Finite Group Databasebeta

← #10728186

Subgroups — #10728186

Order 768.

1622 conjugacy classes of subgroups.

The subgroup lattice has 672 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×C4×C22⋊C4C211yesmaximal, characteristic
2C2×#5210C211yesmaximal, characteristic
3C2×#5212C211yesmaximal, characteristic
4C2×#4661C211yesmaximal, characteristic, complemented
5#71546C212yesmaximal
6#71546C212yesmaximal
7#71536C212yesmaximal, complemented
8#71535C212yesmaximal, complemented
9#69423C211yesmaximal, characteristic, complemented
10#69419C211yesmaximal, characteristic, complemented
11#69410C211yesmaximal, characteristic, complemented
12#14840—31nomaximal, Sylow
13C24×C3×C4C2211yescharacteristic
14C22×C3×C22⋊C4C2212yes—
15C22×C3×C42C2212yes—
16C22×#217C2211yescharacteristic
17C22×#217C2212yes—
18C22×#217C2211yescharacteristic
19C22×C4⋊Dic3C2211yescharacteristic
20C22×C4⋊Dic3C2212yes—
21C22×C4⋊Dic3C2211yescharacteristic
22C22×C4⋊Dic3C2212yes—
23C22×Dic3⋊C4C2211yescharacteristic
24C22×Dic3⋊C4C2211yescharacteristic
25C3×C4×C22⋊C4—24no—
26C2×C3×#95C2211yescharacteristic
27C2×C3×#95C2211yescharacteristic
28#5212—24no—
29#5210—24no—
30C2×C3⋊#95C2211yescharacteristic
31C2×C3⋊#95C2212yes—
32C2×C3⋊#95C2212yes—
33C2×C3⋊#95C2212yes—
34C2×C3⋊#95C2211yescharacteristic
35C2×C3⋊#95C2212yes—
36C2×C3⋊#95C2212yes—
37C2×C3⋊#95C2212yes—
38C2×C3⋊#95C2212yes—
39#4661—24no—
40C2×#398—31no—
41C2×#385—31no—
42C2×#380—31no—
43C2×C4×C22⋊C4D311yescharacteristic
44#1445—31no—
45#1444—31no—
46#1441—31no—
47#1439—32no—
48#1439—32no—
49#1436—32no—
50C4⋊#95—32no—
51C25×C3C2311yescharacteristic
52C23×C3×C4—22no—
53C23×C3×C4—22no—
54C23×C3×C4—24no—
55C23×C3×C4—42no—
56C23×C3×C4D414yes—
57C23×C3×C4C2312yes—
58C23×C3×C4C2311yescharacteristic
59C23×C3×C4C2312yes—
60C23×C3×C4C2311yescharacteristic
61C23×Dic3C2311yescharacteristic
62C23×Dic3C2311yescharacteristic
63C23×Dic3C2312yes—
64C23×Dic3C2312yes—
65C23×Dic3C2311yescharacteristic
66C23×Dic3C2311yescharacteristic
67C2×C3×C22⋊C4—24no—
68C2×C3×C22⋊C4—24no—
69C2×C3×C22⋊C4Q818yes—
70C2×C3×C42—24no—
71C2×C3×C42—24no—
72C2×C3×C42—42no—
73C2×#217—22no—
74C2×#217—22no—
75C2×#217—22no—
76C2×#217—22no—
77C2×#217—24no—
78C2×#217—42no—
79C2×#217—42no—
80C2×#217—42no—
81C2×C4⋊Dic3—24no—
82C2×C4⋊Dic3—22no—
83C2×C4⋊Dic3—22no—
84C2×C4⋊Dic3—22no—
85C2×C4⋊Dic3—24no—
86C2×C4⋊Dic3—22no—
87C2×C4⋊Dic3—24no—
88C2×C4⋊Dic3—42no—
89C2×C4⋊Dic3—42no—
90C2×C4⋊Dic3—42no—
91C2×C4⋊Dic3—42no—
92C2×Dic3⋊C4—22no—
93C2×Dic3⋊C4—22no—
94C2×Dic3⋊C4—22no—
95C2×Dic3⋊C4—22no—
96C2×Dic3⋊C4—24no—
97C3×#95—22no—
98C3×#95D414yes—
99C3⋊#95—24no—
100C3⋊#95—22no—
101C3⋊#95—22no—
102C3⋊#95—24no—
103C3⋊#95—24no—
104C3⋊#95—24no—
105C3⋊#95—42no—
106C3⋊#95—42no—
107C3⋊#95—42no—
108C24×C4C2×D311yescharacteristic
