Finite Group Databasebeta

← #10751722

Subgroups — #10751722

Order 768.

1578 conjugacy classes of subgroups.

The subgroup lattice has 652 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#870C211yesmaximal, characteristic, complemented
2C2×C3×#388C211yesmaximal, characteristic
3C2×#4948C212yesmaximal
4#71568C212yesmaximal, complemented
5#71547C212yesmaximal
6#71546C211yesmaximal, characteristic
7#71546C211yesmaximal, characteristic
8#71535C211yesmaximal, characteristic, complemented
9#69420C211yesmaximal, characteristic, complemented
10#69415C211yesmaximal, characteristic, complemented
11#69410C212yesmaximal, complemented
12#16604—31nomaximal, Sylow
13C22×C3×C4⋊C4C2211yescharacteristic
14C22×C3×C22⋊C4C2211yescharacteristic
15C22×C3×C22⋊C4C2211yescharacteristic
16C22×C3×C22⋊C4C2211yescharacteristic
17C22×C3×C42C2211yescharacteristic
18C22×#217C2212yes—
19C22×#217C2212yes—
20C22×C4⋊Dic3C2211yescharacteristic
21C22×C4⋊Dic3C2211yescharacteristic
22C22×C4⋊Dic3C2212yes—
23C22×Dic3⋊C4C2212yes—
24C22×Dic3⋊C4C2212yes—
25C22×Dic3⋊C4C2212yes—
26C2×#870—24no—
27C2×#870—28no—
28C3×#388—24no—
29C2×C3×#95C2211yescharacteristic
30C2×C3×#95C2211yescharacteristic
31C2×C3⋊#95C2212yes—
32C2×C3⋊#95C2211yescharacteristic
33C2×C3⋊#95C2212yes—
34C2×C3⋊#95C2211yescharacteristic
35C2×C3⋊#95C2212yes—
36C2×C3⋊#95C2212yes—
37C2×C3⋊#95C2212yes—
38C2×C3⋊#95C2212yes—
39#4948—28no—
40C22×#125—31no—
41C2×#396—32no—
42C2×#388D311yescharacteristic
43#1444—31no—
44#1444—32no—
45#1442—31no—
46#1441—32no—
47#1439—31no—
48#1439—31no—
49#1439—32no—
50C4⋊#95—31no—
51C25×C3C2311yescharacteristic
52C23×C3×C4C2311yescharacteristic
53C23×C3×C4C2311yescharacteristic
54C23×C3×C4C2311yescharacteristic
55C23×C3×C4C2311yescharacteristic
56C23×C3×C4C2311yescharacteristic
57C23×C3×C4C2311yescharacteristic
58C23×Dic3C2312yes—
59C23×Dic3C2312yes—
60C23×Dic3C2312yes—
61C23×Dic3C2312yes—
62C2×C3×C4⋊C4—22no—
63C2×C3×C4⋊C4—42no—
64C2×C3×C22⋊C4—22no—
65C2×C3×C22⋊C4—22no—
66C2×C3×C22⋊C4—24no—
67C2×C3×C22⋊C4—22no—
68C2×C3×C22⋊C4—22no—
69C2×C3×C22⋊C4—42no—
70C2×C3×C22⋊C4Q814yes—
71C2×C3×C42—24no—
72C2×C3×C42D414yes—
73C2×#217—24no—
74C2×#217—24no—
75C2×#217—24no—
76C2×#217—42no—
77C2×#217—42no—
78C2×#217—42no—
79C2×C4⋊Dic3—22no—
80C2×C4⋊Dic3—22no—
81C2×C4⋊Dic3—24no—
82C2×C4⋊Dic3—24no—
83C2×C4⋊Dic3—24no—
84C2×C4⋊Dic3—24no—
85C2×C4⋊Dic3—42no—
86C2×Dic3⋊C4—24no—
87C2×Dic3⋊C4—24no—
88C2×Dic3⋊C4—24no—
89C2×Dic3⋊C4—24no—
90C2×Dic3⋊C4—24no—
91C2×Dic3⋊C4—24no—
92C2×Dic3⋊C4—42no—
93C2×Dic3⋊C4—42no—
94C2×Dic3⋊C4D418yes—
95#870—232no—
96C3×#95—22no—
97C3×#95—22no—
98C3⋊#95—22no—
99C3⋊#95—22no—
100C3⋊#95—24no—
101C3⋊#95—24no—
102C3⋊#95—42no—
103C3⋊#95—42no—
104C3⋊#95—42no—
105C3⋊#95—42no—
