Finite Group Databasebeta

← #10727944

Subgroups — #10727944

Order 768.

2190 conjugacy classes of subgroups.

The subgroup lattice has 706 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×C4⋊Dic3C211yesmaximal, characteristic
2C2×C3×#385C211yesmaximal, characteristic
3C2×#5210C211yesmaximal, characteristic
4C2×#4982C212yesmaximal, complemented
5C2×#4942C211yesmaximal, characteristic
6#71565C212yesmaximal, complemented
7#71544C211yesmaximal, characteristic
8#71544C211yesmaximal, characteristic
9#71544C211yesmaximal, characteristic
10#69411C212yesmaximal, complemented
11#69410C212yesmaximal, complemented
12#14821—31nomaximal, Sylow
13C24×C3×C4C2211yescharacteristic
14C24×Dic3C2211yescharacteristic
15C24×Dic3C2211yescharacteristic
16C22×C3×C4⋊C4C2212yes—
17C22×C3×C22⋊C4C2211yescharacteristic
18C22×C3×C22⋊C4C2211yescharacteristic
19C22×#217C2211yescharacteristic
20C22×#217C2211yescharacteristic
21C22×C4⋊Dic3—24no—
22C22×C4⋊Dic3—28no—
23C22×C4⋊Dic3C2212yes—
24C22×C4⋊Dic3C2212yes—
25C22×Dic3⋊C4C2212yes—
26C22×Dic3⋊C4C2211yescharacteristic
27C22×Dic3⋊C4C2211yescharacteristic
28C3×#385—24no—
29C2×C3×#95C2212yes—
30#5210—24no—
31C2×C3⋊#95C2211yescharacteristic
32C2×C3⋊#95C2212yes—
33C2×C3⋊#95C2212yes—
34C2×C3⋊#95C2212yes—
35C2×C3⋊#95C2212yes—
36C2×C3⋊#95C2211yescharacteristic
37C2×C3⋊#95C2212yes—
38C2×C3⋊#95C2212yes—
39C2×C3⋊#95C2212yes—
40#4982—28no—
41#4942—24no—
42C23×C4⋊C4—31no—
43C2×#398—32no—
44C2×#385—31no—
45C2×#385D311yescharacteristic
46C2×#380—31no—
47#1441—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—22no—
56C23×C3×C4—24no—
57C23×C3×C4—24no—
58C23×C3×C4D414yes—
59C23×C3×C4C2311yescharacteristic
60C23×C3×C4C2311yescharacteristic
61C23×C3×C4C2312yes—
62C23×C3×C4C2312yes—
63C23×Dic3—22no—
64C23×Dic3—22no—
65C23×Dic3—22no—
66C23×Dic3—24no—
67C23×Dic3—24no—
68C23×Dic3—22no—
69C23×Dic3—22no—
70C23×Dic3—42no—
71C23×Dic3—42no—
72C23×Dic3D414yes—
73C23×Dic3C2311yescharacteristic
74C23×Dic3C2311yescharacteristic
75C23×Dic3C2311yescharacteristic
76C23×Dic3C2312yes—
77C23×Dic3C2312yes—
78C23×Dic3C2311yescharacteristic
79C2×C3×C4⋊C4—24no—
80C2×C3×C4⋊C4—24no—
81C2×C3×C4⋊C4—24no—
82C2×C3×C22⋊C4—22no—
83C2×C3×C22⋊C4—22no—
84C2×C3×C22⋊C4—22no—
85C2×C3×C22⋊C4—22no—
86C2×C3×C22⋊C4—22no—
87C2×C3×C22⋊C4Q814yes—
88C2×#217—22no—
89C2×#217—22no—
90C2×#217—22no—
91C2×#217—22no—
92C2×#217—22no—
93C2×#217Q814yes—
94C2×C4⋊Dic3—232no—
95C2×C4⋊Dic3—24no—
96C2×C4⋊Dic3—24no—
97C2×C4⋊Dic3—24no—
98C2×C4⋊Dic3—24no—
99C2×C4⋊Dic3—24no—
100C2×C4⋊Dic3D418yes—
101C2×Dic3⋊C4—24no—
102C2×Dic3⋊C4—24no—
103C2×Dic3⋊C4—24no—
104C2×Dic3⋊C4—22no—
105C2×Dic3⋊C4—22no—
106C2×Dic3⋊C4—24no—
107C2×Dic3⋊C4—42no—
108C3×#95—24no—
109C3⋊#95—24no—
110C3⋊#95—22no—
111C3⋊#95—22no—
112C3⋊#95—24no—
113C3⋊#95—24no—
114C3⋊#95—24no—
115C3⋊#95—42no—
116C3⋊#95—42no—
117C3⋊#95—42no—