109C22×C4⋊C4—31no—
110C22×C4⋊C4—31no—
111C22×C4⋊C4—31no—
112C22×C4⋊C4—32no—
113C22×C4⋊C4—32no—
114C22×C4⋊C4—31no—
115C22×C22⋊C4—31no—
116C22×C22⋊C4—32no—
117C22×C22⋊C4—31no—
118C22×C22⋊C4C2×D312yes—
119C22×C42C2×D312yes—
120#398—64no—
121#385—64no—
122#380—64no—
123C4×C22⋊C4—24no—
124C2×#95—32no—
125C2×#95—31no—
126C2×#95—32no—
127C2×#95—32no—
128C2×#95—32no—
129C2×#95—31no—
130C2×#95—32no—
131C2×#95—32no—
132C2×#95—32no—
133C2×#95C2×D311yescharacteristic
134C2×#95C2×D311yescharacteristic
135C24×C3—21no—
136C24×C3—21no—
137C24×C3—21no—
138C24×C3—22no—
139C24×C3—21no—
140C24×C3—22no—
141C24×C3—21no—
142C24×C3—21no—
143C24×C3—22no—
144C24×C3C2×D412yes—
145C24×C3C2×Q814yes—
146C24×C3C2411yescharacteristic
147C22×C3×C4—21no—
148C22×C3×C4—22no—
149C22×C3×C4—22no—
150C22×C3×C4—21no—
151C22×C3×C4—22no—
152C22×C3×C4—22no—
153C22×C3×C4—22no—
154C22×C3×C4—22no—
155C22×C3×C4—22no—
156C22×C3×C4—22no—
157C22×C3×C4—22no—
158C22×C3×C4—22no—
159C22×C3×C4—22no—
160C22×C3×C4—22no—
161C22×C3×C4—22no—
162C22×C3×C4—28no—
163C22×C3×C4—28no—
164C22×C3×C4—44no—
165C22×C3×C4—48no—
166C22×C3×C4—48no—
167C22×C3×C4C2×D412yes—
168C22×C3×C4C2×D412yes—
169C22×C3×C4C2×D412yes—
170C22×C3×C4C2×D414yes—
171C22×C3×C4C2×Q814yes—
172C22×C3×C4C2×Q814yes—
173C22×C3×C4#4112yes—
174C22×C3×C4#4114yes—
175C22×C3×C4#4114yes—
176C22×Dic3—21no—
177C22×Dic3—21no—
178C22×Dic3—21no—
179C22×Dic3—22no—
180C22×Dic3—21no—
181C22×Dic3—22no—
182C22×Dic3—21no—
183C22×Dic3—21no—
184C22×Dic3—21no—
185C22×Dic3—21no—
186C22×Dic3—21no—
187C22×Dic3—21no—
188C22×Dic3—22no—
189C22×Dic3—22no—
190C22×Dic3—22no—
191C22×Dic3—21no—
192C22×Dic3—22no—
193C22×Dic3—22no—
194C22×Dic3—21no—
195C22×Dic3—22no—
196C22×Dic3—22no—
197C22×Dic3—22no—
198C22×Dic3—22no—
199C22×Dic3—22no—
200C22×Dic3—21no—
201C22×Dic3—22no—
202C22×Dic3—22no—
203C22×Dic3—22no—
204C22×Dic3—22no—
205C22×Dic3—22no—
206C22×Dic3—21no—
207C22×Dic3—22no—
208C22×Dic3—22no—
209C22×Dic3—22no—
210C22×Dic3#4112yes—
211C22×Dic3#4112yes—
212C3×C22⋊C4—216no—
213C3×C42—48no—
214#217—28no—
215#217—44no—
216#217—48no—
217C4⋊Dic3—48no—
218C4⋊Dic3—44no—
219C4⋊Dic3—48no—
220C4⋊Dic3—44no—
221Dic3⋊C4—44no—
222Dic3⋊C4—44no—
223C25C22×D311yescharacteristic
224C23×C4—22no—
225C23×C4—24no—
226C23×C4—22no—
227C23×C4—31no—
228C23×C4—31no—
229C23×C4—31no—
230C23×C4—32no—
231C23×C4—32no—
232C23×C4—31no—
233C23×C4—42no—
234C23×C4D1214yes—
235C23×C4C22×D312yes—
236C23×C4C22×D312yes—
237C23×C4C22×D311yescharacteristic
238C23×C4C22×D311yescharacteristic
239C2×C4⋊C4—64no—
240C2×C4⋊C4—62no—
241C2×C4⋊C4—62no—
242C2×C4⋊C4—62no—
243C2×C4⋊C4—62no—
244C2×C4⋊C4—62no—
245C2×C4⋊C4—64no—
246C2×C4⋊C4—62no—
247C2×C4⋊C4—64no—
248C2×C4⋊C4—62no—
249C2×C4⋊C4—64no—
250C2×C4⋊C4—62no—
251C2×C4⋊C4—122no—
252C2×C4⋊C4—122no—
253C2×C4⋊C4—122no—
