Finite Group Databasebeta

← #11233318

Subgroups — #11233318

Order 768.

2218 conjugacy classes of subgroups.

The subgroup lattice has 630 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#557C211yesmaximal, characteristic, complemented
2C2×#5727C211yesmaximal, characteristic
3#83254C212yesmaximal, complemented
4#83232C212yesmaximal, complemented
5#83227C211yesmaximal, characteristic, complemented
6#83068C214yesmaximal, complemented
7#83059C214yesmaximal
8#83048C214yesmaximal, complemented
9#82499C211yesmaximal, characteristic
10#82498C214yesmaximal, complemented
11#82493C212yesmaximal, complemented
12#82492C211yesmaximal, characteristic
13Q8×(C2×D3)⋊C4C212yesmaximal, complemented
14#82480C211yesmaximal, characteristic, complemented
15#82456C211yesmaximal, characteristic, complemented
16#45336—31nomaximal, Sylow
17C22×D3×Q8C2212yes—
18C3×#557—28no—
19C2×C3×C4⋊Q8C2212yes—
20C2×C3×C4⋊Q8C2211yescharacteristic
21C2×C3×#125C2214yes—
22C2×C3×#125C2212yes—
23C2×C3×C4×Q8C2211yescharacteristic
24C2×C3×C4×Q8C2211yescharacteristic
25C2×C3×C4×Q8C2212yes—
26C2×C3×C4×Q8C2212yes—
27C2×#946C2212yes—
28C2×#946C2212yes—
29C2×Q8×Dic3C2212yes—
30C2×#944C2211yescharacteristic
31#5727—28no—
32C2×#896C2214yes—
33C2×#893C2212yes—
34C2×#888C2212yes—
35C2×C4×D12C2211yescharacteristic
36C2×C4×Dic6C2211yescharacteristic
37#5231C2212yes—
38#5230C2212yes—
39#5229C2211yescharacteristic
40#5229C2212yes—
41#4994C2214yes—
42#4992C2212yes—
43#4992C2214yes—
44#4991C2212yes—
45#4991C2212yes—
46#4990C2214yes—
47#4989C2214yes—
48#4989C2214yes—
49#4987C2212yes—
50#4987C2214yes—
51#4986C2212yes—
52#4985C2212yes—
53#4985C2214yes—
54#4982C2214yes—
55#4981C2212yes—
56#4980C2214yes—
57#4980C2214yes—
58#4979C2214yes—
59#4979C2214yes—
60#4978C2214yes—
61#4978C2214yes—
62#4977C2211yescharacteristic
63#4976C2214yes—
64#4973C2211yescharacteristic
65#4973C2214yes—
66#4940C2214yes—
67#4940C2214yes—
68C4×(C2×D3)⋊C4C2212yes—
69C4×(C2×D3)⋊C4C2212yes—
70C4×(C2×D3)⋊C4C2214yes—
71#4936C2211yescharacteristic
72#4933C2214yes—
73#4933C2214yes—
74#4932C2211yescharacteristic
75#4929C2212yes—
76#4929C2214yes—
77#4928C2212yes—
78C2×#557D311yescharacteristic, complemented
79C2×D4×Q8—31no—
80#2713—34no—
81#2690—34no—
82#2581—31no—
83#2536—31no—
84#2465—34no—
85#2460—34no—
86#2434—32no—
87#2429—32no—
88#2421—32no—
89#2398—31no—
90Q8×C22⋊C4—32no—
91#2327—31no—
92#2327—31no—
93C22×C3×Q8C2311yescharacteristic
94C22×C3×Q8C2311yescharacteristic
95C22×C3×Q8C2312yes—
96C2×D3×Q8—28no—
97C2×D3×Q8—28no—
98C2×D3×Q8—44no—
99C22×D12C2311yescharacteristic
100C22×C4×D3C2312yes—
101C22×C4×D3C2314yes—
102C22×Dic6C2314yes—
