Finite Group Databasebeta

← #10972887

Subgroups — #10972887

Order 768.

2230 conjugacy classes of subgroups.

The subgroup lattice has 660 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5688C211yesmaximal, characteristic
2C2×C4×D3×Q8C211yesmaximal, characteristic
3C3×#2634C211yesmaximal, characteristic, complemented
4#83042C211yesmaximal, characteristic, complemented
5#82486C211yesmaximal, characteristic, complemented
6#78345C212yesmaximal
7#78191C212yesmaximal, complemented
8#78181C212yesmaximal, complemented
9#78152C212yesmaximal
10#77954C212yesmaximal
11#77785C212yesmaximal, complemented
12#77191C212yesmaximal, complemented
13#77173C212yesmaximal
14#75448C214yesmaximal
15#74948C214yesmaximal, complemented
16#74278C211yesmaximal, characteristic, complemented
17#74258C211yesmaximal, characteristic
18#35248—31nomaximal, Sylow
19C22×D3×Q8C2211yescharacteristic
20C2×C3×#125C2211yescharacteristic
21C2×C3×C4×Q8C2211yescharacteristic
22C2×Q8×Dic3C2211yescharacteristic
23#5688—28no—
24C4×D3×Q8—28no—
25C2×#896C2212yes—
26C2×#896C2212yes—
27C2×#892C2211yescharacteristic
28C2×#892C2211yescharacteristic
29C2×D3×C4⋊C4C2212yes—
30C2×D3×C4⋊C4C2211yescharacteristic
31C2×#888C2211yescharacteristic
32C2×#888C2211yescharacteristic
33C2×#888C2214yes—
34C2×#887C2212yes—
35C2×#887C2211yescharacteristic
36C2×#887C2212yes—
37C2×#887C2212yes—
38C2×C42×D3C2212yes—
39C2×C42×D3C2211yescharacteristic
40C2×#870C2211yescharacteristic
41C2×C4×Dic6C2211yescharacteristic
42C2×C4×Dic6C2212yes—
43C3×#400C2212yes—
44C3×#400C2212yes—
45C3×#398C2212yes—
46C3×#395C2212yes—
47C3×#387C2214yes—
48C3×#383C2211yescharacteristic
49#5230C2212yes—
50#4989C2214yes—
51#4986C2212yes—
52#4986C2212yes—
53#4985C2212yes—
54#4985C2212yes—
55#4983C2212yes—
56#4981C2212yes—
57#4980C2214yes—
58#4979C2212yes—
59#4978C2214yes—
60#4976C2214yes—
61#4937C2211yescharacteristic
62#4933C2214yes—
63#4668C2214yes—
64#4667C2212yes—
65#4666C2212yes—
66#4664C2212yes—
67#4664C2214yes—
68#4664C2214yes—
69#4661C2212yes—
70#4659C2212yes—
71#4659C2212yes—
72#4656C2212yes—
73#4655C2214yes—
74#4654C2214yes—
75#4653C2214yes—
76#4652C2214yes—
77#4650C2212yes—
78#4650C2214yes—
79#4649C2212yes—
80#4648C2212yes—
81#4647C2214yes—
82#4646C2214yes—
83#4646C2214yes—
84#4646C2214yes—
85#4646C2214yes—
86C2×#533—31no—
87C22×C4×Q8—31no—
88#2855—34no—
89#2634—31no—
90#2634—31no—
91#2634—31no—
92#2634D311yescharacteristic, complemented
93#2581—32no—
94#2581—32no—
95#2568—32no—
96#2546—32no—
97#2503—32no—
98#2484—32no—
99#2484—32no—
100#2434—32no—
101#2327—34no—
102#2283—31no—
103C22×C3×Q8C2311yescharacteristic
104C2×D3×Q8—24no—
105C2×D3×Q8—28no—
106C22×C4×D3C2312yes—
107C22×C4×D3C2312yes—
108C22×C4×D3C2311yescharacteristic