254C2×C4⋊C4—122no—
255C2×C22⋊C4—24no—
256C2×C22⋊C4—24no—
257C2×C22⋊C4—62no—
258C2×C22⋊C4—62no—
259C2×C22⋊C4—62no—
260C2×C22⋊C4—64no—
261C2×C22⋊C4—62no—
262C2×C22⋊C4—122no—
263C2×C22⋊C4—122no—
264C2×C22⋊C4—122no—
265C2×C22⋊C4Dic618yes—
266C2×C42—24no—
267C2×C42—24no—
268C2×C42—42no—
269#95—22no—
270#95—64no—
271#95—64no—
272#95—62no—
273#95—62no—
274#95—64no—
275#95—64no—
276#95—122no—
277#95—122no—
278#95—122no—
279#95#6714yes—
280C23×C3—22no—
281C23×C3—24no—
282C23×C3—22no—
283C23×C3—22no—
284C23×C3—22no—
285C23×C3—24no—
286C23×C3—22no—
287C23×C3—22no—
288C23×C3—24no—
289C23×C3—24no—
290C23×C3—24no—
291C23×C3—24no—
292C23×C3—42no—
293C23×C3—42no—
294C23×C3—42no—
295C23×C3—42no—
296C23×C3—42no—
297C23×C3—41no—
298C23×C3—41no—
299C23×C3—42no—
300C23×C3—41no—
301C23×C3—41no—
302C23×C3C4⋊Q814yes—
303C23×C3C22×D411yescharacteristic
304C23×C3C22×D411yescharacteristic
305C23×C3C22×Q812yes—
306C23×C3C2×#4111yescharacteristic
307C23×C3C2×#4112yes—
308C23×C3C2×#4112yes—
309C23×C3C2×#4111yescharacteristic
310C23×C3C2×#4111yescharacteristic
311C23×C3H(5,𝔽2)12yes—
312C23×C3H(5,𝔽2)11yescharacteristic
313C23×C3H(5,𝔽2)11yescharacteristic
314C2×C3×C4—22no—
315C2×C3×C4—22no—
316C2×C3×C4—24no—
317C2×C3×C4—22no—
318C2×C3×C4—22no—
319C2×C3×C4—24no—
320C2×C3×C4—24no—
321C2×C3×C4—24no—
322C2×C3×C4—24no—
323C2×C3×C4—24no—
324C2×C3×C4—24no—
325C2×C3×C4—24no—
326C2×C3×C4—24no—
327C2×C3×C4—24no—
328C2×C3×C4—24no—
329C2×C3×C4—24no—
330C2×C3×C4—24no—
331C2×C3×C4—416no—
332C2×C3×C4—42no—
333C2×C3×C4—42no—
334C2×C3×C4—42no—
335C2×C3×C4—42no—
336C2×C3×C4#12114yes—
337C2×C3×C4C22⋊Q818yes—
338C2×C3×C4#12314yes—
339C2×Dic3—22no—
340C2×Dic3—22no—
341C2×Dic3—24no—
342C2×Dic3—22no—
343C2×Dic3—22no—
344C2×Dic3—41no—
345C2×Dic3—41no—
346C2×Dic3—42no—
347C2×Dic3—41no—
348C2×Dic3—41no—
349C2×Dic3—41no—
350C2×Dic3—41no—
351C2×Dic3—41no—
352C2×Dic3—42no—
353C2×Dic3—41no—
354C2×Dic3—42no—
355C2×Dic3—41no—
356C2×Dic3—42no—
357C2×Dic3—42no—
358C2×Dic3—41no—
359C2×Dic3—41no—
360C2×Dic3—42no—
361C2×Dic3—42no—
362C2×Dic3—41no—
363C2×Dic3—42no—
364C2×Dic3—42no—
365C2×Dic3—42no—
366C2×Dic3—42no—
367C2×Dic3—42no—
368C2×Dic3—42no—
369C2×Dic3—42no—
370C2×Dic3—42no—
371C2×Dic3—42no—
372C2×Dic3—42no—
373C2×Dic3—42no—
374C2×Dic3—42no—
375C24—22no—
376C24—21no—
377C24—22no—
378C24—22no—
379C24—21no—
380C24—21no—
381C24—21no—
382C24—21no—
383C24—21no—
384C24C2×Dic614yes—
385C24C2×D1212yes—
386C24C23×D311yescharacteristic
387C22×C4—21no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—21no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—28no—
403C22×C4—28no—
404C22×C4—32no—
405C22×C4—32no—
406C22×C4—44no—
407C22×C4—48no—
408C22×C4—48no—
409C22×C4—61no—
410C22×C4—61no—
411C22×C4—62no—
412C22×C4—61no—
413C22×C4—62no—
414C22×C4—61no—
