Finite Group Databasebeta

← C3×#86746 #11906013

Subgroups — C3×#86746 #11906013

Order 1152.

3092 conjugacy classes of subgroups.

The subgroup lattice has 602 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#1007C212yesmaximal, complemented
2C32×D4×Q8C211yesmaximal, characteristic, complemented
3C3×Q8×#67C212yesmaximal, complemented
4C3×#5726C213yesmaximal, complemented
5C3×D3×C4⋊Q8C213yesmaximal, complemented
6C3×#5631C216yesmaximal, complemented
7C3×#5628C216yesmaximal, complemented
8C3×Q8×D12C211yesmaximal, characteristic, complemented
9C3×C4×#239C211yesmaximal, characteristic, complemented
10C3×#5551C213yesmaximal, complemented
11C3×#5548C213yesmaximal, complemented
12C3×#3499—31nomaximal
13#86746C311yesmaximal, characteristic, direct product factor
14C22×C32×Q8C2212yes—
15C3×#1007—216no—
16C2×C3×#239C2216yes—
17C2×C3×#239C2211yescharacteristic
18C2×C3×D3×Q8C2212yes—
19C2×C3×D3×Q8C2216yes—
20C2×C3×#235C2216yes—
21C2×C3×#235C2216yes—
22C32×C4⋊Q8C2213yes—
23C32×C22⋊Q8C2216yes—
24C32×C4×Q8C2211yescharacteristic
25C32×C4×D4C2213yes—
26C3×#946C2216yes—
27C3×Q8×Dic3C2211yescharacteristic
28C3×#944C2216yes—
29C3×#940C2211yescharacteristic
30C3×#938C2213yes—
31C3×C4×#67C2216yes—
32C3×#924C2216yes—
33C3×#897C2216yes—
34C3×#895C2213yes—
35C3×#894C22112yes—
36C3×#893C2213yes—
37C3×#892C2213yes—
38C3×D3×C4⋊C4C2216yes—
39C3×D3×C4⋊C4C2216yes—
40C3×#888C2213yes—
41C3×#888C2216yes—
42C3×#885C2216yes—
43C3×#884C2216yes—
44C3×#883C22112yes—
45C3×C4×D12C2213yes—
46C3×C4×D12C2213yes—
47C3×C42×D3C2213yes—
48C3×#869C2213yes—
49C2×C3×2−1+4—32no—
50C2×#1007C2×C312yescomplemented
51C3×D4×Q8—21no—
52C3×D4×Q8—32no—
53C3×D4×Q8—31no—
54C3×D4×Q8D311yescharacteristic, complemented
55C3×D4×Q8C2×C311yescharacteristic, complemented
56C3×#547—33no—
57C3×#547—36no—
58C3×#547—33no—
59C3×#536—36no—
60C3×#532—33no—
61C2×C3×C4⋊Q8—33no—
62C3×C4×#41—31no—
63Q8×#67C2×C312yescomplemented
64#5726C2×C313yescomplemented
65D3×C4⋊Q8C2×C313yescomplemented
66#5631C2×C316yescomplemented
67#5628C2×C316yescomplemented
68Q8×D12C2×C311yescharacteristic, complemented
69C4×#239C2×C311yescharacteristic, complemented
70#5551C2×C313yescomplemented
71#5548C2×C313yescomplemented
72C2×C32×Q8—28no—
73C2×C32×Q8C2316yes—
74C2×C32×Q8C2311yescharacteristic
75C2×C32×D4C2311yescharacteristic
76C22×C32×C4C2316yes—
77C2×C3×#67C2316yes—
78C2×C3×#67C2312yes—
79C2×C3×#67C2312yes—
80C3×#239—224no—
