Finite Group Databasebeta

← #10729046

Subgroups — #10729046

Order 768.

2150 conjugacy classes of subgroups.

The subgroup lattice has 724 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#878C211yesmaximal, characteristic
2C2×C3×#385C211yesmaximal, characteristic
3C2×#5210C211yesmaximal, characteristic
4C22×C3⋊#95C211yesmaximal, characteristic
5#71568C212yesmaximal, complemented
6#71565C212yesmaximal, complemented
7#71544C211yesmaximal, characteristic
8#71544C211yesmaximal, characteristic
9#71544C211yesmaximal, characteristic
10#69424C212yesmaximal, complemented
11#69409C212yesmaximal, complemented
12#14914—31nomaximal, Sylow
13C24×C3×C4C2211yescharacteristic
14C24×Dic3C2211yescharacteristic
15C24×Dic3C2211yescharacteristic
16C23×Dic6C2211yescharacteristic
17C22×C3×C4⋊C4C2212yes—
18C22×C3×C22⋊C4C2211yescharacteristic
19C22×C3×C22⋊C4C2211yescharacteristic
20C22×#217C2211yescharacteristic
21C22×#217C2211yescharacteristic
22C22×C4⋊Dic3C2211yescharacteristic
23C22×Dic3⋊C4C2211yescharacteristic
24C22×Dic3⋊C4C2211yescharacteristic
25C22×Dic3⋊C4C2212yes—
26C2×#878—24no—
27C2×#878—24no—
28C2×#878—24no—
29C3×#385—24no—
30C2×C3×#95C2212yes—
31#5210—24no—
32C2×C3⋊#95—24no—
33C2×C3⋊#95C2212yes—
34C2×C3⋊#95C2212yes—
35C2×C3⋊#95C2212yes—
36C2×C3⋊#95C2212yes—
37C2×C3⋊#95C2212yes—
38C2×C3⋊#95C2212yes—
39C2×C3⋊#95C2212yes—
40C2×C3⋊#95C2212yes—
41C2×C3⋊#95C2212yes—
42C22×C22⋊Q8—31no—
43C2×#385—31no—
44C2×#385D311yescharacteristic
45C22×#95—31no—
46#1446—32no—
47#1444—32no—
48#1441—32no—
49#1441—32no—
50C22⋊#95—31no—
51C22⋊#95—31no—
52C22⋊#95—31no—
53C25×C3C2311yescharacteristic
54C23×C3×C4—22no—
55C23×C3×C4—24no—
56C23×C3×C4—22no—
57C23×C3×C4—22no—
58C23×C3×C4—42no—
59C23×C3×C4C2311yescharacteristic
60C23×C3×C4C2311yescharacteristic
61C23×C3×C4C2312yes—
62C23×C3×C4C2312yes—
63C23×Dic3—22no—
64C23×Dic3—22no—
65C23×Dic3—22no—
66C23×Dic3—22no—
67C23×Dic3—24no—
68C23×Dic3—42no—
69C23×Dic3—42no—
70C23×Dic3—42no—
71C23×Dic3D414yes—
72C23×Dic3Q814yescomplemented
73C23×Dic3C2311yescharacteristic
74C23×Dic3C2311yescharacteristic
75C23×Dic3C2311yescharacteristic
76C23×Dic3C2312yes—
77C23×Dic3C2312yes—
78C23×Dic3C2311yescharacteristic
79C22×Dic6—22no—
80C22×Dic6—22no—
81C22×Dic6—22no—
82C22×Dic6—24no—
83C22×Dic6—42no—
84C2×C3×C4⋊C4—24no—
85C2×C3×C4⋊C4—24no—
86C2×C3×C4⋊C4—42no—
87C2×C3×C22⋊C4—22no—
88C2×C3×C22⋊C4—22no—
89C2×C3×C22⋊C4—22no—
