Finite Group Databasebeta

← #10728749

Subgroups — #10728749

Order 768.

1894 conjugacy classes of subgroups.

The subgroup lattice has 712 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#872C211yesmaximal, characteristic
2C2×C3×#391C211yesmaximal, characteristic, complemented
3C2×#5231C211yesmaximal, characteristic
4C2×#5230C211yesmaximal, characteristic, complemented
5#71570C211yesmaximal, characteristic
6#71566C211yesmaximal, characteristic, complemented
7#71534C211yesmaximal, characteristic, complemented
8#69428C212yesmaximal
9(C2×D3)⋊#95C212yesmaximal
10#69414C212yesmaximal, complemented
11#69406C212yesmaximal, complemented
12#14887—31nomaximal, Sylow
13C23×C3×Q8C2211yescharacteristic
14C23×C4×D3C2211yescharacteristic
15C22×C3×C4⋊C4C2211yescharacteristic
16C22×C3×C42C2211yescharacteristic
17C22×(C2×D3)⋊C4C2211yescharacteristic
18C22×(C2×D3)⋊C4C2212yes—
19C22×(C2×D3)⋊C4C2212yes—
20C22×(C2×D3)⋊C4C2211yescharacteristic
21C22×Dic3⋊C4C2211yescharacteristic
22C22×Dic3⋊C4C2212yes—
23C22×Dic3⋊C4C2211yescharacteristic
24C22×C4×Dic3C2211yescharacteristic
25C2×#872—24no—
26C2×#872—24no—
27C2×#872—24no—
28C3×#391—24no—
29C2×C3×#95C2212yes—
30C2×C3×#95C2212yes—
31#5231—24no—
32#5230—24no—
33C2×C3⋊#95C2212yes—
34C2×C3⋊#95C2212yes—
35C2×C3⋊#95C2212yes—
36C2×C3⋊#95C2212yes—
37C2×C3⋊#95C2212yes—
38C2×C3⋊#95C2212yes—
39C2×C3⋊#95C2211yescharacteristic
40C2×C3⋊#95C2211yescharacteristic
41C2×C3⋊#95C2212yes—
42C22×#117—31no—
43C2×#395—31no—
44C2×#393—31no—
45C2×#391D311yescharacteristic, complemented
46#1442—32no—
47#1441—31no—
48#1439—32no—
49#1439—31no—
50#1436—32no—
51#1434—31no—
52C22⋊#95—32no—
53C24×D3C2311yescharacteristic
54C22×C3×Q8—22no—
55C22×C3×Q8—22no—
56C22×C3×Q8—22no—
57C22×C3×Q8—24no—
58C22×C3×Q8—42no—
59C23×C3×C4C2312yes—
60C23×C3×C4C2311yescharacteristic
61C23×C3×C4C2312yes—
62C23×C3×C4C2311yescharacteristic
63C23×C3×C4C2311yescharacteristic
64C23×Dic3C2312yes—
65C23×Dic3C2311yescharacteristic
66C23×Dic3C2311yescharacteristic
67C23×Dic3C2312yes—
68C23×Dic3C2311yescharacteristic
69C22×C4×D3—22no—
70C22×C4×D3—22no—
71C22×C4×D3—22no—
72C22×C4×D3—22no—
73C22×C4×D3—22no—
74C22×C4×D3—22no—
75C22×C4×D3Q814yescomplemented
76C2×C3×C4⋊C4—22no—
77C2×C3×C4⋊C4—42no—
78C2×C3×C42—22no—
79C2×C3×C42—22no—
80C2×C3×C42D414yes—
81C2×(C2×D3)⋊C4—22no—
82C2×(C2×D3)⋊C4—22no—
83C2×(C2×D3)⋊C4—22no—
84C2×(C2×D3)⋊C4—22no—
85C2×(C2×D3)⋊C4—22no—
86C2×(C2×D3)⋊C4—24no—
87C2×(C2×D3)⋊C4—24no—
88C2×(C2×D3)⋊C4—24no—
89C2×(C2×D3)⋊C4—24no—
90C2×(C2×D3)⋊C4—42no—
91C2×(C2×D3)⋊C4—42no—
92C2×(C2×D3)⋊C4D414yes—
93C2×Dic3⋊C4—22no—
94C2×Dic3⋊C4—22no—
95C2×Dic3⋊C4—22no—
96C2×Dic3⋊C4—22no—
97C2×Dic3⋊C4—22no—
98C2×Dic3⋊C4—24no—
99C2×Dic3⋊C4—24no—
100C2×Dic3⋊C4—42no—
101C2×Dic3⋊C4D414yes—
102C2×C4×Dic3—22no—
103C2×C4×Dic3—22no—
104C2×C4×Dic3—22no—
105#872—232no—
106C3×#95—24no—
107C3×#95—24no—
