Finite Group Databasebeta

← #10728208

Subgroups — #10728208

Order 768.

1794 conjugacy classes of subgroups.

The subgroup lattice has 748 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×C4×C22⋊C4C211yesmaximal, characteristic, complemented
2C2×#5213C211yesmaximal, characteristic
3C2×#5212C211yesmaximal, characteristic, complemented
4C2×#4674C211yesmaximal, characteristic, complemented
5#71550C212yesmaximal
6#71549C212yesmaximal, complemented
7#71537C212yesmaximal, complemented
8#71534C212yesmaximal, complemented
9#69427C211yesmaximal, characteristic, complemented
10#69423C211yesmaximal, characteristic, complemented
11#69412C211yesmaximal, characteristic, complemented
12#14842—31nomaximal, Sylow
13C24×C3×C4C2211yescharacteristic
14C23×#67C2211yescharacteristic
15C22×C3×C22⋊C4C2212yes—
16C22×C3×C42C2212yes—
17C22×#217C2211yescharacteristic
18C22×#217C2212yes—
19C22×(C2×D3)⋊C4C2211yescharacteristic
20C22×(C2×D3)⋊C4C2211yescharacteristic
21C22×(C2×D3)⋊C4C2212yes—
22C22×(C2×D3)⋊C4C2212yes—
23C22×C4⋊Dic3C2211yescharacteristic
24C22×C4⋊Dic3C2211yescharacteristic
25C3×C4×C22⋊C4—24no—
26C2×C3×#95C2211yescharacteristic
27C2×C3×#95C2211yescharacteristic
28#5213—24no—
29#5212—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#4674—24no—
40C2×#397—31no—
41C2×#386—31no—
42C2×#380—31no—
43C2×C4×C22⋊C4D311yescharacteristic, complemented
44#1445—31no—
45#1442—31no—
46#1440—32no—
47#1439—31no—
48#1438—32no—
49#1437—32no—
50#1434—32no—
51C25×C3C2311yescharacteristic
52C24×D3C2311yescharacteristic
53C23×C3×C4—22no—
54C23×C3×C4—22no—
55C23×C3×C4—24no—
56C23×C3×C4—42no—
57C23×C3×C4D414yes—
58C23×C3×C4C2312yes—
59C23×C3×C4C2312yes—
60C23×C3×C4C2311yescharacteristic
61C23×C3×C4C2311yescharacteristic
62C22×#67—22no—
63C22×#67—22no—
64C22×#67—24no—
65C22×#67—22no—
66C22×#67—42no—
67C23×Dic3C2311yescharacteristic
68C23×Dic3C2311yescharacteristic
69C23×Dic3C2312yes—
70C23×Dic3C2312yes—
71C23×Dic3C2311yescharacteristic
72C2×C3×C22⋊C4—24no—
73C2×C3×C22⋊C4—24no—
74C2×C3×C22⋊C4—24no—
75C2×C3×C42—24no—
76C2×C3×C42—24no—
77C2×C3×C42—42no—
78C2×#217—24no—
79C2×#217—22no—
80C2×#217—24no—
81C2×#217—42no—
82C2×#217—42no—
83C2×(C2×D3)⋊C4—24no—
84C2×(C2×D3)⋊C4—24no—
85C2×(C2×D3)⋊C4—22no—
86C2×(C2×D3)⋊C4—22no—
87C2×(C2×D3)⋊C4—22no—
88C2×(C2×D3)⋊C4—24no—
89C2×(C2×D3)⋊C4—22no—
90C2×(C2×D3)⋊C4—24no—
91C2×(C2×D3)⋊C4—42no—
92C2×(C2×D3)⋊C4—42no—
93C2×(C2×D3)⋊C4—42no—
94C2×C4⋊Dic3—22no—
95C2×C4⋊Dic3—22no—
96C2×C4⋊Dic3—22no—