81C3×#239D418yescomplemented
82C3×D3×Q8—28no—
83C3×D3×Q8—224no—
84C3×#235—224no—
85C3×#235—224no—
86C2×C3×D12C2316yes—
87C2×C3×D12C2311yescharacteristic
88C2×C3×D12C2313yes—
89C2×C3×C4×D3C23112yes—
90C2×C3×C4×D3C2313yes—
91C2×C3×C4×D3C2313yes—
92C2×C3×C4×D3C2316yes—
93C2×C3×C4×D3C2313yes—
94C2×C3×Dic6C2316yes—
95C2×C3×Dic6C2316yes—
96C32×C4⋊C4C2313yes—
97C32×C4⋊C4C2316yes—
98C32×C4⋊C4C2313yes—
99C32×C22⋊C4C2316yes—
100C32×C42C2313yes—
101C3×#217C2316yes—
102C3×(C2×D3)⋊C4C2316yes—
103C3×(C2×D3)⋊C4C2316yes—
104C3×(C2×D3)⋊C4C2316yes—
105C3×C4⋊Dic3C2316yes—
106C3×C4⋊Dic3C2313yes—
107C3×C4⋊Dic3C2313yes—
108C3×Dic3⋊C4C23112yes—
109C3×Dic3⋊C4C2316yes—
110C3×C4×Dic3C2313yes—
111C3×C4×Dic3C2311yescharacteristic
112C3×C4×Dic3C2313yes—
113#3499—31noSylow
114C3×2−1+4—616no—
115C2×C3×#41—31no—
116C2×C3×#41—36no—
117C2×C3×#41—36no—
118C2×C3×#41—36no—
119C22×C3×Q8—22no—
120C22×C3×Q8—36no—
121C22×C3×Q8—32no—
122C22×C3×Q8C2×D312yes—
123C22×C3×Q8C22×C312yes—
124#1007—216no—
125C2×#239C22×C316yes—
126C2×#239C22×C311yescharacteristic
127C2×D3×Q8C22×C316yes—
128C2×D3×Q8C22×C312yes—
129C2×#235C22×C316yes—
130C2×#235C22×C316yes—
131C3×C4⋊Q8—23no—
132C3×C4⋊Q8—36no—
133C3×C4⋊Q8—33no—
134C3×C4⋊Q8—36no—
135C3×C4⋊Q8—33no—
136C3×C4⋊Q8C2×D313yes—
137C3×C4⋊Q8C22×C313yes—
138C3×#127—31no—
139C3×#124—36no—
140C3×#123—312no—
141C3×#123—312no—
142C3×C22⋊Q8—26no—
143C3×C22⋊Q8—36no—
144C3×C22⋊Q8—36no—
145C3×C22⋊Q8—36no—
146C3×C22⋊Q8C2×D316yes—
147C3×C22⋊Q8C22×C316yes—
148C3×#121—33no—
149C3×#121—33no—
150C3×#121—36no—
151C3×C4×Q8—21no—
152C3×C4×Q8—31no—
153C3×C4×Q8C2×D311yescharacteristic
154C3×C4×Q8C22×C311yescharacteristic
155C3×C4×D4—23no—
156C3×C4×D4—36no—
157C3×C4×D4—33no—
158C3×C4×D4—33no—
159C3×C4×D4—33no—
160C3×C4×D4C2×D313yes—
161C3×C4×D4C22×C313yes—
162C3×#117—33no—
163C2×C3×C4⋊C4—36no—
164C2×C3×C4⋊C4—36no—
165C2×C3×C42—33no—
166#946C22×C316yes—
167Q8×Dic3C22×C311yescharacteristic
168#944C22×C316yes—
169#940C22×C311yescharacteristic
170#938C22×C313yes—
171C4×#67C22×C316yes—
172#924C22×C316yes—
173#897C22×C316yes—
174#895C22×C313yes—
175#894C22×C3112yes—
176#893C22×C313yes—
177#892C22×C313yes—
178D3×C4⋊C4C22×C316yes—
179D3×C4⋊C4C22×C316yes—
180#888C22×C313yes—
181#888C22×C316yes—
182#885C22×C316yes—
183#884C22×C316yes—
184#883C22×C3112yes—
185C4×D12C22×C313yes—
186C4×D12C22×C313yes—
187C42×D3C22×C313yes—
188#869C22×C313yes—
189C23×C32C2412yes—
