Finite Group Databasebeta

← #10751726

Subgroups — #10751726

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×C3×#125C211yesmaximal, characteristic, complemented
2C2×#4994C212yesmaximal
3C2×#4940C211yesmaximal, characteristic
4#71570C211yesmaximal, characteristic
5#71570C211yesmaximal, characteristic
6#71570C212yesmaximal
7#71568C212yesmaximal, complemented
8#71568C211yesmaximal, characteristic, complemented
9#71567C211yesmaximal, characteristic, complemented
10#71565C212yesmaximal, complemented
11#71535C211yesmaximal, characteristic, complemented
12#16604—31nomaximal, Sylow
13C2×C3×#125—28no—
14C2×C3×#125—24no—
15C22×C3×C4⋊C4C2211yescharacteristic
16C22×C3×C4⋊C4C2212yes—
17C22×C3×C4⋊C4C2211yescharacteristic
18C22×C3×C4⋊C4C2212yes—
19C22×C3×C42C2211yescharacteristic
20C22×(C2×D3)⋊C4C2211yescharacteristic
21C22×(C2×D3)⋊C4C2212yes—
22C22×(C2×D3)⋊C4C2211yescharacteristic
23C22×(C2×D3)⋊C4C2212yes—
24C22×(C2×D3)⋊C4C2211yescharacteristic
25C22×C4⋊Dic3C2212yes—
26C22×Dic3⋊C4C2212yes—
27C22×Dic3⋊C4C2211yescharacteristic
28C2×C3⋊#95C2212yes—
29C2×C3⋊#95C2211yescharacteristic
30C2×C3⋊#95C2211yescharacteristic
31C2×C3⋊#95C2211yescharacteristic
32C2×C3⋊#95C2212yes—
33C2×C3⋊#95C2211yescharacteristic
34C2×C3⋊#95C2212yes—
35C2×C3⋊#95C2212yes—
36C2×C3⋊#95C2212yes—
37C2×C3⋊#95C2212yes—
38#4994—28no—
39#4940—24no—
40C22×#125D311yescharacteristic, complemented
41C2×#396—32no—
42C2×#388—31no—
43#1444—31no—
44#1444—32no—
45#1442—31no—
46#1441—32no—
47#1439—31no—
48#1439—31no—
49#1439—32no—
50C4⋊#95—31no—
51C24×D3C2311yescharacteristic
52C23×C3×C4C2312yes—
53C23×C3×C4C2311yescharacteristic
54C23×C3×C4C2312yes—
55C23×C3×C4C2311yescharacteristic
56C23×C3×C4C2311yescharacteristic
57C23×Dic3C2311yescharacteristic
58C23×Dic3C2312yes—
59C23×Dic3C2311yescharacteristic
60C23×Dic3C2311yescharacteristic
61C23×Dic3C2312yes—
62C3×#125—232no—
63C2×C3×C4⋊C4—24no—
64C2×C3×C4⋊C4—24no—
65C2×C3×C4⋊C4—24no—
66C2×C3×C4⋊C4—24no—
67C2×C3×C4⋊C4—24no—
68C2×C3×C4⋊C4—22no—
69C2×C3×C4⋊C4—22no—
70C2×C3×C4⋊C4—24no—
71C2×C3×C4⋊C4—24no—
72C2×C3×C4⋊C4D418yes—
73C2×C3×C42—24no—
74C2×C3×C42D414yes—
75C2×(C2×D3)⋊C4—22no—
76C2×(C2×D3)⋊C4—22no—
77C2×(C2×D3)⋊C4—22no—
78C2×(C2×D3)⋊C4—22no—
79C2×(C2×D3)⋊C4—24no—
80C2×(C2×D3)⋊C4—24no—
81C2×(C2×D3)⋊C4—24no—
82C2×(C2×D3)⋊C4—24no—
83C2×(C2×D3)⋊C4—42no—
84C2×(C2×D3)⋊C4—42no—
85C2×(C2×D3)⋊C4—42no—
86C2×(C2×D3)⋊C4—42no—
87C2×(C2×D3)⋊C4Q814yes—
88C2×C4⋊Dic3—24no—
89C2×C4⋊Dic3—24no—
90C2×C4⋊Dic3—42no—
91C2×Dic3⋊C4—22no—
92C2×Dic3⋊C4—24no—
93C2×Dic3⋊C4—42no—
94C2×Dic3⋊C4—42no—
95C2×Dic3⋊C4—42no—
96C3⋊#95—22no—
97C3⋊#95—22no—
