Finite Group Databasebeta

← C3×#86760 #11905886

Subgroups — C3×#86760 #11905886

Order 1152.

3124 conjugacy classes of subgroups.

The subgroup lattice has 606 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×#5817C212yesmaximal, complemented
4C3×#5727C213yesmaximal, complemented
5C3×#5725C213yesmaximal, complemented
6C3×#5633C216yesmaximal, complemented
7C3×#5629C216yesmaximal, complemented
8C3×#5577C211yesmaximal, characteristic, complemented
9C3×C4×D3×Q8C211yesmaximal, characteristic, complemented
10C3×#5553C213yesmaximal, complemented
11C3×#5548C213yesmaximal, complemented
12C3×#3497—31nomaximal
13#86760C311yesmaximal, characteristic, direct product factor
14C22×C32×Q8C2212yes—
15C3×#1007—216no—
16C2×C3×#239C2212yes—
17C2×C3×#239C2216yes—
18C2×C3×D3×Q8C2216yes—
19C2×C3×D3×Q8C2211yescharacteristic
20C2×C3×#235C2216yes—
21C2×C3×#235C2216yes—
22C32×C4⋊Q8C2213yes—
23C32×C22⋊Q8C2216yes—
24C32×C4×Q8C2211yescharacteristic
25C32×C4×D4C2213yes—
26C3×#947C2216yes—
27C3×#946C2216yes—
28C3×Q8×Dic3C2211yescharacteristic
29C3×#940C2213yes—
30C3×#938C2211yescharacteristic
31C3×#930C2216yes—
32C3×C4×#67C2216yes—
33C3×#896C22112yes—
34C3×#895C2213yes—
35C3×#895C2216yes—
36C3×#895C2216yes—
37C3×#893C2216yes—
38C3×#893C2216yes—
39C3×D3×C4⋊C4C2213yes—
40C3×#888C2213yes—
41C3×#887C2213yes—
42C3×#885C2216yes—
43C3×#884C2216yes—
44C3×#884C22112yes—
45C3×#874C2213yes—
46C3×C4×D12C2213yes—
47C3×C42×D3C2213yes—
48C3×C4×Dic6C2213yes—
49C2×C3×2−1+4—32no—
50C2×#1007C2×C312yescomplemented
51C3×#550—31no—
52C3×#550—33no—
53C3×D4×Q8—21no—
54C3×D4×Q8—33no—
55C3×D4×Q8D311yescharacteristic, complemented
56C3×D4×Q8C2×C311yescharacteristic, complemented
57C3×#548—32no—
58C3×#548—36no—
59C3×#537—36no—
60C3×#532—33no—
61C3×#532—33no—
62C2×C3×C4×Q8—31no—
63#5817C2×C312yescomplemented
64#5727C2×C313yescomplemented
65#5725C2×C313yescomplemented
66#5633C2×C316yescomplemented
67#5629C2×C316yescomplemented
68#5577C2×C311yescharacteristic, complemented
69C4×D3×Q8C2×C311yescharacteristic, complemented
70#5553C2×C313yescomplemented
71#5548C2×C313yescomplemented
72C2×C32×Q8—28no—
73C2×C32×Q8C2316yes—
74C2×C32×Q8C2311yescharacteristic
75C2×C32×D4C2311yescharacteristic
76C22×C32×C4C2316yes—
77C2×C3×#67C2312yes—
78C2×C3×#67C2316yes—
79C2×C3×#67C2316yes—
80C3×#239—28no—
81C3×#239—224no—
