Finite Group Databasebeta

← #10943984

Subgroups — #10943984

Order 768.

1576 conjugacy classes of subgroups.

The subgroup lattice has 690 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×D3×Q8C211yesmaximal, characteristic, complemented
2C2×C3×#460C211yesmaximal, characteristic, complemented
3C2×#5170C212yesmaximal, complemented
4#83192C211yesmaximal, characteristic
5C2×#5075C211yesmaximal, characteristic, complemented
6#72048C212yesmaximal, complemented
7#71626C211yesmaximal, characteristic, complemented
8#70485C211yesmaximal, characteristic, complemented
9#70475C211yesmaximal, characteristic, complemented
10#70213C212yesmaximal, complemented
11#70200C212yesmaximal, complemented
12#32903—31nomaximal, Sylow
13C22×D3×Q8C2211yescharacteristic
14C22×C3×SD16C2212yes—
15C2×C3×#127C2211yescharacteristic
16C2×C3×C4×Q8C2211yescharacteristic
17C2×Q8×Dic3C2211yescharacteristic
18C2×#938—21no—
19C2×#938—21no—
20C2×#938C2212yes—
21C22×#214C2212yes—
22C4×D3×Q8—24no—
23C4×D3×Q8—24no—
24C2×D3×C4⋊C4—21no—
25C2×D3×C4⋊C4C2211yescharacteristic
26C2×#887—21no—
27C2×#887C2211yescharacteristic
28C2×C42×D3—21no—
29C2×C42×D3C2211yescharacteristic
30C2×C4×Dic6—21no—
31C2×C4×Dic6C2211yescharacteristic
32C3×#460—24no—
33C2×C3×C4⋊C8C2211yescharacteristic
34C2×C3×#102C2212yes—
35C2×#835C2212yes—
36#5221—21no—
37#5221—21no—
38#5221C2212yes—
39C2×#832C2212yes—
40#5170—28no—
41C2×(C2×D3)⋊C8C2212yes—
42C2×#816C2212yes—
43#5075—24no—
44#4935C2211yescharacteristic
45C2×C4⋊(C3⋊C8)C2211yescharacteristic
46#4551C2212yes—
47#4550C2212yes—
48#4529C2212yes—
49C22×C4×Q8—31no—
50C2×#460—31no—
51C2×#460D311yescharacteristic, complemented
52C2×#448—32no—
53#2856—31no—
54#1880—32no—
55#1880—32no—
56#1874—32no—
57#1803—31no—
58#1803—31no—
59#1797—31no—
60C22×C3×Q8C2311yescharacteristic
61C22×C3×D4—21no—
62C22×C3×D4—21no—
63C22×C3×D4C2312yes—
64C22×#67—22no—
65C2×D3×Q8—22no—
66C2×D3×Q8—22no—
67C2×D3×Q8—22no—
68C2×D3×Q8—22no—
69C2×D3×Q8—22no—
70C2×D3×Q8D414yescomplemented
71C22×C4×D3—21no—
72C22×C4×D3—21no—
73C22×C4×D3D411yescharacteristic
74C22×C4×D3D411yescharacteristic
75C22×C4×D3C2311yescharacteristic
76C22×Dic6—21no—
77C22×Dic6C2311yescharacteristic
78C2×C3×SD16—24no—
79C2×C3×SD16—24no—
80C2×C3×SD16—24no—
81C22×C3×C8C2312yes—
82C3×#127—22no—
83C3×#127—22no—
84C3×C4×Q8—22no—
85C3×C4×Q8D414yescomplemented
86C2×C3×C4⋊C4—21no—
87C2×C3×C4⋊C4C2311yescharacteristic
88C2×C3×C42—21no—
89C2×C3×C42C2311yescharacteristic
