Finite Group Databasebeta

← #10727105

Subgroups — #10727105

Order 768.

2242 conjugacy classes of subgroups.

The subgroup lattice has 714 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#924C211yesmaximal, characteristic
2C22×C3×#95C211yesmaximal, characteristic
3C2×#5210C211yesmaximal, characteristic
4C2×#5212C212yesmaximal
5C2×#4656C211yesmaximal, characteristic, complemented
6#72127C212yesmaximal
7#69424C211yesmaximal, characteristic, complemented
8#69420C212yesmaximal, complemented
9#69419C211yesmaximal, characteristic, complemented
10#69411C212yesmaximal, complemented
11#69410C211yesmaximal, characteristic, complemented
12#14758—31nomaximal, Sylow
13C24×C3×C4C2211yescharacteristic
14C24×C3×C4C2212yes—
15C23×Dic6C2211yescharacteristic
16C22×#217C2211yescharacteristic
17C22×#217C2212yes—
18C22×#217C2211yescharacteristic
19C22×C4⋊Dic3C2211yescharacteristic
20C22×C4⋊Dic3C2212yes—
21C22×C4⋊Dic3C2212yes—
22C2×#924—24no—
23C2×#924—28no—
24C22×Dic3⋊C4C2211yescharacteristic
25C22×Dic3⋊C4C2212yes—
26C22×Dic3⋊C4C2211yescharacteristic
27C2×C3×#95—24no—
28C2×C3×#95C2211yescharacteristic
29C2×C3×#95C2212yes—
30C2×C3×#95C2211yescharacteristic
31#5212—28no—
32#5210—24no—
33C2×C3⋊#95C2212yes—
34C2×C3⋊#95C2211yescharacteristic
35C2×C3⋊#95C2212yes—
36C2×C3⋊#95C2212yes—
37C2×C3⋊#95C2212yes—
38C2×C3⋊#95C2211yescharacteristic
39C2×C3⋊#95C2212yes—
40C2×C3⋊#95C2212yes—
41#4656—24no—
42C22×C22⋊Q8—31no—
43C2×#395—31no—
44C2×#385—31no—
45C2×#380—32no—
46C22×#95D311yescharacteristic
47#1446—31no—
48#1444—31no—
49#1444—32no—
50#1441—31no—
51#1441—32no—
52C22⋊#95—32no—
53C25×C3C2311yescharacteristic
54C23×C3×C4—24no—
55C23×C3×C4—24no—
56C23×C3×C4—24no—
57C23×C3×C4—22no—
58C23×C3×C4—24no—
59C23×C3×C4—22no—
60C23×C3×C4—24no—
61C23×C3×C4—42no—
62C23×C3×C4—42no—
63C23×C3×C4—42no—
64C23×C3×C4D414yes—
65C23×C3×C4Q818yescomplemented
66C23×C3×C4C2311yescharacteristic
67C23×C3×C4C2312yes—
68C23×C3×C4C2311yescharacteristic
69C23×C3×C4C2312yes—
70C23×Dic3C2311yescharacteristic
71C23×Dic3C2311yescharacteristic
72C23×Dic3C2312yes—
73C23×Dic3C2312yes—
74C23×Dic3C2311yescharacteristic
75C23×Dic3C2311yescharacteristic
76C22×Dic6—22no—
77C22×Dic6—24no—
78C22×Dic6—22no—
79C22×Dic6—22no—
80C22×Dic6—42no—
81C2×#217—22no—
82C2×#217—24no—
83C2×#217—24no—
84C2×#217—22no—
85C2×#217—22no—
86C2×#217—24no—
87C2×#217D418yes—
88C2×#217Q814yescomplemented
89C2×C4⋊Dic3—22no—
90C2×C4⋊Dic3—24no—
91C2×C4⋊Dic3—24no—