82C3×D3×Q8—224no—
83C3×D3×Q8D418yescomplemented
84C3×#235—224no—
85C3×#235—224no—
86C2×C3×D12C2316yes—
87C2×C3×D12C2313yes—
88C2×C3×D12C2316yes—
89C2×C3×D12C2316yes—
90C2×C3×C4×D3C23112yes—
91C2×C3×C4×D3C2311yescharacteristic
92C2×C3×C4×D3C2316yes—
93C2×C3×C4×D3C2313yes—
94C2×C3×C4×D3C2313yes—
95C2×C3×Dic6C2316yes—
96C2×C3×Dic6C2313yes—
97C32×C4⋊C4C2313yes—
98C32×C4⋊C4C2313yes—
99C32×C4⋊C4C2316yes—
100C32×C22⋊C4C2316yes—
101C32×C42C2313yes—
102C3×#217C2312yes—
103C3×(C2×D3)⋊C4C2316yes—
104C3×(C2×D3)⋊C4C23112yes—
105C3×(C2×D3)⋊C4C2316yes—
106C3×C4⋊Dic3C2311yescharacteristic
107C3×C4⋊Dic3C2313yes—
108C3×Dic3⋊C4C2316yes—
109C3×Dic3⋊C4C2316yes—
110C3×C4×Dic3C2313yes—
111C3×C4×Dic3C2313yes—
112C3×C4×Dic3C2313yes—
113#3497—31noSylow
114C3×2−1+4—616no—
115C2×C3×#41—36no—
116C2×C3×#41—32no—
117C2×C3×#41—36no—
118C2×C3×#41—36no—
119C22×C3×Q8—22no—
120C22×C3×Q8—31no—
121C22×C3×Q8—36no—
122C22×C3×Q8C2×D312yes—
123C22×C3×Q8C22×C312yes—
124#1007—216no—
125C2×#239C22×C316yes—
126C2×#239C22×C312yes—
127C2×D3×Q8C22×C316yes—
128C2×D3×Q8C22×C311yescharacteristic
129C2×#235C22×C316yes—
130C2×#235C22×C316yes—
131C3×C4⋊Q8—23no—
132C3×C4⋊Q8—33no—
133C3×C4⋊Q8C2×D313yes—
134C3×C4⋊Q8C22×C313yes—
135C3×#127—33no—
136C3×#127—33no—
137C3×#124—36no—
138C3×#124—36no—
139C3×C22⋊Q8—26no—
140C3×C22⋊Q8—36no—
141C3×C22⋊Q8—312no—
142C3×C22⋊Q8C2×D316yes—
143C3×C22⋊Q8C22×C316yes—
144C3×#121—31no—
145C3×#121—36no—
146C3×#121—33no—
147C3×#121—36no—
148C3×#121—36no—
149C3×#121—36no—
150C3×#121—312no—
151C3×C4×Q8—21no—
152C3×C4×Q8—31no—
153C3×C4×Q8—33no—
154C3×C4×Q8—33no—
155C3×C4×Q8C2×D311yescharacteristic
156C3×C4×Q8C22×C311yescharacteristic
157C3×C4×D4—23no—
158C3×C4×D4—36no—
159C3×C4×D4—36no—
160C3×C4×D4—36no—
161C3×C4×D4—33no—
162C3×C4×D4C2×D313yes—
163C3×C4×D4C22×C313yes—
164C2×C3×C4⋊C4—33no—
165C2×C3×C42—33no—
166#947C22×C316yes—
167#946C22×C316yes—
168Q8×Dic3C22×C311yescharacteristic
169#940C22×C313yes—
170#938C22×C311yescharacteristic
171#930C22×C316yes—
172C4×#67C22×C316yes—
173#896C22×C3112yes—
174#895C22×C316yes—
175#895C22×C313yes—
176#895C22×C316yes—
177#893C22×C316yes—
178#893C22×C316yes—
179D3×C4⋊C4C22×C313yes—
180#888C22×C313yes—
181#887C22×C313yes—
182#885C22×C316yes—
183#884C22×C3112yes—
184#884C22×C316yes—
185#874C22×C313yes—
186C4×D12C22×C313yes—
187C42×D3C22×C313yes—