106C2×#125—68no—
107C2×#125—64no—
108C22×C4⋊C4—32no—
109C22×C4⋊C4—32no—
110C22×C4⋊C4—31no—
111C22×C4⋊C4—31no—
112C22×C4⋊C4—32no—
113C22×C4⋊C4—32no—
114C22×C4⋊C4C2×D311yescharacteristic
115C22×C22⋊C4—32no—
116C22×C22⋊C4—32no—
117C22×C22⋊C4C2×D311yescharacteristic
118C22×C22⋊C4C2×D311yescharacteristic
119C22×C22⋊C4C2×D311yescharacteristic
120C22×C42C2×D311yescharacteristic
121#396—68no—
122#388—24no—
123C2×#95—32no—
124C2×#95—31no—
125C2×#95—31no—
126C2×#95—32no—
127C2×#95—32no—
128C2×#95—32no—
129C2×#95—32no—
130C2×#95—32no—
131C2×#95C2×D311yescharacteristic
132C2×#95C2×D311yescharacteristic
133C24×C3—22no—
134C24×C3—21no—
135C24×C3—22no—
136C24×C3—21no—
137C24×C3—21no—
138C24×C3—21no—
139C24×C3—22no—
140C24×C3—21no—
141C24×C3—21no—
142C24×C3—22no—
143C24×C3C2×Q812yes—
144C24×C3C2411yescharacteristic
145C22×C3×C4—22no—
146C22×C3×C4—22no—
147C22×C3×C4—21no—
148C22×C3×C4—22no—
149C22×C3×C4—21no—
150C22×C3×C4—21no—
151C22×C3×C4—22no—
152C22×C3×C4—22no—
153C22×C3×C4—21no—
154C22×C3×C4—21no—
155C22×C3×C4—22no—
156C22×C3×C4—22no—
157C22×C3×C4—21no—
158C22×C3×C4—22no—
159C22×C3×C4—21no—
160C22×C3×C4—21no—
161C22×C3×C4—21no—
162C22×C3×C4—22no—
163C22×C3×C4—21no—
164C22×C3×C4—22no—
165C22×C3×C4—22no—
166C22×C3×C4—21no—
167C22×C3×C4—21no—
168C22×C3×C4C2×D414yes—
169C22×C3×C4C2×D412yes—
170C22×C3×C4C2×D412yes—
171C22×C3×C4C2×D412yes—
172C22×C3×C4C2×Q812yes—
173C22×C3×C4C2×Q812yes—
174C22×C3×C4#4112yes—
175C22×Dic3—22no—
176C22×Dic3—22no—
177C22×Dic3—22no—
178C22×Dic3—22no—
179C22×Dic3—22no—
180C22×Dic3—22no—
181C22×Dic3—22no—
182C22×Dic3—22no—
183C22×Dic3—22no—
184C22×Dic3—22no—
185C22×Dic3—22no—
186C22×Dic3—22no—
187C22×Dic3—22no—
188C22×Dic3—22no—
189C22×Dic3—22no—
190C22×Dic3—22no—
191C22×Dic3—22no—
192C22×Dic3—22no—
193C22×Dic3—22no—
194C22×Dic3—22no—
195C22×Dic3—22no—
196C22×Dic3—22no—
197C22×Dic3—22no—
198C22×Dic3—22no—
199C22×Dic3—22no—
200C22×Dic3C2×D414yes—
201C22×Dic3C2×D414yes—
202C22×Dic3#4114yes—
203C3×C4⋊C4—44no—
204C3×C22⋊C4—28no—
205C3×C22⋊C4—44no—
206C3×C22⋊C4—44no—
207C3×C42—28no—
208#217—48no—
209#217—48no—
210C4⋊Dic3—28no—
211C4⋊Dic3—28no—
212C4⋊Dic3—48no—
213Dic3⋊C4—216no—
214Dic3⋊C4—216no—
215Dic3⋊C4—48no—
216C25C22×D311yescharacteristic
217C23×C4—32no—
218C23×C4—32no—
219C23×C4—32no—
220C23×C4—32no—
221C23×C4C22×D311yescharacteristic
222C23×C4C22×D311yescharacteristic
223C23×C4C22×D311yescharacteristic
224C23×C4C22×D311yescharacteristic
225C23×C4C22×D311yescharacteristic
226C23×C4C22×D311yescharacteristic
227#125—632no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—38no—
230C2×C4⋊C4—42no—
231C2×C4⋊C4—64no—
232C2×C4⋊C4—64no—
233C2×C4⋊C4—64no—
234C2×C4⋊C4—64no—
235C2×C4⋊C4—64no—
236C2×C4⋊C4—64no—
237C2×C4⋊C4—64no—