90C2×C3×C22⋊C4—22no—
91C2×C3×C22⋊C4D414yes—
92C2×C3×C22⋊C4Q814yescomplemented
93C2×#217—22no—
94C2×#217—22no—
95C2×#217—22no—
96C2×#217—22no—
97C2×#217D414yes—
98C2×#217Q814yescomplemented
99C2×C4⋊Dic3—22no—
100C2×C4⋊Dic3—22no—
101C2×C4⋊Dic3—22no—
102C2×Dic3⋊C4—22no—
103C2×Dic3⋊C4—22no—
104C2×Dic3⋊C4—22no—
105C2×Dic3⋊C4—22no—
106C2×Dic3⋊C4—22no—
107C2×Dic3⋊C4—22no—
108C2×Dic3⋊C4—24no—
109C2×Dic3⋊C4—24no—
110C2×Dic3⋊C4—42no—
111#878—232no—
112C3×#95—24no—
113C3⋊#95—24no—
114C3⋊#95—24no—
115C3⋊#95—24no—
116C3⋊#95—24no—
117C3⋊#95—42no—
118C3⋊#95—42no—
119C3⋊#95—42no—
120C3⋊#95—42no—
121C3⋊#95—42no—
122C23×Q8—31no—
123C24×C4—31no—
124C24×C4—31no—
125C24×C4C2×D311yescharacteristic
126C2×C22⋊Q8—64no—
127C2×C22⋊Q8—64no—
128C2×C22⋊Q8—64no—
129C22×C4⋊C4—31no—
130C22×C4⋊C4—31no—
131C22×C4⋊C4—31no—
132C22×C4⋊C4—32no—
133C22×C4⋊C4C2×D312yes—
134C22×C22⋊C4—31no—
135C22×C22⋊C4—31no—
136C22×C22⋊C4C2×D311yescharacteristic
137C22×C22⋊C4C2×D311yescharacteristic
138#385—24no—
139#385—64no—
140C2×#95—32no—
141C2×#95—32no—
142C2×#95—32no—
143C2×#95—32no—
144C2×#95—32no—
145C2×#95—32no—
146C2×#95—32no—
147C2×#95—32no—
148C2×#95—32no—
149C2×#95—64no—
150C2×#95C2×D312yes—
151C24×C3—22no—
152C24×C3—22no—
153C24×C3—22no—
154C24×C3—22no—
155C24×C3C2×D412yes—
156C24×C3C2×D412yes—
157C24×C3C2×D412yes—
158C24×C3C2×Q812yes—
159C24×C3C2×Q812yes—
160C24×C3C2×Q812yes—
161C24×C3#4112yes—
162C24×C3C2411yescharacteristic
163C22×C3×C4—28no—
164C22×C3×C4—22no—
165C22×C3×C4—22no—
166C22×C3×C4—22no—
167C22×C3×C4—22no—
168C22×C3×C4—21no—
169C22×C3×C4—21no—
170C22×C3×C4—21no—
171C22×C3×C4—21no—
172C22×C3×C4—22no—
173C22×C3×C4—22no—
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—48no—
185C22×C3×C4—48no—
186C22×C3×C4—44no—
187C22×C3×C4—44no—
188C22×C3×C4C2×D412yes—
189C22×C3×C4C2×D412yes—
190C22×C3×C4C2×Q814yes—
191C22×C3×C4#4114yes—
192C22×Dic3—21no—
193C22×Dic3—21no—
194C22×Dic3—22no—
195C22×Dic3—28no—
196C22×Dic3—28no—
197C22×Dic3—22no—
198C22×Dic3—22no—
199C22×Dic3—21no—
200C22×Dic3—21no—
201C22×Dic3—21no—
202C22×Dic3—21no—
203C22×Dic3—21no—
204C22×Dic3—21no—
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×Dic3—22no—
220C22×Dic3—22no—
221C22×Dic3—28no—
222C22×Dic3—44no—
223C22×Dic3—48no—
224C22×Dic3—48no—
225C22×Dic3—48no—
226C22×Dic3—44no—
227C22×Dic3—44no—
228C22×Dic3—48no—
229C22×Dic3C2×D412yes—
230C22×Dic3C2×D412yes—
231C22×Dic3C2×D412yes—