188C4×Dic6C22×C313yes—
189C23×C32C2412yes—
190C22×C3×D3C2416yes—
191C22×C3×D3C2411yescharacteristic
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—24no—
201C3×#67—212no—
202C3×#67—46no—
203C2×C3×Dic3C2411yescharacteristic
204C2×C3×Dic3C2413yes—
205C2×C3×Dic3C2412yes—
206C2×C3×Dic3C2413yes—
207C3×D12—212no—
208C3×D12—212no—
209C3×D12—26no—
210C3×D12—46no—
211C3×C4×D3—212no—
212C3×C4×D3—22no—
213C3×C4×D3—224no—
214C3×C4×D3—43no—
215C3×C4×D3C2×D4112yes—
216C3×Dic6—212no—
217C3×Dic6C2×D4112yes—
218C2×2−1+4—32no—
219#550—31no—
220#550—33no—
221D4×Q8—33no—
222D4×Q8C3×D311yescharacteristic, complemented
223#548—32no—
224#548—36no—
225#537—36no—
226#532—33no—
227#532—33no—
228C2×C4×Q8—31no—
229C3×#41—624no—
230C3×#41—624no—
231C3×#41—624no—
232C3×#41—68no—
233C2×C3×Q8—28no—
234C2×C3×Q8—28no—
235C2×C3×Q8—21no—
236C2×C3×Q8—26no—
237C2×C3×Q8—38no—
238C2×C3×Q8—33no—
239C2×C3×Q8—36no—
240C2×C3×Q8—48no—
241C2×C3×Q8—624no—
242C2×C3×Q8C22×D316yes—
243C2×C3×Q8C22×D311yescharacteristic
244C2×C3×Q8C23×C316yes—
245C2×C3×Q8C23×C311yescharacteristic
246C2×C3×D4—21no—
247C2×C3×D4—32no—
248C2×C3×D4—36no—
249C2×C3×D4—36no—
250C2×C3×D4—33no—
251C2×C3×D4—36no—
252C2×C3×D4—36no—
253C2×C3×D4—36no—
254C2×C3×D4C22×D311yescharacteristic
255C2×C3×D4C23×C311yescharacteristic
256C22×C3×C4—26no—
257C22×C3×C4—33no—
258C22×C3×C4—31no—
259C22×C3×C4—312no—
260C22×C3×C4—36no—
261C22×C3×C4—33no—
262C22×C3×C4C22×D316yes—
263C22×C3×C4C23×C316yes—
264C2×#67C23×C312yes—
265C2×#67C23×C316yes—
266C2×#67C23×C316yes—
267#239—28no—
268#239—224no—
269D3×Q8—224no—
270D3×Q8C3×D418yescomplemented
271#235—224no—
272#235—224no—
273C2×D12C23×C313yes—
274C2×D12C23×C316yes—
275C2×D12C23×C316yes—
276C2×D12C23×C316yes—
277C2×C4×D3C23×C311yescharacteristic
278C2×C4×D3C23×C316yes—
279C2×C4×D3C23×C3112yes—
280C2×C4×D3C23×C313yes—
281C2×C4×D3C23×C313yes—
282C2×Dic6C23×C316yes—
283C2×Dic6C23×C313yes—
284C3×C4⋊C4—23no—
285C3×C4⋊C4—23no—
286C3×C4⋊C4—26no—
287C3×C4⋊C4—36no—
288C3×C4⋊C4—36no—
289C3×C4⋊C4—33no—
290C3×C4⋊C4—31no—
291C3×C4⋊C4C22×D313yes—
292C3×C4⋊C4C22×D313yes—
293C3×C4⋊C4C22×D316yes—
294C3×C4⋊C4C23×C313yes—
295C3×C4⋊C4C23×C313yes—
296C3×C4⋊C4C23×C316yes—
297C3×C22⋊C4—26no—
298C3×C22⋊C4—32no—
299C3×C22⋊C4—36no—
300C3×C22⋊C4—312no—
301C3×C22⋊C4—36no—
302C3×C22⋊C4C22×D316yes—
303C3×C22⋊C4C23×C316yes—