109C22×C4×D3C2311yescharacteristic
110C22×C4×D3C2311yescharacteristic
111C22×Dic6C2312yes—
112C22×Dic6C2312yes—
113C22×Dic6C2311yescharacteristic
114C22×Dic6C2312yes—
115C3×#125—24no—
116C3×C4×Q8—24no—
117C2×C3×C4⋊C4C2314yes—
118C2×C3×C4⋊C4C2312yes—
119C2×C3×C4⋊C4C2312yes—
120C2×C3×C4⋊C4C2312yes—
121C2×C3×C4⋊C4C2311yescharacteristic
122C2×C3×C4⋊C4C2311yescharacteristic
123C2×C3×C4⋊C4C2311yescharacteristic
124C2×C3×C42C2312yes—
125C2×C3×C42C2311yescharacteristic
126Q8×Dic3—24no—
127C2×(C2×D3)⋊C4C2314yes—
128C2×(C2×D3)⋊C4C2312yes—
129C2×(C2×D3)⋊C4C2312yes—
130C2×C4⋊Dic3C2314yes—
131C2×C4⋊Dic3C2312yes—
132C2×C4⋊Dic3C2311yescharacteristic
133C2×C4⋊Dic3C2311yescharacteristic
134C2×C4⋊Dic3C2311yescharacteristic
135C2×Dic3⋊C4C2314yes—
136C2×Dic3⋊C4C2312yes—
137C2×Dic3⋊C4C2312yes—
138C2×Dic3⋊C4C2312yes—
139C2×Dic3⋊C4C2314yes—
140C2×Dic3⋊C4C2312yes—
141C2×Dic3⋊C4C2312yes—
142C2×Dic3⋊C4C2314yes—
143C2×Dic3⋊C4C2312yes—
144C2×Dic3⋊C4C2312yes—
145C2×C4×Dic3C2312yes—
146C2×C4×Dic3C2314yes—
147C2×C4×Dic3C2312yes—
148C2×C4×Dic3C2312yes—
149C2×C4×Dic3C2312yes—
150C2×C4×Dic3C2312yes—
151C2×C4×Dic3C2311yescharacteristic
152C2×C4×Dic3C2311yescharacteristic
153C2×C4×Dic3C2311yescharacteristic
154#896—28no—
155#896—28no—
156#892—24no—
157#892—24no—
158D3×C4⋊C4—28no—
159D3×C4⋊C4—24no—
160#888—216no—
161#888—24no—
162#888—24no—
163#887—28no—
164#887—28no—
165#887—24no—
166#887—28no—
167C42×D3—28no—
168C42×D3D418yes—
169#870—24no—
170C4×Dic6—24no—
171C4×Dic6—28no—
172C3×#95C2314yes—
173C3×#95C2314yes—
174C3×#95C2314yes—
175C3×#95C2312yes—
176C3×#95C2312yes—
177C3×#95C2312yes—
178C3⋊#95C2314yes—
179C3⋊#95C2314yes—
180C3⋊#95C2314yes—
181C3⋊#95C2314yes—
182C3⋊#95C2314yes—
183C3⋊#95C2314yes—
184C3⋊#95C2314yes—
185C3⋊#95C2314yes—
186C3⋊#95C2314yes—
187C3⋊#95C2314yes—
188C3⋊#95C2312yes—
189C3⋊#95C2312yes—
190C3⋊#95C2312yes—
191C23×Q8—31no—
192#533—68no—
193C2×C4⋊Q8—31no—
194C2×C4⋊Q8—34no—
195C2×C4⋊Q8—31no—
196C2×#125—31no—
197C2×#125C2×D311yescharacteristic
198C2×C22⋊Q8—32no—
199C2×C22⋊Q8—32no—
200C2×C4×Q8—31no—
201C2×C4×Q8—32no—
202C2×C4×Q8—31no—
203C2×C4×Q8—32no—
204C2×C4×Q8—32no—
205C2×C4×Q8—32no—
206C2×C4×Q8—31no—
207C2×C4×Q8—68no—
208C2×C4×Q8C2×D311yescharacteristic
209C2×#117—31no—
210C2×#117—31no—
211C22×C4⋊C4—32no—
212C22×C4⋊C4—31no—
213C22×C42—31no—
214C22×C42—32no—
215#400—32no—
216#400—32no—
217#400—32no—
218#400—32no—
219#400—32no—
220#400—32no—
221#400C2×D312yes—
222#400C2×D312yes—
223#398—32no—
224#398—32no—
225#398—32no—
226#398C2×D312yes—
227#395—32no—
228#395—32no—
229#395—32no—
230#395C2×D312yes—
231#391—34no—
232#391—34no—
233#391—34no—
234#391—34no—
235#389—34no—
236#389—34no—
237#387—34no—
238#387—34no—
239#387—34no—
240#387—34no—
241#387—34no—
242#387—34no—
