Finite Group Databasebeta

← #10727997

Subgroups — #10727997

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
1C2×#4989C211yesmaximal, characteristic
2C2×#4987C211yesmaximal, characteristic
3C2×#4986C211yesmaximal, characteristic, complemented
4C2×D3×#95C211yesmaximal, characteristic
5C3×#1441C211yesmaximal, characteristic, complemented
6#71567C211yesmaximal, characteristic, complemented
7(C2×D3)⋊#95C212yesmaximal
8(C2×D3)⋊#95C212yesmaximal
9#69422C211yesmaximal, characteristic, complemented
10#69417C212yesmaximal, complemented
11#69414C212yesmaximal, complemented
12#14826—31nomaximal, Sylow
13C23×C4×D3C2212yes—
14C23×C4×D3C2211yescharacteristic
15C22×C3×C4⋊C4C2211yescharacteristic
16C22×C3×C4⋊C4C2211yescharacteristic
17C22×(C2×D3)⋊C4C2212yes—
18C22×(C2×D3)⋊C4C2211yescharacteristic
19C22×(C2×D3)⋊C4C2211yescharacteristic
20C22×C4⋊Dic3C2211yescharacteristic
21C22×Dic3⋊C4C2211yescharacteristic
22C2×C3×#95C2211yescharacteristic
23C2×C3×#95C2212yes—
24C2×C3×#95C2212yes—
25C2×C3⋊#95C2212yes—
26C2×C3⋊#95C2211yescharacteristic
27C2×C3⋊#95C2211yescharacteristic
28C2×C3⋊#95C2212yes—
29C2×C3⋊#95C2212yes—
30C2×C3⋊#95C2212yes—
31C2×C3⋊#95C2212yes—
32C2×C3⋊#95C2212yes—
33C2×C3⋊#95C2212yes—
34C2×C3⋊#95C2212yes—
35C2×C3⋊#95C2211yescharacteristic
36#4989—24no—
37#4987—24no—
38#4986—24no—
39D3×#95—24no—
40C2×#400—31no—
41C2×#385—31no—
42C2×#380—31no—
43C22×#95—31no—
44#1445—31no—
45#1443—32no—
46#1442—32no—
47#1442—31no—
48#1441D311yescharacteristic, complemented
49C22⋊#95—32no—
50C22⋊#95—32no—
51C24×D3C2311yescharacteristic
52C23×C3×C4C2312yes—
53C23×C3×C4C2311yescharacteristic
54C23×C3×C4C2311yescharacteristic
55C23×C3×C4C2312yes—
56C23×C3×C4C2311yescharacteristic
57C23×Dic3C2311yescharacteristic
58C23×Dic3C2312yes—
59C23×Dic3C2312yes—
60C23×Dic3C2311yescharacteristic
61C23×Dic3C2311yescharacteristic
62C22×C4×D3—24no—
63C22×C4×D3—24no—
64C22×C4×D3—22no—
65C22×C4×D3—22no—
66C22×C4×D3—24no—
67C22×C4×D3—24no—
68C22×C4×D3—24no—
69C22×C4×D3—24no—
70C22×C4×D3—42no—
71C22×C4×D3—42no—
72C22×C4×D3—42no—
73C22×C4×D3D414yes—
74C2×C3×C4⋊C4—22no—
75C2×C3×C4⋊C4—22no—
76C2×C3×C4⋊C4—22no—
77C2×C3×C4⋊C4—22no—
78C2×C3×C4⋊C4—42no—
79C2×(C2×D3)⋊C4—22no—
80C2×(C2×D3)⋊C4—22no—
81C2×(C2×D3)⋊C4—24no—
82C2×(C2×D3)⋊C4—22no—
83C2×(C2×D3)⋊C4—24no—
84C2×(C2×D3)⋊C4—24no—
85C2×(C2×D3)⋊C4—24no—
86C2×(C2×D3)⋊C4Q814yes—
87C2×C4⋊Dic3—22no—
88C2×C4⋊Dic3—22no—
89C2×C4⋊Dic3—22no—
90C2×Dic3⋊C4—22no—
91C2×Dic3⋊C4—42no—
92C3×#95—22no—
93C3×#95—42no—
94C3×#95—42no—
95C3⋊#95—22no—
96C3⋊#95—24no—
97C3⋊#95—24no—
98C3⋊#95—24no—
99C3⋊#95—24no—
100C3⋊#95—24no—
101C3⋊#95—24no—
102C3⋊#95—22no—
103C3⋊#95—42no—
104C3⋊#95—42no—