97C2×C4⋊Dic3—22no—
98C2×C4⋊Dic3—42no—
99C3×#95—22no—
100C3×#95D414yescomplemented
101C3⋊#95—24no—
102C3⋊#95—24no—
103C3⋊#95—22no—
104C3⋊#95—22no—
105C3⋊#95—24no—
106C3⋊#95—24no—
107C3⋊#95—42no—
108C3⋊#95—42no—
109C3⋊#95—42no—
110C23×D4—31no—
111C24×C4C2×D311yescharacteristic
112C22×C4⋊C4—31no—
113C22×C4⋊C4—31no—
114C22×C22⋊C4—31no—
115C22×C22⋊C4—31no—
116C22×C22⋊C4—31no—
117C22×C22⋊C4—32no—
118C22×C22⋊C4—32no—
119C22×C22⋊C4—32no—
120C22×C22⋊C4C2×D312yes—
121C22×C42C2×D312yes—
122#397—64no—
123#386—64no—
124#380—64no—
125C4×C22⋊C4—24no—
126C2×#95—32no—
127C2×#95—31no—
128C2×#95—32no—
129C2×#95—32no—
130C2×#95—32no—
131C2×#95—31no—
132C2×#95—32no—
133C2×#95—32no—
134C2×#95—32no—
135C2×#95C2×D311yescharacteristic
136C2×#95C2×D311yescharacteristic
137C24×C3—21no—
138C24×C3—22no—
139C24×C3—21no—
140C24×C3—21no—
141C24×C3—21no—
142C24×C3—22no—
143C24×C3—21no—
144C24×C3—21no—
145C24×C3—22no—
146C24×C3C2×D412yes—
147C24×C3#4114yes—
148C24×C3C2411yescharacteristic
149C23×D3—21no—
150C23×D3—21no—
151C23×D3—22no—
152C23×D3—21no—
153C23×D3—22no—
154C23×D3—22no—
155C23×D3—21no—
156C23×D3—21no—
157C23×D3—21no—
158C23×D3—22no—
159C23×D3#4112yes—
160C22×C3×C4—21no—
161C22×C3×C4—21no—
162C22×C3×C4—22no—
163C22×C3×C4—22no—
164C22×C3×C4—22no—
165C22×C3×C4—22no—
166C22×C3×C4—22no—
167C22×C3×C4—22no—
168C22×C3×C4—22no—
169C22×C3×C4—22no—
170C22×C3×C4—22no—
171C22×C3×C4—22no—
172C22×C3×C4—22no—
173C22×C3×C4—22no—
174C22×C3×C4—22no—
175C22×C3×C4—22no—
176C22×C3×C4—22no—
177C22×C3×C4—28no—
178C22×C3×C4—28no—
179C22×C3×C4—44no—
180C22×C3×C4—48no—
181C22×C3×C4—48no—
182C22×C3×C4C2×D412yes—
183C22×C3×C4C2×D412yes—
184C22×C3×C4C2×D414yes—
185C22×C3×C4C2×D412yes—
186C22×C3×C4#4114yes—
187C22×C3×C4#4112yes—
188C22×C3×C4#4114yes—
189C2×#67—28no—
190C2×#67—48no—
191C2×#67—48no—
192C2×#67—44no—
193C2×#67—44no—
194C22×Dic3—21no—
195C22×Dic3—22no—
196C22×Dic3—21no—
197C22×Dic3—22no—
198C22×Dic3—21no—
199C22×Dic3—22no—
200C22×Dic3—22no—
201C22×Dic3—22no—
202C22×Dic3—21no—
203C22×Dic3—21no—
204C22×Dic3—22no—
205C22×Dic3—22no—
206C22×Dic3—22no—
207C22×Dic3—22no—
208C22×Dic3—21no—
209C22×Dic3—22no—
210C22×Dic3—22no—
211C22×Dic3—22no—
212C22×Dic3—22no—
213C22×Dic3—22no—
214C22×Dic3—21no—
215C22×Dic3—21no—
216C22×Dic3—22no—
217C22×Dic3—22no—
218C22×Dic3—22no—
219C22×Dic3—22no—
220C22×Dic3—22no—
221C22×Dic3—22no—
222C22×Dic3#4112yes—
223C3×C22⋊C4—216no—
224C3×C42—48no—
225#217—48no—
226#217—44no—
227(C2×D3)⋊C4—48no—
228(C2×D3)⋊C4—48no—
229(C2×D3)⋊C4—44no—