415C22×C4—61no—
416C22×C4—61no—
417C22×C4—61no—
418C22×C4—61no—
419C22×C4—61no—
420C22×C4—62no—
421C22×C4—62no—
422C22×C4—61no—
423C22×C4—62no—
424C22×C4—62no—
425C22×C4—62no—
426C22×C4—61no—
427C22×C4—62no—
428C22×C4—62no—
429C22×C4—62no—
430C22×C4—62no—
431C22×C4—62no—
432C22×C4—62no—
433C22×C4—62no—
434C22×C4—62no—
435C22×C4—61no—
436C22×C4—62no—
437C22×C4—62no—
438C22×C4—62no—
439C22×C4—62no—
440C22×C4—61no—
441C22×C4—61no—
442C22×C4—62no—
443C22×C4C2×Dic614yes—
444C22×C4C2×Dic614yes—
445C22×C4C2×D1212yes—
446C22×C4C2×D1212yes—
447C22×C4#23514yes—
448C22×C4#23512yes—
449C22×C4#23714yes—
450C22×C4C2×#6714yes—
451C22×C4C2×#6712yes—
452C4⋊C4—128no—
453C4⋊C4—124no—
454C4⋊C4—128no—
455C4⋊C4—124no—
456C4⋊C4—124no—
457C4⋊C4—124no—
458C22⋊C4—216no—
459C22⋊C4—68no—
460C22⋊C4—124no—
461C22⋊C4—128no—
462C42—48no—
463C22×C3—24no—
464C22×C3—24no—
465C22×C3—24no—
466C22×C3—44no—
467C22×C3—42no—
468C22×C3—42no—
469C22×C3—42no—
470C22×C3—42no—
471C22×C3—44no—
472C22×C3—44no—
473C22×C3—44no—
474C22×C3C2×#12111yescharacteristic
475C22×C3C2×C22⋊Q812yes—
476C22×C3C2×#12311yescharacteristic
477C22×C3#52511yescharacteristic
478C22×C3#52912yes—
479C22×C3#52912yes—
480C22×C3C2×C4⋊Q811yescharacteristic
481C22×C3#53211yescharacteristic
482C22×C3#53312yes—
483C22×C3#53411yescharacteristic
484C22×C3#53511yescharacteristic
485C22×C3#53812yes—
486C22×C3#54012yes—
487C22×C3#54612yes—
488C22×C3#54611yescharacteristic
489C22×C3#55011yescharacteristic
490C22×C3#55112yes—
491C22×C3#55311yescharacteristic
492C22×C3#55412yes—
493C22×C3#55412yes—
494C22×C3#55412yes—
495C22×C3#55412yes—
496C22×C3#55511yescharacteristic
497C3×C4—24no—
498C3×C4—24no—
499C3×C4—28no—
500C3×C4—44no—
501Dic3—42no—
502Dic3—42no—
503Dic3—82no—
504Dic3—81no—
505Dic3—81no—
506Dic3—82no—
507C23—24no—
508C23—22no—
509C23—22no—
510C23—22no—
511C23—22no—
512C23—22no—
513C23—24no—
514C23—22no—
515C23—24no—
516C23—24no—
517C23—24no—
518C23—24no—
519C23—42no—
520C23—41no—
521C23—41no—
522C23—42no—
523C23—42no—
524C23—42no—
525C23—42no—
526C23—42no—
527C23—41no—
528C23—41no—
529C23#86914yes—
530C23C22×Dic612yes—
531C23C22×D1211yescharacteristic
532C23C2×#23511yescharacteristic
533C23C2×#23512yes—
534C23C2×#23712yes—
535C23#100412yes—
536C23D3×#4111yescharacteristic
537C23D3×#4111yescharacteristic
538C23#100911yescharacteristic
539C23#100911yescharacteristic
540C23C22×#6711yescharacteristic
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—24no—
545C2×C4—24no—
546C2×C4—22no—
547C2×C4—24no—
548C2×C4—24no—
549C2×C4—24no—
550C2×C4—24no—
551C2×C4—24no—
552C2×C4—24no—
553C2×C4—24no—
554C2×C4—24no—
555C2×C4—24no—
556C2×C4—24no—
557C2×C4—24no—
558C2×C4—416no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—62no—
564C2×C4—62no—
565C2×C4—62no—
566C2×C4—64no—
