Finite Group Databasebeta

← #10720179

Subgroups — #10720179

Order 768.

2742 conjugacy classes of subgroups.

The subgroup lattice has 734 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#938C211yesmaximal, characteristic
2C22×#892C211yesmaximal, characteristic, complemented
3C2×C3×#390C211yesmaximal, characteristic, complemented
4C2×#5221C211yesmaximal, characteristic, complemented
5C2×C4×(C2×D3)⋊C4C211yesmaximal, characteristic, complemented
6#71563C211yesmaximal, characteristic, complemented
7#71550C214yesmaximal, complemented
8#71545C214yesmaximal, complemented
9#71535C211yesmaximal, characteristic, complemented
10#14136—31nomaximal, Sylow
11C23×C3×D4C2211yescharacteristic
12C23×#67C2212yes—
13C23×C4×D3C2211yescharacteristic
14C22×C3×C4⋊C4C2211yescharacteristic
15C22×C3×C22⋊C4C2214yes—
16C22×C3×C42C2211yescharacteristic
17C22×#217C2214yes—
18C22×#217C2212yes—
19C2×#938—24no—
20C2×#938—24no—
21C2×#938C418yescomplemented
22C22×(C2×D3)⋊C4C2212yes—
23C22×(C2×D3)⋊C4C2212yes—
24C22×C4⋊Dic3C2211yescharacteristic
25C22×C4⋊Dic3C2211yescharacteristic
26C22×C4×Dic3C2211yescharacteristic
27C22×C4×Dic3C2212yes—
28C2×#892—24no—
29C2×#892—24no—
30C2×#892—24no—
31C3×#390—24no—
32#5221—24no—
33C2×C3⋊#95C2212yes—
34C2×C3⋊#95C2214yes—
35C2×C3⋊#95C2214yes—
36C4×(C2×D3)⋊C4—24no—
37C22×#121—31no—
38C22×#117—31no—
39C2×#390—31no—
40C2×#390D311yescharacteristic, complemented
41C2×C4×C22⋊C4—31no—
42#1438—34no—
43#1437—34no—
44C4⋊#95—31no—
45C4⋊#95—31no—
46C25×C3C2312yes—
47C24×D3C2311yescharacteristic
48C22×C3×D4—22no—
49C22×C3×D4—22no—
50C22×C3×D4—22no—
51C22×C3×D4—22no—
52C22×C3×D4—22no—
53C22×C3×D4—22no—
54C22×C3×D4C2×C414yes—
55C23×C3×C4C2312yes—
56C23×C3×C4C2312yes—
57C23×C3×C4C2311yescharacteristic
58C22×#67—24no—
59C22×#67—24no—
60C22×#67—44no—
61C22×#67—44no—
62C22×#67C2×C418yes—
63C23×Dic3C2312yes—
64C23×Dic3C2312yes—
65C23×Dic3C2312yes—
66C23×Dic3C2311yescharacteristic
67C22×C4×D3—22no—
68C22×C4×D3—22no—
69C22×C4×D3—22no—
70C22×C4×D3—22no—
71C22×C4×D3C2×C414yes—
72C22×C4×D3D414yes—
73C22×C4×D3D414yes—
74C2×C3×C4⋊C4—22no—
75C2×C3×C4⋊C4—22no—
76C2×C3×C4⋊C4D414yescomplemented
77C2×C3×C22⋊C4—28no—
78C2×C3×C22⋊C4—28no—
79C2×C3×C22⋊C4—44no—
80C2×C3×C42—22no—
81C2×C3×C42—22no—
82C2×C3×C42—22no—
83C2×#217—24no—
84C2×#217—24no—
85C2×#217—28no—
86C2×#217—28no—
87C2×#217—44no—
88C2×#217C2×C418yes—
89#938—232no—
90C2×(C2×D3)⋊C4—24no—
91C2×(C2×D3)⋊C4—24no—
92C2×(C2×D3)⋊C4—24no—
93C2×(C2×D3)⋊C4—24no—
94C2×(C2×D3)⋊C4—24no—
95C2×(C2×D3)⋊C4—24no—
96C2×C4⋊Dic3—22no—
97C2×C4⋊Dic3—22no—
98C2×C4⋊Dic3—22no—
99C2×C4⋊Dic3—22no—
100C2×C4⋊Dic3C2×C414yes—
101C2×C4⋊Dic3D414yescomplemented
102C2×C4×Dic3—24no—
103C2×C4×Dic3—24no—
104C2×C4×Dic3—24no—
105C2×C4×Dic3—22no—
106C2×C4×Dic3—22no—
107C2×C4×Dic3—22no—
108#892—232no—
109C3⋊#95—24no—
110C3⋊#95—44no—
111C3⋊#95—44no—
112C23×D4—32no—
113C23×D4C2×D311yescharacteristic
114C24×C4—31no—
115C2×#121—38no—
