Finite Group Databasebeta

← #10727998

Subgroups — #10727998

Order 768.

1794 conjugacy classes of subgroups.

The subgroup lattice has 676 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×#95C211yesmaximal, characteristic
2C2×#5210C211yesmaximal, characteristic
3C2×#5212C211yesmaximal, characteristic
4C2×#4659C211yesmaximal, characteristic, complemented
5#72127C212yesmaximal
6#72127C212yesmaximal
7#69422C211yesmaximal, characteristic, complemented
8#69418C212yesmaximal, complemented
9#69413C212yesmaximal, complemented
10#69412C211yesmaximal, characteristic, complemented
11#69410C211yesmaximal, characteristic, complemented
12#14826—31nomaximal, Sylow
13C24×C3×C4C2212yes—
14C24×C3×C4C2211yescharacteristic
15C22×#217C2212yes—
16C22×#217C2211yescharacteristic
17C22×#217C2211yescharacteristic
18C22×C4⋊Dic3C2211yescharacteristic
19C22×C4⋊Dic3C2211yescharacteristic
20C22×Dic3⋊C4C2211yescharacteristic
21C22×Dic3⋊C4C2211yescharacteristic
22C2×C3×#95—24no—
23C2×C3×#95C2211yescharacteristic
24C2×C3×#95C2212yes—
25C2×C3×#95C2211yescharacteristic
26#5212—24no—
27#5210—24no—
28C2×C3⋊#95C2211yescharacteristic
29C2×C3⋊#95C2212yes—
30C2×C3⋊#95C2212yes—
31C2×C3⋊#95C2212yes—
32C2×C3⋊#95C2211yescharacteristic
33C2×C3⋊#95C2212yes—
34C2×C3⋊#95C2212yes—
35C2×C3⋊#95C2212yes—
36C2×C3⋊#95C2212yes—
37C2×C3⋊#95C2212yes—
38C2×C3⋊#95C2212yes—
39#4659—24no—
40C2×#400—31no—
41C2×#385—31no—
42C2×#380—31no—
43C22×#95D311yescharacteristic
44#1445—31no—
45#1443—32no—
46#1442—31no—
47#1442—32no—
48#1441—31no—
49C22⋊#95—32no—
50C22⋊#95—32no—
51C25×C3C2311yescharacteristic
52C23×C3×C4—24no—
53C23×C3×C4—24no—
54C23×C3×C4—24no—
55C23×C3×C4—22no—
56C23×C3×C4—24no—
57C23×C3×C4—22no—
58C23×C3×C4—24no—
59C23×C3×C4—24no—
60C23×C3×C4—42no—
61C23×C3×C4—42no—
62C23×C3×C4—42no—
63C23×C3×C4D414yes—
64C23×C3×C4C2312yes—
65C23×C3×C4C2311yescharacteristic
66C23×C3×C4C2312yes—
67C23×C3×C4C2311yescharacteristic
68C23×Dic3C2311yescharacteristic
69C23×Dic3C2312yes—
70C23×Dic3C2312yes—
71C23×Dic3C2311yescharacteristic
72C23×Dic3C2311yescharacteristic
73C23×Dic3C2311yescharacteristic
74C2×#217—22no—
75C2×#217—22no—
76C2×#217—22no—
77C2×#217—24no—
78C2×#217—24no—
79C2×#217—24no—
80C2×#217—24no—
81C2×#217Q814yes—
82C2×C4⋊Dic3—22no—
83C2×C4⋊Dic3—22no—
84C2×C4⋊Dic3—22no—
85C2×C4⋊Dic3—22no—
86C2×C4⋊Dic3—42no—
87C2×Dic3⋊C4—22no—
88C2×Dic3⋊C4—22no—
89C2×Dic3⋊C4—22no—
90C2×Dic3⋊C4—22no—
91C2×Dic3⋊C4—42no—
92C3×#95—24no—
93C3×#95—22no—
94C3×#95D414yes—
95C3⋊#95—24no—
96C3⋊#95—24no—
97C3⋊#95—22no—
98C3⋊#95—22no—
99C3⋊#95—24no—
100C3⋊#95—24no—
101C3⋊#95—24no—
102C3⋊#95—42no—
103C3⋊#95—42no—
104C3⋊#95—42no—
105C3⋊#95—42no—
106C24×C4C2×D312yes—
107C24×C4C2×D311yescharacteristic