92C2×C4⋊Dic3—24no—
93C2×C4⋊Dic3—24no—
94C2×C4⋊Dic3—24no—
95C2×C4⋊Dic3—42no—
96C2×C4⋊Dic3—42no—
97#924—232no—
98C2×Dic3⋊C4—24no—
99C2×Dic3⋊C4—24no—
100C2×Dic3⋊C4—24no—
101C2×Dic3⋊C4—22no—
102C2×Dic3⋊C4—22no—
103C2×Dic3⋊C4—22no—
104C2×Dic3⋊C4—22no—
105C2×Dic3⋊C4—42no—
106C3×#95—22no—
107C3×#95—24no—
108C3×#95D414yes—
109C3⋊#95—24no—
110C3⋊#95—22no—
111C3⋊#95—22no—
112C3⋊#95—24no—
113C3⋊#95—24no—
114C3⋊#95—24no—
115C3⋊#95—42no—
116C3⋊#95—42no—
117C23×Q8—31no—
118C24×C4C2×D312yes—
119C24×C4C2×D311yescharacteristic
120C2×C22⋊Q8—68no—
121C2×C22⋊Q8—64no—
122C22×C4⋊C4—31no—
123C22×C4⋊C4—32no—
124C22×C4⋊C4—31no—
125C22×C4⋊C4—32no—
126C22×C4⋊C4—32no—
127C22×C4⋊C4—31no—
128C22×C22⋊C4—32no—
129C22×C22⋊C4—31no—
130C22×C22⋊C4—31no—
131#395—64no—
132#385—64no—
133#380—68no—
134C2×#95—24no—
135C2×#95—32no—
136C2×#95—32no—
137C2×#95—31no—
138C2×#95—32no—
139C2×#95—32no—
140C2×#95—31no—
141C2×#95—32no—
142C2×#95—32no—
143C2×#95C2×D312yes—
144C2×#95C2×D311yescharacteristic
145C2×#95C2×D311yescharacteristic
146C24×C3—21no—
147C24×C3—21no—
148C24×C3—22no—
149C24×C3—22no—
150C24×C3—21no—
151C24×C3—21no—
152C24×C3C2×D412yes—
153C24×C3C2×D414yes—
154C24×C3C2×Q814yes—
155C24×C3C2×Q812yes—
156C24×C3#4112yes—
157C24×C3C2411yescharacteristic
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—28no—
165C22×C3×C4—216no—
166C22×C3×C4—22no—
167C22×C3×C4—21no—
168C22×C3×C4—22no—
169C22×C3×C4—22no—
170C22×C3×C4—21no—
171C22×C3×C4—22no—
172C22×C3×C4—22no—
173C22×C3×C4—22no—
174C22×C3×C4—22no—
175C22×C3×C4—22no—
176C22×C3×C4—28no—
177C22×C3×C4—216no—
178C22×C3×C4—44no—
179C22×C3×C4—48no—
180C22×C3×C4—48no—
181C22×C3×C4—48no—
182C22×C3×C4—48no—
183C22×C3×C4—48no—
184C22×C3×C4—48no—
185C22×C3×C4—48no—
186C22×C3×C4C2×D412yes—
187C22×C3×C4C2×D412yes—
188C22×C3×C4C2×D412yes—
189C22×C3×C4C2×D414yes—
190C22×C3×C4C2×Q814yes—
191C22×C3×C4C2×Q814yes—
192C22×C3×C4#4112yes—
193C22×C3×C4#4114yes—
194C22×Dic3—21no—
195C22×Dic3—21no—
196C22×Dic3—21no—
197C22×Dic3—22no—
198C22×Dic3—22no—
199C22×Dic3—22no—
200C22×Dic3—22no—
201C22×Dic3—22no—
202C22×Dic3—22no—
203C22×Dic3—22no—
204C22×Dic3—21no—
205C22×Dic3—22no—
206C22×Dic3—21no—
207C22×Dic3—21no—
208C22×Dic3—22no—
209C22×Dic3—21no—
210C22×Dic3—22no—
211C22×Dic3—22no—
212C22×Dic3—22no—
213C22×Dic3—22no—
214C22×Dic3—22no—
215C22×Dic3—21no—
216C22×Dic3—21no—
217C22×Dic3—22no—
218C22×Dic3—22no—
219C22×Dic3—21no—
220C22×Dic3—21no—
221C22×Dic3—21no—
222C22×Dic3—22no—