98C3⋊#95—24no—
99C3⋊#95—22no—
100C3⋊#95—22no—
101C3⋊#95—24no—
102C3⋊#95—42no—
103C3⋊#95—42no—
104C3⋊#95—42no—
105C3⋊#95—42no—
106C2×#125—28no—
107C2×#125—24no—
108C22×C4⋊C4—32no—
109C22×C4⋊C4—32no—
110C22×C4⋊C4—31no—
111C22×C4⋊C4C2×D311yescharacteristic
112C22×C4⋊C4C2×D311yescharacteristic
113C22×C4⋊C4C2×D312yes—
114C22×C4⋊C4C2×D312yes—
115C22×C22⋊C4—31no—
116C22×C22⋊C4—31no—
117C22×C22⋊C4—32no—
118C22×C22⋊C4—32no—
119C22×C22⋊C4—31no—
120C22×C42C2×D311yescharacteristic
121#396—68no—
122#388—64no—
123C2×#95—32no—
124C2×#95—31no—
125C2×#95—31no—
126C2×#95—32no—
127C2×#95—31no—
128C2×#95—31no—
129C2×#95—32no—
130C2×#95—32no—
131C2×#95—32no—
132C2×#95—32no—
133C24×C3C2411yescharacteristic
134C23×D3—21no—
135C23×D3—22no—
136C23×D3—21no—
137C23×D3—22no—
138C23×D3—21no—
139C23×D3—21no—
140C23×D3—22no—
141C23×D3—22no—
142C23×D3—21no—
143C23×D3—21no—
144C23×D3C2×Q812yes—
145C22×C3×C4—22no—
146C22×C3×C4—22no—
147C22×C3×C4—21no—
148C22×C3×C4—22no—
149C22×C3×C4—21no—
150C22×C3×C4—22no—
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—21no—
158C22×C3×C4—22no—
159C22×C3×C4—22no—
160C22×C3×C4—22no—
161C22×C3×C4—22no—
162C22×C3×C4—22no—
163C22×C3×C4—21no—
164C22×C3×C4—22no—
165C22×C3×C4—22no—
166C22×C3×C4C2×D414yes—
167C22×C3×C4C2×D412yes—
168C22×C3×C4C2×D414yes—
169C22×C3×C4C2×D412yes—
170C22×C3×C4C2×D414yes—
171C22×C3×C4C2×D412yes—
172C22×C3×C4C2×Q812yes—
173C22×C3×C4#4112yes—
174C22×Dic3—21no—
175C22×Dic3—22no—
176C22×Dic3—22no—
177C22×Dic3—22no—
178C22×Dic3—21no—
179C22×Dic3—22no—
180C22×Dic3—22no—
181C22×Dic3—22no—
182C22×Dic3—22no—
183C22×Dic3—22no—
184C22×Dic3—21no—
185C22×Dic3—21no—
186C22×Dic3—22no—
187C22×Dic3—22no—
188C22×Dic3—22no—
189C22×Dic3—22no—
190C22×Dic3—21no—
191C22×Dic3—22no—
192C22×Dic3—22no—
193C22×Dic3—22no—
194C22×Dic3—22no—
195C22×Dic3—21no—
196C22×Dic3—22no—
197C22×Dic3—22no—
198C22×Dic3—22no—
199C22×Dic3—21no—
200C22×Dic3—21no—
201C22×Dic3C2×Q812yes—
202C22×Dic3#4114yes—
203C3×C4⋊C4—28no—
204C3×C4⋊C4—28no—
205C3×C4⋊C4—216no—
206C3×C4⋊C4—216no—
207C3×C42—28no—
208(C2×D3)⋊C4—28no—
209(C2×D3)⋊C4—48no—
210(C2×D3)⋊C4—44no—
211(C2×D3)⋊C4—48no—
212(C2×D3)⋊C4—44no—
213C4⋊Dic3—48no—
214Dic3⋊C4—44no—
215Dic3⋊C4—48no—
216C25—31no—
217C23×C4—32no—
218C23×C4—31no—
219C23×C4—31no—
220C23×C4—32no—
221C23×C4—31no—
222C23×C4C22×D312yes—
223C23×C4C22×D311yescharacteristic
224C23×C4C22×D312yes—
225C23×C4C22×D311yescharacteristic
226C23×C4C22×D311yescharacteristic
227#125—232no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—64no—
238C2×C4⋊C4—62no—
239C2×C4⋊C4—64no—
240C2×C4⋊C4—64no—
241C2×C4⋊C4—122no—