108C22×C4⋊C4—31no—
109C22×C4⋊C4—31no—
110C22×C4⋊C4—31no—
111C22×C4⋊C4—31no—
112C22×C22⋊C4—31no—
113C22×C22⋊C4—32no—
114C22×C22⋊C4—31no—
115#400—64no—
116#385—64no—
117#380—64no—
118C2×#95—24no—
119C2×#95—32no—
120C2×#95—32no—
121C2×#95—32no—
122C2×#95—32no—
123C2×#95—32no—
124C2×#95—31no—
125C2×#95—31no—
126C2×#95—32no—
127C2×#95—32no—
128C2×#95—32no—
129C2×#95—32no—
130C2×#95C2×D311yescharacteristic
131C2×#95C2×D312yes—
132C2×#95C2×D311yescharacteristic
133C24×C3—21no—
134C24×C3—21no—
135C24×C3—22no—
136C24×C3—22no—
137C24×C3—21no—
138C24×C3—21no—
139C24×C3C2×D412yes—
140C24×C3C2×Q812yes—
141C24×C3#4114yes—
142C24×C3#4112yes—
143C24×C3#4114yes—
144C24×C3C2411yescharacteristic
145C22×C3×C4—22no—
146C22×C3×C4—22no—
147C22×C3×C4—216no—
148C22×C3×C4—28no—
149C22×C3×C4—21no—
150C22×C3×C4—22no—
151C22×C3×C4—21no—
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—22no—
163C22×C3×C4—22no—
164C22×C3×C4—22no—
165C22×C3×C4—28no—
166C22×C3×C4—44no—
167C22×C3×C4—48no—
168C22×C3×C4—48no—
169C22×C3×C4—48no—
170C22×C3×C4—48no—
171C22×C3×C4—48no—
172C22×C3×C4—48no—
173C22×C3×C4—48no—
174C22×C3×C4—48no—
175C22×C3×C4C2×D412yes—
176C22×C3×C4C2×D412yes—
177C22×C3×C4C2×D414yes—
178C22×C3×C4C2×D412yes—
179C22×C3×C4#4114yes—
180C22×C3×C4#4112yes—
181C22×Dic3—21no—
182C22×Dic3—21no—
183C22×Dic3—21no—
184C22×Dic3—22no—
185C22×Dic3—21no—
186C22×Dic3—21no—
187C22×Dic3—21no—
188C22×Dic3—21no—
189C22×Dic3—22no—
190C22×Dic3—21no—
191C22×Dic3—21no—
192C22×Dic3—22no—
193C22×Dic3—21no—
194C22×Dic3—22no—
195C22×Dic3—22no—
196C22×Dic3—21no—
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—22no—
207C22×Dic3—22no—
208C22×Dic3—22no—
209C22×Dic3—21no—
210C22×Dic3—22no—
211C22×Dic3—22no—
212C22×Dic3—22no—
213C22×Dic3—22no—
214C22×Dic3—21no—
215C22×Dic3C2×Q812yes—
216C22×Dic3C2×Q812yes—
217#217—28no—
218#217—44no—
219#217—48no—
220C4⋊Dic3—44no—
221C4⋊Dic3—44no—
222Dic3⋊C4—44no—
223Dic3⋊C4—44no—
224C25C22×D311yescharacteristic
225C23×C4—24no—
226C23×C4—24no—
227C23×C4—24no—
228C23×C4—22no—
229C23×C4—24no—
230C23×C4—22no—
231C23×C4—24no—
232C23×C4—24no—
233C23×C4—31no—
234C23×C4—31no—
235C23×C4—31no—
236C23×C4—32no—
237C23×C4—32no—
238C23×C4—31no—
239C23×C4—42no—
240C23×C4—42no—
241C23×C4—42no—
242C23×C4D1214yes—
243C23×C4C22×D311yescharacteristic
244C23×C4C22×D312yes—
245C23×C4C22×D312yes—
246C23×C4C22×D311yescharacteristic
247C2×C4⋊C4—62no—
248C2×C4⋊C4—62no—
249C2×C4⋊C4—62no—
250C2×C4⋊C4—62no—
251C2×C4⋊C4—62no—
252C2×C4⋊C4—62no—
253C2×C4⋊C4—62no—
254C2×C4⋊C4—62no—
255C2×C4⋊C4—122no—
256C2×C4⋊C4—122no—
257C2×C22⋊C4—34no—
258C2×C22⋊C4—62no—
259C2×C22⋊C4—62no—
260C2×C22⋊C4—64no—
261C2×C22⋊C4—62no—
262C2×C22⋊C4—64no—