103C22×Dic6C2311yescharacteristic
104C3×C4⋊Q8—28no—
105C3×C4⋊Q8D418yes—
106C3×#125—216no—
107C3×#125—28no—
108C3×C4×Q8—28no—
109C3×C4×Q8—28no—
110C3×C4×Q8—24no—
111C3×C4×Q8—24no—
112C2×C3×C4⋊C4C2311yescharacteristic
113C2×C3×C4⋊C4C2314yes—
114C2×C3×C4⋊C4C2314yes—
115C2×C3×C4⋊C4C2312yes—
116C2×C3×C4⋊C4C2312yes—
117C2×C3×C4⋊C4C2314yes—
118C2×C3×C4⋊C4C2311yescharacteristic
119C2×C3×C4⋊C4C2312yes—
120C2×C3×C4⋊C4C2312yes—
121C2×C3×C42C2312yes—
122C2×C3×C42C2314yes—
123C2×C3×C42C2312yes—
124C2×C3×C42C2311yescharacteristic
125#946—28no—
126#946—44no—
127Q8×Dic3—44no—
128#944—24no—
129C2×(C2×D3)⋊C4C2312yes—
130C2×(C2×D3)⋊C4C2312yes—
131C2×(C2×D3)⋊C4C2314yes—
132C2×(C2×D3)⋊C4C2312yes—
133C2×(C2×D3)⋊C4C2312yes—
134C2×(C2×D3)⋊C4C2312yes—
135C2×(C2×D3)⋊C4C2314yes—
136C2×(C2×D3)⋊C4C2314yes—
137C2×C4⋊Dic3C2311yescharacteristic
138C2×C4⋊Dic3C2312yes—
139C2×C4⋊Dic3C2311yescharacteristic
140C2×C4⋊Dic3C2314yes—
141C2×Dic3⋊C4C2314yes—
142C2×Dic3⋊C4C2312yes—
143C2×Dic3⋊C4C2314yes—
144C2×Dic3⋊C4C2314yes—
145C2×Dic3⋊C4C2314yes—
146C2×C4×Dic3C2311yescharacteristic
147C2×C4×Dic3C2311yescharacteristic
148C2×C4×Dic3C2312yes—
149C2×C4×Dic3C2314yes—
150#896—216no—
151#893—28no—
152#888—28no—
153C4×D12—24no—
154C4×Dic6—24no—
155C3⋊#95C2314yes—
156C3⋊#95C2314yes—
157C3⋊#95C2312yes—
158C3⋊#95C2314yes—
159C3⋊#95C2314yes—
160C3⋊#95C2314yes—
161C3⋊#95C2312yes—
162C3⋊#95C2312yes—
163C3⋊#95C2312yes—
164C3⋊#95C2314yes—
165C3⋊#95C2314yes—
166C3⋊#95C2314yes—
167C3⋊#95C2312yes—
168C3⋊#95C2312yes—
169C3⋊#95C2312yes—
170C3⋊#95C2312yes—
171C3⋊#95C2314yes—
172C23×Q8—32no—
173#557—28no—
174D4×Q8—68no—
175C2×C4⋊Q8—31no—
176C2×C4⋊Q8—32no—
177C2×C4⋊Q8C2×D312yes—
178C2×C4⋊Q8C2×D311yescharacteristic
179C2×#125C2×D312yes—
180C2×#125C2×D314yes—
181C2×C22⋊Q8—32no—
182C2×C22⋊Q8—32no—
183C2×C22⋊Q8—34no—
184C2×C4×Q8—32no—
185C2×C4×Q8—31no—
186C2×C4×Q8C2×D311yescharacteristic
187C2×C4×Q8C2×D311yescharacteristic
188C2×C4×Q8C2×D312yes—
189C2×C4×Q8C2×D312yes—
190C2×C4×D4—32no—
191C2×C4×D4—31no—
192#400—32no—
193#400—32no—
194#400—34no—
195#398—32no—
196#398—34no—
197#397—32no—
198#397—34no—
199#396—34no—
200#395—32no—
201#395—34no—
202#395—34no—
203#394—32no—
204#394—32no—
205#393—32no—
206#391—32no—
207#391—31no—
208#391—31no—
209#391—34no—
210#391—32no—
211#391—34no—
212#389—34no—
213#389—32no—
214#388—34no—
215#388—34no—
216#388—34no—
217#387—34no—
218#387—34no—
219#387—34no—
220#387—34no—
221#387—34no—
222#387—34no—
223#385—34no—
224#385—34no—
225#382—31no—
226#382—31no—
227#380—32no—
228#380—34no—
229C4×C22⋊C4—32no—
230C4×C22⋊C4—34no—
231C4×C22⋊C4—32no—
232#376—31no—
233C23×D3C2412yes—
234C2×C3×Q8—22no—