304C3×C42—23no—
305C3×C42—33no—
306C3×C42—33no—
307C3×C42—33no—
308C3×C42C22×D313yes—
309C3×C42C23×C313yes—
310#217C23×C312yes—
311(C2×D3)⋊C4C23×C316yes—
312(C2×D3)⋊C4C23×C3112yes—
313(C2×D3)⋊C4C23×C316yes—
314C4⋊Dic3C23×C313yes—
315C4⋊Dic3C23×C311yescharacteristic
316Dic3⋊C4C23×C316yes—
317Dic3⋊C4C23×C316yes—
318C4×Dic3C23×C313yes—
319C4×Dic3C23×C313yes—
320C4×Dic3C23×C313yes—
321C22×C32—22no—
322C22×C32—22no—
323C22×C32—22no—
324C22×C32C2511yescharacteristic
325C2×C3×D3—21no—
326C2×C3×D3—21no—
327C2×C3×D3—26no—
328C2×C3×D3—26no—
329C2×C3×D3—26no—
330C2×C3×D3C22×D412yes—
331C32×C4—26no—
332C32×C4—23no—
333C32×C4C22×D416yes—
334C32×C42−1+412yescomplemented
335C3×Dic3—22no—
336C3×Dic3—21no—
337C3×Dic3—23no—
338C3×Dic3C22×D416yes—
3392−1+4—616no—
340C2×#41—36no—
341C2×#41—32no—
342C2×#41—36no—
343C2×#41—36no—
344C22×Q8—31no—
345C22×Q8—36no—
346C22×Q8C2×C3×D312yes—
347C4⋊Q8—33no—
348C4⋊Q8C2×C3×D313yes—
349#127—33no—
350#127—33no—
351#124—36no—
352#124—36no—
353C22⋊Q8—36no—
354C22⋊Q8—312no—
355C22⋊Q8C2×C3×D316yes—
356#121—31no—
357#121—36no—
358#121—33no—
359#121—36no—
360#121—36no—
361#121—36no—
362#121—312no—
363C4×Q8—31no—
364C4×Q8—33no—
365C4×Q8—33no—
366C4×Q8C2×C3×D311yescharacteristic
367C4×D4—36no—
368C4×D4—36no—
369C4×D4—36no—
370C4×D4—33no—
371C4×D4C2×C3×D313yes—
372C2×C4⋊C4—33no—
373C2×C42—33no—
374C23×C3—22no—
375C23×C3—31no—
376C23×C3—36no—
377C23×C3C23×D312yes—
378C23×C3C24×C312yes—
379C22×D3C24×C316yes—
380C22×D3C24×C311yescharacteristic
381C3×Q8—212no—
382C3×Q8—212no—
383C3×Q8—24no—
384C3×Q8—312no—
385C3×Q8—412no—
386C3×Q8—612no—
387C3×Q8D3×D414yes—
388C3×Q8C2×C3×D414yes—
389C3×D4—24no—
390C3×D4—22no—
391C3×D4—22no—
392C3×D4—612no—
393C3×D4—612no—
394C3×D4—64no—
395C3×D4—612no—
396C3×D4—66no—
397C3×D4—126no—
398C3×D4—126no—
399C2×C3×C4—212no—
400C2×C3×C4—212no—
401C2×C3×C4—21no—
402C2×C3×C4—23no—
403C2×C3×C4—23no—
404C2×C3×C4—26no—
405C2×C3×C4—33no—
406C2×C3×C4—312no—
407C2×C3×C4—31no—
408C2×C3×C4—33no—
409C2×C3×C4—32no—
410C2×C3×C4—412no—
411C2×C3×C4—624no—
412C2×C3×C4—612no—
413C2×C3×C4—62no—
414C2×C3×C4—123no—
415C2×C3×C4C23×D313yes—
416C2×C3×C4C23×D316yes—
417C2×C3×C4C23×D313yes—
418C2×C3×C4C23×D311yescharacteristic
419C2×C3×C4C24×C313yes—
420C2×C3×C4C24×C316yes—
421C2×C3×C4C24×C313yes—
422C2×C3×C4C24×C311yescharacteristic
423#67—24no—