190C22×C3×D3C2412yes—
191C22×C3×D3C2413yes—
192C32×Q8—212no—
193C32×Q8C2×D414yes—
194C32×D4—22no—
195C2×C32×C4—212no—
196C2×C32×C4C2413yes—
197C2×C32×C4C2416yes—
198C2×C32×C4C2413yes—
199C2×C32×C4C2411yescharacteristic
200C3×#67—212no—
201C3×#67—24no—
202C3×#67—24no—
203C2×C3×Dic3C2413yes—
204C2×C3×Dic3C2416yes—
205C2×C3×Dic3C2411yescharacteristic
206C2×C3×Dic3C2411yescharacteristic
207C3×D12—212no—
208C3×D12—22no—
209C3×D12C2×D4112yes—
210C3×C4×D3—224no—
211C3×C4×D3—212no—
212C3×C4×D3—26no—
213C3×C4×D3—26no—
214C3×C4×D3C2×D4112yes—
215C3×Dic6—212no—
216C3×Dic6—212no—
217C2×2−1+4—32no—
218D4×Q8—32no—
219D4×Q8—31no—
220D4×Q8C3×D311yescharacteristic, complemented
221#547—33no—
222#547—36no—
223#547—33no—
224#536—36no—
225#532—33no—
226C2×C4⋊Q8—33no—
227C4×#41—31no—
228C3×#41—38no—
229C3×#41—624no—
230C3×#41—624no—
231C3×#41—624no—
232C2×C3×Q8—28no—
233C2×C3×Q8—28no—
234C2×C3×Q8—21no—
235C2×C3×Q8—26no—
236C2×C3×Q8—36no—
237C2×C3×Q8—36no—
238C2×C3×Q8—48no—
239C2×C3×Q8—624no—
240C2×C3×Q8—68no—
241C2×C3×Q8C22×D316yes—
242C2×C3×Q8C22×D311yescharacteristic
243C2×C3×Q8C23×C316yes—
244C2×C3×Q8C23×C311yescharacteristic
245C2×C3×D4—21no—
246C2×C3×D4—32no—
247C2×C3×D4—36no—
248C2×C3×D4—32no—
249C2×C3×D4—33no—
250C2×C3×D4—31no—
251C2×C3×D4—36no—
252C2×C3×D4C22×D311yescharacteristic
253C2×C3×D4C23×C311yescharacteristic
254C22×C3×C4—26no—
255C22×C3×C4—33no—
256C22×C3×C4—33no—
257C22×C3×C4—33no—
258C22×C3×C4—312no—
259C22×C3×C4—36no—
260C22×C3×C4C22×D316yes—
261C22×C3×C4C23×C316yes—
262C2×#67C23×C316yes—
263C2×#67C23×C312yes—
264C2×#67C23×C312yes—
265#239—224no—
266#239C3×D418yescomplemented
267D3×Q8—28no—
268D3×Q8—224no—
269#235—224no—
270#235—224no—
271C2×D12C23×C311yescharacteristic
272C2×D12C23×C316yes—
273C2×D12C23×C313yes—
274C2×C4×D3C23×C313yes—
275C2×C4×D3C23×C313yes—
276C2×C4×D3C23×C316yes—
277C2×C4×D3C23×C3112yes—
278C2×C4×D3C23×C313yes—
279C2×Dic6C23×C316yes—
280C2×Dic6C23×C316yes—
281C3×C4⋊C4—23no—
282C3×C4⋊C4—23no—
283C3×C4⋊C4—26no—
284C3×C4⋊C4—36no—
285C3×C4⋊C4—312no—
286C3×C4⋊C4—33no—
287C3×C4⋊C4—36no—
288C3×C4⋊C4—33no—
289C3×C4⋊C4C22×D313yes—
290C3×C4⋊C4C22×D316yes—
291C3×C4⋊C4C22×D313yes—
292C3×C4⋊C4C23×C313yes—
293C3×C4⋊C4C23×C316yes—
294C3×C4⋊C4C23×C313yes—
295C3×C22⋊C4—26no—
296C3×C22⋊C4—36no—
297C3×C22⋊C4—36no—
298C3×C22⋊C4—36no—
299C3×C22⋊C4—36no—
300C3×C22⋊C4C22×D316yes—
301C3×C22⋊C4C23×C316yes—
302C3×C42—23no—
303C3×C42—31no—
