Finite Group Databasebeta

← #10751697

Subgroups — #10751697

Order 768.

1750 conjugacy classes of subgroups.

The subgroup lattice has 688 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×C22⋊Q8C211yesmaximal, characteristic
2C2×#5230C211yesmaximal, characteristic, complemented
3C2×#5210C211yesmaximal, characteristic
4C2×#4985C211yesmaximal, characteristic, complemented
5#72134C211yesmaximal, characteristic, complemented
6#72127C211yesmaximal, characteristic
7#71569C211yesmaximal, characteristic, complemented
8#71568C212yesmaximal, complemented
9#71567C212yesmaximal, complemented
10#71549C212yesmaximal
11#71547C212yesmaximal
12#16601—31nomaximal, Sylow
13C23×C3×Q8C2211yescharacteristic
14C24×C3×C4C2211yescharacteristic
15C2×C3×C22⋊Q8—24no—
16C2×C3×C22⋊Q8—24no—
17C2×C3×C22⋊Q8—24no—
18C22×C3×C4⋊C4C2212yes—
19C22×C3×C4⋊C4C2211yescharacteristic
20C22×C3×C22⋊C4C2212yes—
21C22×#217C2212yes—
22C22×#217C2211yescharacteristic
23C22×#217C2211yescharacteristic
24C22×Dic3⋊C4C2211yescharacteristic
25C22×Dic3⋊C4C2211yescharacteristic
26C22×Dic3⋊C4C2212yes—
27#5230—24no—
28#5210—24no—
29C2×C3⋊#95C2211yescharacteristic
30C2×C3⋊#95C2211yescharacteristic
31C2×C3⋊#95C2212yes—
32C2×C3⋊#95C2211yescharacteristic
33C2×C3⋊#95C2212yes—
34C2×C3⋊#95C2212yes—
35C2×C3⋊#95C2212yes—
36C2×C3⋊#95C2212yes—
37C2×C3⋊#95C2211yescharacteristic
38C2×C3⋊#95C2212yes—
39C2×C3⋊#95C2212yes—
40C2×C3⋊#95C2212yes—
41#4985—24no—
42C22×C22⋊Q8D311yescharacteristic
43C2×#400—31no—
44C2×#395—31no—
45C2×#385—31no—
46#1446—31no—
47#1445—31no—
48#1444—32no—
49#1442—32no—
50#1440—32no—
51#1439—32no—
52C22⋊#95—31no—
53C25×C3C2311yescharacteristic
54C22×C3×Q8—22no—
55C22×C3×Q8—22no—
56C22×C3×Q8—42no—
57C22×C3×Q8—42no—
58C22×C3×Q8D414yes—
59C23×C3×C4—22no—
60C23×C3×C4—22no—
61C23×C3×C4—42no—
62C23×C3×C4—42no—
63C23×C3×C4D414yes—
64C23×C3×C4C2311yescharacteristic
65C23×C3×C4C2312yes—
66C23×C3×C4C2312yes—
67C23×C3×C4C2311yescharacteristic
68C23×Dic3C2312yes—
69C23×Dic3C2311yescharacteristic
70C23×Dic3C2311yescharacteristic
71C23×Dic3C2311yescharacteristic
72C23×Dic3C2311yescharacteristic
73C23×Dic3C2312yes—
74C3×C22⋊Q8—232no—
75C2×C3×C4⋊C4—24no—
76C2×C3×C4⋊C4—24no—
77C2×C3×C4⋊C4—24no—
78C2×C3×C4⋊C4—22no—
79C2×C3×C4⋊C4—22no—
80C2×C3×C4⋊C4D414yes—
81C2×C3×C22⋊C4—24no—
82C2×C3×C22⋊C4—24no—
83C2×C3×C22⋊C4—24no—
84C2×#217—22no—
85C2×#217—22no—
86C2×#217—22no—
87C2×#217—24no—
88C2×#217—24no—
89C2×#217—42no—
90C2×#217—42no—
91C2×#217Q814yescomplemented
92C2×Dic3⋊C4—22no—
93C2×Dic3⋊C4—22no—
94C2×Dic3⋊C4—24no—
95C2×Dic3⋊C4—22no—
96C2×Dic3⋊C4—22no—
97C2×Dic3⋊C4—42no—
98C2×Dic3⋊C4—42no—
99C2×Dic3⋊C4—42no—
100C3⋊#95—22no—
101C3⋊#95—22no—
102C3⋊#95—24no—
103C3⋊#95—22no—
104C3⋊#95—24no—
105C3⋊#95—22no—