230(C2×D3)⋊C4—44no—
231C4⋊Dic3—44no—
232C4⋊Dic3—44no—
233C25—31no—
234C25C22×D311yescharacteristic
235C22×D4—62no—
236C22×D4—62no—
237C22×D4—62no—
238C22×D4—64no—
239C22×D4—122no—
240C23×C4—22no—
241C23×C4—24no—
242C23×C4—22no—
243C23×C4—31no—
244C23×C4—32no—
245C23×C4—32no—
246C23×C4—31no—
247C23×C4—31no—
248C23×C4—42no—
249C23×C4D1214yes—
250C23×C4C22×D312yes—
251C23×C4C22×D312yes—
252C23×C4C22×D311yescharacteristic
253C23×C4C22×D311yescharacteristic
254C2×C4⋊C4—62no—
255C2×C4⋊C4—62no—
256C2×C4⋊C4—62no—
257C2×C4⋊C4—62no—
258C2×C4⋊C4—122no—
259C2×C22⋊C4—24no—
260C2×C22⋊C4—24no—
261C2×C22⋊C4—24no—
262C2×C22⋊C4—64no—
263C2×C22⋊C4—64no—
264C2×C22⋊C4—62no—
265C2×C22⋊C4—64no—
266C2×C22⋊C4—62no—
267C2×C22⋊C4—62no—
268C2×C22⋊C4—64no—
269C2×C22⋊C4—62no—
270C2×C22⋊C4—64no—
271C2×C22⋊C4—64no—
272C2×C22⋊C4—62no—
273C2×C22⋊C4—122no—
274C2×C22⋊C4—122no—
275C2×C22⋊C4—122no—
276C2×C22⋊C4—122no—
277C2×C22⋊C4—122no—
278C2×C42—24no—
279C2×C42—24no—
280C2×C42—42no—
281#95—22no—
282#95—64no—
283#95—64no—
284#95—62no—
285#95—62no—
286#95—64no—
287#95—64no—
288#95—122no—
289#95—122no—
290#95—122no—
291#95#6714yescomplemented
292C23×C3—22no—
293C23×C3—22no—
294C23×C3—24no—
295C23×C3—22no—
296C23×C3—24no—
297C23×C3—22no—
298C23×C3—22no—
299C23×C3—22no—
300C23×C3—24no—
301C23×C3—24no—
302C23×C3—24no—
303C23×C3—24no—
304C23×C3—42no—
305C23×C3—42no—
306C23×C3—42no—
307C23×C3—42no—
308C23×C3—42no—
309C23×C3—41no—
310C23×C3—41no—
311C23×C3—42no—
312C23×C3—41no—
313C23×C3—41no—
314C23×C3#12414yes—
315C23×C3C22×D411yescharacteristic
316C23×C3C22×D411yescharacteristic
317C23×C3C2×#4111yescharacteristic
318C23×C3C2×#4112yes—
319C23×C3C2×#4112yes—
320C23×C3C2×#4112yes—
321C23×C3C2×#4111yescharacteristic
322C23×C3C2×#4111yescharacteristic
323C23×C3H(5,𝔽2)12yes—
324C23×C3H(5,𝔽2)11yescharacteristic
325C23×C3H(5,𝔽2)11yescharacteristic
326C22×D3—22no—
327C22×D3—24no—
328C22×D3—22no—
329C22×D3—24no—
330C22×D3—22no—
331C22×D3—41no—
332C22×D3—41no—
333C22×D3—42no—
334C22×D3—41no—
335C22×D3—42no—
336C22×D3—42no—
337C22×D3—41no—
338C22×D3—41no—
339C22×D3—41no—
340C22×D3—42no—
341C22×D3—42no—
342C22×D3—42no—
343C22×D3—41no—
344C22×D3—42no—
345C22×D3—42no—
346C22×D3—42no—
347C22×D3—42no—
348C22×D3—41no—
349C2×C3×C4—22no—
350C2×C3×C4—22no—
351C2×C3×C4—22no—
352C2×C3×C4—22no—
353C2×C3×C4—24no—
354C2×C3×C4—24no—
355C2×C3×C4—24no—
356C2×C3×C4—24no—
357C2×C3×C4—24no—
358C2×C3×C4—24no—
359C2×C3×C4—24no—
360C2×C3×C4—24no—
361C2×C3×C4—24no—