242C2×C4⋊C4—122no—
243C2×C4⋊C4—122no—
244C2×C4⋊C4—122no—
245C2×C4⋊C4#6718yes—
246C2×C22⋊C4—34no—
247C2×C22⋊C4—62no—
248C2×C22⋊C4—62no—
249C2×C22⋊C4—64no—
250C2×C22⋊C4—62no—
251C2×C22⋊C4—62no—
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⋊C4—122no—
259C2×C42—24no—
260C2×C42#6714yes—
261#95—64no—
262#95—62no—
263#95—62no—
264#95—62no—
265#95—62no—
266#95—64no—
267#95—122no—
268#95—122no—
269#95—122no—
270#95—122no—
271C23×C3C22×D411yescharacteristic
272C23×C3C22×D412yes—
273C23×C3C22×Q811yescharacteristic
274C23×C3C2×#4111yescharacteristic
275C23×C3C2×#4112yes—
276C23×C3H(5,𝔽2)11yescharacteristic
277C23×C3H(5,𝔽2)12yes—
278C23×C3H(5,𝔽2)11yescharacteristic
279C23×C32−1+412yes—
280C23×C32−1+411yescharacteristic
281C23×C32−1+411yescharacteristic
282C22×D3—22no—
283C22×D3—22no—
284C22×D3—22no—
285C22×D3—24no—
286C22×D3—24no—
287C22×D3—41no—
288C22×D3—41no—
289C22×D3—41no—
290C22×D3—42no—
291C22×D3—41no—
292C22×D3—41no—
293C22×D3—41no—
294C22×D3—42no—
295C22×D3—42no—
296C22×D3—41no—
297C22×D3—42no—
298C22×D3—41no—
299C22×D3—42no—
300C22×D3—42no—
301C22×D3—42no—
302C22×D3—42no—
303C22×D3—42no—
304C22×D3—42no—
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—24no—
315C2×C3×C4—24no—
316C2×C3×C4—22no—
317C2×C3×C4—24no—
318C2×C3×C4—22no—
319C2×C3×C4—24no—
320C2×C3×C4—24no—
321C2×C3×C4—24no—
322C2×C3×C4—42no—
323C2×C3×C4—42no—
324C2×C3×C4—42no—
325C2×C3×C4—42no—
326C2×C3×C4—42no—
327C2×C3×C4—42no—
328C2×C3×C4—42no—
329C2×C3×C4—42no—
330C2×C3×C4—42no—
331C2×C3×C4—42no—
332C2×C3×C4#12014yes—
333C2×C3×C4C22⋊Q814yes—
334C2×Dic3—22no—
335C2×Dic3—24no—
336C2×Dic3—24no—
337C2×Dic3—24no—
338C2×Dic3—24no—
339C2×Dic3—41no—
340C2×Dic3—41no—
341C2×Dic3—42no—
342C2×Dic3—42no—
343C2×Dic3—42no—
344C2×Dic3—42no—
345C2×Dic3—42no—
346C2×Dic3—41no—
347C2×Dic3—41no—
348C2×Dic3—42no—
349C2×Dic3—41no—
350C2×Dic3—42no—
351C2×Dic3—41no—
352C2×Dic3—41no—
353C2×Dic3—41no—
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—32no—
364C24—61no—
365C24—62no—
366C24—62no—
367C24—61no—
368C24—61no—
369C24—62no—
370C24—61no—
371C24—61no—
372C24—61no—
373C24—62no—
374C24C23×D311yescharacteristic
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—21no—
378C22×C4—21no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—21no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—21no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—32no—
397C22×C4—34no—
398C22×C4—62no—
399C22×C4—61no—
400C22×C4—62no—
401C22×C4—62no—
402C22×C4—62no—
403C22×C4—62no—
404C22×C4—62no—
405C22×C4—62no—
406C22×C4—61no—
407C22×C4—62no—
408C22×C4—62no—
409C22×C4—62no—