106C3⋊#95—24no—
107C3⋊#95—24no—
108C3⋊#95—24no—
109C3⋊#95—42no—
110C3⋊#95—42no—
111C3⋊#95—42no—
112C23×Q8C2×D311yescharacteristic
113C24×C4C2×D311yescharacteristic
114C2×C22⋊Q8—24no—
115C2×C22⋊Q8—24no—
116C2×C22⋊Q8—24no—
117C22×C4⋊C4—31no—
118C22×C4⋊C4—32no—
119C22×C4⋊C4—31no—
120C22×C4⋊C4C2×D312yes—
121C22×C4⋊C4C2×D311yescharacteristic
122C22×C22⋊C4—31no—
123C22×C22⋊C4—32no—
124C22×C22⋊C4—31no—
125C22×C22⋊C4C2×D312yes—
126#400—64no—
127#395—64no—
128#385—64no—
129C2×#95—31no—
130C2×#95—32no—
131C2×#95—31no—
132C2×#95—32no—
133C2×#95—32no—
134C2×#95—32no—
135C2×#95—32no—
136C2×#95—31no—
137C2×#95—31no—
138C2×#95—32no—
139C2×#95—32no—
140C2×#95—32no—
141C24×C3—21no—
142C24×C3—21no—
143C24×C3—22no—
144C24×C3—22no—
145C24×C3—22no—
146C24×C3—21no—
147C24×C3—21no—
148C24×C3—22no—
149C24×C3C2×D412yes—
150C24×C3C2×Q812yes—
151C24×C3#4112yes—
152C24×C3C2411yescharacteristic
153C2×C3×Q8—28no—
154C2×C3×Q8—48no—
155C2×C3×Q8—44no—
156C2×C3×Q8—44no—
157C2×C3×Q8—48no—
158C22×C3×C4—22no—
159C22×C3×C4—22no—
160C22×C3×C4—22no—
161C22×C3×C4—22no—
162C22×C3×C4—22no—
163C22×C3×C4—22no—
164C22×C3×C4—21no—
165C22×C3×C4—21no—
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—28no—
177C22×C3×C4—44no—
178C22×C3×C4—44no—
179C22×C3×C4—48no—
180C22×C3×C4—48no—
181C22×C3×C4C2×D414yes—
182C22×C3×C4C2×D414yes—
183C22×C3×C4C2×D412yes—
184C22×C3×C4C2×D412yes—
185C22×C3×C4C2×D412yes—
186C22×C3×C4C2×D412yes—
187C22×Dic3—22no—
188C22×Dic3—21no—
189C22×Dic3—21no—
190C22×Dic3—21no—
191C22×Dic3—21no—
192C22×Dic3—22no—
193C22×Dic3—21no—
194C22×Dic3—21no—
195C22×Dic3—22no—
196C22×Dic3—22no—
197C22×Dic3—22no—
198C22×Dic3—22no—
199C22×Dic3—22no—
200C22×Dic3—21no—
201C22×Dic3—21no—
202C22×Dic3—22no—
203C22×Dic3—22no—
204C22×Dic3—21no—
205C22×Dic3—21no—
206C22×Dic3—21no—
207C22×Dic3—21no—
208C22×Dic3—22no—
209C22×Dic3—22no—
210C22×Dic3—22no—
211C22×Dic3—22no—
212C22×Dic3—22no—
213C22×Dic3—22no—
214C22×Dic3—22no—
215C22×Dic3—22no—
216C22×Dic3—21no—
217C22×Dic3—21no—
218C22×Dic3—22no—
219C22×Dic3—22no—
220C22×Dic3C2×Q812yes—
221C22×Dic3C2×Q812yes—
222C22×Dic3#4114yes—
223C3×C4⋊C4—216no—
224C3×C4⋊C4—28no—
225C3×C22⋊C4—216no—
226#217—28no—
227#217—48no—
228#217—44no—
229Dic3⋊C4—44no—
230Dic3⋊C4—48no—
231Dic3⋊C4—44no—
232C25C22×D311yescharacteristic
233C22×Q8—22no—
234C22×Q8—22no—
235C22×Q8—42no—
236C22×Q8—42no—
237C22×Q8#6714yes—
238C23×C4—22no—
239C23×C4—22no—
240C23×C4—31no—
241C23×C4—31no—
242C23×C4—32no—
243C23×C4—31no—
244C23×C4—32no—
245C23×C4—31no—
246C23×C4—42no—
247C23×C4—42no—
248C23×C4#6714yes—
249C23×C4C22×D312yes—
250C23×C4C22×D312yes—
251C23×C4C22×D311yescharacteristic
252C23×C4C22×D311yescharacteristic
253C22⋊Q8—232no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—62no—