116C2×#121—64no—
117C2×#121—64no—
118C2×#117—64no—
119C2×#117—64no—
120C2×#117—64no—
121C22×C4⋊C4—31no—
122C22×C4⋊C4—31no—
123C22×C4⋊C4C2×D311yescharacteristic
124C22×C22⋊C4—32no—
125C22×C22⋊C4—32no—
126C22×C22⋊C4—32no—
127C22×C22⋊C4—34no—
128C22×C22⋊C4C2×D314yes—
129C22×C42—32no—
130C22×C42—31no—
131C22×C42C2×D311yescharacteristic
132#390—24no—
133#390—64no—
134C4×C22⋊C4—64no—
135C2×#95—32no—
136C2×#95—34no—
137C2×#95—34no—
138C24×C3—22no—
139C24×C3—22no—
140C24×C3—22no—
141C24×C3—22no—
142C24×C3—24no—
143C24×C3—24no—
144C24×C3—22no—
145C24×C3—22no—
146C24×C3—24no—
147C24×C3—24no—
148C24×C3C22×C414yes—
149C24×C3C2411yescharacteristic
150C23×D3—21no—
151C23×D3—21no—
152C23×D3—22no—
153C23×D3—22no—
154C23×D3—21no—
155C23×D3—21no—
156C23×D3—22no—
157C23×D3—22no—
158C23×D3C22×C412yes—
159C23×D3C2×D412yes—
160C23×D3C2×D412yes—
161C2×C3×D4—28no—
162C2×C3×D4—28no—
163C2×C3×D4—28no—
164C2×C3×D4—44no—
165C2×C3×D4—44no—
166C2×C3×D4—44no—
167C2×C3×D4—44no—
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—24no—
175C22×C3×C4—22no—
176C22×C3×C4—24no—
177C22×C3×C4—22no—
178C22×C3×C4C22×C412yes—
179C22×C3×C4C2×D412yes—
180C22×C3×C4C2×D412yes—
181C22×C3×C4C2×D412yes—
182C22×C3×C4C2×D414yes—
183C22×C3×C4C2×D412yes—
184C22×C3×C4#4112yes—
185C22×C3×C4#4114yes—
186C22×C3×C4#4112yes—
187C2×#67—216no—
188C2×#67—416no—
189C2×#67—48no—
190C2×#67—48no—
191C2×#67—416no—
192C22×Dic3—21no—
193C22×Dic3—21no—
194C22×Dic3—21no—
195C22×Dic3—21no—
196C22×Dic3—22no—
197C22×Dic3—22no—
198C22×Dic3—22no—
199C22×Dic3—22no—
200C22×Dic3—22no—
201C22×Dic3—22no—
202C22×Dic3—22no—
203C22×Dic3—22no—
204C22×Dic3—24no—
205C22×Dic3—22no—
206C22×Dic3—24no—
207C22×Dic3—24no—
208C22×Dic3—24no—
209C22×Dic3—22no—
210C22×Dic3C22×C414yes—
211C22×Dic3C22×C412yes—
212C22×Dic3C2×D414yes—
213C22×Dic3C2×D412yes—
214C22×Dic3C2×D412yes—
215C22×Dic3#4114yes—
216C2×C4×D3—28no—
217C2×C4×D3—28no—
218C2×C4×D3—28no—
219C2×C4×D3—28no—
220C2×C4×D3—28no—
221C2×C4×D3—28no—
222C2×C4×D3C22⋊C4116yescomplemented
223C3×C4⋊C4—28no—
224C3×C22⋊C4—416no—
225C3×C42—44no—
226#217—216no—
227#217—416no—
228(C2×D3)⋊C4—216no—
229(C2×D3)⋊C4—216no—
230C4⋊Dic3—28no—
231C4⋊Dic3—28no—
232C4×Dic3—216no—
233C4×Dic3—44no—
234C25—31no—
235C25C22×D312yes—
236C22×D4—22no—
237C22×D4—22no—
238C22×D4—22no—
239C22×D4—22no—
240C22×D4—22no—
241C22×D4—22no—
242C22×D4—38no—
243C22×D4—64no—
244C22×D4—64no—
245C22×D4—124no—
246C22×D4—124no—
247C22×D4C4×D314yes—
248C23×C4—31no—
249C23×C4—34no—
250C23×C4—34no—
251C23×C4—34no—
252C23×C4—32no—
253C23×C4—32no—
254C23×C4—32no—
255C23×C4—62no—
256C23×C4—62no—
257C23×C4—62no—
258C23×C4—62no—
259C23×C4C22×D312yes—
260C23×C4C22×D312yes—
261C23×C4C22×D311yescharacteristic
262#121—632no—
263#117—632no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—34no—
267C2×C4⋊C4—34no—
268C2×C4⋊C4—62no—
269C2×C4⋊C4—62no—
270C2×C4⋊C4—62no—
271C2×C4⋊C4—62no—
272C2×C4⋊C4#6714yescomplemented