105C3⋊#95D414yes—
106C24×C4—31no—
107C24×C4—32no—
108C22×C4⋊C4—31no—
109C22×C4⋊C4—31no—
110C22×C4⋊C4C2×D311yescharacteristic
111C22×C4⋊C4C2×D311yescharacteristic
112C22×C22⋊C4—31no—
113C22×C22⋊C4—32no—
114C22×C22⋊C4—31no—
115#400—64no—
116#385—64no—
117#380—64no—
118C2×#95—31no—
119C2×#95—32no—
120C2×#95—32no—
121C2×#95—32no—
122C2×#95—32no—
123C2×#95—32no—
124C2×#95—32no—
125C2×#95—31no—
126C2×#95—31no—
127C2×#95—32no—
128C2×#95—32no—
129C2×#95—64no—
130C2×#95C2×D311yescharacteristic
131C2×#95C2×D312yes—
132C2×#95C2×D312yes—
133C24×C3C2411yescharacteristic
134C23×D3—21no—
135C23×D3—21no—
136C23×D3—22no—
137C23×D3—22no—
138C23×D3—21no—
139C23×D3—21no—
140C23×D3C2×D412yes—
141C23×D3C2×Q812yes—
142C23×D3#4112yes—
143C23×D3#4114yes—
144C23×D3#4114yes—
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—21no—
152C22×C3×C4—22no—
153C22×C3×C4—22no—
154C22×C3×C4—22no—
155C22×C3×C4—21no—
156C22×C3×C4—22no—
157C22×C3×C4—22no—
158C22×C3×C4—21no—
159C22×C3×C4—21no—
160C22×C3×C4—22no—
161C22×C3×C4—21no—
162C22×C3×C4—22no—
163C22×C3×C4—22no—
164C22×C3×C4—22no—
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×C4C2×D412yes—
172C22×C3×C4C2×D412yes—
173C22×C3×C4C2×Q812yes—
174C22×C3×C4#4112yes—
175C22×C3×C4#4114yes—
176C22×Dic3—21no—
177C22×Dic3—21no—
178C22×Dic3—22no—
179C22×Dic3—21no—
180C22×Dic3—21no—
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—21no—
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×Dic3—21no—
201C22×Dic3—21no—
202C22×Dic3C2×D414yes—
203C22×Dic3C2×D412yes—
204C22×Dic3C2×Q812yes—
205C2×C4×D3—216no—
206C2×C4×D3—28no—
207C2×C4×D3—28no—
208C2×C4×D3—44no—
209C2×C4×D3—48no—
210C2×C4×D3—48no—
211C2×C4×D3—48no—
212C2×C4×D3—48no—
213C2×C4×D3—48no—
214C2×C4×D3—48no—
215C2×C4×D3—48no—
216C2×C4×D3—48no—
217C3×C4⋊C4—44no—
218C3×C4⋊C4—44no—
219(C2×D3)⋊C4—28no—
220(C2×D3)⋊C4—44no—
221(C2×D3)⋊C4—48no—
222C4⋊Dic3—44no—
223Dic3⋊C4—44no—
224C25—31no—
225C23×C4—31no—
226C23×C4—31no—
227C23×C4—32no—
228C23×C4—34no—
229C23×C4—32no—
230C23×C4—31no—
231C23×C4—64no—
232C23×C4—64no—
233C23×C4—64no—
234C23×C4—64no—
235C23×C4—64no—
236C23×C4—62no—
237C23×C4—62no—
238C23×C4—64no—
239C23×C4—122no—
240C23×C4—122no—
241C23×C4—122no—
242C23×C4C22×D311yescharacteristic
243C23×C4C22×D312yes—
244C23×C4C22×D311yescharacteristic
245C23×C4C22×D312yes—
246C23×C4C22×D311yescharacteristic
247C2×C4⋊C4—22no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—42no—
252C2×C4⋊C4—62no—
253C2×C4⋊C4—62no—
254C2×C4⋊C4—62no—
255C2×C4⋊C4—62no—
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—22no—
266#95—34no—
267#95—42no—
268#95—42no—
269#95—64no—
270#95—64no—
271#95—62no—
272#95—64no—