567C2×C4—62no—
568C2×C4—121no—
569C2×C4—121no—
570C2×C4—121no—
571C2×C4—121no—
572C2×C4—121no—
573C2×C4—122no—
574C2×C4—121no—
575C2×C4—121no—
576C2×C4—122no—
577C2×C4—121no—
578C2×C4—121no—
579C2×C4—121no—
580C2×C4—122no—
581C2×C4—122no—
582C2×C4—122no—
583C2×C4—122no—
584C2×C4—122no—
585C2×C4—121no—
586C2×C4—122no—
587C2×C4—122no—
588C2×C4—122no—
589C2×C4—121no—
590C2×C4—122no—
591C2×C4—122no—
592C2×C4—122no—
593C2×C4—122no—
594C2×C4—122no—
595C2×C4—122no—
596C2×C4—122no—
597C2×C4—122no—
598C2×C4—122no—
599C2×C4#88614yes—
600C2×C4#92418yes—
601C2×C4#93014yes—
602C2×C3—44no—
603C2×C3#239511yescharacteristic
604C2×C3#241211yescharacteristic
605C2×C3#241411yescharacteristic
606C2×C3#245012yes—
607C2×C3#245412yes—
608C2×C3#250211yescharacteristic
609C2×C3#252212yes—
610C2×C3#257412yes—
611C2×C3#258311yescharacteristic
612C2×C3#266111yescharacteristic
613C2×C3#285711yescharacteristic
614C22—24no—
615C22—24no—
616C22—24no—
617C22—44no—
618C22—44no—
619C22—42no—
620C22—42no—
621C22—42no—
622C22—44no—
623C22—44no—
624C22—42no—
625C22C2×#86911yescharacteristic
626C22#547012yes—
627C22#547711yescharacteristic
628C22C2×#88611yescharacteristic
629C22#549311yescharacteristic
630C22#549512yes—
631C22#552411yescharacteristic
632C22#552511yescharacteristic
633C22#553011yescharacteristic
634C22#553211yescharacteristic
635C22#553812yes—
636C22#553912yes—
637C22#553912yes—
638C22#554312yes—
639C22#554312yes—
640C22#555712yes—
641C22#556012yes—
642C22#556212yes—
643C22C2×#92412yes—
644C22C2×#93011yescharacteristic
645C22#580612yes—
646C22#582911yescharacteristic
647C22#583411yescharacteristic
648C4—28no—
649C4—24no—
650C4—24no—
651C4—44no—
652C4—122no—
653C4—122no—
654C4—242no—
655C4—241no—
656C4—242no—
657C4—241no—
658C3#1484011yescharacteristic, complemented, Sylow
659C2—44no—
660C2#7351711yescharacteristic
661C2#7352812yes—
662C2#7354212yes—
663C2#7354711yescharacteristic
664C2#7358311yescharacteristic
665C2#7361111yescharacteristic
666C2#8207611yescharacteristic
667C2#8210811yescharacteristic
668C2#8227611yescharacteristic
669C2#8236112yes—
670C2#8236312yes—
subgroupG/H
center, FrattiniC24C23×D3
commutator, socle, cosocleC24×C3C24
FittingC2×C3×C4×C22⋊C4C2
prime radicalC25×C3C23
hypercenterC2×C4×C22⋊C4D3
hypocenterC3#14840
LCSG ▷ C24×C3 ▷ C3
UCSC1 ◁ C24 ◁ C2×C4×C22⋊C4

Complements

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

NH ≅ G/Ntype
4. C2×#4661C2semidirect
7. #71536C2semidirect
8. #71535C2semidirect
9. #69423C2semidirect
10. #69419C2semidirect
11. #69410C2semidirect
658. C3#14840semidirect

Maximal subgroups

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

subgroupindex
C2×C3×C4×C22⋊C42
C2×#52102
C2×#52122
C2×#46612
#715462
#715362
#715352
#694232
#694192
#694102
#148403