118C24×C4—31no—
119C24×C4—31no—
120C24×C4C2×D311yescharacteristic
121C22×C4⋊C4—32no—
122C22×C4⋊C4—31no—
123C22×C4⋊C4—32no—
124C22×C4⋊C4—32no—
125C22×C4⋊C4—31no—
126C22×C4⋊C4—68no—
127C22×C4⋊C4—64no—
128C22×C4⋊C4C2×D312yes—
129C22×C22⋊C4—31no—
130C22×C22⋊C4—31no—
131C22×C22⋊C4C2×D311yescharacteristic
132C22×C22⋊C4C2×D311yescharacteristic
133#398—68no—
134#385—24no—
135#385—64no—
136#380—64no—
137C2×#95—32no—
138C2×#95—32no—
139C2×#95—32no—
140C2×#95—31no—
141C2×#95—31no—
142C2×#95—32no—
143C2×#95—32no—
144C2×#95—32no—
145C2×#95—32no—
146C2×#95C2×D312yes—
147C24×C3—22no—
148C24×C3—22no—
149C24×C3—22no—
150C24×C3—22no—
151C24×C3C2×D412yes—
152C24×C3C2×D412yes—
153C24×C3C2×Q812yes—
154C24×C3C2×Q812yes—
155C24×C3#4112yes—
156C24×C3#4112yes—
157C24×C3#4112yes—
158C24×C3C2411yescharacteristic
159C22×C3×C4—28no—
160C22×C3×C4—216no—
161C22×C3×C4—28no—
162C22×C3×C4—28no—
163C22×C3×C4—22no—
164C22×C3×C4—22no—
165C22×C3×C4—22no—
166C22×C3×C4—22no—
167C22×C3×C4—216no—
168C22×C3×C4—22no—
169C22×C3×C4—22no—
170C22×C3×C4—22no—
171C22×C3×C4—22no—
172C22×C3×C4—22no—
173C22×C3×C4—22no—
174C22×C3×C4—21no—
175C22×C3×C4—21no—
176C22×C3×C4—22no—
177C22×C3×C4—21no—
178C22×C3×C4—21no—
179C22×C3×C4—21no—
180C22×C3×C4—21no—
181C22×C3×C4—22no—
182C22×C3×C4—22no—
183C22×C3×C4—22no—
184C22×C3×C4C2×D414yes—
185C22×C3×C4C2×D414yes—
186C22×C3×C4C2×Q814yes—
187C22×C3×C4#4114yes—
188C22×Dic3—21no—
189C22×Dic3—21no—
190C22×Dic3—22no—
191C22×Dic3—28no—
192C22×Dic3—28no—
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—22no—
202C22×Dic3—21no—
203C22×Dic3—22no—
204C22×Dic3—22no—
205C22×Dic3—22no—
206C22×Dic3—22no—
207C22×Dic3—22no—
208C22×Dic3—22no—
209C22×Dic3—28no—
210C22×Dic3—21no—
211C22×Dic3—22no—
212C22×Dic3—21no—
213C22×Dic3—21no—
214C22×Dic3—21no—
215C22×Dic3—22no—
216C22×Dic3—21no—
217C22×Dic3—21no—
218C22×Dic3—22no—
219C22×Dic3—22no—
220C22×Dic3—44no—
221C22×Dic3—48no—
222C22×Dic3—48no—
223C22×Dic3—48no—
224C22×Dic3—48no—
225C22×Dic3—44no—
226C22×Dic3—44no—
227C22×Dic3C2×D412yes—
228C22×Dic3C2×D414yes—
229C22×Dic3C2×D414yes—
230C22×Dic3C2×D412yes—
231C22×Dic3C2×Q814yes—
232C3×C4⋊C4—48no—
233C3×C22⋊C4—28no—
234C3×C22⋊C4—44no—
235#217—28no—
236#217—44no—
237C4⋊Dic3—216no—
238C4⋊Dic3—216no—
239Dic3⋊C4—44no—
240Dic3⋊C4—48no—
241Dic3⋊C4—44no—
242C25C22×D311yescharacteristic
243C23×C4—22no—
244C23×C4—22no—
245C23×C4—24no—
246C23×C4—24no—
247C23×C4—31no—
248C23×C4—32no—
249C23×C4—32no—
250C23×C4—31no—
251C23×C4—34no—
252C23×C4—31no—
253C23×C4—31no—
254C23×C4—64no—
255C23×C4—64no—
256C23×C4—62no—
257C23×C4—62no—
258C23×C4—62no—
259C23×C4—62no—
260C23×C4—62no—
261C23×C4—122no—
262C23×C4—122no—
263C23×C4#6714yes—
264C23×C4C22×D311yescharacteristic
265C23×C4C22×D311yescharacteristic
266C23×C4C22×D312yes—
267C23×C4C22×D312yes—
268C2×C4⋊C4—24no—
269C2×C4⋊C4—24no—
270C2×C4⋊C4—24no—
