C2×C3×Q8—36no—
225C2×C3×Q8—64no—
226C2×C3×Q8C22×D311yescharacteristic
227C2×C3×Q8C23×C311yescharacteristic
228C2×C3×D4—23no—
229C2×C3×D4—26no—
230C2×C3×D4—26no—
231C2×C3×D4—36no—
232C2×C3×D4C22×D313yes—
233C2×C3×D4C22×D316yes—
234C2×C3×D4C22×D316yes—
235C2×C3×D4C23×C313yes—
236C2×C3×D4C23×C316yes—
237C2×C3×D4C23×C316yes—
238C22×C3×C4—26no—
239C22×C3×C4—312no—
240C22×C3×C4—31no—
241C22×C3×C4—33no—
242C22×C3×C4—33no—
243C22×C3×C4C22×D316yes—
244C22×C3×C4C23×C316yes—
245C2×#67C23×C316yes—
246C22×Dic3C23×C3112yes—
247D3×Q8—24no—
248#237—412no—
249C2×C4×D3C23×C311yescharacteristic
250C2×C4×D3C23×C313yes—
251C2×C4×D3C23×C313yes—
252C2×Dic6C23×C311yescharacteristic
253C2×Dic6C23×C313yes—
254C2×Dic6C23×C316yes—
255C3×C4⋊C4—21no—
256C3×C4⋊C4—23no—
257C3×C4⋊C4—36no—
258C3×C4⋊C4—36no—
259C3×C4⋊C4—36no—
260C3×C4⋊C4—33no—
261C3×C4⋊C4—36no—
262C3×C4⋊C4—36no—
263C3×C4⋊C4—32no—
264C3×C4⋊C4—33no—
265C3×C4⋊C4C22×D313yes—
266C3×C4⋊C4C22×D311yescharacteristic
267C3×C4⋊C4C23×C313yes—
268C3×C4⋊C4C23×C311yescharacteristic
269C3×C22⋊C4—26no—
270C3×C22⋊C4—36no—
271C3×C22⋊C4—36no—
272C3×C22⋊C4—312no—
273C3×C22⋊C4—36no—
274C3×C22⋊C4—32no—
275C3×C22⋊C4C22×D316yes—
276C3×C22⋊C4C23×C316yes—
277C3×C42—23no—
278C3×C42—33no—
279C3×C42—33no—
280C3×C42—33no—
281C3×C42—36no—
282C3×C42C22×D313yes—
283C3×C42C23×C313yes—
284#217C23×C3112yes—
285#217C23×C316yes—
286#217C23×C316yes—
287#217C23×C316yes—
288(C2×D3)⋊C4C23×C312yes—
289C4⋊Dic3C23×C313yes—
290C4⋊Dic3C23×C313yes—
291C4⋊Dic3C23×C316yes—
292C4⋊Dic3C23×C316yes—
293C4⋊Dic3C23×C312yes—
294Dic3⋊C4C23×C316yes—
295Dic3⋊C4C23×C316yes—
296Dic3⋊C4C23×C316yes—
297C4×Dic3C23×C313yes—
298C4×Dic3C23×C316yes—
299C4×Dic3C23×C313yes—
300C4×Dic3C23×C313yes—
301C22×C32—26no—
302C22×C32—26no—
303C22×C32—26no—
304C22×C32C2511yescharacteristic
305C2×C3×D3—21no—
306C2×C3×D3—21no—
307C2×C3×D3H(5,𝔽2)12yes—
308C32×C4—22no—
309C32×C4—23no—
310C32×C4C2×#4116yes—
311C32×C42−1+412yes—
312C3×Dic3—23no—
313C3×Dic3—26no—
314C3×Dic3—23no—
315C3×Dic3—22no—
316C3×Dic3—21no—
317C2×#41—36no—
318C2×#41C2×C3×D312yes—
319C22×Q8—31no—
320C4⋊Q8—36no—
321C4⋊Q8—33no—
322#127C2×C3×D313yes—
323#126—36no—
324#125—33no—
325#124—33no—
326#124—36no—
327#124—36no—
328#123—312no—
329#123—36no—
330C22⋊Q8—36no—
331C22⋊Q8—31no—
332C22⋊Q8—312no—
333#121—33no—
334#121—36no—
335#121C2×C3×D316yes—
336C4×Q8—31no—
337C4×Q8—32no—
338C4×Q8—33no—
339C4×Q8—33no—
340C4×Q8—33no—
341C4×Q8C2×C3×D311yescharacteristic
342C4×D4—36no—
343C4×D4—36no—
344C4×D4—36no—
345C4×D4—36no—
346C4×D4C2×C3×D313yes—
347#117—36no—
348C2×C4⋊C4—36no—
349C2×C4⋊C4—33no—
350C2×C42—33no—
351C23×C3—26no—