90C2×#217—22no—
91C2×#217—22no—
92C2×#217—22no—
93Q8×Dic3—22no—
94Q8×Dic3—22no—
95#938—24no—
96#938—24no—
97#938—44no—
98#938—44no—
99C2×#214—24no—
100C2×#214—24no—
101C2×#214—24no—
102C2×C4⋊Dic3—21no—
103C2×C4⋊Dic3—21no—
104C2×C4⋊Dic3—21no—
105C2×C4⋊Dic3C2312yes—
106C2×C4⋊Dic3C2311yescharacteristic
107C2×Dic3⋊C4—21no—
108C2×Dic3⋊C4—21no—
109C2×Dic3⋊C4—21no—
110C2×C4×Dic3—21no—
111C2×C4×Dic3—21no—
112C2×C4×Dic3—21no—
113C2×C4×Dic3D411yescharacteristic
114C2×C4×Dic3D411yescharacteristic
115C2×C4×Dic3C2311yescharacteristic
116C22×C3⋊C8C2312yes—
117D3×C4⋊C4—28no—
118D3×C4⋊C4—22no—
119D3×C4⋊C4—22no—
120#887—28no—
121#887—22no—
122#887—22no—
123C42×D3—28no—
124C42×D3—22no—
125C42×D3—22no—
126C4×Dic6—28no—
127C4×Dic6—22no—
128C4×Dic6D414yescomplemented
129C3×C4⋊C8—22no—
130C3×#102—24no—
131#835—24no—
132#832—24no—
133(C2×D3)⋊C8—24no—
134#816—24no—
135C4⋊(C3⋊C8)—22no—
136C23×Q8—31no—
137C22×SD16—32no—
138C22×SD16C2×D312yes—
139C2×#127C2×D311yescharacteristic
140C2×#121—32no—
141C2×#121—61no—
142C2×#121—61no—
143C2×C4×Q8—31no—
144C2×C4×Q8—31no—
145C2×C4×Q8—31no—
146C2×C4×Q8—61no—
147C2×C4×Q8—61no—
148C2×C4×Q8—64no—
149C2×C4×Q8—64no—
150C2×C4×Q8C2×D311yescharacteristic
151C22×C4⋊C4—31no—
152C22×C4⋊C4—61no—
153C22×C42—31no—
154C22×C42—61no—
155#460—24no—
156#460—64no—
157#448—68no—
158C2×C4⋊C8—31no—
159C2×C4⋊C8C2×D311yescharacteristic
160C2×#103—32no—
161C2×#103—32no—
162C2×#102—32no—
163C2×#102C2×D312yes—
164C2×C22⋊C8—32no—
165#390—32no—
166#390—61no—
167#390—61no—
168#384—31no—
169#340—32no—
170#340—32no—
171#336—32no—
172C24×C3—22no—
173C23×D3C2×D411yescharacteristic
174C2×C3×Q8—21no—
175C2×C3×Q8—21no—
176C2×C3×Q8—21no—
177C2×C3×Q8C2×D412yes—
178C2×C3×Q8C2×D412yes—
179C2×C3×Q8#4112yes—
180C2×C3×D4—22no—
181C2×C3×D4—22no—
182C2×C3×D4—22no—
183C2×C3×D4—22no—
184C2×C3×D4—22no—
185C2×C3×D4—22no—
186C2×C3×D4—42no—
187C2×C3×D4—42no—
188C2×C3×D4—42no—
189C2×C3×D4—42no—
190C2×C3×D4—42no—
191C2×C3×D4—42no—
192C22×C3×C4—21no—
193C22×C3×C4—21no—
194C22×C3×C4C2×D411yescharacteristic
195C22×C3×C4C2×D411yescharacteristic
196C22×C3×C4C2411yescharacteristic
197C2×#67—44no—
198C2×#67—44no—
199C2×#67—44no—
200C22×Dic3—21no—
201C22×Dic3—21no—
202C22×Dic3—22no—
203C22×Dic3C2×D411yescharacteristic
204C22×Dic3C2×D411yescharacteristic
205C22×Dic3C2×D411yescharacteristic
206D3×Q8—28no—
207D3×Q8—28no—
208D3×Q8—28no—
209D3×Q8—28no—
210C2×C4×D3—24no—
211C2×C4×D3—24no—
212C2×C4×D3—24no—
213C2×C4×D3—22no—