304C3×C42—33no—
305C3×C42—33no—
306C3×C42C22×D313yes—
307C3×C42C23×C313yes—
308#217C23×C316yes—
309(C2×D3)⋊C4C23×C316yes—
310(C2×D3)⋊C4C23×C316yes—
311(C2×D3)⋊C4C23×C316yes—
312C4⋊Dic3C23×C313yes—
313C4⋊Dic3C23×C316yes—
314C4⋊Dic3C23×C313yes—
315Dic3⋊C4C23×C3112yes—
316Dic3⋊C4C23×C316yes—
317C4×Dic3C23×C311yescharacteristic
318C4×Dic3C23×C313yes—
319C4×Dic3C23×C313yes—
320C22×C32—22no—
321C22×C32—22no—
322C22×C32—22no—
323C22×C32C2511yescharacteristic
324C2×C3×D3—23no—
325C2×C3×D3—23no—
326C2×C3×D3—22no—
327C2×C3×D3—22no—
328C2×C3×D3—22no—
329C2×C3×D3C22×D416yes—
330C32×C4—26no—
331C32×C4—23no—
332C32×C4C22×D416yes—
333C32×C42−1+412yescomplemented
334C3×Dic3—26no—
335C3×Dic3—23no—
336C3×Dic3C22×D412yes—
337C3×Dic32−1+412yescomplemented
3382−1+4—616no—
339C2×#41—31no—
340C2×#41—36no—
341C2×#41—36no—
342C2×#41—36no—
343C22×Q8—36no—
344C22×Q8—32no—
345C22×Q8C2×C3×D312yes—
346C4⋊Q8—36no—
347C4⋊Q8—33no—
348C4⋊Q8—36no—
349C4⋊Q8—33no—
350C4⋊Q8C2×C3×D313yes—
351#127—31no—
352#124—36no—
353#123—312no—
354#123—312no—
355C22⋊Q8—36no—
356C22⋊Q8—36no—
357C22⋊Q8—36no—
358C22⋊Q8C2×C3×D316yes—
359#121—33no—
360#121—33no—
361#121—36no—
362C4×Q8—31no—
363C4×Q8C2×C3×D311yescharacteristic
364C4×D4—36no—
365C4×D4—33no—
366C4×D4—33no—
367C4×D4—33no—
368C4×D4C2×C3×D313yes—
369#117—33no—
370C2×C4⋊C4—36no—
371C2×C4⋊C4—36no—
372C2×C42—33no—
373C23×C3—22no—
374C23×C3—33no—
375C23×C3—32no—
376C23×C3C23×D312yes—
377C23×C3C24×C312yes—
378C22×D3C24×C312yes—
379C22×D3C24×C313yes—
380C3×Q8—212no—
381C3×Q8—212no—
382C3×Q8—24no—
383C3×Q8—412no—
384C3×Q8—612no—
385C3×Q8—612no—
386C3×Q8D3×D414yes—
387C3×Q8C2×C3×D414yes—
388C3×D4—24no—
389C3×D4—22no—
390C3×D4—22no—
391C3×D4—312no—
392C3×D4—612no—
393C3×D4—64no—
394C3×D4—64no—
395C3×D4—612no—
396C3×D4—62no—
397C2×C3×C4—212no—
398C2×C3×C4—212no—
399C2×C3×C4—21no—
400C2×C3×C4—23no—
401C2×C3×C4—23no—
402C2×C3×C4—26no—
403C2×C3×C4—31no—
404C2×C3×C4—31no—
405C2×C3×C4—312no—
406C2×C3×C4—33no—
407C2×C3×C4—36no—
408C2×C3×C4—412no—
409C2×C3×C4—66no—
410C2×C3×C4—624no—
411C2×C3×C4—612no—
412C2×C3×C4—66no—
413C2×C3×C4C23×D313yes—
414C2×C3×C4C23×D316yes—
415C2×C3×C4C23×D313yes—
416C2×C3×C4C23×D311yescharacteristic
417C2×C3×C4C24×C313yes—
418C2×C3×C4C24×C316yes—
419C2×C3×C4C24×C313yes—
420C2×C3×C4C24×C311yescharacteristic
421#67—212no—
422#67—24no—
423#67—24no—
424C2×Dic3C24×C313yes—
425C2×Dic3C24×C316yes—
426C2×Dic3C24×C311yescharacteristic