352C23×C3—31no—
353C23×C3C23×D316yes—
354C23×C3C24×C316yes—
355C22×D3C24×C311yescharacteristic
356C3×Q8—24no—
357C3×Q8—66no—
358C3×Q8—62no—
359C3×Q8—126no—
360C3×Q8#23714yescomplemented
361C3×Q8C3×#4114yescomplemented
362C3×D4—212no—
363C3×D4—26no—
364C3×D4—212no—
365C3×D4—26no—
366C3×D4—46no—
367C3×D4—412no—
368C3×D4—46no—
369C3×D4—46no—
370C3×D4—612no—
371C3×D4—86no—
372C2×C3×C4—212no—
373C2×C3×C4—212no—
374C2×C3×C4—21no—
375C2×C3×C4—23no—
376C2×C3×C4—23no—
377C2×C3×C4—22no—
378C2×C3×C4—31no—
379C2×C3×C4—32no—
380C2×C3×C4—33no—
381C2×C3×C4—33no—
382C2×C3×C4—36no—
383C2×C3×C4—412no—
384C2×C3×C4—66no—
385C2×C3×C4—66no—
386C2×C3×C4—62no—
387C2×C3×C4—1212no—
388C2×C3×C4C23×D313yes—
389C2×C3×C4C23×D312yes—
390C2×C3×C4C23×D313yes—
391C2×C3×C4C23×D311yescharacteristic
392C2×C3×C4C24×C313yes—
393C2×C3×C4C24×C312yes—
394C2×C3×C4C24×C313yes—
395C2×C3×C4C24×C311yescharacteristic
396#67—212no—
397C2×Dic3—412no—
398C2×Dic3C24×C311yescharacteristic
399C2×Dic3C24×C312yes—
400C2×Dic3C24×C313yes—
401C2×Dic3C24×C313yes—
402C2×Dic3C24×C316yes—
403C4×D3—26no—
404C4×D3—26no—
405C4×D3—22no—
406Dic6—26no—
407Dic6—22no—
408Dic6—46no—
409C2×C32—46no—
410C2×C32C22×#4111yescharacteristic
411C2×C32C2×H(5,𝔽2)11yescharacteristic
412C2×C32C2×2−1+411yescharacteristic
413C3×D3—22no—
414#41—28no—
415#41—1212no—
416C2×Q8—31no—
417C2×Q8—33no—
418C2×Q8—36no—
419C2×Q8—64no—
420C2×Q8C22×C3×D311yescharacteristic
421C2×D4—36no—
422C2×D4C22×C3×D313yes—
423C2×D4C22×C3×D316yes—
424C2×D4C22×C3×D316yes—
425C22×C4—312no—
426C22×C4—31no—
427C22×C4—33no—
428C22×C4—33no—
429C22×C4C22×C3×D316yes—
430C4⋊C4—33no—
431C4⋊C4—36no—
432C4⋊C4—36no—
433C4⋊C4—36no—
434C4⋊C4—32no—
435C4⋊C4—33no—
436C4⋊C4—36no—
437C4⋊C4—36no—
438C4⋊C4C22×C3×D313yes—
439C4⋊C4C22×C3×D311yescharacteristic
440C22⋊C4—36no—
441C22⋊C4—36no—
442C22⋊C4—312no—
443C22⋊C4—36no—
444C22⋊C4—32no—
445C22⋊C4C22×C3×D316yes—
446C42—33no—
447C42—33no—
448C42—33no—
449C42—36no—
450C42C22×C3×D313yes—
451C22×C3—26no—
452C22×C3—26no—
453C22×C3—26no—
454C22×C3—26no—
455C22×C3—26no—
456C22×C3—26no—
457C22×C3—21no—
458C22×C3—32no—
459C22×C3—46no—
460C22×C3—46no—
461C22×C3—46no—
462C22×C3—61no—
463C22×C3—61no—
464C22×C3C24×D311yescharacteristic
465C22×C3C25×C311yescharacteristic
466C2×D3—21no—
467C2×D3—21no—
468C2×D3C3×H(5,𝔽2)12yes—
469C3×C4—22no—
470C3×C4—23no—
471C3×C4—22no—
472C3×C4—23no—
473C3×C4—26no—
474C3×C4—22no—
475C3×C4—42no—
476C3×C4—43no—
477C3×C4—63no—
478C3×C4—61no—
479C3×C4—62no—
480C3×C4—63no—
481C3×C4—66no—
482C3×C4C2×#23716yes—
483C3×C4#101012yes—
484C3×C4C2×C3×#4116yes—
485C3×C4C3×2−1+412yes—
486Dic3—23no—