260C2×C4⋊C4—62no—
261C2×C4⋊C4—64no—
262C2×C4⋊C4—62no—
263C2×C4⋊C4—62no—
264C2×C4⋊C4—122no—
265C2×C4⋊C4—122no—
266C2×C4⋊C4—122no—
267C2×C4⋊C4#6714yes—
268C2×C22⋊C4—24no—
269C2×C22⋊C4—24no—
270C2×C22⋊C4—24no—
271C2×C22⋊C4—34no—
272C2×C22⋊C4—62no—
273C2×C22⋊C4—62no—
274C2×C22⋊C4—64no—
275C2×C22⋊C4—64no—
276C2×C22⋊C4—62no—
277C2×C22⋊C4—122no—
278C2×C22⋊C4—122no—
279#95—64no—
280#95—64no—
281#95—62no—
282#95—62no—
283#95—64no—
284#95—62no—
285#95—64no—
286#95—62no—
287#95—64no—
288#95—122no—
289#95—122no—
290#95—122no—
291C23×C3—22no—
292C23×C3—22no—
293C23×C3—22no—
294C23×C3—22no—
295C23×C3—22no—
296C23×C3—22no—
297C23×C3—24no—
298C23×C3—24no—
299C23×C3—22no—
300C23×C3—22no—
301C23×C3—22no—
302C23×C3—22no—
303C23×C3—24no—
304C23×C3—24no—
305C23×C3—42no—
306C23×C3—42no—
307C23×C3—42no—
308C23×C3—42no—
309C23×C3—42no—
310C23×C3—42no—
311C23×C3—42no—
312C23×C3—42no—
313C23×C3C22⋊Q814yes—
314C23×C3C22×D411yescharacteristic
315C23×C3C22×D411yescharacteristic
316C23×C3C22×D411yescharacteristic
317C23×C3C22×Q811yescharacteristic
318C23×C3C2×#4111yescharacteristic
319C23×C3C2×#4112yes—
320C23×C3H(5,𝔽2)12yes—
321C23×C3H(5,𝔽2)12yes—
322C23×C32−1+411yescharacteristic
323C23×C32−1+411yescharacteristic
324C23×C32−1+412yes—
325C3×Q8—416no—
326C2×C3×C4—22no—
327C2×C3×C4—22no—
328C2×C3×C4—24no—
329C2×C3×C4—24no—
330C2×C3×C4—24no—
331C2×C3×C4—24no—
332C2×C3×C4—22no—
333C2×C3×C4—22no—
334C2×C3×C4—24no—
335C2×C3×C4—22no—
336C2×C3×C4—24no—
337C2×C3×C4—24no—
338C2×C3×C4—24no—
339C2×C3×C4—416no—
340C2×C3×C4—42no—
341C2×C3×C4—42no—
342C2×C3×C4—42no—
343C2×C3×C4—42no—
344C2×C3×C4—42no—
345C2×C3×C4—42no—
346C2×C3×C4—42no—
347C2×C3×C4—42no—
348C2×C3×C4—42no—
349C2×C3×C4—42no—
350C2×C3×C4#12014yes—
351C2×Dic3—22no—
352C2×Dic3—22no—
353C2×Dic3—24no—
354C2×Dic3—24no—
355C2×Dic3—22no—
356C2×Dic3—22no—
357C2×Dic3—24no—
358C2×Dic3—24no—
359C2×Dic3—41no—
360C2×Dic3—41no—
361C2×Dic3—41no—
362C2×Dic3—42no—
363C2×Dic3—41no—
364C2×Dic3—42no—
365C2×Dic3—41no—
366C2×Dic3—41no—
367C2×Dic3—42no—
368C2×Dic3—42no—
369C2×Dic3—42no—
370C2×Dic3—41no—
371C2×Dic3—41no—
372C2×Dic3—41no—
373C2×Dic3—42no—
374C2×Dic3—41no—
375C2×Dic3—41no—
376C2×Dic3—42no—
377C2×Dic3—42no—
378C2×Dic3—41no—
379C2×Dic3—42no—
380C2×Dic3—42no—
381C2×Dic3—42no—
382C2×Dic3—42no—
383C2×Dic3—42no—
384C2×Dic3—42no—
385C2×Dic3—42no—
386C2×Dic3—42no—
387C24—22no—
388C24—22no—
389C24—21no—
390C24—21no—
391C24—22no—
392C24—22no—
393C24—21no—
394C24—21no—
395C24D3×Q812yes—
396C24#23912yes—
397C24C2×#6712yes—
398C24C23×D311yescharacteristic
399C2×Q8—28no—
400C2×Q8—48no—
401C2×Q8—44no—
402C2×Q8—44no—
403C2×Q8—48no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—28no—
423C22×C4—32no—
424C22×C4—32no—
425C22×C4—34no—
426C22×C4—44no—