238C2×C4⋊C4—62no—
239C2×C4⋊C4—62no—
240C2×C4⋊C4—64no—
241C2×C4⋊C4—64no—
242C2×C4⋊C4—64no—
243C2×C4⋊C4—122no—
244C2×C4⋊C4—122no—
245C2×C4⋊C4—122no—
246C2×C22⋊C4—22no—
247C2×C22⋊C4—22no—
248C2×C22⋊C4—24no—
249C2×C22⋊C4—22no—
250C2×C22⋊C4—22no—
251C2×C22⋊C4—42no—
252C2×C22⋊C4—64no—
253C2×C22⋊C4—64no—
254C2×C22⋊C4—64no—
255C2×C22⋊C4—122no—
256C2×C22⋊C4—122no—
257C2×C22⋊C4—122no—
258C2×C22⋊C4Dic614yes—
259C2×C42—24no—
260C2×C42#6714yes—
261#95—22no—
262#95—22no—
263#95—64no—
264#95—64no—
265#95—62no—
266#95—62no—
267#95—122no—
268#95—122no—
269#95—122no—
270#95—122no—
271C23×C3—22no—
272C23×C3—22no—
273C23×C3—22no—
274C23×C3—24no—
275C23×C3—24no—
276C23×C3—41no—
277C23×C3—41no—
278C23×C3—42no—
279C23×C3—41no—
280C23×C3—41no—
281C23×C3—41no—
282C23×C3—41no—
283C23×C3—42no—
284C23×C3—42no—
285C23×C3—41no—
286C23×C3—42no—
287C23×C3—41no—
288C23×C3—42no—
289C23×C3—42no—
290C23×C3—42no—
291C23×C3—42no—
292C23×C3—42no—
293C23×C3—42no—
294C23×C3C22×D412yes—
295C23×C3C22×D411yescharacteristic
296C23×C3C22×Q811yescharacteristic
297C23×C3C2×#4111yescharacteristic
298C23×C3C2×#4112yes—
299C23×C3H(5,𝔽2)12yes—
300C23×C3H(5,𝔽2)11yescharacteristic
301C23×C3H(5,𝔽2)11yescharacteristic
302C23×C32−1+412yes—
303C23×C32−1+411yescharacteristic
304C23×C32−1+411yescharacteristic
305C2×C3×C4—22no—
306C2×C3×C4—22no—
307C2×C3×C4—22no—
308C2×C3×C4—22no—
309C2×C3×C4—24no—
310C2×C3×C4—24no—
311C2×C3×C4—24no—
312C2×C3×C4—24no—
313C2×C3×C4—22no—
314C2×C3×C4—22no—
315C2×C3×C4—22no—
316C2×C3×C4—22no—
317C2×C3×C4—24no—
318C2×C3×C4—41no—
319C2×C3×C4—41no—
320C2×C3×C4—42no—
321C2×C3×C4—42no—
322C2×C3×C4—42no—
323C2×C3×C4—41no—
324C2×C3×C4—42no—
325C2×C3×C4—41no—
326C2×C3×C4—41no—
327C2×C3×C4—41no—
328C2×C3×C4—41no—
329C2×C3×C4—41no—
330C2×C3×C4—42no—
331C2×C3×C4—42no—
332C2×C3×C4—42no—
333C2×C3×C4#12014yes—
334C2×C3×C4C22⋊Q814yes—
335C2×Dic3—24no—
336C2×Dic3—24no—
337C2×Dic3—24no—
338C2×Dic3—24no—
339C2×Dic3—24no—
340C2×Dic3—24no—
341C2×Dic3—24no—
342C2×Dic3—24no—
343C2×Dic3—24no—
344C2×Dic3—42no—
345C2×Dic3—42no—
346C2×Dic3—42no—
347C2×Dic3—42no—
348C2×Dic3—42no—
349C2×Dic3—42no—
350C2×Dic3—42no—
351C2×Dic3—42no—
352C2×Dic3—42no—
353C2×Dic3—42no—
354C2×Dic3—42no—
355C2×Dic3—42no—
356C2×Dic3—42no—
357C2×Dic3—42no—
358C2×Dic3—42no—
359C2×Dic3—42no—
360C2×Dic3—42no—
361C2×Dic3—42no—
362C2×Dic3—42no—
363C24—21no—
364C24—22no—
365C24—22no—
366C24—22no—
367C24—21no—
368C24—21no—
369C24—21no—
370C24—21no—
371C24—22no—
372C24—21no—
373C24C2×Dic612yes—
374C24C23×D311yescharacteristic
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—21no—
378C22×C4—21no—
379C22×C4—22no—
380C22×C4—21no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—21no—
386C22×C4—21no—