223C22×Dic3—22no—
224C22×Dic3—21no—
225C22×Dic3—21no—
226C22×Dic3C2×D414yes—
227C22×Dic3C2×D414yes—
228C22×Dic3C2×Q812yes—
229C22×Dic3C2×Q812yes—
230C2×Dic6—28no—
231C2×Dic6—48no—
232C2×Dic6—48no—
233C2×Dic6—44no—
234C2×Dic6—44no—
235#217—216no—
236#217—28no—
237#217—44no—
238C4⋊Dic3—28no—
239C4⋊Dic3—48no—
240C4⋊Dic3—48no—
241Dic3⋊C4—216no—
242Dic3⋊C4—44no—
243Dic3⋊C4—44no—
244C25C22×D311yescharacteristic
245C22×Q8—64no—
246C22×Q8—62no—
247C22×Q8—62no—
248C22×Q8—62no—
249C22×Q8—122no—
250C23×C4—24no—
251C23×C4—24no—
252C23×C4—24no—
253C23×C4—24no—
254C23×C4—22no—
255C23×C4—24no—
256C23×C4—22no—
257C23×C4—32no—
258C23×C4—32no—
259C23×C4—31no—
260C23×C4—31no—
261C23×C4—31no—
262C23×C4—31no—
263C23×C4—42no—
264C23×C4—42no—
265C23×C4—42no—
266C23×C4Dic618yescomplemented
267C23×C4D1214yes—
268C23×C4C22×D312yes—
269C23×C4C22×D311yescharacteristic
270C23×C4C22×D312yes—
271C23×C4C22×D311yescharacteristic
272C22⋊Q8—632no—
273C2×C4⋊C4—64no—
274C2×C4⋊C4—64no—
275C2×C4⋊C4—64no—
276C2×C4⋊C4—64no—
277C2×C4⋊C4—62no—
278C2×C4⋊C4—64no—
279C2×C4⋊C4—64no—
280C2×C4⋊C4—62no—
281C2×C4⋊C4—62no—
282C2×C4⋊C4—62no—
283C2×C4⋊C4—64no—
284C2×C4⋊C4—62no—
285C2×C4⋊C4—64no—
286C2×C4⋊C4—122no—
287C2×C4⋊C4—122no—
288C2×C4⋊C4—122no—
289C2×C22⋊C4—38no—
290C2×C22⋊C4—34no—
291C2×C22⋊C4—64no—
292C2×C22⋊C4—62no—
293C2×C22⋊C4—64no—
294C2×C22⋊C4—62no—
295C2×C22⋊C4—62no—
296C2×C22⋊C4—64no—
297#95—24no—
298#95—22no—
299#95—64no—
300#95—64no—
301#95—62no—
302#95—62no—
303#95—64no—
304#95—64no—
305#95—122no—
306#95—122no—
307#95#6714yes—
308C23×C3—22no—
309C23×C3—24no—
310C23×C3—22no—
311C23×C3—24no—
312C23×C3—22no—
313C23×C3—22no—
314C23×C3—24no—
315C23×C3—24no—
316C23×C3—24no—
317C23×C3—22no—
318C23×C3—24no—
319C23×C3—24no—
320C23×C3—24no—
321C23×C3—22no—
322C23×C3—24no—
323C23×C3—22no—
324C23×C3—24no—
325C23×C3—22no—
326C23×C3#12014yes—
327C23×C3C22⋊Q818yes—
328C23×C3C22⋊Q814yes—
329C23×C3C4⋊Q814yes—
330C23×C3C4⋊Q818yes—
331C23×C3C22×D411yescharacteristic
332C23×C3C22×D412yes—
333C23×C3C22×D411yescharacteristic
334C23×C3C22×Q812yes—
335C23×C3C22×Q811yescharacteristic
336C23×C3C2×#4111yescharacteristic
337C23×C3C2×#4112yes—
338C23×C3C2×#4111yescharacteristic
339C23×C3H(5,𝔽2)12yes—
340C23×C3H(5,𝔽2)11yescharacteristic
341C23×C32−1+411yescharacteristic
342C2×C3×C4—22no—
343C2×C3×C4—24no—
344C2×C3×C4—22no—
345C2×C3×C4—24no—
346C2×C3×C4—24no—
347C2×C3×C4—24no—
348C2×C3×C4—24no—
349C2×C3×C4—24no—