410C22×C4—62no—
411C22×C4—62no—
412C22×C4—61no—
413C22×C4—62no—
414C22×C4—62no—
415C22×C4—62no—
416C22×C4—62no—
417C22×C4—61no—
418C22×C4—62no—
419C22×C4—61no—
420C22×C4—61no—
421C22×C4—61no—
422C22×C4—62no—
423C22×C4—61no—
424C22×C4—62no—
425C22×C4D3×Q812yes—
426C22×C4#23912yes—
427C22×C4C2×#6714yes—
428C22×C4C2×#6712yes—
429C22×C4C2×#6714yes—
430C22×C4C2×#6712yes—
431C22×C4C2×#6712yes—
432C22×C4C2×#6714yes—
433C4⋊C4—28no—
434C4⋊C4—28no—
435C4⋊C4—216no—
436C4⋊C4—216no—
437C4⋊C4—128no—
438C4⋊C4—124no—
439C4⋊C4—128no—
440C22⋊C4—68no—
441C22⋊C4—128no—
442C22⋊C4—124no—
443C22⋊C4—128no—
444C22⋊C4—124no—
445C42—28no—
446C22×C3C2×#12011yescharacteristic
447C22×C3C2×C22⋊Q811yescharacteristic
448C22×C3#53211yescharacteristic
449C22×C3#53511yescharacteristic
450C22×C3#53512yes—
451C22×C3#53611yescharacteristic
452C22×C3#53711yescharacteristic
453C22×C3#53711yescharacteristic
454C22×C3#53712yes—
455C22×C3#53912yes—
456C22×C3#54011yescharacteristic
457C22×C3#54112yes—
458C22×C3#54111yescharacteristic
459C22×C3#54411yescharacteristic
460C22×C3#54411yescharacteristic
461C22×C3#54612yes—
462C22×C3#54712yes—
463C22×C3#54812yes—
464C22×C3D4×Q812yes—
465C22×C3#55312yes—
466C22×C3#55412yes—
467C22×C3#56212yes—
468C22×C3#56312yes—
469C2×D3—42no—
470C2×D3—42no—
471C2×D3—42no—
472C2×D3—44no—
473C2×D3—44no—
474C2×D3—81no—
475C2×D3—81no—
476C2×D3—81no—
477C2×D3—81no—
478C2×D3—82no—
479C2×D3—82no—
480C3×C4—24no—
481C3×C4—24no—
482C3×C4—42no—
483C3×C4—44no—
484C3×C4—44no—
485Dic3—42no—
486Dic3—44no—
487Dic3—81no—
488Dic3—81no—
489Dic3—82no—
490C23—62no—
491C23—62no—
492C23—64no—
493C23—62no—
494C23—64no—
495C23—122no—
496C23—122no—
497C23—122no—
498C23—121no—
499C23—121no—
500C23—122no—
501C23—122no—
502C23—121no—
503C23—122no—
504C23—122no—
505C23—122no—
506C23—121no—
507C23—122no—
508C23—121no—
509C23—121no—
510C23—121no—
511C23—122no—
512C23—121no—
513C23C2×D3×Q811yescharacteristic
514C23C2×#23911yescharacteristic
515C23#100712yes—
516C23D3×#4112yes—
517C23#100911yescharacteristic
518C23#100911yescharacteristic
519C23#100912yes—
520C23#101011yescharacteristic
521C23#101011yescharacteristic
522C23C22×#6711yescharacteristic
523C23C22×#6712yes—
524C2×C4—24no—
525C2×C4—22no—
526C2×C4—24no—
527C2×C4—24no—
528C2×C4—24no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—24no—
533C2×C4—24no—
534C2×C4—22no—
535C2×C4—24no—
536C2×C4—22no—
537C2×C4—24no—
538C2×C4—22no—
539C2×C4—24no—
540C2×C4—24no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—64no—
552C2×C4—64no—
553C2×C4—64no—
554C2×C4—62no—
555C2×C4—64no—
556C2×C4—122no—
557C2×C4—122no—