273#95—64no—
274#95—62no—
275#95—64no—
276#95—64no—
277#95—122no—
278#95—122no—
279C23×C3C22×D411yescharacteristic
280C23×C3C22×D411yescharacteristic
281C23×C3C22×Q811yescharacteristic
282C23×C3C2×#4111yescharacteristic
283C23×C3C2×#4112yes—
284C23×C3C2×#4111yescharacteristic
285C23×C3C2×#4112yes—
286C23×C3C2×#4112yes—
287C23×C3H(5,𝔽2)12yes—
288C23×C3H(5,𝔽2)11yescharacteristic
289C23×C32−1+411yescharacteristic
290C22×D3—22no—
291C22×D3—24no—
292C22×D3—24no—
293C22×D3—22no—
294C22×D3—22no—
295C22×D3—22no—
296C22×D3—24no—
297C22×D3—24no—
298C22×D3—24no—
299C22×D3—22no—
300C22×D3—22no—
301C22×D3—24no—
302C22×D3—24no—
303C22×D3—24no—
304C22×D3—24no—
305C22×D3—22no—
306C22×D3—24no—
307C22×D3—22no—
308C22×D3C22⋊Q814yes—
309C22×D3#12314yes—
310C22×D3#12414yes—
311C22×D3#12518yes—
312C22×D3#12618yes—
313C2×C3×C4—22no—
314C2×C3×C4—22no—
315C2×C3×C4—24no—
316C2×C3×C4—24no—
317C2×C3×C4—22no—
318C2×C3×C4—24no—
319C2×C3×C4—22no—
320C2×C3×C4—24no—
321C2×C3×C4—24no—
322C2×C3×C4—22no—
323C2×C3×C4—41no—
324C2×C3×C4—42no—
325C2×C3×C4—41no—
326C2×C3×C4—42no—
327C2×C3×C4—42no—
328C2×C3×C4—42no—
329C2×C3×C4—41no—
330C2×C3×C4—41no—
331C2×C3×C4—41no—
332C2×C3×C4—41no—
333C2×C3×C4—42no—
334C2×C3×C4—41no—
335C2×C3×C4—42no—
336C2×C3×C4—42no—
337C2×C3×C4—42no—
338C2×C3×C4—42no—
339C2×C3×C4—42no—
340C2×C3×C4—42no—
341C2×C3×C4—42no—
342C2×C3×C4—42no—
343C2×C3×C4#12114yes—
344C2×Dic3—22no—
345C2×Dic3—22no—
346C2×Dic3—22no—
347C2×Dic3—24no—
348C2×Dic3—22no—
349C2×Dic3—24no—
350C2×Dic3—24no—
351C2×Dic3—24no—
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—42no—
364C2×Dic3—42no—
365C2×Dic3—42no—
366C2×Dic3—42no—
367C2×Dic3—42no—
368C2×Dic3—42no—
369C2×Dic3—41no—
370C2×Dic3—41no—
371C2×Dic3—42no—
372C2×Dic3—42no—
373C4×D3—416no—
374C4×D3—432no—
375C24—32no—
376C24—32no—
377C24—34no—
378C24—34no—
379C24—32no—
380C24—61no—
381C24—61no—
382C24—61no—
383C24—62no—
384C24—62no—
385C24—61no—
386C24C23×D311yescharacteristic
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—21no—
390C22×C4—22no—
391C22×C4—21no—
392C22×C4—21no—
393C22×C4—21no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—21no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—21no—
401C22×C4—21no—
402C22×C4—21no—
403C22×C4—22no—
404C22×C4—21no—
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—34no—
414C22×C4—32no—
415C22×C4—32no—
416C22×C4—68no—
417C22×C4—616no—
418C22×C4—61no—
419C22×C4—61no—
420C22×C4—61no—
421C22×C4—62no—
422C22×C4—62no—
423C22×C4—62no—
424C22×C4—62no—
425C22×C4—62no—
426C22×C4—62no—
427C22×C4—61no—
428C22×C4—62no—
429C22×C4—62no—
430C22×C4—62no—
431C22×C4—62no—
432C22×C4—68no—
433C22×C4—62no—