243#387—34no—
244#387C2×D314yes—
245#385—34no—
246#385—34no—
247#383—31no—
248#383C2×D311yescharacteristic
249#382—34no—
250#382—32no—
251#381—34no—
252#381—34no—
253#381—32no—
254#380—32no—
255C4×C4⋊C4—32no—
256C4×C22⋊C4—32no—
257#376—32no—
258C23×D3C2411yescharacteristic
259C2×C3×Q8—22no—
260C2×C3×Q8—24no—
261C22×C3×C4C2414yes—
262C22×C3×C4C2412yes—
263C22×C3×C4C2412yes—
264C22×C3×C4C2412yes—
265C22×C3×C4C2411yescharacteristic
266C22×C3×C4C2411yescharacteristic
267C22×C3×C4C2411yescharacteristic
268C22×C3×C4C2412yes—
269C22×Dic3C2414yes—
270C22×Dic3C2412yes—
271C22×Dic3C2412yes—
272C22×Dic3C2412yes—
273C22×Dic3C2411yescharacteristic
274C22×Dic3C2412yes—
275C22×Dic3C2411yescharacteristic
276C22×Dic3C2411yescharacteristic
277D3×Q8—232no—
278C2×C4×D3—24no—
279C2×C4×D3—24no—
280C2×C4×D3—24no—
281C2×C4×D3—24no—
282C2×C4×D3—24no—
283C2×C4×D3—24no—
284C2×C4×D3—24no—
285C2×C4×D3—28no—
286C2×C4×D3C2×D414yes—
287C2×C4×D3C2×D414yes—
288C2×C4×D3C2×D414yes—
289C2×Dic6—22no—
290C2×Dic6—24no—
291C2×Dic6—24no—
292C2×Dic6—24no—
293C2×Dic6—24no—
294C2×Dic6—24no—
295C2×Dic6—28no—
296C2×Dic6—28no—
297C2×Dic6#4118yes—
298C3×C4⋊C4—24no—
299C3×C4⋊C4—22no—
300C3×C4⋊C4—24no—
301C3×C4⋊C4—24no—
302C3×C4⋊C4—22no—
303C3×C4⋊C4—22no—
304C3×C4⋊C4—28no—
305C3×C42—24no—
306C3×C42C2×D414yes—
307(C2×D3)⋊C4—24no—
308(C2×D3)⋊C4—24no—
309(C2×D3)⋊C4—28no—
310C4⋊Dic3—22no—
311C4⋊Dic3—22no—
312C4⋊Dic3—22no—
313C4⋊Dic3—24no—
314C4⋊Dic3—28no—
315Dic3⋊C4—24no—
316Dic3⋊C4—24no—
317Dic3⋊C4—24no—
318Dic3⋊C4—28no—
319Dic3⋊C4—24no—
320Dic3⋊C4—24no—
321Dic3⋊C4—24no—
322Dic3⋊C4—24no—
323Dic3⋊C4—28no—
324Dic3⋊C4—28no—
325C4×Dic3—24no—
326C4×Dic3—24no—
327C4×Dic3—24no—
328C4×Dic3—24no—
329C4×Dic3—24no—
330C4×Dic3—28no—
331C4×Dic3C2×D414yes—
332C4×Dic3C2×D414yes—
333C4×Dic3C2×D414yes—
334C22×Q8—32no—
335C22×Q8—31no—
336C22×Q8—32no—
337C22×Q8—32no—
338C22×Q8—64no—
339C22×Q8—68no—
340C22×Q8C22×D311yescharacteristic
341C23×C4—31no—
342C23×C4—31no—
343C23×C4—31no—
344C23×C4—32no—
345C23×C4—32no—
346C4⋊Q8—64no—
347C4⋊Q8—64no—
348C4⋊Q8—616no—
349#125—24no—
350#125—64no—
351C22⋊Q8—68no—
352C22⋊Q8—68no—
353C4×Q8—24no—
354C4×Q8—68no—
355C4×Q8—64no—
356C4×Q8—68no—
357C4×Q8—68no—
358C4×Q8—68no—
359C4×Q8—64no—
360C4×Q8—64no—
361#117—64no—
362#117—64no—
363C2×C4⋊C4—32no—
364C2×C4⋊C4—32no—
365C2×C4⋊C4—32no—
366C2×C4⋊C4—32no—
367C2×C4⋊C4—31no—
368C2×C4⋊C4—31no—
369C2×C4⋊C4—31no—
370C2×C4⋊C4—32no—
371C2×C4⋊C4—32no—
372C2×C4⋊C4—32no—
373C2×C4⋊C4—32no—
374C2×C4⋊C4—34no—
375C2×C4⋊C4—34no—
376C2×C4⋊C4—34no—
377C2×C4⋊C4—34no—
378C2×C4⋊C4—64no—
379C2×C4⋊C4—68no—
380C2×C4⋊C4C22×D312yes—
381C2×C4⋊C4C22×D311yescharacteristic
382C2×C4⋊C4C22×D311yescharacteristic
383C2×C4⋊C4C22×D311yescharacteristic
384C2×C4⋊C4C22×D312yes—