362C2×C3×C4—24no—
363C2×C3×C4—416no—
364C2×C3×C4—42no—
365C2×C3×C4—42no—
366C2×C3×C4—42no—
367C2×C3×C4—42no—
368C2×C3×C4—42no—
369C2×C3×C4—42no—
370C2×C3×C4—42no—
371C2×C3×C4—42no—
372C2×C3×C4#12114yes—
373C2×C3×C4#12314yes—
374#67—416no—
375C2×Dic3—22no—
376C2×Dic3—24no—
377C2×Dic3—41no—
378C2×Dic3—41no—
379C2×Dic3—42no—
380C2×Dic3—42no—
381C2×Dic3—41no—
382C2×Dic3—41no—
383C2×Dic3—42no—
384C2×Dic3—42no—
385C2×Dic3—41no—
386C2×Dic3—41no—
387C2×Dic3—42no—
388C2×Dic3—41no—
389C2×Dic3—42no—
390C2×Dic3—42no—
391C2×Dic3—42no—
392C2×Dic3—42no—
393C2×Dic3—41no—
394C2×Dic3—42no—
395C2×Dic3—42no—
396C2×Dic3—42no—
397C2×Dic3—42no—
398C2×Dic3—42no—
399C2×Dic3—42no—
400C2×Dic3—42no—
401C2×Dic3—42no—
402C2×Dic3—42no—
403C2×Dic3—42no—
404C24—22no—
405C24—21no—
406C24—22no—
407C24—22no—
408C24—21no—
409C24—21no—
410C24—21no—
411C24—21no—
412C24—21no—
413C24—32no—
414C24—61no—
415C24—61no—
416C24—62no—
417C24—62no—
418C24—62no—
419C24—62no—
420C24—61no—
421C24—61no—
422C24—61no—
423C24—61no—
424C24C2×D1212yes—
425C24#23514yes—
426C24C23×D311yescharacteristic
427C2×D4—68no—
428C2×D4—128no—
429C2×D4—128no—
430C2×D4—124no—
431C2×D4—124no—
432C22×C4—21no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—21no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—28no—
450C22×C4—28no—
451C22×C4—32no—
452C22×C4—44no—
453C22×C4—48no—
454C22×C4—48no—
455C22×C4—61no—
456C22×C4—62no—
457C22×C4—62no—
458C22×C4—61no—
459C22×C4—61no—
460C22×C4—61no—
461C22×C4—62no—
462C22×C4—62no—
463C22×C4—62no—
464C22×C4—62no—
465C22×C4—62no—
466C22×C4—62no—
467C22×C4—62no—
468C22×C4—61no—
469C22×C4—62no—
470C22×C4—62no—
471C22×C4—62no—
472C22×C4—62no—
473C22×C4—62no—
474C22×C4—62no—
475C22×C4—61no—
476C22×C4—61no—
477C22×C4—62no—
478C22×C4—62no—
479C22×C4—62no—
480C22×C4—62no—
481C22×C4—61no—
482C22×C4—62no—
483C22×C4C2×D1212yes—
484C22×C4C2×D1212yes—
485C22×C4#23514yes—
486C22×C4#23512yes—
487C22×C4#23714yes—
488C22×C4C2×#6714yes—
489C22×C4C2×#6712yes—
490C4⋊C4—124no—
491C4⋊C4—124no—
492C22⋊C4—216no—
493C22⋊C4—128no—
494C22⋊C4—128no—
495C22⋊C4—128no—
496C22⋊C4—124no—
497C22⋊C4—124no—
498C22⋊C4—124no—
499C42—48no—
500C22×C3—24no—
501C22×C3—24no—
502C22×C3—24no—
503C22×C3—44no—
504C22×C3—42no—
505C22×C3—42no—
506C22×C3—42no—
507C22×C3—42no—
508C22×C3—44no—
509C22×C3—44no—
510C22×C3—44no—
511C22×C3C2×#12111yescharacteristic
512C22×C3C2×#12311yescharacteristic
513C22×C3#52511yescharacteristic
514C22×C3#52512yes—
515C22×C3C2×#12411yescharacteristic