434C22×C4—61no—
435C22×C4—62no—
436C22×C4—62no—
437C22×C4—61no—
438C22×C4—62no—
439C22×C4—62no—
440C22×C4—62no—
441C22×C4—62no—
442C22×C4—62no—
443C22×C4—62no—
444C22×C4—61no—
445C22×C4—124no—
446C22×C4—128no—
447C22×C4—128no—
448C22×C4—128no—
449C22×C4—128no—
450C22×C4—128no—
451C22×C4—128no—
452C22×C4—128no—
453C22×C4—128no—
454C22×C4#23512yes—
455C22×C4D3×D412yes—
456C22×C4D3×D412yes—
457C22×C4D3×Q812yes—
458C22×C4#23914yes—
459C4⋊C4—44no—
460C4⋊C4—44no—
461C4⋊C4—124no—
462C4⋊C4—124no—
463C22⋊C4—68no—
464C22⋊C4—124no—
465C22⋊C4—128no—
466C22×C3C2×#12111yescharacteristic
467C22×C3C2×C22⋊Q811yescharacteristic
468C22×C3C2×#12311yescharacteristic
469C22×C3#52512yes—
470C22×C3C2×#12411yescharacteristic
471C22×C3C2×#12512yes—
472C22×C3C2×#12612yes—
473C22×C3#52912yes—
474C22×C3#53211yescharacteristic
475C22×C3#53411yescharacteristic
476C22×C3#53711yescharacteristic
477C22×C3#53812yes—
478C22×C3#53912yes—
479C22×C3#54612yes—
480C22×C3#54611yescharacteristic
481C22×C3D4×Q811yescharacteristic
482C22×C3#55112yes—
483C22×C3#55112yes—
484C22×C3#55112yes—
485C22×C3#55212yes—
486C22×C3#55211yescharacteristic
487C22×C3#55412yes—
488C22×C3#55411yescharacteristic
489C2×D3—24no—
490C2×D3—24no—
491C2×D3—24no—
492C2×D3—24no—
493C2×D3—24no—
494C2×D3—28no—
495C2×D3—28no—
496C2×D3—28no—
497C2×D3—24no—
498C2×D3—28no—
499C2×D3#40018yescomplemented
500C3×C4—24no—
501C3×C4—42no—
502C3×C4—44no—
503C3×C4—82no—
504C3×C4—81no—
505Dic3—42no—
506Dic3—42no—
507Dic3—44no—
508Dic3—82no—
509Dic3—81no—
510C23—34no—
511C23—34no—
512C23—38no—
513C23—38no—
514C23—34no—
515C23—62no—
516C23—64no—
517C23—64no—
518C23—64no—
519C23—64no—
520C23—62no—
521C23—64no—
522C23—62no—
523C23—62no—
524C23—62no—
525C23—64no—
526C23—64no—
527C23—62no—
528C23—62no—
529C23—64no—
530C23—62no—
531C23—64no—
532C23—64no—
533C23C2×#23511yescharacteristic
534C23C2×D3×D411yescharacteristic
535C23C2×D3×D411yescharacteristic
536C23#100411yescharacteristic
537C23#100412yes—
538C23C2×D3×Q811yescharacteristic
539C23C2×#23912yes—
540C23D3×#4111yescharacteristic
541C23D3×#4112yes—
542C23D3×#4112yes—
543C23#101011yescharacteristic
544C2×C4—22no—
545C2×C4—24no—
546C2×C4—22no—
547C2×C4—24no—
548C2×C4—22no—
549C2×C4—24no—
550C2×C4—24no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—24no—
554C2×C4—42no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—41no—
561C2×C4—42no—
562C2×C4—41no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—41no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—41no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—62no—
575C2×C4—64no—
576C2×C4—62no—
577C2×C4—62no—
578C2×C4—64no—
579C2×C4—64no—