232C22×Dic3C2×D412yes—
233C22×Dic3C2×Q814yes—
234C22×Dic3C2×Q814yes—
235C22×Dic3#4114yes—
236C2×Dic6—28no—
237C2×Dic6—48no—
238C2×Dic6—48no—
239C2×Dic6—44no—
240C2×Dic6—44no—
241C3×C4⋊C4—48no—
242C3×C22⋊C4—28no—
243C3×C22⋊C4—28no—
244#217—28no—
245#217—28no—
246C4⋊Dic3—28no—
247Dic3⋊C4—28no—
248Dic3⋊C4—28no—
249Dic3⋊C4—48no—
250C25C22×D311yescharacteristic
251C22×Q8—62no—
252C22×Q8—62no—
253C22×Q8—64no—
254C22×Q8—62no—
255C22×Q8—122no—
256C23×C4—22no—
257C23×C4—24no—
258C23×C4—22no—
259C23×C4—22no—
260C23×C4—31no—
261C23×C4—31no—
262C23×C4—31no—
263C23×C4—32no—
264C23×C4—31no—
265C23×C4—32no—
266C23×C4—34no—
267C23×C4—34no—
268C23×C4—42no—
269C23×C4—64no—
270C23×C4—62no—
271C23×C4—62no—
272C23×C4—62no—
273C23×C4—62no—
274C23×C4—122no—
275C23×C4—122no—
276C23×C4—122no—
277C23×C4C22×D312yes—
278C23×C4C22×D312yes—
279C23×C4C22×D311yescharacteristic
280C23×C4C22×D311yescharacteristic
281C22⋊Q8—632no—
282C2×C4⋊C4—24no—
283C2×C4⋊C4—24no—
284C2×C4⋊C4—42no—
285C2×C4⋊C4—62no—
286C2×C4⋊C4—62no—
287C2×C4⋊C4—62no—
288C2×C4⋊C4—62no—
289C2×C4⋊C4—62no—
290C2×C4⋊C4—62no—
291C2×C4⋊C4—62no—
292C2×C4⋊C4—62no—
293C2×C4⋊C4—62no—
294C2×C4⋊C4—64no—
295C2×C4⋊C4—64no—
296C2×C4⋊C4—122no—
297C2×C22⋊C4—22no—
298C2×C22⋊C4—22no—
299C2×C22⋊C4—22no—
300C2×C22⋊C4—22no—
301C2×C22⋊C4—34no—
302C2×C22⋊C4—34no—
303C2×C22⋊C4—62no—
304C2×C22⋊C4—62no—
305C2×C22⋊C4—62no—
306C2×C22⋊C4—62no—
307C2×C22⋊C4Dic614yescomplemented
308C2×C22⋊C4#6714yes—
309#95—24no—
310#95—64no—
311#95—64no—
312#95—64no—
313#95—64no—
314#95—122no—
315#95—122no—
316#95—122no—
317#95—122no—
318#95—122no—
319C23×C3—22no—
320C23×C3—22no—
321C23×C3—24no—
322C23×C3—24no—
323C23×C3—22no—
324C23×C3—22no—
325C23×C3—24no—
326C23×C3—24no—
327C23×C3—24no—
328C23×C3—24no—
329C23×C3—24no—
330C23×C3—24no—
331C23×C3—24no—
332C23×C3—24no—
333C23×C3—24no—
334C23×C3—24no—
335C23×C3#12014yes—
336C23×C3C22⋊Q814yes—
337C23×C3C22⋊Q814yes—
338C23×C3C22⋊Q814yes—
339C23×C3C4⋊Q814yes—
340C23×C3C4⋊Q814yes—
341C23×C3C4⋊Q814yes—
342C23×C3C22×D411yescharacteristic
343C23×C3C22×D411yescharacteristic
344C23×C3C22×D411yescharacteristic
345C23×C3C22×Q811yescharacteristic
346C23×C3C22×Q811yescharacteristic
347C23×C3C22×Q811yescharacteristic
348C23×C3C2×#4111yescharacteristic
349C23×C3C2×#4112yes—
350C23×C3C2×#4112yes—
351C23×C3H(5,𝔽2)12yes—
352C23×C3H(5,𝔽2)12yes—
353C2×C3×C4—22no—
354C2×C3×C4—22no—
355C2×C3×C4—24no—