424#67—212no—
425#67—46no—
426C2×Dic3C24×C311yescharacteristic
427C2×Dic3C24×C313yes—
428C2×Dic3C24×C312yes—
429C2×Dic3C24×C313yes—
430D12—212no—
431D12—212no—
432D12—26no—
433D12—46no—
434C4×D3—212no—
435C4×D3—22no—
436C4×D3—224no—
437C4×D3—43no—
438C4×D3C2×C3×D4112yes—
439Dic6—212no—
440Dic6C2×C3×D4112yes—
441C2×C32—42no—
442C2×C32C23×D411yescharacteristic
443C2×C32C2×2−1+411yescharacteristic
444C2×C32#58511yescharacteristic
445C3×D3—22no—
446C3×D3—46no—
447#41—624no—
448#41—624no—
449#41—624no—
450#41—68no—
451C2×Q8—28no—
452C2×Q8—38no—
453C2×Q8—33no—
454C2×Q8—36no—
455C2×Q8—624no—
456C2×Q8C22×C3×D316yes—
457C2×Q8C22×C3×D311yescharacteristic
458C2×D4—32no—
459C2×D4—36no—
460C2×D4—36no—
461C2×D4—33no—
462C2×D4—36no—
463C2×D4—36no—
464C2×D4—36no—
465C2×D4C22×C3×D311yescharacteristic
466C22×C4—33no—
467C22×C4—31no—
468C22×C4—312no—
469C22×C4—36no—
470C22×C4—33no—
471C22×C4C22×C3×D316yes—
472C4⋊C4—36no—
473C4⋊C4—33no—
474C4⋊C4—31no—
475C4⋊C4—36no—
476C4⋊C4C22×C3×D313yes—
477C4⋊C4C22×C3×D313yes—
478C4⋊C4C22×C3×D316yes—
479C22⋊C4—32no—
480C22⋊C4—36no—
481C22⋊C4—312no—
482C22⋊C4—36no—
483C22⋊C4C22×C3×D316yes—
484C42—33no—
485C42—33no—
486C42—33no—
487C42C22×C3×D313yes—
488C22×C3—24no—
489C22×C3—22no—
490C22×C3—22no—
491C22×C3—22no—
492C22×C3—22no—
493C22×C3—22no—
494C22×C3—22no—
495C22×C3—21no—
496C22×C3—32no—
497C22×C3—42no—
498C22×C3—42no—
499C22×C3—61no—
500C22×C3—61no—
501C22×C3—66no—
502C22×C3—66no—
503C22×C3—66no—
504C22×C3C24×D311yescharacteristic
505C22×C3C25×C311yescharacteristic
506C2×D3—21no—
507C2×D3—21no—
508C2×D3—26no—
509C2×D3—26no—
510C2×D3—26no—
511C2×D3C22×C3×D412yes—
512C3×C4—26no—
513C3×C4—23no—
514C3×C4—26no—
515C3×C4—23no—
516C3×C4—26no—
517C3×C4—22no—
518C3×C4—36no—
519C3×C4—46no—
520C3×C4—43no—
521C3×C4—62no—
522C3×C4—61no—
523C3×C4—63no—
524C3×C4C2×D3×D416yes—
525C3×C4#100712yescomplemented
526C3×C4C22×C3×D416yes—
527C3×C4C3×2−1+412yescomplemented
528Dic3—22no—
529Dic3—21no—
530Dic3—23no—
531Dic3C22×C3×D416yes—
532C32#349711yescharacteristic, complemented, Sylow
533C23—31no—
534C23—36no—
535C23C23×C3×D312yes—
536Q8—212no—
537Q8—312no—
538Q8—612no—
539Q8C3×D3×D414yes—
540D4—22no—
541D4—612no—
542D4—612no—
543D4—64no—
544D4—612no—
545D4—66no—
546D4—126no—
547D4—126no—
548C2×C4—212no—
549C2×C4—33no—
550C2×C4—312no—
551C2×C4—31no—