108C3⋊#95—22no—
109C3⋊#95—22no—
110C3⋊#95—24no—
111C3⋊#95—24no—
112C3⋊#95—24no—
113C3⋊#95—24no—
114C3⋊#95—24no—
115C3⋊#95—24no—
116C3⋊#95—42no—
117C23×Q8C2×D311yescharacteristic
118C24×C4—31no—
119C2×#117—64no—
120C2×#117—64no—
121C2×#117—64no—
122C22×C4⋊C4—31no—
123C22×C4⋊C4—31no—
124C22×C4⋊C4—32no—
125C22×C4⋊C4C2×D311yescharacteristic
126C22×C22⋊C4—31no—
127C22×C22⋊C4—31no—
128C22×C22⋊C4—32no—
129C22×C22⋊C4—32no—
130C22×C42—31no—
131C22×C42C2×D311yescharacteristic
132#395—64no—
133#393—64no—
134#391—24no—
135C2×#95—31no—
136C2×#95—32no—
137C2×#95—32no—
138C2×#95—31no—
139C2×#95—32no—
140C2×#95—32no—
141C2×#95—32no—
142C2×#95—32no—
143C2×#95—32no—
144C2×#95C2×D312yes—
145C2×#95C2×D312yes—
146C24×C3C2411yescharacteristic
147C23×D3—21no—
148C23×D3—21no—
149C23×D3—22no—
150C23×D3—22no—
151C23×D3—21no—
152C23×D3—21no—
153C23×D3—22no—
154C23×D3—22no—
155C23×D3C2×D412yes—
156C23×D3C2×Q812yes—
157C23×D3#4112yes—
158C2×C3×Q8—28no—
159C2×C3×Q8—48no—
160C2×C3×Q8—48no—
161C2×C3×Q8—44no—
162C2×C3×Q8—44no—
163C22×C3×C4—21no—
164C22×C3×C4—21no—
165C22×C3×C4—21no—
166C22×C3×C4—21no—
167C22×C3×C4—22no—
168C22×C3×C4—22no—
169C22×C3×C4—22no—
170C22×C3×C4—22no—
171C22×C3×C4—22no—
172C22×C3×C4—21no—
173C22×C3×C4—21no—
174C22×C3×C4—22no—
175C22×C3×C4—22no—
176C22×C3×C4—22no—
177C22×C3×C4—22no—
178C22×C3×C4—22no—
179C22×C3×C4—22no—
180C22×C3×C4—22no—
181C22×C3×C4—22no—
182C22×C3×C4—22no—
183C22×C3×C4—22no—
184C22×C3×C4—22no—
185C22×C3×C4—22no—
186C22×C3×C4C2×D412yes—
187C22×C3×C4C2×D412yes—
188C22×C3×C4C2×D412yes—
189C22×C3×C4C2×D412yes—
190C22×C3×C4C2×D412yes—
191C22×C3×C4C2×Q812yes—
192C22×C3×C4#4114yes—
193C22×C3×C4#4112yes—
194C22×Dic3—21no—
195C22×Dic3—21no—
196C22×Dic3—21no—
197C22×Dic3—21no—
198C22×Dic3—22no—
199C22×Dic3—21no—
200C22×Dic3—21no—
201C22×Dic3—21no—
202C22×Dic3—21no—
203C22×Dic3—22no—
204C22×Dic3—22no—
205C22×Dic3—22no—
206C22×Dic3—22no—
207C22×Dic3—22no—
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—22no—
217C22×Dic3—22no—
218C22×Dic3—22no—
219C22×Dic3C2×D412yes—
220C22×Dic3C2×D412yes—
221C22×Dic3C2×D412yes—
222C22×Dic3C2×Q812yes—
223C22×Dic3#4114yes—
224C22×Dic3#4112yes—
225C2×C4×D3—28no—
226C2×C4×D3—28no—
227C2×C4×D3—28no—
228C2×C4×D3—28no—
229C2×C4×D3—28no—
230C2×C4×D3—28no—
231C2×C4×D3—28no—
232C3×C4⋊C4—44no—
233C3×C42—28no—
234(C2×D3)⋊C4—28no—
235(C2×D3)⋊C4—28no—
236(C2×D3)⋊C4—48no—
237(C2×D3)⋊C4—48no—
238Dic3⋊C4—28no—
239Dic3⋊C4—28no—
240Dic3⋊C4—48no—
241C4×Dic3—28no—
242C25—31no—
243C22×Q8—24no—
244C22×Q8—22no—
245C22×Q8—22no—
246C22×Q8—22no—
247C22×Q8—42no—
248C23×C4—31no—
249C23×C4—31no—
250C23×C4—32no—
251C23×C4—31no—
252C23×C4—34no—
253C23×C4—32no—
254C23×C4—62no—
255C23×C4—62no—
256C23×C4—62no—
257C23×C4—62no—
258C23×C4—62no—
259C23×C4—62no—
260C23×C4C22×D312yes—
261C23×C4C22×D312yes—
262C23×C4C22×D311yescharacteristic