356C2×C3×C4—24no—
357C2×C3×C4—24no—
358C2×C3×C4—24no—
359C2×C3×C4—24no—
360C2×C3×C4—24no—
361C2×C3×C4—24no—
362C2×C3×C4—24no—
363C2×C3×C4—416no—
364C2×C3×C4—42no—
365C2×C3×C4—41no—
366C2×C3×C4—41no—
367C2×C3×C4—42no—
368C2×C3×C4—41no—
369C2×C3×C4—41no—
370C2×C3×C4—42no—
371C2×C3×C4—42no—
372C2×C3×C4—42no—
373C2×C3×C4—42no—
374C2×C3×C4—42no—
375C2×C3×C4—42no—
376C2×C3×C4—42no—
377C2×C3×C4—42no—
378C2×Dic3—22no—
379C2×Dic3—22no—
380C2×Dic3—24no—
381C2×Dic3—24no—
382C2×Dic3—22no—
383C2×Dic3—22no—
384C2×Dic3—22no—
385C2×Dic3—24no—
386C2×Dic3—24no—
387C2×Dic3—22no—
388C2×Dic3—22no—
389C2×Dic3—24no—
390C2×Dic3—24no—
391C2×Dic3—24no—
392C2×Dic3—24no—
393C2×Dic3—24no—
394C2×Dic3—24no—
395C2×Dic3—416no—
396C2×Dic3—416no—
397C2×Dic3—41no—
398C2×Dic3—41no—
399C2×Dic3—42no—
400C2×Dic3—41no—
401C2×Dic3—42no—
402C2×Dic3—41no—
403C2×Dic3—42no—
404C2×Dic3—42no—
405C2×Dic3—42no—
406C2×Dic3—42no—
407C2×Dic3—42no—
408C2×Dic3—42no—
409C2×Dic3—42no—
410C2×Dic3—42no—
411C2×Dic3—42no—
412C2×Dic3—42no—
413C2×Dic3#12314yes—
414Dic6—416no—
415C24—22no—
416C24—22no—
417C24—22no—
418C24—22no—
419C24C2×Dic612yes—
420C24#23512yes—
421C24D3×D412yes—
422C24D3×D412yes—
423C24D3×Q812yes—
424C24D3×Q812yes—
425C24C2×#6712yes—
426C24C23×D311yescharacteristic
427C2×Q8—68no—
428C2×Q8—128no—
429C2×Q8—124no—
430C2×Q8—128no—
431C2×Q8—124no—
432C22×C4—28no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—32no—
454C22×C4—34no—
455C22×C4—32no—
456C22×C4—32no—
457C22×C4—32no—
458C22×C4—34no—
459C22×C4—34no—
460C22×C4—48no—
461C22×C4—48no—
462C22×C4—44no—
463C22×C4—44no—
464C22×C4—68no—
465C22×C4—61no—
466C22×C4—62no—
467C22×C4—61no—
468C22×C4—68no—
469C22×C4—68no—
470C22×C4—62no—
471C22×C4—62no—
472C22×C4—62no—
473C22×C4—62no—
474C22×C4—61no—
475C22×C4—61no—
476C22×C4—61no—
477C22×C4—62no—
478C22×C4—62no—
479C22×C4—62no—
480C22×C4—62no—
481C22×C4—62no—
482C22×C4—62no—
483C22×C4—62no—
484C22×C4—62no—
485C22×C4—62no—
486C22×C4—62no—
487C22×C4—62no—
488C22×C4—62no—
489C22×C4—62no—
490C22×C4—62no—
491C22×C4—61no—
492C22×C4—61no—
493C22×C4—61no—
494C22×C4—124no—
495C22×C4—128no—
496C22×C4—128no—
497C22×C4—128no—
498C22×C4—124no—
499C22×C4—128no—
500C22×C4—124no—
501C22×C4C2×Dic614yes—
502C22×C4#23714yes—
503C22×C4C2×#6712yes—
504C22×C4C2×#6712yes—
505C4⋊C4—48no—
506C4⋊C4—68no—
507C4⋊C4—68no—
508C4⋊C4—68no—
509C4⋊C4—128no—
510C22⋊C4—28no—
511C22⋊C4—28no—