516C22×C3#52912yes—
517C22×C3#52912yes—
518C22×C3#52912yes—
519C22×C3#53211yescharacteristic
520C22×C3#53411yescharacteristic
521C22×C3#53511yescharacteristic
522C22×C3#53812yes—
523C22×C3#54012yes—
524C22×C3#54612yes—
525C22×C3#54611yescharacteristic
526C22×C3#55011yescharacteristic
527C22×C3#55112yes—
528C22×C3#55112yes—
529C22×C3#55112yes—
530C22×C3#55311yescharacteristic
531C22×C3#55312yes—
532C22×C3#55312yes—
533C22×C3#55511yescharacteristic
534C2×D3—42no—
535C2×D3—44no—
536C2×D3—42no—
537C2×D3—42no—
538C2×D3—44no—
539C2×D3—81no—
540C2×D3—81no—
541C2×D3—81no—
542C2×D3—82no—
543C2×D3—81no—
544C2×D3—82no—
545C3×C4—24no—
546C3×C4—24no—
547C3×C4—44no—
548C3×C4—44no—
549Dic3—42no—
550Dic3—81no—
551Dic3—82no—
552Dic3—82no—
553Dic3—81no—
554C23—22no—
555C23—24no—
556C23—22no—
557C23—22no—
558C23—22no—
559C23—22no—
560C23—22no—
561C23—24no—
562C23—24no—
563C23—24no—
564C23—24no—
565C23—24no—
566C23—42no—
567C23—41no—
568C23—41no—
569C23—42no—
570C23—42no—
571C23—42no—
572C23—42no—
573C23—42no—
574C23—41no—
575C23—41no—
576C23—62no—
577C23—62no—
578C23—64no—
579C23—64no—
580C23—62no—
581C23—121no—
582C23—121no—
583C23—122no—
584C23—121no—
585C23—121no—
586C23—122no—
587C23—121no—
588C23—121no—
589C23—121no—
590C23—122no—
591C23—122no—
592C23—122no—
593C23—122no—
594C23—122no—
595C23—122no—
596C23—122no—
597C23—122no—
598C23—121no—
599C23#87514yes—
600C23C22×D1211yescharacteristic
601C23C2×#23512yes—
602C23C2×#23512yes—
603C23C2×#23511yescharacteristic
604C23C2×#23712yes—
605C23#100412yes—
606C23D3×#4111yescharacteristic
607C23D3×#4111yescharacteristic
608C23#100911yescharacteristic
609C23#100911yescharacteristic
610C23C22×#6711yescharacteristic
611D4—1216no—
612C2×C4—22no—
613C2×C4—22no—
614C2×C4—22no—
615C2×C4—24no—
616C2×C4—24no—
617C2×C4—22no—
618C2×C4—24no—
619C2×C4—24no—
620C2×C4—24no—
621C2×C4—24no—
622C2×C4—24no—
623C2×C4—24no—
624C2×C4—24no—
625C2×C4—24no—
626C2×C4—416no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—62no—
636C2×C4—64no—
637C2×C4—121no—
638C2×C4—121no—
639C2×C4—122no—
640C2×C4—121no—
641C2×C4—121no—
642C2×C4—122no—
643C2×C4—121no—
644C2×C4—122no—
645C2×C4—121no—
646C2×C4—122no—
647C2×C4—122no—
648C2×C4—122no—
649C2×C4—122no—
650C2×C4—121no—
651C2×C4—122no—
652C2×C4—122no—
653C2×C4—121no—
654C2×C4—122no—
655C2×C4—122no—
656C2×C4—122no—
657C2×C4—122no—
658C2×C4—122no—
659C2×C4—122no—
660C2×C4—122no—
661C2×C4—122no—
662C2×C4—122no—
663C2×C4—122no—
664C2×C4#88614yes—