263C23×C4C22×D311yescharacteristic
264C23×C4C22×D311yescharacteristic
265#117—632no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—34no—
268C2×C4⋊C4—42no—
269C2×C4⋊C4—62no—
270C2×C4⋊C4—62no—
271C2×C4⋊C4—62no—
272C2×C4⋊C4—62no—
273C2×C4⋊C4—62no—
274C2×C4⋊C4—64no—
275C2×C4⋊C4—64no—
276C2×C4⋊C4—122no—
277C2×C22⋊C4—34no—
278C2×C22⋊C4—62no—
279C2×C22⋊C4—62no—
280C2×C22⋊C4—62no—
281C2×C22⋊C4—62no—
282C2×C22⋊C4—62no—
283C2×C22⋊C4—64no—
284C2×C22⋊C4—64no—
285C2×C22⋊C4—64no—
286C2×C22⋊C4—64no—
287C2×C22⋊C4—122no—
288C2×C22⋊C4—122no—
289C2×C42—22no—
290C2×C42—22no—
291C2×C42—62no—
292C2×C42—62no—
293C2×C42—62no—
294C2×C42#6714yes—
295#95—24no—
296#95—24no—
297#95—64no—
298#95—64no—
299#95—64no—
300#95—64no—
301#95—64no—
302#95—62no—
303#95—62no—
304#95—64no—
305#95—122no—
306C23×C3C22×D411yescharacteristic
307C23×C3C22×D411yescharacteristic
308C23×C3C22×D411yescharacteristic
309C23×C3C22×Q811yescharacteristic
310C23×C3C2×#4112yes—
311C23×C3C2×#4111yescharacteristic
312C23×C3C2×#4111yescharacteristic
313C23×C3C2×#4111yescharacteristic
314C23×C3C2×#4112yes—
315C23×C3H(5,𝔽2)12yes—
316C23×C32−1+412yes—
317C22×D3—22no—
318C22×D3—22no—
319C22×D3—22no—
320C22×D3—22no—
321C22×D3—22no—
322C22×D3—22no—
323C22×D3—24no—
324C22×D3—24no—
325C22×D3—24no—
326C22×D3—24no—
327C22×D3—24no—
328C22×D3—24no—
329C22×D3—41no—
330C22×D3—42no—
331C22×D3—41no—
332C22×D3—41no—
333C22×D3—41no—
334C22×D3—42no—
335C22×D3—42no—
336C22×D3—42no—
337C22×D3—42no—
338C22×D3—42no—
339C22×D3C22⋊Q814yes—
340C3×Q8—416no—
341C2×C3×C4—22no—
342C2×C3×C4—22no—
343C2×C3×C4—22no—
344C2×C3×C4—22no—
345C2×C3×C4—22no—
346C2×C3×C4—22no—
347C2×C3×C4—22no—
348C2×C3×C4—22no—
349C2×C3×C4—24no—
350C2×C3×C4—24no—
351C2×C3×C4—22no—
352C2×C3×C4—24no—
353C2×C3×C4—22no—
354C2×C3×C4—22no—
355C2×C3×C4—24no—
356C2×C3×C4—24no—
357C2×C3×C4—42no—
358C2×C3×C4—42no—
359C2×C3×C4—42no—
360C2×C3×C4—42no—
361C2×C3×C4—42no—
362C2×C3×C4—42no—
363C2×C3×C4—42no—
364C2×C3×C4—42no—
365C2×C3×C4—42no—
366C2×C3×C4—42no—
367C2×C3×C4—42no—
368C2×C3×C4—42no—
369C2×C3×C4—42no—
370C2×C3×C4#12014yes—
371C2×C3×C4C22⋊Q814yes—
372C2×Dic3—22no—
373C2×Dic3—22no—
374C2×Dic3—22no—
375C2×Dic3—22no—
376C2×Dic3—22no—
377C2×Dic3—22no—
378C2×Dic3—22no—
379C2×Dic3—22no—
380C2×Dic3—24no—
381C2×Dic3—24no—
382C2×Dic3—24no—
383C2×Dic3—24no—
384C2×Dic3—24no—
385C2×Dic3—41no—
386C2×Dic3—42no—
387C2×Dic3—41no—
388C2×Dic3—41no—
389C2×Dic3—41no—
390C2×Dic3—42no—
391C2×Dic3—42no—
392C2×Dic3—42no—
393C2×Dic3—42no—
394C2×Dic3—42no—
395C2×Dic3—42no—
396C2×Dic3—42no—
397C2×Dic3—42no—
398C2×Dic3—42no—
399C2×Dic3—42no—
400C2×Dic3—42no—
401C2×Dic3—42no—
402C2×Dic3C22⋊Q814yes—
403C4×D3—232no—
404C24—32no—
405C24—32no—
406C24—32no—
407C24—62no—
408C24—61no—
409C24—62no—
410C24—61no—
411C24—62no—
412C24—62no—
413C24—61no—
414C24—61no—
415C24C23×D311yescharacteristic
416C2×Q8—28no—
417C2×Q8—48no—