271C2×C4⋊C4—38no—
272C2×C4⋊C4—64no—
273C2×C4⋊C4—64no—
274C2×C4⋊C4—64no—
275C2×C4⋊C4—64no—
276C2×C4⋊C4—632no—
277C2×C4⋊C4—64no—
278C2×C4⋊C4—62no—
279C2×C4⋊C4—62no—
280C2×C4⋊C4—64no—
281C2×C4⋊C4—64no—
282C2×C4⋊C4—64no—
283C2×C4⋊C4—64no—
284C2×C4⋊C4—122no—
285C2×C22⋊C4—22no—
286C2×C22⋊C4—22no—
287C2×C22⋊C4—22no—
288C2×C22⋊C4—22no—
289C2×C22⋊C4—22no—
290C2×C22⋊C4—34no—
291C2×C22⋊C4—62no—
292C2×C22⋊C4—62no—
293C2×C22⋊C4—62no—
294C2×C22⋊C4—62no—
295C2×C22⋊C4—62no—
296C2×C22⋊C4Dic614yes—
297#95—24no—
298#95—62no—
299#95—62no—
300#95—64no—
301#95—64no—
302#95—64no—
303#95—64no—
304#95—122no—
305#95—122no—
306#95—122no—
307C23×C3—22no—
308C23×C3—24no—
309C23×C3—22no—
310C23×C3—24no—
311C23×C3—22no—
312C23×C3—22no—
313C23×C3—24no—
314C23×C3—24no—
315C23×C3—24no—
316C23×C3—24no—
317C23×C3—24no—
318C23×C3—24no—
319C23×C3—24no—
320C23×C3—24no—
321C23×C3—24no—
322C23×C3—24no—
323C23×C3#12114yes—
324C23×C3C22⋊Q814yes—
325C23×C3C22⋊Q814yes—
326C23×C3#12314yes—
327C23×C3#12514yes—
328C23×C3#12514yes—
329C23×C3C4⋊Q814yes—
330C23×C3C22×D411yescharacteristic
331C23×C3C22×D412yes—
332C23×C3C22×D411yescharacteristic
333C23×C3C22×Q811yescharacteristic
334C23×C3C22×Q811yescharacteristic
335C23×C3C2×#4111yescharacteristic
336C23×C3C2×#4111yescharacteristic
337C23×C3C2×#4112yes—
338C23×C3C2×#4111yescharacteristic
339C23×C3H(5,𝔽2)12yes—
340C23×C32−1+412yes—
341C2×C3×C4—232no—
342C2×C3×C4—24no—
343C2×C3×C4—24no—
344C2×C3×C4—24no—
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—42no—
352C2×C3×C4—42no—
353C2×C3×C4—42no—
354C2×C3×C4—42no—
355C2×C3×C4—41no—
356C2×C3×C4—41no—
357C2×C3×C4—41no—
358C2×C3×C4—42no—
359C2×C3×C4—41no—
360C2×C3×C4—42no—
361C2×C3×C4—41no—
362C2×C3×C4—41no—
363C2×C3×C4—42no—
364C2×C3×C4—42no—
365C2×C3×C4#12118yes—
366C2×Dic3—22no—
367C2×Dic3—22no—
368C2×Dic3—24no—
369C2×Dic3—24no—
370C2×Dic3—22no—
371C2×Dic3—24no—
372C2×Dic3—24no—
373C2×Dic3—22no—
374C2×Dic3—22no—
375C2×Dic3—24no—
376C2×Dic3—24no—
377C2×Dic3—24no—
378C2×Dic3—24no—
379C2×Dic3—416no—
380C2×Dic3—416no—
381C2×Dic3—42no—
382C2×Dic3—42no—
383C2×Dic3—42no—
384C2×Dic3—42no—
385C2×Dic3—42no—
386C2×Dic3—42no—
387C2×Dic3—42no—
388C2×Dic3—42no—
389C2×Dic3—42no—
390C2×Dic3—42no—
391C2×Dic3—41no—
392C2×Dic3—42no—
393C2×Dic3—41no—
394C2×Dic3—41no—
395C2×Dic3—41no—
396C2×Dic3—42no—
397C2×Dic3—41no—
398C2×Dic3—41no—
399C2×Dic3—42no—
400C2×Dic3—42no—
401C2×Dic3#12014yes—
402C24—22no—
403C24—22no—
404C24—22no—
405C24—22no—
406C24C2×Dic612yes—
407C24#23512yes—
408C24D3×D412yes—
409C24#23712yes—
410C24D3×Q812yes—
411C24#23912yes—
412C24C2×#6712yes—
413C24C23×D311yescharacteristic
414C22×C4—28no—
415C22×C4—216no—
416C22×C4—28no—
417C22×C4—28no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—216no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—22no—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—34no—