512C22⋊C4—68no—
513C22⋊C4—68no—
514C22×C3—24no—
515C22×C3—24no—
516C22×C3—24no—
517C22×C3—24no—
518C22×C3—24no—
519C22×C3—28no—
520C22×C3—28no—
521C22×C3—24no—
522C22×C3—28no—
523C22×C3—28no—
524C22×C3#39518yes—
525C22×C3C2×#12011yescharacteristic
526C22×C3C2×C22⋊Q811yescharacteristic
527C22×C3C2×C22⋊Q811yescharacteristic
528C22×C3C2×C22⋊Q811yescharacteristic
529C22×C3C2×#12311yescharacteristic
530C22×C3#52511yescharacteristic
531C22×C3#52912yes—
532C22×C3C2×C4⋊Q811yescharacteristic
533C22×C3C2×C4⋊Q811yescharacteristic
534C22×C3C2×C4⋊Q811yescharacteristic
535C22×C3#53312yes—
536C22×C3#53411yescharacteristic
537C22×C3#53511yescharacteristic
538C22×C3#53812yes—
539C22×C3#54312yes—
540C22×C3#54612yes—
541C22×C3#54612yes—
542C22×C3#54612yes—
543C22×C3#54612yes—
544C22×C3#55412yes—
545C22×C3#55412yes—
546C22×C3#55412yes—
547C22×C3#55412yes—
548C3×C4—42no—
549C3×C4—44no—
550C3×C4—42no—
551C3×C4—44no—
552Dic3—24no—
553Dic3—44no—
554Dic3—42no—
555Dic3—42no—
556Dic3—42no—
557Dic3—44no—
558C23—24no—
559C23—24no—
560C23—24no—
561C23—22no—
562C23—22no—
563C23—24no—
564C23—24no—
565C23—24no—
566C23—24no—
567C23—24no—
568C23—24no—
569C23—24no—
570C23—22no—
571C23—24no—
572C23—22no—
573C23—24no—
574C23#88814yes—
575C23#88814yes—
576C23#89614yes—
577C23#89614yes—
578C23#92414yes—
579C23#93714yes—
580C23#94414yes—
581C23C22×Dic611yescharacteristic
582C23C2×#23511yescharacteristic
583C23C2×D3×D411yescharacteristic
584C23C2×D3×D411yescharacteristic
585C23C2×#23712yes—
586C23#100412yes—
587C23C2×D3×Q811yescharacteristic
588C23C2×D3×Q811yescharacteristic
589C23D3×#4112yes—
590C23#100912yes—
591C23C22×#6711yescharacteristic
592Q8—1216no—
593C2×C4—24no—
594C2×C4—24no—
595C2×C4—24no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—24no—
599C2×C4—24no—
600C2×C4—24no—
601C2×C4—24no—
602C2×C4—24no—
603C2×C4—34no—
604C2×C4—416no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—41no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—41no—
619C2×C4—62no—
620C2×C4—64no—
621C2×C4—62no—
622C2×C4—64no—
623C2×C4—64no—
624C2×C4—64no—
625C2×C4—64no—
626C2×C4—64no—
627C2×C4—62no—
628C2×C4—62no—
629C2×C4—64no—
630C2×C4—64no—
631C2×C4—64no—
632C2×C4—62no—
633C2×C4—62no—
634C2×C4—62no—
635C2×C4—64no—
636C2×C4—1216no—
637C2×C4—1216no—
638C2×C4—121no—
639C2×C4—122no—
640C2×C4—121no—
641C2×C4—122no—
642C2×C4—122no—
643C2×C4—122no—
644C2×C4—122no—