214C2×C4×D3—22no—
215C2×C4×D3—21no—
216C2×C4×D3—21no—
217C2×C4×D3—21no—
218C2×C4×D3—21no—
219C2×C4×D3—24no—
220C2×C4×D3—22no—
221C2×C4×D3—22no—
222C2×C4×D3—41no—
223C2×C4×D3—42no—
224C2×C4×D3—42no—
225C2×C4×D3SD1614yescomplemented
226C2×C4×D3C2×D412yes—
227C2×C4×D3#4112yes—
228C2×Dic6—24no—
229C2×Dic6—24no—
230C2×Dic6—21no—
231C2×Dic6—21no—
232C2×Dic6—21no—
233C2×Dic6—24no—
234C2×Dic6C2×D412yes—
235C2×Dic6C2×D412yes—
236C2×Dic6#4112yes—
237C3×SD16—48no—
238C2×C3×C8—22no—
239C2×C3×C8—22no—
240C2×C3×C8—22no—
241C3×C4⋊C4—24no—
242C3×C4⋊C4—21no—
243C3×C4⋊C4C2×D412yes—
244C3×C42—24no—
245C3×C42C2×D412yes—
246C3×C42C2×D412yes—
247#217—44no—
248#217—44no—
249#217—44no—
250#214—48no—
251C4⋊Dic3—24no—
252C4⋊Dic3—21no—
253C4⋊Dic3—22no—
254C4⋊Dic3—22no—
255C4⋊Dic3—42no—
256C4⋊Dic3—42no—
257C4⋊Dic3C2×D412yes—
258Dic3⋊C4—24no—
259Dic3⋊C4—24no—
260Dic3⋊C4—24no—
261C4×Dic3—24no—
262C4×Dic3—24no—
263C4×Dic3—22no—
264C4×Dic3—24no—
265C4×Dic3—22no—
266C4×Dic3—21no—
267C4×Dic3—21no—
268C2×C3⋊C8—22no—
269C2×C3⋊C8—22no—
270C2×C3⋊C8—22no—
271C22×Q8—34no—
272C22×Q8—31no—
273C22×Q8—61no—
274C22×Q8—62no—
275C22×Q8—62no—
276C22×Q8—62no—
277C22×Q8—62no—
278C22×Q8—62no—
279C22×Q8C22×D311yescharacteristic
280C22×D4—21no—
281C22×D4—21no—
282C22×D4—62no—
283C22×D4C22×D312yes—
284C23×C4—31no—
285C23×C4—31no—
286C23×C4—31no—
287C23×C4—61no—
288C23×C4—61no—
289C2×SD16—24no—
290C2×SD16—24no—
291C2×SD16—24no—
292C2×SD16—64no—
293C2×SD16—64no—
294C2×SD16—64no—
295C22×C8—32no—
296C22×C8C22×D312yes—
297#127—22no—
298#127—22no—
299#121—64no—
300#121—64no—
301#121—124no—
302#121—124no—
303C4×Q8—22no—
304C4×Q8—34no—
305C4×Q8—68no—
306C4×Q8—62no—
307C4×Q8—68no—
308C4×Q8—62no—
309C4×Q8—62no—
310C4×Q8—62no—
311C4×Q8—62no—
312C4×Q8#6714yescomplemented
313C2×C4⋊C4—21no—
314C2×C4⋊C4—31no—
315C2×C4⋊C4—32no—
316C2×C4⋊C4—62no—
317C2×C4⋊C4—68no—
318C2×C4⋊C4—61no—
319C2×C4⋊C4—61no—
320C2×C4⋊C4—61no—
321C2×C4⋊C4—61no—
322C2×C4⋊C4—62no—
323C2×C4⋊C4—61no—
324C2×C4⋊C4—61no—
325C2×C4⋊C4C22×D311yescharacteristic
326C2×C22⋊C4—62no—
327C2×C22⋊C4—62no—
328C2×C22⋊C4—62no—
329C2×C42—21no—
330C2×C42—31no—
331C2×C42—31no—
332C2×C42—31no—
333C2×C42—68no—
334C2×C42—61no—
335C2×C42—61no—
336C2×C42—61no—
337C2×C42—62no—
338C2×C42—62no—
339C2×C42C22×D311yescharacteristic
340C4⋊C8—22no—
341C4⋊C8—62no—
342#103—64no—
343#103—64no—
344#102—24no—
345#102—64no—
346C22⋊C8—64no—
347C23×C3—42no—
348C23×C3—42no—
349C23×C3—42no—
350C23×C3—42no—
351C23×C3—42no—