235C2×C3×Q8—22no—
236C2×C3×Q8—24no—
237C2×C3×Q8—22no—
238C2×C3×Q8—24no—
239C2×C3×Q8—22no—
240C2×C3×Q8—24no—
241C2×C3×Q8C2×D418yes—
242C22×C3×C4C2412yes—
243C22×C3×C4C2412yes—
244C22×C3×C4C2412yes—
245C22×C3×C4C2411yescharacteristic
246C22×C3×C4C2411yescharacteristic
247C22×C3×C4C2414yes—
248C22×C3×C4C2412yes—
249C22×C3×C4C2411yescharacteristic
250C22×Dic3C2412yes—
251C22×Dic3C2412yes—
252C22×Dic3C2414yes—
253C22×Dic3C2412yes—
254C22×Dic3C2412yes—
255C22×Dic3C2412yes—
256D3×Q8—432no—
257C2×D12—24no—
258C2×D12—22no—
259C2×C4×D3—24no—
260C2×C4×D3—28no—
261C2×C4×D3—24no—
262C2×C4×D3—24no—
263C2×C4×D3—28no—
264C2×C4×D3—44no—
265C2×Dic6—22no—
266C2×Dic6—22no—
267C2×Dic6—28no—
268C2×Dic6—28no—
269C2×Dic6—44no—
270C2×Dic6#4114yes—
271C3×C4⋊C4—22no—
272C3×C4⋊C4—24no—
273C3×C4⋊C4—28no—
274C3×C4⋊C4—24no—
275C3×C4⋊C4—22no—
276C3×C4⋊C4—28no—
277C3×C4⋊C4—28no—
278C3×C4⋊C4C2×D418yes—
279C3×C4⋊C4C2×D418yes—
280C3×C42—28no—
281C3×C42—24no—
282C3×C42—24no—
283C3×C42C2×D414yes—
284(C2×D3)⋊C4—28no—
285(C2×D3)⋊C4—24no—
286(C2×D3)⋊C4—24no—
287(C2×D3)⋊C4—24no—
288(C2×D3)⋊C4—28no—
289(C2×D3)⋊C4—24no—
290(C2×D3)⋊C4—24no—
291(C2×D3)⋊C4—44no—
292C4⋊Dic3—22no—
293C4⋊Dic3—22no—
294C4⋊Dic3—24no—
295C4⋊Dic3—44no—
296Dic3⋊C4—24no—
297Dic3⋊C4—28no—
298Dic3⋊C4—28no—
299Dic3⋊C4—28no—
300Dic3⋊C4—44no—
301C4×Dic3—24no—
302C4×Dic3—22no—
303C4×Dic3—22no—
304C4×Dic3—44no—
305C22×Q8—31no—
306C22×Q8—34no—
307C22×Q8—68no—
308C22×Q8—68no—
309C22×Q8—124no—
310C22×Q8C22×D311yescharacteristic
311C22×Q8C22×D311yescharacteristic
312C22×Q8C22×D312yes—
313C22×D4—31no—
314C23×C4—32no—
315C23×C4—34no—
316C4⋊Q8—28no—
317C4⋊Q8—64no—
318C4⋊Q8—68no—
319C4⋊Q8#6718yes—
320#125—216no—
321#125—28no—
322C22⋊Q8—68no—
323C22⋊Q8—616no—
324C22⋊Q8—124no—
325C4×Q8—28no—
326C4×Q8—28no—
327C4×Q8—24no—
328C4×Q8—24no—
329C4×Q8—64no—
330C4×Q8—124no—
331C4×D4—68no—
332C4×D4—64no—
333C2×C4⋊C4—31no—
334C2×C4⋊C4—31no—
335C2×C4⋊C4—32no—
336C2×C4⋊C4—32no—
337C2×C4⋊C4—34no—
338C2×C4⋊C4—34no—
339C2×C4⋊C4—34no—
340C2×C4⋊C4—34no—
341C2×C4⋊C4—34no—
342C2×C4⋊C4C22×D311yescharacteristic
343C2×C4⋊C4C22×D314yes—
344C2×C4⋊C4C22×D312yes—
345C2×C4⋊C4C22×D312yes—
346C2×C4⋊C4C22×D311yescharacteristic
347C2×C4⋊C4C22×D314yes—
348C2×C4⋊C4C22×D314yes—
349C2×C4⋊C4C22×D312yes—
350C2×C4⋊C4C22×D312yes—
351C2×C22⋊C4—32no—
352C2×C22⋊C4—34no—
353C2×C22⋊C4—32no—
354C2×C22⋊C4—32no—
355C2×C22⋊C4—34no—
356C2×C22⋊C4—32no—
357C2×C22⋊C4—32no—
358C2×C22⋊C4—34no—
359C2×C42—32no—
360C2×C42—31no—
361C2×C42—31no—
362C2×C42—34no—
363C2×C42C22×D314yes—
364C2×C42C22×D312yes—
365C2×C42C22×D312yes—
366C2×C42C22×D311yescharacteristic
367#95—32no—
368#95—32no—
369#95—34no—
370#95—34no—