440C22×C4—32no—
441C22×C4—34no—
442C22×C4—32no—
443C22×C4—34no—
444C22×C4—68no—
445C22×C4—61no—
446C22×C4—61no—
447C22×C4—62no—
448C22×C4—68no—
449C22×C4—68no—
450C22×C4—61no—
451C22×C4—62no—
452C22×C4—62no—
453C22×C4—62no—
454C22×C4—62no—
455C22×C4—62no—
456C22×C4—62no—
457C22×C4—62no—
458C22×C4—61no—
459C22×C4—62no—
460C22×C4—62no—
461C22×C4—62no—
462C22×C4—62no—
463C22×C4—62no—
464C22×C4—62no—
465C22×C4—62no—
466C22×C4—61no—
467C22×C4—61no—
468C22×C4—62no—
469C22×C4—61no—
470C22×C4—61no—
471C22×C4—62no—
472C22×C4—62no—
473C22×C4—62no—
474C22×C4—61no—
475C22×C4—61no—
476C22×C4—124no—
477C22×C4—128no—
478C22×C4—128no—
479C22×C4—128no—
480C22×C4—128no—
481C22×C4—124no—
482C22×C4—124no—
483C22×C4C2×Dic614yes—
484C22×C4D3×D414yes—
485C22×C4#23714yes—
486C22×C4C2×#6714yes—
487C4⋊C4—48no—
488C4⋊C4—616no—
489C4⋊C4—616no—
490C4⋊C4—124no—
491C4⋊C4—124no—
492C4⋊C4—128no—
493C22⋊C4—28no—
494C22⋊C4—44no—
495C22⋊C4—68no—
496C22⋊C4—124no—
497C22×C3—24no—
498C22×C3—24no—
499C22×C3—24no—
500C22×C3—24no—
501C22×C3—28no—
502C22×C3—28no—
503C22×C3—24no—
504C22×C3—28no—
505C22×C3—28no—
506C22×C3—24no—
507C22×C3#39818yescomplemented
508C22×C3C2×#12011yescharacteristic
509C22×C3C2×#12111yescharacteristic
510C22×C3C2×#12112yes—
511C22×C3C2×C22⋊Q811yescharacteristic
512C22×C3C2×C22⋊Q811yescharacteristic
513C22×C3C2×#12311yescharacteristic
514C22×C3#52511yescharacteristic
515C22×C3#52512yes—
516C22×C3C2×#12511yescharacteristic
517C22×C3C2×#12511yescharacteristic
518C22×C3C2×C4⋊Q811yescharacteristic
519C22×C3#53411yescharacteristic
520C22×C3#53611yescharacteristic
521C22×C3#53712yes—
522C22×C3#53912yes—
523C22×C3#54612yes—
524C22×C3#54612yes—
525C22×C3D4×Q812yes—
526C22×C3D4×Q812yes—
527C22×C3#55212yes—
528C22×C3#55212yes—
529C22×C3#55412yes—
530C22×C3#55412yes—
531C3×C4—28no—
532C3×C4—44no—
533C3×C4—81no—
534C3×C4—81no—
535Dic3—24no—
536Dic3—42no—
537Dic3—44no—
538Dic3—44no—
539Dic3—81no—
540Dic3—81no—
541C23—24no—
542C23—24no—
543C23—24no—
544C23—22no—
545C23—22no—
546C23—24no—
547C23—24no—
548C23—24no—
549C23—24no—
550C23—24no—
551C23—24no—
552C23—24no—
553C23—22no—
554C23—24no—
555C23—22no—
556C23—24no—
557C23#88814yes—
558C23#88914yes—
559C23#89014yes—
560C23#89414yes—
561C23#92414yes—
562C23#93914yes—
563C23#94614yes—
564C23C22×Dic611yescharacteristic
565C23C2×#23511yescharacteristic
566C23C2×D3×D412yes—
567C23C2×D3×D411yescharacteristic
568C23C2×#23712yes—
569C23C2×#23711yescharacteristic
570C23C2×D3×Q811yescharacteristic
571C23C2×#23911yescharacteristic
572C23#100912yes—
573C23#101012yes—
574C23C22×#6711yescharacteristic
575C2×C4—232no—
576C2×C4—24no—
577C2×C4—24no—
578C2×C4—24no—
579C2×C4—24no—
580C2×C4—24no—
581C2×C4—24no—
582C2×C4—24no—
583C2×C4—24no—
584C2×C4—24no—
585C2×C4—34no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—42no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—62no—