350C2×C3×C4—24no—
351C2×C3×C4—24no—
352C2×C3×C4—22no—
353C2×C3×C4—22no—
354C2×C3×C4—24no—
355C2×C3×C4—24no—
356C2×C3×C4—22no—
357C2×C3×C4—24no—
358C2×C3×C4—416no—
359C2×C3×C4—432no—
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#12114yes—
367C2×C3×C4C22⋊Q818yescomplemented
368C2×Dic3—24no—
369C2×Dic3—24no—
370C2×Dic3—22no—
371C2×Dic3—22no—
372C2×Dic3—24no—
373C2×Dic3—24no—
374C2×Dic3—24no—
375C2×Dic3—24no—
376C2×Dic3—22no—
377C2×Dic3—24no—
378C2×Dic3—22no—
379C2×Dic3—41no—
380C2×Dic3—42no—
381C2×Dic3—41no—
382C2×Dic3—42no—
383C2×Dic3—41no—
384C2×Dic3—42no—
385C2×Dic3—41no—
386C2×Dic3—42no—
387C2×Dic3—42no—
388C2×Dic3—42no—
389C2×Dic3—42no—
390C2×Dic3—41no—
391C2×Dic3—42no—
392C2×Dic3—42no—
393C2×Dic3—41no—
394C2×Dic3—41no—
395C2×Dic3—41no—
396C2×Dic3—41no—
397C2×Dic3—41no—
398C2×Dic3—41no—
399C2×Dic3—41no—
400C2×Dic3—42no—
401C2×Dic3—42no—
402C2×Dic3—42no—
403C2×Dic3—42no—
404Dic6—416no—
405C24—21no—
406C24—22no—
407C24—21no—
408C24—22no—
409C24—21no—
410C24—21no—
411C24C2×Dic614yes—
412C24C2×D1212yes—
413C24D3×D414yes—
414C24#23712yes—
415C24D3×Q812yes—
416C24C23×D311yescharacteristic
417C2×Q8—68no—
418C2×Q8—128no—
419C2×Q8—128no—
420C2×Q8—124no—
421C2×Q8—124no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—216no—
429C22×C4—28no—
430C22×C4—22no—
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—21no—
440C22×C4—216no—
441C22×C4—28no—
442C22×C4—34no—
443C22×C4—34no—
444C22×C4—32no—
445C22×C4—32no—
446C22×C4—48no—
447C22×C4—44no—
448C22×C4—48no—
449C22×C4—48no—
450C22×C4—48no—
451C22×C4—48no—
452C22×C4—48no—
453C22×C4—48no—
454C22×C4—61no—
455C22×C4—61no—
456C22×C4—62no—
457C22×C4—62no—
458C22×C4—62no—
459C22×C4—62no—
460C22×C4—62no—
461C22×C4—61no—
462C22×C4—62no—
463C22×C4—62no—
464C22×C4—62no—
465C22×C4—61no—
466C22×C4—61no—
467C22×C4—61no—
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—61no—
477C22×C4—61no—
478C22×C4—62no—
479C22×C4—61no—
480C22×C4—62no—
481C22×C4—62no—
482C22×C4—61no—
483C22×C4—62no—
484C22×C4—61no—
485C22×C4—61no—
486C22×C4C2×Dic614yes—
487C22×C4C2×Dic614yes—
488C22×C4C2×D1212yes—
489C22×C4C2×D1212yes—
490C22×C4#23514yes—
491C22×C4#23512yes—
492C22×C4C2×#6714yes—
493C22×C4C2×#6712yes—
494C4⋊C4—616no—
495C4⋊C4—68no—
496C4⋊C4—128no—
497C4⋊C4—128no—
498C4⋊C4—124no—
499C4⋊C4—124no—
500C22⋊C4—616no—
501C22⋊C4—68no—
502C22⋊C4—124no—
503C22×C3—24no—
504C22×C3—24no—
505C22×C3—24no—
506C22×C3—24no—
507C22×C3—24no—
508C22×C3—24no—