427C2×Dic3C24×C311yescharacteristic
428D12—212no—
429D12—22no—
430D12C2×C3×D4112yes—
431C4×D3—224no—
432C4×D3—212no—
433C4×D3—26no—
434C4×D3—26no—
435C4×D3C2×C3×D4112yes—
436Dic6—212no—
437Dic6—212no—
438C2×C32—42no—
439C2×C32C23×D411yescharacteristic
440C2×C32C2×2−1+411yescharacteristic
441C2×C32C2×2−1+411yescharacteristic
442C3×D3—26no—
443C3×D3—42no—
444#41—38no—
445#41—624no—
446#41—624no—
447#41—624no—
448C2×Q8—28no—
449C2×Q8—36no—
450C2×Q8—36no—
451C2×Q8—624no—
452C2×Q8—68no—
453C2×Q8C22×C3×D316yes—
454C2×Q8C22×C3×D311yescharacteristic
455C2×D4—32no—
456C2×D4—36no—
457C2×D4—32no—
458C2×D4—33no—
459C2×D4—31no—
460C2×D4—36no—
461C2×D4C22×C3×D311yescharacteristic
462C22×C4—33no—
463C22×C4—33no—
464C22×C4—33no—
465C22×C4—312no—
466C22×C4—36no—
467C22×C4C22×C3×D316yes—
468C4⋊C4—36no—
469C4⋊C4—33no—
470C4⋊C4—312no—
471C4⋊C4—36no—
472C4⋊C4—33no—
473C4⋊C4C22×C3×D313yes—
474C4⋊C4C22×C3×D316yes—
475C4⋊C4C22×C3×D313yes—
476C22⋊C4—36no—
477C22⋊C4—36no—
478C22⋊C4—36no—
479C22⋊C4—36no—
480C22⋊C4C22×C3×D316yes—
481C42—31no—
482C42—33no—
483C42—33no—
484C42C22×C3×D313yes—
485C22×C3—24no—
486C22×C3—22no—
487C22×C3—22no—
488C22×C3—22no—
489C22×C3—22no—
490C22×C3—22no—
491C22×C3—22no—
492C22×C3—21no—
493C22×C3—36no—
494C22×C3—42no—
495C22×C3—42no—
496C22×C3—63no—
497C22×C3—63no—
498C22×C3—62no—
499C22×C3—62no—
500C22×C3—62no—
501C22×C3C24×D311yescharacteristic
502C22×C3C25×C311yescharacteristic
503C2×D3—23no—
504C2×D3—23no—
505C2×D3—22no—
506C2×D3—22no—
507C2×D3—22no—
508C2×D3C22×C3×D416yes—
509C3×C4—26no—
510C3×C4—23no—
511C3×C4—26no—
512C3×C4—23no—
513C3×C4—26no—
514C3×C4—22no—
515C3×C4—32no—
516C3×C4—32no—
517C3×C4—46no—
518C3×C4—43no—
519C3×C4—66no—
520C3×C4—63no—
521C3×C4C2×D3×D416yes—
522C3×C4#100712yescomplemented
523C3×C4C22×C3×D416yes—
524C3×C4C3×2−1+412yescomplemented
525Dic3—26no—
526Dic3—23no—
527Dic3C22×C3×D412yes—
528Dic3C3×2−1+412yescomplemented
529C32#349911yescharacteristic, complemented, Sylow
530C23—33no—
531C23—32no—
532C23C23×C3×D312yes—
533Q8—212no—
534Q8—612no—
535Q8—612no—
536Q8C3×D3×D414yes—
537D4—22no—
538D4—312no—
539D4—612no—
540D4—64no—
541D4—64no—
542D4—612no—
543D4—62no—
544C2×C4—212no—
545C2×C4—31no—
546C2×C4—31no—
547C2×C4—312no—
548C2×C4—33no—
549C2×C4—36no—
550C2×C4—66no—
551C2×C4—624no—
552C2×C4—612no—
553C2×C4—66no—
554C2×C4C23×C3×D313yes—
555C2×C4C23×C3×D316yes—
556C2×C4C23×C3×D313yes—