558C2×C4—122no—
559C2×C4—122no—
560C2×C4—121no—
561C2×C4—122no—
562C2×C4—121no—
563C2×C4—122no—
564C2×C4—122no—
565C2×C4—122no—
566C2×C4—122no—
567C2×C4—122no—
568C2×C4—122no—
569C2×C4—121no—
570C2×C4—122no—
571C2×C4—122no—
572C2×C4—122no—
573C2×C4—121no—
574C2×C4—121no—
575C2×C4—122no—
576C2×C4—121no—
577C2×C4—121no—
578C2×C4—122no—
579C2×C4—121no—
580C2×C4#94614yes—
581C2×C4#95314yes—
582C2×C3#240911yescharacteristic
583C2×C3#243011yescharacteristic
584C2×C3#256611yescharacteristic
585C2×C3#261511yescharacteristic
586C2×C3#262911yescharacteristic
587C2×C3#266411yescharacteristic
588C2×C3#270412yes—
589C2×C3#271112yes—
590C2×C3#272012yes—
591C2×C3#272512yes—
592C2×C3#282011yescharacteristic
593D3—82no—
594C22—122no—
595C22—122no—
596C22—124no—
597C22—122no—
598C22—124no—
599C22—241no—
600C22—242no—
601C22—241no—
602C22—242no—
603C22—241no—
604C22—241no—
605C22#568911yescharacteristic
606C22#568911yescharacteristic
607C22#569012yes—
608C22#569212yes—
609C22#569512yes—
610C22#569612yes—
611C22#569711yescharacteristic
612C22#569811yescharacteristic
613C22#569912yes—
614C22#570012yes—
615C22C2×#94611yescharacteristic
616C22Q8×#6712yes—
617C22#581611yescharacteristic
618C22#581712yes—
619C22#582412yes—
620C22#582912yes—
621C22#583011yescharacteristic
622C22#583012yes—
623C22#583211yescharacteristic
624C22#583311yescharacteristic
625C22#583312yes—
626C22#583311yescharacteristic
627C22C2×#95311yescharacteristic
628C4—24no—
629C4—24no—
630C4—44no—
631C4—42no—
632C4—44no—
633C4—122no—
634C4—124no—
635C4—242no—
636C4—241no—
637C4—241no—
638C3#1660411yescharacteristic, complemented, Sylow
639C2—242no—
640C2#8304011yescharacteristic
641C2#8304511yescharacteristic
642C2#8305112yes—
643C2#8305312yes—
644C2#8306511yescharacteristic
645C2#8306912yes—
646C2#8307012yes—
647C2#8307111yescharacteristic
648C2#8406811yescharacteristic
649C2#8409111yescharacteristic
650C2#8409811yescharacteristic
subgroupG/H
center, FrattiniC24C23×D3
commutator, socle, cosocleC24×C3C24
FittingC22×C3×#125C2
prime radicalC24×D3C23
hypercenterC22×#125D3
hypocenterC3#16604
LCSG ▷ C24×C3 ▷ C3
UCSC1 ◁ C24 ◁ C22×#125

Complements

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

NH ≅ G/Ntype
1. C22×C3×#125C2semidirect
7. #71568C2semidirect
8. #71568C2semidirect
9. #71567C2semidirect
10. #71565C2semidirect
11. #71535C2semidirect
40. C22×#125D3semidirect
638. C3#16604semidirect

Maximal subgroups

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

subgroupindex
C22×C3×#1252
C2×#49942
C2×#49402
#715702
#715682
#715672
#715652
#715352
#166043