352C23×C3—42no—
353C23×C3—42no—
354C23×C3C22×D411yescharacteristic
355C22×D3—21no—
356C22×D3—21no—
357C22×D3—21no—
358C22×D3—21no—
359C22×D3#12112yes—
360C22×D3C2×SD1612yes—
361C22×D3AGL(1,ℤ/8)12yes—
362C3×Q8—22no—
363C3×Q8—22no—
364C3×Q8—22no—
365C3×Q8#12114yescomplemented
366C3×D4—42no—
367C3×D4—42no—
368C3×D4—42no—
369C3×D4—42no—
370C3×D4—84no—
371C3×D4—84no—
372C2×C3×C4—22no—
373C2×C3×C4—22no—
374C2×C3×C4—22no—
375C2×C3×C4—21no—
376C2×C3×C4—21no—
377C2×C3×C4—22no—
378C2×C3×C4—41no—
379C2×C3×C4—41no—
380C2×C3×C4#12012yes—
381C2×C3×C4#12012yes—
382C2×C3×C4C2×SD1612yes—
383C2×C3×C4#13712yes—
384C2×C3×C4C22×D411yescharacteristic
385C2×C3×C4C22×D411yescharacteristic
386C2×C3×C4C22×D411yescharacteristic
387C2×C3×C4C2×#4111yescharacteristic
388C2×C3×C4C2×#4111yescharacteristic
389C2×C3×C4C2×#4111yescharacteristic
390#67—88no—
391C2×Dic3—21no—
392C2×Dic3—21no—
393C2×Dic3—22no—
394C2×Dic3—22no—
395C2×Dic3—21no—
396C2×Dic3—22no—
397C2×Dic3—21no—
398C2×Dic3—21no—
399C2×Dic3—22no—
400C2×Dic3—21no—
401C2×Dic3—21no—
402C2×Dic3—41no—
403C2×Dic3—41no—
404C2×Dic3—42no—
405C2×Dic3—42no—
406C2×Dic3—42no—
407C2×Dic3#12112yes—
408C2×Dic3C2×SD1612yes—
409C4×D3—216no—
410C4×D3—22no—
411C4×D3—22no—
412C4×D3—22no—
413C4×D3—22no—
414C4×D3—28no—
415C4×D3—44no—
416C4×D3—48no—
417Dic6—216no—
418Dic6—22no—
419Dic6—22no—
420Dic6—22no—
421Dic6#12114yescomplemented
422C3×C8—42no—
423C3⋊C8—42no—
424C24—22no—
425C24—31no—
426C2×Q8—21no—
427C2×Q8—21no—
428C2×Q8—21no—
429C2×Q8—32no—
430C2×Q8—32no—
431C2×Q8—32no—
432C2×Q8—64no—
433C2×Q8—64no—
434C2×Q8—64no—
435C2×Q8—68no—
436C2×Q8—68no—
437C2×Q8—68no—
438C2×Q8—68no—
439C2×Q8—61no—
440C2×Q8—61no—
441C2×Q8—61no—
442C2×Q8D3×D412yes—
443C2×Q8#23712yes—
444C2×Q8C2×#6712yes—
445C2×D4—22no—
446C2×D4—22no—
447C2×D4—22no—
448C2×D4—22no—
449C2×D4—22no—
450C2×D4—22no—
451C2×D4—42no—
452C2×D4—42no—
453C2×D4—42no—
454C2×D4—42no—
455C2×D4—42no—
456C2×D4—42no—
457C2×D4—124no—
458C2×D4—124no—
459C2×D4—124no—
460C22×C4—21no—
461C22×C4—21no—
462C22×C4—31no—
463C22×C4—31no—
464C22×C4—34no—
465C22×C4—32no—
466C22×C4—32no—
467C22×C4—31no—
468C22×C4—64no—
469C22×C4—64no—
470C22×C4—64no—
471C22×C4—62no—
472C22×C4—64no—
473C22×C4—61no—
474C22×C4—62no—
475C22×C4—62no—
476C22×C4—61no—
477C22×C4—61no—
478C22×C4—61no—
479C22×C4—61no—
480C22×C4—61no—
481C22×C4—62no—
482C22×C4—62no—
483C22×C4—121no—
484C22×C4—122no—
485C22×C4—122no—
486C22×C4D3×D411yescharacteristic
487C22×C4D3×D411yescharacteristic