263C2×C22⋊C4—64no—
264C2×C22⋊C4—64no—
265#95—24no—
266#95—22no—
267#95—64no—
268#95—62no—
269#95—62no—
270#95—64no—
271#95—64no—
272#95—64no—
273#95—64no—
274#95—122no—
275#95—122no—
276#95—122no—
277#95—122no—
278#95#6714yes—
279C23×C3—22no—
280C23×C3—24no—
281C23×C3—24no—
282C23×C3—22no—
283C23×C3—22no—
284C23×C3—22no—
285C23×C3—24no—
286C23×C3—24no—
287C23×C3—24no—
288C23×C3—22no—
289C23×C3—22no—
290C23×C3—24no—
291C23×C3—24no—
292C23×C3—24no—
293C23×C3—22no—
294C23×C3—24no—
295C23×C3—24no—
296C23×C3—22no—
297C23×C3C22⋊Q814yes—
298C23×C3#12314yes—
299C23×C3#12414yes—
300C23×C3#12518yes—
301C23×C3#12618yes—
302C23×C3C22×D411yescharacteristic
303C23×C3C22×D411yescharacteristic
304C23×C3C22×Q811yescharacteristic
305C23×C3C2×#4111yescharacteristic
306C23×C3C2×#4112yes—
307C23×C3C2×#4112yes—
308C23×C3C2×#4111yescharacteristic
309C23×C3C2×#4112yes—
310C23×C3H(5,𝔽2)12yes—
311C23×C3H(5,𝔽2)11yescharacteristic
312C23×C32−1+411yescharacteristic
313C2×C3×C4—22no—
314C2×C3×C4—22no—
315C2×C3×C4—24no—
316C2×C3×C4—24no—
317C2×C3×C4—22no—
318C2×C3×C4—22no—
319C2×C3×C4—24no—
320C2×C3×C4—24no—
321C2×C3×C4—22no—
322C2×C3×C4—24no—
323C2×C3×C4—24no—
324C2×C3×C4—24no—
325C2×C3×C4—24no—
326C2×C3×C4—416no—
327C2×C3×C4—432no—
328C2×C3×C4—42no—
329C2×C3×C4—42no—
330C2×C3×C4—42no—
331C2×C3×C4—42no—
332C2×C3×C4—42no—
333C2×C3×C4—42no—
334C2×C3×C4—42no—
335C2×C3×C4—42no—
336C2×C3×C4—42no—
337C2×C3×C4—42no—
338C2×C3×C4#12114yes—
339C2×Dic3—22no—
340C2×Dic3—22no—
341C2×Dic3—22no—
342C2×Dic3—24no—
343C2×Dic3—22no—
344C2×Dic3—41no—
345C2×Dic3—41no—
346C2×Dic3—41no—
347C2×Dic3—41no—
348C2×Dic3—41no—
349C2×Dic3—42no—
350C2×Dic3—41no—
351C2×Dic3—41no—
352C2×Dic3—42no—
353C2×Dic3—41no—
354C2×Dic3—41no—
355C2×Dic3—42no—
356C2×Dic3—42no—
357C2×Dic3—42no—
358C2×Dic3—42no—
359C2×Dic3—42no—
360C2×Dic3—42no—
361C2×Dic3—42no—
362C2×Dic3—41no—
363C2×Dic3—41no—
364C2×Dic3—42no—
365C2×Dic3—42no—
366C2×Dic3—41no—
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—21no—
379C24—21no—
380C24—21no—
381C24C2×D1212yes—
382C24#23514yes—
383C24#23712yes—
384C24#23714yes—
385C24D3×Q812yes—
386C24C23×D311yescharacteristic
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—216no—
390C22×C4—28no—
391C22×C4—21no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—21no—
406C22×C4—22no—
407C22×C4—28no—
408C22×C4—32no—
409C22×C4—32no—
410C22×C4—44no—
411C22×C4—48no—
412C22×C4—48no—
413C22×C4—48no—
414C22×C4—48no—
415C22×C4—48no—
416C22×C4—48no—
417C22×C4—48no—
418C22×C4—48no—
419C22×C4—61no—
420C22×C4—61no—
421C22×C4—62no—
422C22×C4—61no—
423C22×C4—61no—
424C22×C4—61no—
425C22×C4—61no—
426C22×C4—62no—
427C22×C4—61no—
428C22×C4—61no—
429C22×C4—61no—