387C22×C4—21no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—21no—
392C22×C4—21no—
393C22×C4—21no—
394C22×C4—22no—
395C22×C4—21no—
396C22×C4—21no—
397C22×C4—21no—
398C22×C4—34no—
399C22×C4—34no—
400C22×C4—34no—
401C22×C4—62no—
402C22×C4—62no—
403C22×C4—62no—
404C22×C4—62no—
405C22×C4—62no—
406C22×C4—62no—
407C22×C4—62no—
408C22×C4—62no—
409C22×C4—62no—
410C22×C4—62no—
411C22×C4—62no—
412C22×C4—62no—
413C22×C4—62no—
414C22×C4—62no—
415C22×C4—62no—
416C22×C4—62no—
417C22×C4—62no—
418C22×C4—62no—
419C22×C4—62no—
420C22×C4—62no—
421C22×C4—62no—
422C22×C4—62no—
423C22×C4—62no—
424C22×C4—62no—
425C22×C4—62no—
426C22×C4C2×Dic612yes—
427C22×C4C2×Dic612yes—
428C22×C4#23512yes—
429C22×C4D3×D414yes—
430C22×C4C2×#6712yes—
431C22×C4C2×#6712yes—
432C22×C4C2×#6712yes—
433C4⋊C4—44no—
434C4⋊C4—616no—
435C4⋊C4—68no—
436C4⋊C4—68no—
437C4⋊C4—616no—
438C4⋊C4—128no—
439C4⋊C4—128no—
440C22⋊C4—28no—
441C22⋊C4—44no—
442C22⋊C4—44no—
443C22⋊C4—128no—
444C22⋊C4—128no—
445C42—28no—
446C22×C3—42no—
447C22×C3—42no—
448C22×C3—42no—
449C22×C3—44no—
450C22×C3—44no—
451C22×C3—81no—
452C22×C3—81no—
453C22×C3—81no—
454C22×C3—81no—
455C22×C3—82no—
456C22×C3—82no—
457C22×C3C2×#12011yescharacteristic
458C22×C3C2×C22⋊Q811yescharacteristic
459C22×C3#53211yescharacteristic
460C22×C3#53511yescharacteristic
461C22×C3#53512yes—
462C22×C3#53611yescharacteristic
463C22×C3#53711yescharacteristic
464C22×C3#53711yescharacteristic
465C22×C3#53712yes—
466C22×C3#53912yes—
467C22×C3#54011yescharacteristic
468C22×C3#54112yes—
469C22×C3#54111yescharacteristic
470C22×C3#54411yescharacteristic
471C22×C3#54411yescharacteristic
472C22×C3#54612yes—
473C22×C3#54712yes—
474C22×C3#54812yes—
475C22×C3D4×Q812yes—
476C22×C3#55312yes—
477C22×C3#55412yes—
478C22×C3#56212yes—
479C22×C3#56312yes—
480C3×C4—24no—
481C3×C4—24no—
482C3×C4—42no—
483C3×C4—42no—
484C3×C4—81no—
485C3×C4—81no—
486Dic3—44no—
487Dic3—44no—
488Dic3—44no—
489Dic3—82no—
490C23—22no—
491C23—22no—
492C23—22no—
493C23—24no—
494C23—24no—
495C23—42no—
496C23—42no—
497C23—42no—
498C23—42no—
499C23—42no—
500C23—42no—
501C23—41no—
502C23—41no—
503C23—41no—
504C23—41no—
505C23—42no—
506C23—42no—
507C23—41no—
508C23—41no—
509C23—41no—
510C23—41no—
511C23—42no—
512C23—42no—
513C23C22×Dic611yescharacteristic
514C23C2×#23511yescharacteristic
515C23C2×D3×D412yes—
516C23#100412yes—
517C23D3×#4112yes—
518C23#100911yescharacteristic
519C23#100911yescharacteristic
520C23#101011yescharacteristic
521C23#101012yes—
522C23#101011yescharacteristic
523C23C22×#6711yescharacteristic
524C2×C4—24no—
525C2×C4—24no—
526C2×C4—24no—
527C2×C4—22no—
528C2×C4—24no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—24no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—41no—