488C22×C4C23×D311yescharacteristic
489SD16—48no—
490SD16—128no—
491C2×C8—22no—
492C2×C8—22no—
493C2×C8—22no—
494C2×C8—62no—
495C2×C8—62no—
496C2×C8—62no—
497C4⋊C4—24no—
498C4⋊C4—21no—
499C4⋊C4—32no—
500C4⋊C4—64no—
501C4⋊C4—64no—
502C4⋊C4—64no—
503C4⋊C4—64no—
504C4⋊C4—61no—
505C4⋊C4—62no—
506C4⋊C4—62no—
507C4⋊C4—122no—
508C4⋊C4—122no—
509C4⋊C4C2×#6712yes—
510C22⋊C4—124no—
511C22⋊C4—124no—
512C22⋊C4—124no—
513C42—24no—
514C42—64no—
515C42—64no—
516C42—62no—
517C42—62no—
518C42—64no—
519C42—61no—
520C42—61no—
521C42D3×D412yes—
522C42C2×#6712yes—
523C22×C3—82no—
524C22×C3—82no—
525C22×C3—82no—
526C22×C3—82no—
527C22×C3—82no—
528C22×C3—82no—
529C22×C3—82no—
530C22×C3C2×#12011yescharacteristic
531C22×C3C2×#12111yescharacteristic
532C22×C3#52511yescharacteristic
533C22×C3C22×SD1611yescharacteristic
534C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
535C22×C3C2×#13711yescharacteristic
536C22×C3#57511yescharacteristic
537C2×D3—22no—
538C2×D3—22no—
539C2×D3—22no—
540C2×D3—22no—
541C2×D3—22no—
542C2×D3—22no—
543C2×D3#46014yescomplemented
544C3×C4—24no—
545C3×C4—22no—
546C3×C4—42no—
547C3×C4#44814yescomplemented
548C3×C4C2×#12111yescharacteristic
549C3×C4C2×#12111yescharacteristic
550C3×C4#52511yescharacteristic
551C3×C4#52511yescharacteristic
552Dic3—22no—
553Dic3—24no—
554Dic3—22no—
555Dic3—41no—
556Dic3—42no—
557Dic3—82no—
558C23—32no—
559C23—32no—
560C23—32no—
561C23—42no—
562C23—42no—
563C23—42no—
564C23—42no—
565C23—42no—
566C23—42no—
567C23—42no—
568C23—61no—
569C23—61no—
570C23—61no—
571C23—61no—
572C23C2×D3×D411yescharacteristic
573Q8—22no—
574Q8—22no—
575Q8—22no—
576Q8—34no—
577Q8—616no—
578Q8—62no—
579Q8—62no—
580Q8—62no—
581Q8#93814yescomplemented
582D4—42no—
583D4—42no—
584D4—42no—
585D4—42no—
586D4—84no—
587D4—84no—
588D4—248no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—32no—
596C2×C4—32no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—616no—
600C2×C4—61no—
601C2×C4—61no—
602C2×C4—61no—
603C2×C4—62no—
604C2×C4—62no—
605C2×C4—61no—
606C2×C4—62no—
607C2×C4—68no—
608C2×C4—62no—
609C2×C4—62no—
610C2×C4—62no—
611C2×C4—62no—
612C2×C4—61no—
613C2×C4—61no—
614C2×C4—62no—
615C2×C4—61no—
616C2×C4—124no—
617C2×C4—128no—
618C2×C4—121no—
619C2×C4—121no—
620C2×C4—122no—
621C2×C4—122no—
622C2×C4—122no—
623C2×C4D3×SD1612yes—
624C2×C4#91512yes—
625C2×C4#93712yes—
626C2×C4#93712yes—
627C2×C4C2×D3×D411yescharacteristic
628C2×C4C2×D3×D411yescharacteristic