385C2×C4⋊C4C22×D312yes—
386C2×C4⋊C4C22×D314yes—
387C2×C22⋊C4—32no—
388C2×C22⋊C4—32no—
389C2×C22⋊C4—34no—
390C2×C42—31no—
391C2×C42—31no—
392C2×C42—31no—
393C2×C42—38no—
394C2×C42—32no—
395C2×C42—32no—
396C2×C42—32no—
397C2×C42—32no—
398C2×C42—32no—
399C2×C42—34no—
400C2×C42—68no—
401C2×C42C22×D312yes—
402C2×C42C22×D311yescharacteristic
403#95—32no—
404#95—34no—
405#95—32no—
406#95—32no—
407#95—34no—
408#95—34no—
409#95—34no—
410#95—34no—
411#95—34no—
412#95—34no—
413#95—34no—
414#95—34no—
415#95—34no—
416#95C22×D312yes—
417#95C22×D312yes—
418#95C22×D312yes—
419#95C22×D314yes—
420#95C22×D314yes—
421#95C22×D314yes—
422C23×C3C2511yescharacteristic
423C22×D3—22no—
424C22×D3—22no—
425C22×D3C22×D412yes—
426C22×D32−1+414yes—
427C3×Q8—28no—
428C2×C3×C4—22no—
429C2×C3×C4—22no—
430C2×C3×C4—22no—
431C2×C3×C4—22no—
432C2×C3×C4—22no—
433C2×C3×C4—22no—
434C2×C3×C4—22no—
435C2×C3×C4—22no—
436C2×C3×C4—24no—
437C2×C3×C4—24no—
438C2×C3×C4—24no—
439C2×C3×C4—24no—
440C2×C3×C4—24no—
441C2×C3×C4—24no—
442C2×C3×C4C22×D412yes—
443C2×C3×C4C22×D412yes—
444C2×C3×C4C22×D412yes—
445C2×C3×C4C2×#4114yes—
446C2×Dic3—22no—
447C2×Dic3—22no—
448C2×Dic3—22no—
449C2×Dic3—22no—
450C2×Dic3—22no—
451C2×Dic3—22no—
452C2×Dic3—24no—
453C2×Dic3—24no—
454C2×Dic3—24no—
455C2×Dic3—24no—
456C2×Dic3—24no—
457C2×Dic3—24no—
458C2×Dic3C22×D412yes—
459C2×Dic3C22×D412yes—
460C2×Dic3C22×D412yes—
461C2×Dic3C2×#4114yes—
462C2×Dic3C2×#4114yes—
463C2×Dic32−1+414yes—
464C4×D3—216no—
465C4×D3—28no—
466C4×D3—28no—
467C4×D3—28no—
468C4×D3—48no—
469Dic6—28no—
470Dic6—216no—
471Dic6—48no—
472Dic6—48no—
473C24—31no—
474C2×Q8—22no—
475C2×Q8—24no—
476C2×Q8—38no—
477C2×Q8—64no—
478C2×Q8—64no—
479C2×Q8—64no—
480C2×Q8—64no—
481C2×Q8—64no—
482C2×Q8—68no—
483C2×Q8—68no—
484C2×Q8—632no—
485C2×Q8—62no—
486C22×C4—31no—
487C22×C4—31no—
488C22×C4—31no—
489C22×C4—32no—
490C22×C4—34no—
491C22×C4—34no—
492C22×C4—34no—
493C22×C4—34no—
494C22×C4—32no—
495C22×C4—32no—
496C22×C4—32no—
497C22×C4—64no—
498C22×C4—64no—
499C22×C4—64no—
500C22×C4—64no—
501C22×C4—64no—
502C22×C4—64no—
503C22×C4—68no—
504C22×C4—64no—
505C22×C4C23×D314yes—
506C22×C4C23×D312yes—
507C22×C4C23×D312yes—
508C22×C4C23×D312yes—
509C22×C4C23×D311yescharacteristic
510C22×C4C23×D311yescharacteristic
511C22×C4C23×D311yescharacteristic
512C22×C4C23×D312yes—
513C4⋊C4—24no—
514C4⋊C4—22no—
515C4⋊C4—22no—
516C4⋊C4—22no—
517C4⋊C4—24no—
518C4⋊C4—24no—
519C4⋊C4—28no—
520C4⋊C4—64no—
521C4⋊C4—64no—
522C4⋊C4—64no—
523C4⋊C4—64no—
524C4⋊C4—64no—
525C4⋊C4—68no—
526C4⋊C4—68no—
527C4⋊C4—68no—
528C4⋊C4—64no—
529C4⋊C4—62no—
530C4⋊C4—62no—
531C4⋊C4—62no—
532C4⋊C4—64no—
533C4⋊C4—64no—
534C4⋊C4—68no—
535C22⋊C4—64no—
536C22⋊C4—64no—
537C22⋊C4—68no—
538C42—24no—
539C42—34no—