557C2×C4C23×C3×D311yescharacteristic
558C2×C3—21no—
559C2×C3—21no—
560C2×C3—21no—
561C2×C3—44no—
562C2×C3—42no—
563C2×C3—42no—
564C2×C3—66no—
565C2×C3—122no—
566C2×C3C22×D3×D411yescharacteristic
567C2×C3C2×#100711yescharacteristic
568C2×C3D3×2−1+411yescharacteristic
569C2×C3C23×C3×D411yescharacteristic
570C2×C3C2×C3×2−1+411yescharacteristic
571C2×C3C2×C3×2−1+411yescharacteristic
572D3—26no—
573D3—42no—
574C22—22no—
575C22—22no—
576C22—22no—
577C22—36no—
578C22—63no—
579C22—63no—
580C22—62no—
581C22—62no—
582C22—62no—
583C22C24×C3×D311yescharacteristic
584C4—26no—
585C4—23no—
586C4—32no—
587C4—32no—
588C4—66no—
589C4—63no—
590C4C2×C3×D3×D416yes—
591C4C3×#100712yescomplemented
592C3—21no—
593C3#8674611yescharacteristic, direct product factor
594C3C3×#349911yescharacteristic, complemented
595C2—42no—
596C2—66no—
597C2—122no—
598C2C22×C3×D3×D411yescharacteristic
599C2C2×C3×#100711yescharacteristic
600C2C3×D3×2−1+411yescharacteristic
subgroupG/H
centerC22×C3C24×D3
commutator, cosocleC22×C3C25×C3
FrattiniC22C24×C3×D3
FittingC32×D4×Q8C2
socleC22×C32C25
hypercenterC3×D4×Q8D3
hypocenterC3C3×#3499
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C22×C3 ◁ C3×D4×Q8

Complements

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

NH ≅ G/Ntype
1. C2×C3×#1007C2semidirect
2. C32×D4×Q8C2semidirect
3. C3×Q8×#67C2semidirect
4. C3×#5726C2semidirect
5. C3×D3×C4⋊Q8C2semidirect
6. C3×#5631C2semidirect
7. C3×#5628C2semidirect
8. C3×Q8×D12C2semidirect
9. C3×C4×#239C2semidirect
10. C3×#5551C2semidirect
11. C3×#5548C2semidirect
13. #86746C3direct (N × H)
50. C2×#1007C2×C3semidirect
54. C3×D4×Q8D3semidirect
55. C3×D4×Q8C2×C3semidirect
63. Q8×#67C2×C3semidirect
64. #5726C2×C3semidirect
65. D3×C4⋊Q8C2×C3semidirect
66. #5631C2×C3semidirect
67. #5628C2×C3semidirect
68. Q8×D12C2×C3semidirect
69. C4×#239C2×C3semidirect
70. #5551C2×C3semidirect
71. #5548C2×C3semidirect
81. C3×#239D4semidirect
220. D4×Q8C3×D3semidirect
266. #239C3×D4semidirect
333. C32×C42−1+4semidirect
337. C3×Dic32−1+4semidirect
522. C3×C4#1007semidirect
524. C3×C4C3×2−1+4semidirect
528. Dic3C3×2−1+4semidirect
529. C32#3499semidirect
591. C4C3×#1007semidirect
593. C3#86746direct (N × H)
594. C3C3×#3499semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#10072
C32×D4×Q82
C3×Q8×#672
C3×#57262
C3×D3×C4⋊Q82
C3×#56312
C3×#56282
C3×Q8×D122
C3×C4×#2392
C3×#55512
C3×#55482
C3×#34993
#867463