629C2×C4C2×#23711yescharacteristic
630C2×C4D3×#4111yescharacteristic
631C2×C4D3×#4111yescharacteristic
632C2×C4C22×#6711yescharacteristic
633C8—42no—
634C8—122no—
635C2×C3—162no—
636C2×C3#284711yescharacteristic
637C2×C3C2×#44811yescharacteristic
638C2×C3#300611yescharacteristic
639C2×C3C2×#46011yescharacteristic
640C2×C3#303711yescharacteristic
641C2×C3#310311yescharacteristic
642C2×C3#311211yescharacteristic
643D3—24no—
644C22—34no—
645C22—62no—
646C22—62no—
647C22—62no—
648C22—62no—
649C22—62no—
650C22—62no—
651C22—82no—
652C22—82no—
653C22—82no—
654C22—82no—
655C22—82no—
656C22—82no—
657C22—82no—
658C22D3×#12111yescharacteristic
659C22#560611yescharacteristic
660C22C2×D3×SD1611yescharacteristic
661C22C2×#91511yescharacteristic
662C22D3×AGL(1,ℤ/8)11yescharacteristic
663C22#577311yescharacteristic
664C22C2×#93711yescharacteristic
665C4—24no—
666C4—22no—
667C4—42no—
668C4—62no—
669C4—64no—
670C4—62no—
671C4—121no—
672C4—122no—
673C4—242no—
674C4#517014yescomplemented
675C4D3×#12111yescharacteristic
676C4#560611yescharacteristic
677C4C2×#93811yescharacteristic
678C4#582811yescharacteristic
679C3#3290311yescharacteristic, complemented, Sylow
680C2—64no—
681C2—162no—
682C2#8036011yescharacteristic
683C2#8036211yescharacteristic
684C2D3×#46011yescharacteristic
685C2#8038311yescharacteristic
686C2#8272311yescharacteristic
687C2C2×#517011yescharacteristic
688C2#8394611yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#460C2
socleC23×C3C22×D4
prime radical#83192C2
hypercenterC2×#460D3
hypocenterC3#32903
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#460

Complements

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

NH ≅ G/Ntype
1. C2×C4×D3×Q8C2semidirect
2. C2×C3×#460C2semidirect
3. C2×#5170C2semidirect
5. C2×#5075C2semidirect
6. #72048C2semidirect
7. #71626C2semidirect
8. #70485C2semidirect
9. #70475C2semidirect
10. #70213C2semidirect
11. #70200C2semidirect
51. C2×#460D3semidirect
70. C2×D3×Q8D4semidirect
85. C3×C4×Q8D4semidirect
128. C4×Dic6D4semidirect
225. C2×C4×D3SD16semidirect
312. C4×Q8#67semidirect
365. C3×Q8#121semidirect
421. Dic6#121semidirect
543. C2×D3#460semidirect
547. C3×C4#448semidirect
581. Q8#938semidirect
674. C4#5170semidirect
679. C3#32903semidirect

Maximal subgroups

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

subgroupindex
C2×C4×D3×Q82
C2×C3×#4602
C2×#51702
#831922
C2×#50752
#720482
#716262
#704852
#704752
#702132
#702002
#329033