665C2×C4#93014yes—
666C2×C3—44no—
667C2×C3#239511yescharacteristic
668C2×C3#241211yescharacteristic
669C2×C3#241411yescharacteristic
670C2×C3#245712yes—
671C2×C3#246712yes—
672C2×C3#250411yescharacteristic
673C2×C3#253112yes—
674C2×C3#256111yescharacteristic
675C2×C3#257812yes—
676C2×C3#266111yescharacteristic
677C2×C3#285011yescharacteristic
678D3—82no—
679C22—24no—
680C22—24no—
681C22—24no—
682C22—44no—
683C22—44no—
684C22—42no—
685C22—42no—
686C22—42no—
687C22—42no—
688C22—44no—
689C22—44no—
690C22—124no—
691C22—122no—
692C22—122no—
693C22—122no—
694C22—124no—
695C22—241no—
696C22—242no—
697C22—241no—
698C22—241no—
699C22—241no—
700C22—242no—
701C22C2×#87511yescharacteristic
702C22#547711yescharacteristic
703C22#547912yes—
704C22C2×#88611yescharacteristic
705C22#549311yescharacteristic
706C22#549512yes—
707C22#552411yescharacteristic
708C22#552511yescharacteristic
709C22#553011yescharacteristic
710C22#553211yescharacteristic
711C22#553812yes—
712C22#555612yes—
713C22#555612yes—
714C22#555712yes—
715C22#556012yes—
716C22#556212yes—
717C22#556512yes—
718C22#556512yes—
719C22C2×#93011yescharacteristic
720C22#579112yes—
721C22#580612yes—
722C22#582911yescharacteristic
723C22#583411yescharacteristic
724C4—24no—
725C4—24no—
726C4—44no—
727C4—44no—
728C4—122no—
729C4—241no—
730C4—242no—
731C4—242no—
732C4—241no—
733C3#1484211yescharacteristic, complemented, Sylow
734C2—44no—
735C2—242no—
736C2#7351911yescharacteristic
737C2#7353312yes—
738C2#7354612yes—
739C2#7358311yescharacteristic
740C2#7360511yescharacteristic
741C2#7361111yescharacteristic
742C2#8207711yescharacteristic
743C2#8210811yescharacteristic
744C2#8227611yescharacteristic
745C2#8237012yes—
746C2#8241212yes—
subgroupG/H
center, FrattiniC24C23×D3
commutator, socle, cosocleC24×C3C24
FittingC2×C3×C4×C22⋊C4C2
prime radicalC23×#67C22
hypercenterC2×C4×C22⋊C4D3
hypocenterC3#14842
LCSG ▷ C24×C3 ▷ C3
UCSC1 ◁ C24 ◁ C2×C4×C22⋊C4

Complements

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

NH ≅ G/Ntype
1. C2×C3×C4×C22⋊C4C2semidirect
3. C2×#5212C2semidirect
4. C2×#4674C2semidirect
6. #71549C2semidirect
7. #71537C2semidirect
8. #71534C2semidirect
9. #69427C2semidirect
10. #69423C2semidirect
11. #69412C2semidirect
43. C2×C4×C22⋊C4D3semidirect
100. C3×#95D4semidirect
291. #95#67semidirect
733. C3#14842semidirect

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×#52132
C2×#52122
C2×#46742
#715502
#715492
#715372
#715342
#694272
#694232
#694122
#148423