487Dic3—26no—
488Dic3—23no—
489Dic3—22no—
490Dic3—21no—
491C32#352211yescharacteristic, complemented, Sylow
492C23—31no—
493C23C23×C3×D316yes—
494Q8—66no—
495Q8—62no—
496Q8—126no—
497Q8C3×#23714yescomplemented
498D4—212no—
499D4—26no—
500D4—46no—
501D4—612no—
502C2×C4—212no—
503C2×C4—31no—
504C2×C4—32no—
505C2×C4—33no—
506C2×C4—33no—
507C2×C4—36no—
508C2×C4—66no—
509C2×C4—66no—
510C2×C4—62no—
511C2×C4—1212no—
512C2×C4C23×C3×D313yes—
513C2×C4C23×C3×D312yes—
514C2×C4C23×C3×D313yes—
515C2×C4C23×C3×D311yescharacteristic
516C2×C3—21no—
517C2×C3—21no—
518C2×C3—21no—
519C2×C3—46no—
520C2×C3—46no—
521C2×C3—62no—
522C2×C3—86no—
523C2×C3C22×#23711yescharacteristic
524C2×C3C2×#101011yescharacteristic
525C2×C3D3×H(5,𝔽2)11yescharacteristic
526C2×C3C22×C3×#4111yescharacteristic
527C2×C3C2×C3×H(5,𝔽2)11yescharacteristic
528C2×C3C2×C3×2−1+411yescharacteristic
529D3—22no—
530C22—26no—
531C22—26no—
532C22—26no—
533C22—32no—
534C22—61no—
535C22—61no—
536C22C24×C3×D311yescharacteristic
537C4—22no—
538C4—23no—
539C4—63no—
540C4—61no—
541C4—62no—
542C4—63no—
543C4—66no—
544C4C2×C3×#23716yes—
545C4C3×#101012yes—
546C3—21no—
547C3#8677611yescharacteristic, direct product factor
548C3C3×#352211yescharacteristic, complemented
549C2—46no—
550C2—62no—
551C2C22×C3×#23711yescharacteristic
552C2C2×C3×#101011yescharacteristic
553C2C3×D3×H(5,𝔽2)11yescharacteristic
subgroupG/H
centerC22×C3C24×D3
commutator, cosocleC22×C3C25×C3
FrattiniC22C24×C3×D3
FittingC32×#550C2
socleC22×C32C25
hypercenterC3×#550D3
hypocenterC3C3×#3522
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×#5831C2semidirect
3. C3×#5827C2semidirect
4. C3×#5716C2semidirect
5. C3×#5713C2semidirect
6. C3×#5611C2semidirect
7. C3×#5603C2semidirect
8. C3×C4×D3×Q8C2semidirect
9. C3×Q8×Dic6C2semidirect
10. C3×#5557C2semidirect
11. C3×#5543C2semidirect
13. #86776C3direct (N × H)
52. C3×#550D3semidirect
53. C3×#550C2×C3semidirect
61. #5831C2×C3semidirect
62. #5827C2×C3semidirect
63. #5716C2×C3semidirect
64. #5713C2×C3semidirect
65. #5611C2×C3semidirect
66. #5603C2×C3semidirect
67. C4×D3×Q8C2×C3semidirect
68. Q8×Dic6C2×C3semidirect
69. #5557C2×C3semidirect
70. #5543C2×C3semidirect
184. C32×Q8#41semidirect
209. #550C3×D3semidirect
360. C3×Q8#237semidirect
361. C3×Q8C3×#41semidirect
491. C32#3522semidirect
497. Q8C3×#237semidirect
547. C3#86776direct (N × H)
548. C3C3×#3522semidirect

Maximal subgroups

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

subgroupindex
C32×#5502
C3×#58312
C3×#58272
C3×#57162
C3×#57132
C3×#56112
C3×#56032
C3×C4×D3×Q82
C3×Q8×Dic62
C3×#55572
C3×#55432
C3×#35223
#867763