418C2×Q8—48no—
419C2×Q8—44no—
420C2×Q8—44no—
421C22×C4—22no—
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—22no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—32no—
445C22×C4—32no—
446C22×C4—32no—
447C22×C4—32no—
448C22×C4—32no—
449C22×C4—34no—
450C22×C4—68no—
451C22×C4—68no—
452C22×C4—68no—
453C22×C4—68no—
454C22×C4—68no—
455C22×C4—68no—
456C22×C4—61no—
457C22×C4—61no—
458C22×C4—61no—
459C22×C4—61no—
460C22×C4—68no—
461C22×C4—62no—
462C22×C4—62no—
463C22×C4—62no—
464C22×C4—61no—
465C22×C4—61no—
466C22×C4—62no—
467C22×C4—62no—
468C22×C4—62no—
469C22×C4—62no—
470C22×C4—62no—
471C22×C4—62no—
472C22×C4—62no—
473C22×C4—61no—
474C22×C4—61no—
475C22×C4—62no—
476C22×C4—62no—
477C22×C4—62no—
478C22×C4—62no—
479C22×C4—62no—
480C22×C4—62no—
481C22×C4—62no—
482C22×C4#23514yes—
483C22×C4D3×D412yes—
484C22×C4D3×D412yes—
485C22×C4D3×Q812yes—
486C22×C4#23912yes—
487C22×C4C2×#6712yes—
488C22×C4C2×#6712yes—
489C22×C4C2×#6712yes—
490C4⋊C4—44no—
491C4⋊C4—68no—
492C4⋊C4—68no—
493C4⋊C4—128no—
494C22⋊C4—68no—
495C22⋊C4—68no—
496C22⋊C4—128no—
497C22⋊C4—128no—
498C42—28no—
499C42—68no—
500C22×C3C2×#12011yescharacteristic
501C22×C3C2×C22⋊Q811yescharacteristic
502C22×C3C2×C22⋊Q811yescharacteristic
503C22×C3C2×C22⋊Q811yescharacteristic
504C22×C3#52511yescharacteristic
505C22×C3#52511yescharacteristic
506C22×C3#52511yescharacteristic
507C22×C3#52511yescharacteristic
508C22×C3#52912yes—
509C22×C3#53211yescharacteristic
510C22×C3#53312yes—
511C22×C3#53511yescharacteristic
512C22×C3#53611yescharacteristic
513C22×C3#54212yes—
514C22×C3#54412yes—
515C22×C3#54612yes—
516C22×C3#54612yes—
517C22×C3#54712yes—
518C22×C3#54812yes—
519C22×C3#55112yes—
520C22×C3#55212yes—
521C22×C3#55512yes—
522C22×C3#55612yes—
523C2×D3—24no—
524C2×D3—28no—
525C2×D3—42no—
526C2×D3—42no—
527C2×D3—42no—
528C2×D3—42no—
529C2×D3—42no—
530C2×D3—42no—
531C2×D3—44no—
532C2×D3—44no—
533C2×D3—44no—
534C3×C4—24no—
535C3×C4—24no—
536C3×C4—44no—
537C3×C4—42no—
538C3×C4—82no—
539Dic3—24no—
540Dic3—42no—
541Dic3—42no—
542Dic3—44no—
543Dic3—82no—
544C23—34no—
545C23—64no—
546C23—64no—
547C23—64no—
548C23—62no—
549C23—62no—
550C23—62no—
551C23—64no—
552C23—64no—
553C23—62no—
554C23—62no—
555C23—62no—
556C23—64no—
557C23—122no—
558C23—122no—
559C23—122no—
560C23—122no—
561C23—121no—
562C23—121no—
563C23—121no—
564C23—122no—
565C23—121no—
566C23—122no—
567C23C2×#23512yes—
568C23C2×D3×D411yescharacteristic
569C23C2×D3×D411yescharacteristic
570C23#100412yes—
571C23C2×D3×Q811yescharacteristic
572C23C2×#23911yescharacteristic
573C23#100712yes—
574C23D3×#4112yes—
575C23D3×#4111yescharacteristic
576C23D3×#4111yescharacteristic
577C23C22×#6711yescharacteristic
578Q8—416no—
579C2×C4—24no—
580C2×C4—24no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—24no—
590C2×C4—24no—