427C22×C4—44no—
428C22×C4—48no—
429C22×C4—48no—
430C22×C4—62no—
431C22×C4—62no—
432C22×C4—62no—
433C22×C4—62no—
434C22×C4—61no—
435C22×C4—61no—
436C22×C4—62no—
437C22×C4—61no—
438C22×C4—61no—
439C22×C4—62no—
440C22×C4—62no—
441C22×C4—61no—
442C22×C4—61no—
443C22×C4—62no—
444C22×C4—61no—
445C22×C4—61no—
446C22×C4—62no—
447C22×C4—62no—
448C22×C4—62no—
449C22×C4—62no—
450C22×C4—62no—
451C22×C4—62no—
452C22×C4—61no—
453C22×C4—61no—
454C22×C4—62no—
455C22×C4—62no—
456C22×C4—62no—
457C22×C4—62no—
458C22×C4—62no—
459C22×C4—61no—
460C22×C4—61no—
461C22×C4—61no—
462C22×C4—61no—
463C22×C4C2×#6714yes—
464C22×C4C2×#6714yes—
465C22×C4C2×#6712yes—
466C22×C4C2×#6712yes—
467C22×C4C2×#6712yes—
468C22×C4C2×#6712yes—
469C4⋊C4—216no—
470C4⋊C4—28no—
471C4⋊C4—124no—
472C4⋊C4—128no—
473C4⋊C4—124no—
474C22⋊C4—216no—
475C22⋊C4—68no—
476C22⋊C4—128no—
477C22⋊C4—124no—
478C22×C3—24no—
479C22×C3—24no—
480C22×C3—24no—
481C22×C3—42no—
482C22×C3—42no—
483C22×C3—42no—
484C22×C3—42no—
485C22×C3—44no—
486C22×C3—44no—
487C22×C3—44no—
488C22×C3—44no—
489C22×C3C2×#12011yescharacteristic
490C22×C3C2×C22⋊Q811yescharacteristic
491C22×C3#53211yescharacteristic
492C22×C3#53411yescharacteristic
493C22×C3#53411yescharacteristic
494C22×C3#53412yes—
495C22×C3#53512yes—
496C22×C3#53611yescharacteristic
497C22×C3#53611yescharacteristic
498C22×C3#53912yes—
499C22×C3#53912yes—
500C22×C3#54212yes—
501C22×C3#54412yes—
502C22×C3#54712yes—
503C22×C3#54811yescharacteristic
504C22×C3#54811yescharacteristic
505C22×C3#54812yes—
506C22×C3D4×Q811yescharacteristic
507C22×C3D4×Q811yescharacteristic
508C22×C3#55112yes—
509C22×C3#55412yes—
510C22×C3#56312yes—
511C22×C3#56312yes—
512C3×C4—24no—
513C3×C4—42no—
514C3×C4—44no—
515C3×C4—44no—
516Dic3—42no—
517Dic3—42no—
518Dic3—44no—
519Dic3—81no—
520Dic3—81no—
521Dic3—82no—
522C23—24no—
523C23—24no—
524C23—24no—
525C23—22no—
526C23—22no—
527C23—22no—
528C23—22no—
529C23—22no—
530C23—22no—
531C23—22no—
532C23—22no—
533C23—22no—
534C23—22no—
535C23—24no—
536C23—42no—
537C23—42no—
538C23—42no—
539C23—42no—
540C23—42no—
541C23—42no—
542C23—42no—
543C23—42no—
544C23#94614yes—
545C23#100412yes—
546C23#100412yes—
547C23C2×D3×Q811yescharacteristic
548C23C2×#23911yescharacteristic
549C23#100711yescharacteristic
550C23#100711yescharacteristic
551C23D3×#4112yes—
552C23#101012yes—
553C23C22×#6711yescharacteristic
554C23C22×#6711yescharacteristic
555C23C22×#6711yescharacteristic
556Q8—416no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—22no—
562C2×C4—24no—
563C2×C4—22no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—22no—
570C2×C4—416no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—64no—
582C2×C4—64no—
583C2×C4—64no—
584C2×C4—62no—
585C2×C4—62no—