538C2×C4—42no—
539C2×C4—41no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—64no—
553C2×C4—64no—
554C2×C4—64no—
555C2×C4—64no—
556C2×C4—64no—
557C2×C4—64no—
558C2×C4—64no—
559C2×C4—64no—
560C2×C4—64no—
561C2×C4—122no—
562C2×C4—122no—
563C2×C4—122no—
564C2×C4—122no—
565C2×C4—122no—
566C2×C4—122no—
567C2×C4—122no—
568C2×C4—122no—
569C2×C4—122no—
570C2×C4—122no—
571C2×C4—122no—
572C2×C4—122no—
573C2×C4—122no—
574C2×C4—122no—
575C2×C4—122no—
576C2×C4—122no—
577C2×C4—122no—
578C2×C4—122no—
579C2×C4—122no—
580C2×C4#92414yes—
581C2×C4#93714yes—
582C2×C3—82no—
583C2×C3#240911yescharacteristic
584C2×C3#243011yescharacteristic
585C2×C3#256611yescharacteristic
586C2×C3#261511yescharacteristic
587C2×C3#262911yescharacteristic
588C2×C3#266411yescharacteristic
589C2×C3#270412yes—
590C2×C3#271112yes—
591C2×C3#272012yes—
592C2×C3#272512yes—
593C2×C3#282011yescharacteristic
594C22—42no—
595C22—44no—
596C22—42no—
597C22—44no—
598C22—42no—
599C22—81no—
600C22—81no—
601C22—81no—
602C22—82no—
603C22—81no—
604C22—82no—
605C22#551811yescharacteristic
606C22#551911yescharacteristic
607C22#551911yescharacteristic
608C22#553511yescharacteristic
609C22D4×Dic612yes—
610C22#553912yes—
611C22#555212yes—
612C22#555812yes—
613C22#560712yes—
614C22#560812yes—
615C22#565412yes—
616C22#565512yes—
617C22#567212yes—
618C22#568412yes—
619C22#570112yes—
620C22#570712yes—
621C22C2×#92411yescharacteristic
622C22C2×#93711yescharacteristic
623C22#580511yescharacteristic
624C22#582511yescharacteristic
625C22#583211yescharacteristic
626C22#583311yescharacteristic
627C22#583311yescharacteristic
628C4—24no—
629C4—24no—
630C4—42no—
631C4—42no—
632C4—81no—
633C4—81no—
634C4—124no—
635C4—124no—
636C4—124no—
637C4—242no—
638C3#1660411yescharacteristic, complemented, Sylow
639C2—82no—
640C2#7568412yes—
641C2#7569712yes—
642C2#7570212yes—
643C2#7592912yes—
644C2#8226511yescharacteristic
645C2#8234911yescharacteristic
646C2#8238611yescharacteristic
647C2#8270911yescharacteristic
648C2#8284811yescharacteristic
649C2#8299211yescharacteristic
650C2#8309011yescharacteristic
subgroupG/H
center, FrattiniC24C23×D3
commutator, socle, cosocleC24×C3C24
FittingC2×C3×#388C2
prime radicalC25×C3C23
hypercenterC2×#388D3
hypocenterC3#16604
LCSG ▷ C24×C3 ▷ C3
UCSC1 ◁ C24 ◁ C2×#388

Complements

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

NH ≅ G/Ntype
1. C22×#870C2semidirect
4. #71568C2semidirect
8. #71535C2semidirect
9. #69420C2semidirect
10. #69415C2semidirect
11. #69410C2semidirect
638. C3#16604semidirect

Maximal subgroups

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

subgroupindex
C22×#8702
C2×C3×#3882
C2×#49482
#715682
#715472
#715462
#715352
#694202
#694152
#694102
#166043