430C22×C4—62no—
431C22×C4—62no—
432C22×C4—61no—
433C22×C4—62no—
434C22×C4—62no—
435C22×C4—62no—
436C22×C4—62no—
437C22×C4—62no—
438C22×C4—62no—
439C22×C4—62no—
440C22×C4—62no—
441C22×C4—61no—
442C22×C4—62no—
443C22×C4—62no—
444C22×C4—62no—
445C22×C4—61no—
446C22×C4—62no—
447C22×C4—62no—
448C22×C4—62no—
449C22×C4—61no—
450C22×C4—62no—
451C22×C4—62no—
452C22×C4—61no—
453C22×C4C2×D1212yes—
454C22×C4C2×D1212yes—
455C22×C4#23514yes—
456C22×C4#23512yes—
457C22×C4C2×#6714yes—
458C22×C4C2×#6712yes—
459C4⋊C4—124no—
460C4⋊C4—124no—
461C4⋊C4—124no—
462C4⋊C4—124no—
463C22⋊C4—68no—
464C22⋊C4—128no—
465C22⋊C4—124no—
466C22×C3—24no—
467C22×C3—24no—
468C22×C3—24no—
469C22×C3—24no—
470C22×C3—24no—
471C22×C3—28no—
472C22×C3—28no—
473C22×C3—28no—
474C22×C3—24no—
475C22×C3—28no—
476C22×C3#40018yescomplemented
477C22×C3C2×#12111yescharacteristic
478C22×C3C2×C22⋊Q811yescharacteristic
479C22×C3C2×#12311yescharacteristic
480C22×C3#52512yes—
481C22×C3C2×#12411yescharacteristic
482C22×C3C2×#12512yes—
483C22×C3C2×#12612yes—
484C22×C3#52912yes—
485C22×C3#53211yescharacteristic
486C22×C3#53411yescharacteristic
487C22×C3#53711yescharacteristic
488C22×C3#53812yes—
489C22×C3#53912yes—
490C22×C3#54612yes—
491C22×C3#54611yescharacteristic
492C22×C3D4×Q811yescharacteristic
493C22×C3#55112yes—
494C22×C3#55112yes—
495C22×C3#55112yes—
496C22×C3#55211yescharacteristic
497C22×C3#55212yes—
498C22×C3#55412yes—
499C22×C3#55411yescharacteristic
500C3×C4—24no—
501C3×C4—42no—
502C3×C4—44no—
503C3×C4—44no—
504Dic3—42no—
505Dic3—42no—
506Dic3—82no—
507Dic3—82no—
508Dic3—81no—
509Dic3—81no—
510C23—22no—
511C23—24no—
512C23—22no—
513C23—24no—
514C23—24no—
515C23—24no—
516C23—24no—
517C23—22no—
518C23—22no—
519C23—24no—
520C23—22no—
521C23—24no—
522C23—24no—
523C23—22no—
524C23—24no—
525C23—22no—
526C23—24no—
527C23—22no—
528C23#87514yes—
529C23#87918yes—
530C23#88614yes—
531C23#88918yes—
532C23#89714yes—
533C23C22×D1211yescharacteristic
534C23C2×#23512yes—
535C23C2×#23512yes—
536C23C2×#23511yescharacteristic
537C23C2×#23711yescharacteristic
538C23C2×#23712yes—
539C23#100412yes—
540C23#100411yescharacteristic
541C23C2×D3×Q811yescharacteristic
542C23#100711yescharacteristic
543C23C22×#6711yescharacteristic
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—24no—
547C2×C4—24no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—24no—
551C2×C4—24no—
552C2×C4—22no—
553C2×C4—24no—
554C2×C4—24no—
555C2×C4—24no—
556C2×C4—24no—
557C2×C4—416no—
558C2×C4—432no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—62no—
570C2×C4—62no—
571C2×C4—62no—
572C2×C4—64no—
573C2×C4—62no—
574C2×C4—121no—
575C2×C4—121no—
576C2×C4—122no—
577C2×C4—121no—
578C2×C4—121no—