552C2×C4—33no—
553C2×C4—32no—
554C2×C4—624no—
555C2×C4—612no—
556C2×C4—62no—
557C2×C4—123no—
558C2×C4C23×C3×D313yes—
559C2×C4C23×C3×D316yes—
560C2×C4C23×C3×D313yes—
561C2×C4C23×C3×D311yescharacteristic
562C2×C3—21no—
563C2×C3—21no—
564C2×C3—21no—
565C2×C3—44no—
566C2×C3—42no—
567C2×C3—42no—
568C2×C3—62no—
569C2×C3—126no—
570C2×C3C22×D3×D411yescharacteristic
571C2×C3C2×#100711yescharacteristic
572C2×C3#596811yescharacteristic
573C2×C3C23×C3×D411yescharacteristic
574C2×C3C2×C3×2−1+411yescharacteristic
575C2×C3C3×#58511yescharacteristic
576D3—22no—
577D3—46no—
578C22—22no—
579C22—22no—
580C22—22no—
581C22—32no—
582C22—61no—
583C22—61no—
584C22—66no—
585C22—66no—
586C22—66no—
587C22C24×C3×D311yescharacteristic
588C4—26no—
589C4—23no—
590C4—36no—
591C4—62no—
592C4—61no—
593C4—63no—
594C4C2×C3×D3×D416yes—
595C4C3×#100712yescomplemented
596C3—21no—
597C3#8676011yescharacteristic, direct product factor
598C3C3×#349711yescharacteristic, complemented
599C2—42no—
600C2—62no—
601C2—126no—
602C2C22×C3×D3×D411yescharacteristic
603C2C2×C3×#100711yescharacteristic
604C2C3×#596811yescharacteristic
subgroupG/H
centerC22×C3C24×D3
commutator, cosocleC22×C3C25×C3
FrattiniC22C24×C3×D3
FittingC32×D4×Q8C2
socleC22×C32C25
hypercenterC3×D4×Q8D3
hypocenterC3C3×#3497
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×#5817C2semidirect
4. C3×#5727C2semidirect
5. C3×#5725C2semidirect
6. C3×#5633C2semidirect
7. C3×#5629C2semidirect
8. C3×#5577C2semidirect
9. C3×C4×D3×Q8C2semidirect
10. C3×#5553C2semidirect
11. C3×#5548C2semidirect
13. #86760C3direct (N × H)
50. C2×#1007C2×C3semidirect
55. C3×D4×Q8D3semidirect
56. C3×D4×Q8C2×C3semidirect
63. #5817C2×C3semidirect
64. #5727C2×C3semidirect
65. #5725C2×C3semidirect
66. #5633C2×C3semidirect
67. #5629C2×C3semidirect
68. #5577C2×C3semidirect
69. C4×D3×Q8C2×C3semidirect
70. #5553C2×C3semidirect
71. #5548C2×C3semidirect
83. C3×D3×Q8D4semidirect
222. D4×Q8C3×D3semidirect
270. D3×Q8C3×D4semidirect
334. C32×C42−1+4semidirect
525. C3×C4#1007semidirect
527. C3×C4C3×2−1+4semidirect
532. C32#3497semidirect
595. C4C3×#1007semidirect
597. C3#86760direct (N × H)
598. C3C3×#3497semidirect

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×#58172
C3×#57272
C3×#57252
C3×#56332
C3×#56292
C3×#55772
C3×C4×D3×Q82
C3×#55532
C3×#55482
C3×#34973
#867603