645C2×C4—122no—
646C2×C4—122no—
647C2×C4—122no—
648C2×C4—122no—
649C2×C4—122no—
650C2×C4—121no—
651C2×C4—122no—
652C2×C4—122no—
653C2×C4—121no—
654C2×C3—28no—
655C2×C3C2×#39511yescharacteristic
656C2×C3#240411yescharacteristic
657C2×C3#240511yescharacteristic
658C2×C3#245012yes—
659C2×C3#245012yes—
660C2×C3#250211yescharacteristic
661C2×C3#254912yes—
662C2×C3#254912yes—
663C2×C3#258311yescharacteristic
664C2×C3#262611yescharacteristic
665C2×C3#285511yescharacteristic
666C22—24no—
667C22—28no—
668C22—24no—
669C22—24no—
670C22—28no—
671C22—28no—
672C22—24no—
673C22—24no—
674C22—28no—
675C22—24no—
676C22#497918yes—
677C22C2×#88811yescharacteristic
678C22C2×#88811yescharacteristic
679C22C2×#89611yescharacteristic
680C22C2×#89611yescharacteristic
681C22#553812yes—
682C22#553912yes—
683C22#554312yes—
684C22#556012yes—
685C22#558611yescharacteristic
686C22#559112yes—
687C22#559211yescharacteristic
688C22#562212yes—
689C22#563412yes—
690C22#563512yes—
691C22#563612yes—
692C22D3×#12311yescharacteristic
693C22#565311yescharacteristic
694C22#565412yes—
695C22C2×#92411yescharacteristic
696C22C2×#93711yescharacteristic
697C22#580612yes—
698C22C2×#94411yescharacteristic
699C22#582912yes—
700C4—44no—
701C4—42no—
702C4—44no—
703C4—42no—
704C4—64no—
705C4—124no—
706C4—122no—
707C4—124no—
708C4—122no—
709C4—122no—
710C3#1491411yescharacteristic, complemented, Sylow
711C2—28no—
712C2#7442312yes—
713C2#7446011yescharacteristic
714C2#7447212yes—
715C2#7448811yescharacteristic
716C2#7458911yescharacteristic
717C2#7462111yescharacteristic
718C2C2×#497911yescharacteristic
719C2#8236112yes—
720C2#8258111yescharacteristic
721C2#8278112yes—
722C2#8289311yescharacteristic
subgroupG/H
center, FrattiniC24C23×D3
commutator, socle, cosocleC24×C3C24
FittingC2×C3×#385C2
prime radicalC25×C3C23
hypercenterC2×#385D3
hypocenterC3#14914
LCSG ▷ C24×C3 ▷ C3
UCSC1 ◁ C24 ◁ C2×#385

Complements

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

NH ≅ G/Ntype
5. #71568C2semidirect
6. #71565C2semidirect
10. #69424C2semidirect
11. #69409C2semidirect
72. C23×Dic3Q8semidirect
92. C2×C3×C22⋊C4Q8semidirect
98. C2×#217Q8semidirect
307. C2×C22⋊C4Dic6semidirect
710. C3#14914semidirect

Maximal subgroups

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

subgroupindex
C22×#8782
C2×C3×#3852
C2×#52102
C22×C3⋊#952
#715682
#715652
#715442
#694242
#694092
#149143