371#95—32no—
372#95—32no—
373#95—32no—
374#95—32no—
375#95—34no—
376#95—34no—
377#95—32no—
378#95—34no—
379#95—32no—
380#95—34no—
381#95—34no—
382#95—34no—
383#95—34no—
384C23×C3C2511yescharacteristic
385C22×D3—22no—
386C22×D3—22no—
387C22×D3—22no—
388C22×D3—22no—
389C22×D3—22no—
390C22×D3—22no—
391C22×D32−1+414yes—
392C3×Q8—28no—
393C3×Q8—216no—
394C3×Q8—44no—
395C2×C3×C4—22no—
396C2×C3×C4—24no—
397C2×C3×C4—24no—
398C2×C3×C4—22no—
399C2×C3×C4—22no—
400C2×C3×C4—22no—
401C2×C3×C4—22no—
402C2×C3×C4—22no—
403C2×C3×C4—22no—
404C2×C3×C4—22no—
405C2×C3×C4—22no—
406C2×C3×C4—24no—
407C2×C3×C4—22no—
408C2×C3×C4—22no—
409C2×C3×C4C22×D412yes—
410C2×C3×C4C22×D412yes—
411C2×C3×C4C22×D418yes—
412C2×C3×C4C22×D412yes—
413C2×C3×C4C2×#4112yes—
414C2×C3×C4H(5,𝔽2)12yes—
415C2×C3×C42−1+414yes—
416C2×Dic3—22no—
417C2×Dic3—22no—
418C2×Dic3—22no—
419C2×Dic3—22no—
420C2×Dic3—22no—
421C2×Dic3—22no—
422C2×Dic3—22no—
423C2×Dic3—24no—
424C2×Dic3—22no—
425C2×Dic3—24no—
426C2×Dic3—24no—
427C2×Dic3—22no—
428C2×Dic3—22no—
429C2×Dic3—22no—
430C2×Dic3—22no—
431C2×Dic3—24no—
432C2×Dic3C2×#4114yes—
433D12—28no—
434C4×D3—416no—
435C4×D3—48no—
436Dic6—28no—
437Dic6—416no—
438C24—32no—
439C2×Q8—24no—
440C2×Q8—24no—
441C2×Q8—22no—
442C2×Q8—24no—
443C2×Q8—22no—
444C2×Q8—22no—
445C2×Q8—22no—
446C2×Q8—34no—
447C2×Q8—68no—
448C2×Q8—62no—
449C2×Q8—62no—
450C2×Q8—68no—
451C2×Q8—124no—
452C2×Q8—1232no—
453C2×Q8C2×#6718yes—
454C2×D4—64no—
455C2×D4—62no—
456C22×C4—32no—
457C22×C4—32no—
458C22×C4—32no—
459C22×C4—34no—
460C22×C4—32no—
461C22×C4—32no—
462C22×C4—68no—
463C22×C4—64no—
464C22×C4—64no—
465C22×C4—68no—
466C22×C4—64no—
467C22×C4—124no—
468C22×C4C23×D312yes—
469C22×C4C23×D312yes—
470C22×C4C23×D312yes—
471C22×C4C23×D312yes—
472C22×C4C23×D311yescharacteristic
473C22×C4C23×D311yescharacteristic
474C22×C4C23×D314yes—
475C22×C4C23×D311yescharacteristic
476C4⋊C4—22no—
477C4⋊C4—24no—
478C4⋊C4—28no—
479C4⋊C4—24no—
480C4⋊C4—22no—
481C4⋊C4—28no—
482C4⋊C4—28no—
483C4⋊C4—68no—
484C4⋊C4—68no—
485C4⋊C4—64no—
486C4⋊C4—62no—
487C4⋊C4—64no—
488C4⋊C4—62no—
489C4⋊C4—68no—
490C4⋊C4—124no—
491C4⋊C4—124no—
492C4⋊C4C2×#6718yes—
493C4⋊C4C2×#6718yes—
494C22⋊C4—68no—
495C22⋊C4—64no—
496C22⋊C4—64no—
497C22⋊C4—64no—
498C22⋊C4—64no—
499C22⋊C4—68no—
500C22⋊C4—64no—
501C22⋊C4—124no—
502C42—28no—
503C42—24no—
504C42—24no—
505C42—62no—
506C42—62no—
507C42—64no—
508C42—124no—
509C42C2×#6714yes—
510C22×C3C23×D411yescharacteristic
511C22×C3C22×#4111yescharacteristic
512C22×C3C2×H(5,𝔽2)11yescharacteristic
513C22×C3C2×2−1+412yes—
514C22×C3C2×2−1+412yes—
515C2×D3—24no—
516C2×D3—24no—
517C2×D3—24no—
518C2×D3—42no—
519C2×D3—42no—