586C2×C4—62no—
587C2×C4—62no—
588C2×C4—64no—
589C2×C4—122no—
590C2×C4—122no—
591C2×C4—122no—
592C2×C4—121no—
593C2×C4—122no—
594C2×C4—121no—
595C2×C4—121no—
596C2×C4—121no—
597C2×C4—122no—
598C2×C4—121no—
599C2×C4—121no—
600C2×C4—122no—
601C2×C4—122no—
602C2×C4—122no—
603C2×C4—122no—
604C2×C4—122no—
605C2×C4—122no—
606C2×C4—121no—
607C2×C4—121no—
608C2×C4—121no—
609C2×C4—122no—
610C2×C4—122no—
611C2×C4—122no—
612C2×C4—121no—
613C2×C4—122no—
614C2×C4—122no—
615C2×C4—121no—
616C2×C4—121no—
617C2×C4#95314yes—
618C2×C3—44no—
619C2×C3#240911yescharacteristic
620C2×C3#243211yescharacteristic
621C2×C3#256411yescharacteristic
622C2×C3#261311yescharacteristic
623C2×C3#262711yescharacteristic
624C2×C3#266311yescharacteristic
625C2×C3#272012yes—
626C2×C3#272312yes—
627C2×C3#272812yes—
628C2×C3#272912yes—
629C2×C3#281711yescharacteristic
630C22—24no—
631C22—24no—
632C22—24no—
633C22—42no—
634C22—44no—
635C22—44no—
636C22—44no—
637C22—42no—
638C22—42no—
639C22—42no—
640C22—44no—
641C22#562312yes—
642C22#563612yes—
643C22#563712yes—
644C22#563812yes—
645C22#564012yes—
646C22#564512yes—
647C22#564612yes—
648C22#564612yes—
649C22#580411yescharacteristic
650C22#580411yescharacteristic
651C22#580412yes—
652C22#580512yes—
653C22C2×#94611yescharacteristic
654C22Q8×#6711yescharacteristic
655C22Q8×#6711yescharacteristic
656C22#581611yescharacteristic
657C22#581711yescharacteristic
658C22#581711yescharacteristic
659C22#582412yes—
660C22#582511yescharacteristic
661C22#583112yes—
662C22#583211yescharacteristic
663C22C2×#95311yescharacteristic
664C4—24no—
665C4—42no—
666C4—44no—
667C4—44no—
668C4—122no—
669C4—122no—
670C4—124no—
671C4—241no—
672C4—242no—
673C4—241no—
674C3#1660111yescharacteristic, complemented, Sylow
675C2—44no—
676C2#8278411yescharacteristic
677C2#8278811yescharacteristic
678C2#8280312yes—
679C2#8280412yes—
680C2#8280512yes—
681C2#8280811yescharacteristic
682C2#8282511yescharacteristic
683C2#8283112yes—
684C2#8405411yescharacteristic
685C2#8406811yescharacteristic
686C2#8409311yescharacteristic
subgroupG/H
center, FrattiniC24C23×D3
commutator, socle, cosocleC24×C3C24
FittingC22×C3×C22⋊Q8C2
prime radicalC25×C3C23
hypercenterC22×C22⋊Q8D3
hypocenterC3#16601
LCSG ▷ C24×C3 ▷ C3
UCSC1 ◁ C24 ◁ C22×C22⋊Q8

Complements

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

NH ≅ G/Ntype
2. C2×#5230C2semidirect
4. C2×#4985C2semidirect
5. #72134C2semidirect
7. #71569C2semidirect
8. #71568C2semidirect
9. #71567C2semidirect
91. C2×#217Q8semidirect
674. C3#16601semidirect

Maximal subgroups

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

subgroupindex
C22×C3×C22⋊Q82
C2×#52302
C2×#52102
C2×#49852
#721342
#721272
#715692
#715682
#715672
#715492
#715472
#166013