509C22×C3—28no—
510C22×C3—28no—
511C22×C3—28no—
512C22×C3—28no—
513C22×C3#39518yescomplemented
514C22×C3C2×#12011yescharacteristic
515C22×C3C2×#12111yescharacteristic
516C22×C3C2×C22⋊Q812yes—
517C22×C3C2×C22⋊Q811yescharacteristic
518C22×C3C2×C22⋊Q812yes—
519C22×C3C2×C4⋊Q811yescharacteristic
520C22×C3C2×C4⋊Q812yes—
521C22×C3#53211yescharacteristic
522C22×C3#53312yes—
523C22×C3#53411yescharacteristic
524C22×C3#53711yescharacteristic
525C22×C3#54312yes—
526C22×C3#54412yes—
527C22×C3D4212yes—
528C22×C3#54612yes—
529C22×C3#54612yes—
530C22×C3#54611yescharacteristic
531C22×C3#54712yes—
532C22×C3D4×Q811yescharacteristic
533C22×C3#55112yes—
534C22×C3#55211yescharacteristic
535C22×C3#55412yes—
536C22×C3#55411yescharacteristic
537C3×C4—24no—
538C3×C4—28no—
539C3×C4—42no—
540C3×C4—44no—
541Dic3—42no—
542Dic3—44no—
543Dic3—42no—
544Dic3—44no—
545Dic3—81no—
546Dic3—81no—
547C23—24no—
548C23—22no—
549C23—24no—
550C23—24no—
551C23—24no—
552C23—22no—
553C23—24no—
554C23—24no—
555C23—24no—
556C23—22no—
557C23—22no—
558C23—24no—
559C23—22no—
560C23—24no—
561C23—22no—
562C23—24no—
563C23—22no—
564C23—22no—
565C23#86914yes—
566C23#87818yes—
567C23#88214yes—
568C23#88818yes—
569C23#89714yes—
570C23C22×Dic612yes—
571C23C22×D1211yescharacteristic
572C23C2×#23511yescharacteristic
573C23C2×#23512yes—
574C23C2×D3×D412yes—
575C23C2×#23711yescharacteristic
576C23#100411yescharacteristic
577C23#100412yes—
578C23C2×D3×Q811yescharacteristic
579C23#100711yescharacteristic
580C23C22×#6711yescharacteristic
581Q8—1216no—
582C2×C4—24no—
583C2×C4—24no—
584C2×C4—22no—
585C2×C4—24no—
586C2×C4—24no—
587C2×C4—22no—
588C2×C4—24no—
589C2×C4—24no—
590C2×C4—24no—
591C2×C4—24no—
592C2×C4—24no—
593C2×C4—24no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—24no—
597C2×C4—22no—
598C2×C4—432no—
599C2×C4—416no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—64no—
607C2×C4—64no—
608C2×C4—64no—
609C2×C4—64no—
610C2×C4—64no—
611C2×C4—62no—
612C2×C4—62no—
613C2×C4—62no—
614C2×C4—64no—
615C2×C4—64no—
616C2×C4—62no—
617C2×C4—121no—
618C2×C4—122no—
619C2×C4—122no—
620C2×C4—122no—
621C2×C4—121no—
622C2×C4—122no—
623C2×C4—122no—
624C2×C4—122no—
625C2×C4—121no—
626C2×C4—122no—
627C2×C4—121no—
628C2×C4—121no—
629C2×C4—122no—
630C2×C4—122no—
631C2×C4—121no—
632C2×C4—121no—
633C2×C4—121no—
634C2×C4—121no—
635C2×C4—122no—
636C2×C4—122no—
637C2×C4—121no—
638C2×C4—122no—
639C2×C4—121no—
640C2×C4—121no—
641C2×C4—122no—