520C2×D3—42no—
521C2×D3—42no—
522C3×C4—28no—
523C3×C4—22no—
524C3×C4—22no—
525C3×C4—24no—
526C3×C4—42no—
527C3×C4—42no—
528C3×C4—42no—
529C3×C4#54614yes—
530Dic3—24no—
531Dic3—42no—
532Dic3—42no—
533Dic3—42no—
534Dic3—42no—
535Dic3—44no—
536C23—34no—
537C23—62no—
538C23—62no—
539C23—62no—
540C23—62no—
541C23—62no—
542C23—62no—
543C23C24×D311yescharacteristic
544Q8—28no—
545Q8—216no—
546Q8—44no—
547Q8—68no—
548Q8—1216no—
549D4—68no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—24no—
553C2×C4—24no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—24no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—34no—
565C2×C4—62no—
566C2×C4—62no—
567C2×C4—62no—
568C2×C4—62no—
569C2×C4—64no—
570C2×C4—64no—
571C2×C4—62no—
572C2×C4—62no—
573C2×C4—62no—
574C2×C4—64no—
575C2×C4—62no—
576C2×C4—62no—
577C2×C4—62no—
578C2×C4—64no—
579C2×C4—62no—
580C2×C4—62no—
581C2×C4—1216no—
582C2×C4—128no—
583C2×C4#100714yes—
584C2×C4D3×#4112yes—
585C2×C4#100912yes—
586C2×C4C22×#6712yes—
587C2×C4C22×#6718yes—
588C2×C4C22×#6712yes—
589C2×C4C22×#6712yes—
590C2×C3C2×#54611yescharacteristic
591C2×C3#348611yescharacteristic
592C2×C3#348611yescharacteristic
593C2×C3#352312yes—
594C2×C3#355212yes—
595D3—44no—
596C22—64no—
597C22—64no—
598C22—64no—
599C22—122no—
600C22—122no—
601C22—122no—
602C22—122no—
603C22C2×#100712yes—
604C22C2×D3×#4111yescharacteristic
605C22C2×#100911yescharacteristic
606C22D3×2−1+412yes—
607C22C23×#6711yescharacteristic
608C4—24no—
609C4—22no—
610C4—28no—
611C4—22no—
612C4—42no—
613C4—42no—
614C4—42no—
615C4—64no—
616C4—122no—
617C4—122no—
618C4—122no—
619C4—122no—
620C4—124no—
621C4#582914yes—
622C3#4533611yescharacteristic, complemented, Sylow
623C2—124no—
624C2#8702512yes—
625C2#8703412yes—
626C2#8725111yescharacteristic
627C2C2×#582911yescharacteristic
628C2#8729411yescharacteristic
subgroupG/H
center, FrattiniC23C24×D3
commutator, socle, cosocleC23×C3C25
FittingC2×C3×#557C2
prime radicalC22×D12C23
hypercenterC2×#557D3
hypocenterC3#45336
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×#557

Complements

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

NH ≅ G/Ntype
1. C2×C3×#557C2semidirect
3. #83254C2semidirect
4. #83232C2semidirect
5. #83227C2semidirect
6. #83068C2semidirect
8. #83048C2semidirect
10. #82498C2semidirect
11. #82493C2semidirect
13. Q8×(C2×D3)⋊C4C2semidirect
14. #82480C2semidirect
15. #82456C2semidirect
78. C2×#557D3semidirect
622. C3#45336semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#5572
C2×#57272
#832542
#832322
#832272
#830682
#830592
#830482
#824992
#824982
#824932
#824922
Q8×(C2×D3)⋊C42
#824802
#824562
#453363