540C42—34no—
541C42—34no—
542C42—64no—
543C42—64no—
544C42—64no—
545C42—64no—
546C42—68no—
547C42—64no—
548C42D3×D414yes—
549C22×C3C23×D411yescharacteristic
550C22×C3C22×#4112yes—
551C22×C3C2×2−1+412yes—
552C22×C3C2×2−1+412yes—
553C2×D3—22no—
554C2×D3—22no—
555C2×D3—28no—
556C2×D3#53614yes—
557C3×C4—22no—
558C3×C4—24no—
559C3×C4—22no—
560C3×C4—22no—
561C3×C4—42no—
562C3×C4—42no—
563C3×C4—42no—
564C3×C4—44no—
565Dic3—24no—
566Dic3—22no—
567Dic3—42no—
568Dic3—42no—
569Dic3—42no—
570Dic3—44no—
571Dic3#52514yes—
572Dic3#53614yes—
573C23—32no—
574C23—34no—
575C23—62no—
576C23—62no—
577C23C24×D311yescharacteristic
578Q8—28no—
579Q8—616no—
580Q8—68no—
581Q8—128no—
582Q8—128no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—24no—
592C2×C4—24no—
593C2×C4—24no—
594C2×C4—24no—
595C2×C4—24no—
596C2×C4—24no—
597C2×C4—32no—
598C2×C4—32no—
599C2×C4—34no—
600C2×C4—34no—
601C2×C4—32no—
602C2×C4—34no—
603C2×C4—68no—
604C2×C4—68no—
605C2×C4—68no—
606C2×C4—616no—
607C2×C4—62no—
608C2×C4—62no—
609C2×C4—62no—
610C2×C4—62no—
611C2×C4—62no—
612C2×C4—62no—
613C2×C4—64no—
614C2×C4—64no—
615C2×C4—64no—
616C2×C4—64no—
617C2×C4—64no—
618C2×C4—64no—
619C2×C4—128no—
620C2×C4C2×D3×D412yes—
621C2×C4C2×D3×D412yes—
622C2×C4C2×D3×D412yes—
623C2×C4D3×#4114yes—
624C2×C3C2×#52511yescharacteristic
625C2×C3C2×#53611yescharacteristic
626C2×C3C2×#53611yescharacteristic
627C2×C3#352314yes—
628D3—24no—
629C22—34no—
630C22—62no—
631C22—62no—
632C22—68no—
633C22C22×D3×D411yescharacteristic
634C22C2×D3×#4112yes—
635C22D3×2−1+412yes—
636C22D3×2−1+412yes—
637C4—22no—
638C4—24no—
639C4—22no—
640C4—22no—
641C4—34no—
642C4—34no—
643C4—42no—
644C4—42no—
645C4—42no—
646C4—44no—
647C4—64no—
648C4—62no—
649C4—122no—
650C4—122no—
651C4—122no—
652C4—124no—
653C3#3524811yescharacteristic, complemented, Sylow
654C2—64no—
655C2D3×#52511yescharacteristic
656C2D3×#53611yescharacteristic
657C2D3×#53611yescharacteristic
658C2#8651414yes—
subgroupG/H
center, FrattiniC23C24×D3
commutator, socle, cosocleC23×C3C25
FittingC3×#2634C2
prime radicalC23×D3C24
hypercenter#2634D3
hypocenterC3#35248
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #2634

Complements

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

NH ≅ G/Ntype
3. C3×#2634C2semidirect
4. #83042C2semidirect
5. #82486C2semidirect
7. #78191C2semidirect
8. #78181C2semidirect
11. #77785C2semidirect
12. #77191C2semidirect
15. #74948C2semidirect
16. #74278C2semidirect
92. #2634D3semidirect
653. C3#35248semidirect

Maximal subgroups

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

subgroupindex
C2×#56882
C2×C4×D3×Q82
C3×#26342
#830422
#824862
#783452
#781912
#781812
#781522
#779542
#777852
#771912
#771732
#754482
#749482
#742782
#742582
#352483