579C2×C4—121no—
580C2×C4—121no—
581C2×C4—122no—
582C2×C4—122no—
583C2×C4—121no—
584C2×C4—122no—
585C2×C4—122no—
586C2×C4—122no—
587C2×C4—121no—
588C2×C4—121no—
589C2×C4—122no—
590C2×C4—121no—
591C2×C4—122no—
592C2×C4—122no—
593C2×C4—122no—
594C2×C4—122no—
595C2×C4—121no—
596C2×C4—122no—
597C2×C4—122no—
598C2×C4—122no—
599C2×C4—122no—
600C2×C4—122no—
601C2×C4—121no—
602C2×C4—122no—
603C2×C4—122no—
604C2×C4—122no—
605C2×C4#93014yes—
606C2×C3—28no—
607C2×C3C2×#40011yescharacteristic
608C2×C3#241211yescharacteristic
609C2×C3#241711yescharacteristic
610C2×C3#247312yes—
611C2×C3#248012yes—
612C2×C3#249312yes—
613C2×C3#249811yescharacteristic
614C2×C3#250512yes—
615C2×C3#256411yescharacteristic
616C2×C3#263911yescharacteristic
617C2×C3#285011yescharacteristic
618C22—24no—
619C22—28no—
620C22—24no—
621C22—28no—
622C22—24no—
623C22—24no—
624C22—24no—
625C22—24no—
626C22—28no—
627C22—28no—
628C22#465918yescomplemented
629C22C2×#87511yescharacteristic
630C22#547711yescharacteristic
631C22#547912yes—
632C22C2×#87912yes—
633C22C2×#88611yescharacteristic
634C22#549311yescharacteristic
635C22#549412yes—
636C22#549512yes—
637C22#549512yes—
638C22#549512yes—
639C22C2×#88912yes—
640C22C2×#89711yescharacteristic
641C22#551011yescharacteristic
642C22#551111yescharacteristic
643C22#551112yes—
644C22#551311yescharacteristic
645C22#551312yes—
646C22#551512yes—
647C22C2×#93011yescharacteristic
648C22#579112yes—
649C22#580611yescharacteristic
650C22#580612yes—
651C22Q8×#6711yescharacteristic
652C4—24no—
653C4—42no—
654C4—44no—
655C4—44no—
656C4—122no—
657C4—122no—
658C4—241no—
659C4—242no—
660C4—242no—
661C4—241no—
662C3#1482611yescharacteristic, complemented, Sylow
663C2—28no—
664C2C2×#465911yescharacteristic
665C2#7335312yes—
666C2#7335411yescharacteristic
667C2#7335512yes—
668C2#7335811yescharacteristic
669C2#7337911yescharacteristic
670C2#8207711yescharacteristic
671C2#8210811yescharacteristic
672C2#8211212yes—
673C2#8217412yes—
674C2#8219411yescharacteristic
subgroupG/H
center, FrattiniC24C23×D3
commutator, socle, cosocleC24×C3C24
FittingC22×C3×#95C2
prime radicalC25×C3C23
hypercenterC22×#95D3
hypocenterC3#14826
LCSG ▷ C24×C3 ▷ C3
UCSC1 ◁ C24 ◁ C22×#95

Complements

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

NH ≅ G/Ntype
4. C2×#4659C2semidirect
7. #69422C2semidirect
8. #69418C2semidirect
9. #69413C2semidirect
10. #69412C2semidirect
11. #69410C2semidirect
476. C22×C3#400semidirect
628. C22#4659semidirect
662. C3#14826semidirect

Maximal subgroups

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

subgroupindex
C22×C3×#952
C2×#52102
C2×#52122
C2×#46592
#721272
#694222
#694182
#694132
#694122
#694102
#148263