580C2×C4—62no—
581C2×C4—64no—
582C2×C4—1216no—
583C2×C4—1232no—
584C2×C4—122no—
585C2×C4—122no—
586C2×C4—122no—
587C2×C4—122no—
588C2×C4—121no—
589C2×C4—122no—
590C2×C4—121no—
591C2×C4—122no—
592C2×C4—122no—
593C2×C4—122no—
594C2×C4—121no—
595C2×C4—122no—
596C2×C4—122no—
597C2×C4—122no—
598C2×C4—121no—
599C2×C4—122no—
600C2×C4—122no—
601C2×C4—122no—
602C2×C4—121no—
603C2×C4—122no—
604C2×C4—122no—
605C2×C4#88414yes—
606C2×C3C2×#40011yescharacteristic
607C2×C3#241211yescharacteristic
608C2×C3#241711yescharacteristic
609C2×C3#247312yes—
610C2×C3#248012yes—
611C2×C3#249312yes—
612C2×C3#249811yescharacteristic
613C2×C3#250512yes—
614C2×C3#256411yescharacteristic
615C2×C3#263911yescharacteristic
616C2×C3#285011yescharacteristic
617D3—28no—
618C22—38no—
619C22—64no—
620C22—68no—
621C22—64no—
622C22—64no—
623C22—64no—
624C22—68no—
625C22—64no—
626C22—64no—
627C22—68no—
628C22—68no—
629C22C2×#88411yescharacteristic
630C22#551111yescharacteristic
631C22#552911yescharacteristic
632C22#555612yes—
633C22#558012yes—
634C22#558911yescharacteristic
635C22#560811yescharacteristic
636C22#561012yes—
637C22#561211yescharacteristic
638C22#561312yes—
639C22D3×C22⋊Q811yescharacteristic
640C22#562712yes—
641C22#563211yescharacteristic
642C22#563712yes—
643C22#563712yes—
644C22D3×#12311yescharacteristic
645C22#566012yes—
646C22D3×#12411yescharacteristic
647C22D3×#12512yes—
648C22#569412yes—
649C22D3×#12612yes—
650C22#572511yescharacteristic
651C22#573012yes—
652C4—24no—
653C4—42no—
654C4—44no—
655C4—82no—
656C4—81no—
657C4—122no—
658C4—122no—
659C4—124no—
660C4—242no—
661C4—241no—
662C3#1482611yescharacteristic, complemented, Sylow
663C2—68no—
664C2#7388811yescharacteristic
665C2#7469011yescharacteristic
666C2#7533812yes—
667C2#7624112yes—
668C2#7666211yescharacteristic
669C2#7712611yescharacteristic
670C2#7725411yescharacteristic
671C2#7730312yes—
672C2#7775812yes—
673C2#7795511yescharacteristic
674C2D3×#40011yescharacteristic
subgroupG/H
center, FrattiniC24C23×D3
commutator, socle, cosocleC24×C3C24
FittingC3×#1441C2
prime radicalC24×D3C23
hypercenter#1441D3
hypocenterC3#14826
LCSG ▷ C24×C3 ▷ C3
UCSC1 ◁ C24 ◁ #1441

Complements

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

NH ≅ G/Ntype
3. C2×#4986C2semidirect
5. C3×#1441C2semidirect
6. #71567C2semidirect
9. #69422C2semidirect
10. #69417C2semidirect
11. #69414C2semidirect
48. #1441D3semidirect
499. C2×D3#400semidirect
662. C3#14826semidirect

Maximal subgroups

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

subgroupindex
C2×#49892
C2×#49872
C2×#49862
C2×D3×#952
C3×#14412
#715672
(C2×D3)⋊#952
#694222
#694172
#694142
#148263