591C2×C4—24no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—34no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—632no—
610C2×C4—62no—
611C2×C4—62no—
612C2×C4—62no—
613C2×C4—62no—
614C2×C4—62no—
615C2×C4—62no—
616C2×C4—64no—
617C2×C4—64no—
618C2×C4—64no—
619C2×C4—62no—
620C2×C4—62no—
621C2×C4—64no—
622C2×C4—64no—
623C2×C4—122no—
624C2×C4—122no—
625C2×C4—122no—
626C2×C4—122no—
627C2×C4—121no—
628C2×C4—121no—
629C2×C4—121no—
630C2×C4—121no—
631C2×C4—122no—
632C2×C4—122no—
633C2×C4—122no—
634C2×C4—122no—
635C2×C4—122no—
636C2×C4—122no—
637C2×C4—122no—
638C2×C4—122no—
639C2×C4—122no—
640C2×C4#93714yes—
641C2×C4#94614yes—
642C2×C3#239611yescharacteristic
643C2×C3#240411yescharacteristic
644C2×C3#243211yescharacteristic
645C2×C3#243812yes—
646C2×C3#243912yes—
647C2×C3#247412yes—
648C2×C3#247712yes—
649C2×C3#259911yescharacteristic
650C2×C3#261011yescharacteristic
651C2×C3#262811yescharacteristic
652C2×C3#284811yescharacteristic
653D3—44no—
654C22—68no—
655C22—64no—
656C22—122no—
657C22—122no—
658C22—122no—
659C22—122no—
660C22—122no—
661C22—122no—
662C22—124no—
663C22—124no—
664C22—124no—
665C22#549012yes—
666C22#549012yes—
667C22#556812yes—
668C22#557812yes—
669C22#558012yes—
670C22#558112yes—
671C22D3×C22⋊Q811yescharacteristic
672C22D3×C22⋊Q811yescharacteristic
673C22#562711yescharacteristic
674C22#562711yescharacteristic
675C22#563112yes—
676C22#563312yes—
677C22#567012yes—
678C22#567912yes—
679C22#572912yes—
680C22#573312yes—
681C22#579111yescharacteristic
682C22C2×#93711yescharacteristic
683C22#580511yescharacteristic
684C22C2×#94611yescharacteristic
685C22#581611yescharacteristic
686C22#582811yescharacteristic
687C22#583211yescharacteristic
688C4—24no—
689C4—24no—
690C4—42no—
691C4—44no—
692C4—64no—
693C4—82no—
694C4—122no—
695C4—122no—
696C4—124no—
697C4—242no—
698C3#1488711yescharacteristic, complemented, Sylow
699C2—124no—
700C2#7650112yes—
701C2#7651512yes—
702C2#7651612yes—
703C2#7662712yes—
704C2#8212511yescharacteristic
705C2#8249111yescharacteristic
706C2#8250211yescharacteristic
707C2#8275911yescharacteristic
708C2#8283411yescharacteristic
709C2#8300011yescharacteristic
710C2#8326211yescharacteristic
subgroupG/H
center, FrattiniC24C23×D3
commutator, socle, cosocleC24×C3C24
FittingC2×C3×#391C2
prime radicalC24×D3C23
hypercenterC2×#391D3
hypocenterC3#14887
LCSG ▷ C24×C3 ▷ C3
UCSC1 ◁ C24 ◁ C2×#391

Complements

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

NH ≅ G/Ntype
2. C2×C3×#391C2semidirect
4. C2×#5230C2semidirect
6. #71566C2semidirect
7. #71534C2semidirect
10. #69414C2semidirect
11. #69406C2semidirect
45. C2×#391D3semidirect
75. C22×C4×D3Q8semidirect
698. C3#14887semidirect

Maximal subgroups

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

subgroupindex
C22×#8722
C2×C3×#3912
C2×#52312
C2×#52302
#715702
#715662
#715342
#694282
(C2×D3)⋊#952
#694142
#694062
#148873