642C2×C4#92418yescomplemented
643C2×C4#93014yes—
644C2×C3—28no—
645C2×C3C2×#39511yescharacteristic
646C2×C3#241011yescharacteristic
647C2×C3#241711yescharacteristic
648C2×C3#243011yescharacteristic
649C2×C3#243712yes—
650C2×C3#243812yes—
651C2×C3#244012yes—
652C2×C3#254912yes—
653C2×C3#256411yescharacteristic
654C2×C3#265911yescharacteristic
655C2×C3#285711yescharacteristic
656C22—24no—
657C22—28no—
658C22—28no—
659C22—24no—
660C22—24no—
661C22—24no—
662C22—24no—
663C22—28no—
664C22—24no—
665C22—28no—
666C22#465618yescomplemented
667C22C2×#86911yescharacteristic
668C22#547012yes—
669C22#547711yescharacteristic
670C22C2×#87812yes—
671C22#548312yes—
672C22C2×#88211yescharacteristic
673C22#549012yes—
674C22#549012yes—
675C22#549311yescharacteristic
676C22#549512yes—
677C22C2×#88812yes—
678C22#550012yes—
679C22#550712yes—
680C22C2×#89711yescharacteristic
681C22#551011yescharacteristic
682C22#551111yescharacteristic
683C22#551112yes—
684C22#551311yescharacteristic
685C22C2×#92412yes—
686C22C2×#93011yescharacteristic
687C22D4×#6712yes—
688C22#580611yescharacteristic
689C22Q8×#6711yescharacteristic
690C4—28no—
691C4—24no—
692C4—44no—
693C4—42no—
694C4—122no—
695C4—122no—
696C4—124no—
697C4—124no—
698C4—241no—
699C4—241no—
700C3#1475811yescharacteristic, complemented, Sylow
701C2—28no—
702C2C2×#465611yescharacteristic
703C2#7335012yes—
704C2#7335712yes—
705C2#7335811yescharacteristic
706C2#7336111yescharacteristic
707C2#7337611yescharacteristic
708C2#8207611yescharacteristic
709C2#8211012yes—
710C2#8212711yescharacteristic
711C2#8217112yes—
712C2#8219411yescharacteristic
subgroupG/H
center, FrattiniC24C23×D3
commutator, socle, cosocleC24×C3C24
FittingC22×C3×#95C2
prime radicalC25×C3C23
hypercenterC22×#95D3
hypocenterC3#14758
LCSG ▷ C24×C3 ▷ C3
UCSC1 ◁ C24 ◁ C22×#95

Complements

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

NH ≅ G/Ntype
5. C2×#4656C2semidirect
7. #69424C2semidirect
8. #69420C2semidirect
9. #69419C2semidirect
10. #69411C2semidirect
11. #69410C2semidirect
65. C23×C3×C4Q8semidirect
88. C2×#217Q8semidirect
266. C23×C4Dic6semidirect
367. C2×C3×C4C22⋊Q8semidirect
513. C22×C3#395semidirect
642. C2×C4#924semidirect
666. C22#4656semidirect
700. C3#14758semidirect

Maximal subgroups

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

subgroupindex
C22×#9242
C22×C3×#952
C2×#52102
C2×#52122
C2×#46562
#721272
#694242
#694202
#694192
#694112
#694102
#147583