601C2×C4—62no—
602C2×C4—64no—
603C2×C4—64no—
604C2×C4—64no—
605C2×C4—64no—
606C2×C4—62no—
607C2×C4—62no—
608C2×C4—64no—
609C2×C4—64no—
610C2×C4—64no—
611C2×C4—62no—
612C2×C4—64no—
613C2×C4—1216no—
614C2×C4—1216no—
615C2×C4—122no—
616C2×C4—122no—
617C2×C4—122no—
618C2×C4—122no—
619C2×C4—122no—
620C2×C4—122no—
621C2×C4—122no—
622C2×C4—122no—
623C2×C4—122no—
624C2×C4—122no—
625C2×C4—121no—
626C2×C4—121no—
627C2×C4—122no—
628C2×C4—121no—
629C2×C4—121no—
630C2×C4—122no—
631C2×C4—122no—
632C2×C4—122no—
633C2×C4—121no—
634C2×C4—121no—
635C2×C4#93918yes—
636C2×C3—28no—
637C2×C3C2×#39811yescharacteristic
638C2×C3#240511yescharacteristic
639C2×C3#241712yes—
640C2×C3#241712yes—
641C2×C3#243411yescharacteristic
642C2×C3#247312yes—
643C2×C3#247312yes—
644C2×C3#248411yescharacteristic
645C2×C3#258311yescharacteristic
646C2×C3#261811yescharacteristic
647C2×C3#284711yescharacteristic
648C22—24no—
649C22—28no—
650C22—24no—
651C22—24no—
652C22—28no—
653C22—24no—
654C22—28no—
655C22—28no—
656C22—24no—
657C22—24no—
658C22#498218yescomplemented
659C22C2×#88811yescharacteristic
660C22C2×#88911yescharacteristic
661C22C2×#89011yescharacteristic
662C22C2×#89411yescharacteristic
663C22D4×Dic612yes—
664C22#554112yes—
665C22#554312yes—
666C22#555012yes—
667C22#558612yes—
668C22#558611yescharacteristic
669C22D3×#12011yescharacteristic
670C22#563012yes—
671C22#563212yes—
672C22#563512yes—
673C22#564212yes—
674C22#564811yescharacteristic
675C22#565311yescharacteristic
676C22#566512yes—
677C22C2×#92411yescharacteristic
678C22C2×#93912yes—
679C22C2×#93911yescharacteristic
680C22C2×#94611yescharacteristic
681C22#583312yes—
682C4—28no—
683C4—44no—
684C4—64no—
685C4—81no—
686C4—81no—
687C4—122no—
688C4—124no—
689C4—124no—
690C4—241no—
691C4—241no—
692C3#1482111yescharacteristic, complemented, Sylow
693C2—28no—
694C2#7441211yescharacteristic
695C2#7444112yes—
696C2#7444212yes—
697C2#7445611yescharacteristic
698C2#7448811yescharacteristic
699C2#7459911yescharacteristic
700C2C2×#498211yescharacteristic
701C2#8235312yes—
702C2#8260111yescharacteristic
703C2#8281512yes—
704C2#8286511yescharacteristic
subgroupG/H
center, FrattiniC24C23×D3
commutator, socle, cosocleC24×C3C24
FittingC2×C3×#385C2
prime radicalC25×C3C23
hypercenterC2×#385D3
hypocenterC3#14821
LCSG ▷ C24×C3 ▷ C3
UCSC1 ◁ C24 ◁ C2×#385

Complements

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

NH ≅ G/Ntype
4. C2×#4982C2semidirect
6. #71565C2semidirect
10. #69411C2semidirect
11. #69410C2semidirect
507. C22×C3#398semidirect
658. C22#4982semidirect
692. C3#14821semidirect

Maximal subgroups

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

subgroupindex
C23×C4⋊Dic32
C2×C3×#3852
C2×#52102
C2×#49822
C2×#49422
#715652
#715442
#694112
#694102
#148213