273C2×C22⋊C4—28no—
274C2×C22⋊C4—28no—
275C2×C22⋊C4—38no—
276C2×C22⋊C4—44no—
277C2×C22⋊C4—64no—
278C2×C22⋊C4—64no—
279C2×C22⋊C4—64no—
280C2×C22⋊C4—64no—
281C2×C22⋊C4—64no—
282C2×C22⋊C4—64no—
283C2×C22⋊C4—64no—
284C2×C22⋊C4—64no—
285C2×C22⋊C4—68no—
286C2×C22⋊C4—68no—
287C2×C22⋊C4—124no—
288C2×C42—22no—
289C2×C42—22no—
290C2×C42—22no—
291C2×C42—64no—
292C2×C42—64no—
293C2×C42—64no—
294C2×C42—62no—
295C2×C42—62no—
296C2×C42—62no—
297#95—64no—
298#95—124no—
299#95—124no—
300C23×C3—24no—
301C23×C3—24no—
302C23×C3—24no—
303C23×C3—28no—
304C23×C3—28no—
305C23×C3—42no—
306C23×C3—42no—
307C23×C3—42no—
308C23×C3—42no—
309C23×C3—44no—
310C23×C3—42no—
311C23×C3—42no—
312C23×C3—42no—
313C23×C3—42no—
314C23×C3—44no—
315C23×C3—44no—
316C23×C3—44no—
317C23×C3—44no—
318C23×C3—44no—
319C23×C3—44no—
320C23×C3—44no—
321C23×C3—44no—
322C23×C3—44no—
323C23×C3C23×C411yescharacteristic
324C23×C3C22×D411yescharacteristic
325C23×C3C22×D411yescharacteristic
326C23×C3C22×D411yescharacteristic
327C23×C3C22×D411yescharacteristic
328C23×C3C2×#4111yescharacteristic
329C23×C3C2×#4111yescharacteristic
330C23×C3C2×#4112yes—
331C23×C3C2×#4112yes—
332C23×C3H(5,𝔽2)12yes—
333C23×C3H(5,𝔽2)12yes—
334C22×D3—22no—
335C22×D3—22no—
336C22×D3—24no—
337C22×D3—24no—
338C22×D3—22no—
339C22×D3—22no—
340C22×D3—22no—
341C22×D3—22no—
342C22×D3—24no—
343C22×D3—22no—
344C22×D3—22no—
345C22×D3—22no—
346C22×D3—22no—
347C22×D3—24no—
348C22×D3—42no—
349C22×D3—42no—
350C22×D3—42no—
351C22×D3—42no—
352C22×D3—42no—
353C22×D3—42no—
354C22×D3—42no—
355C22×D3—42no—
356C22×D3C2×C22⋊C414yes—
357C3×D4—416no—
358C2×C3×C4—24no—
359C2×C3×C4—24no—
360C2×C3×C4—24no—
361C2×C3×C4—28no—
362C2×C3×C4—24no—
363C2×C3×C4—42no—
364C2×C3×C4—42no—
365C2×C3×C4—42no—
366C2×C3×C4—42no—
367C2×C3×C4—44no—
368C2×C3×C4—42no—
369C2×C3×C4—42no—
370C2×C3×C4C2×C22⋊C414yes—
371C2×C3×C4C4×D414yes—
372C2×C3×C4C4×D414yes—
373C2×C3×C4#12014yes—
374C2×C3×C4#12114yes—
375C2×C3×C4#12114yes—
376C2×C3×C4#12314yes—
377#67—432no—
378C2×Dic3—22no—
379C2×Dic3—22no—
380C2×Dic3—22no—
381C2×Dic3—22no—
382C2×Dic3—22no—
383C2×Dic3—22no—
384C2×Dic3—24no—
385C2×Dic3—28no—
386C2×Dic3—24no—
387C2×Dic3—24no—
388C2×Dic3—24no—
389C2×Dic3—28no—
390C2×Dic3—24no—
391C2×Dic3—42no—
392C2×Dic3—42no—
393C2×Dic3—42no—
394C2×Dic3—42no—
395C2×Dic3—42no—
396C2×Dic3—42no—
397C2×Dic3—44no—
398C2×Dic3—42no—
399C2×Dic3—44no—
400C2×Dic3—42no—
401C2×Dic3C2×C22⋊C414yes—
402C4×D3—232no—
403C24—22no—
404C24—22no—
405C24—22no—
406C24—22no—
407C24—24no—
408C24—22no—
409C24—24no—
410C24—24no—
411C24—24no—
412C24—22no—
413C24—32no—
414C24—32no—
415C24—32no—
416C24—61no—
417C24—61no—
418C24—62no—
419C24—61no—
420C24—62no—
421C24—62no—
422C24—62no—
423C24—61no—
424C24C2×C4×D314yes—
425C24C23×D311yescharacteristic
426C2×D4—28no—
427C2×D4—28no—
428C2×D4—28no—
429C2×D4—44no—
430C2×D4—44no—
431C2×D4—44no—
432C2×D4—44no—
433C2×D4—616no—
434C2×D4—1216no—
435C2×D4—128no—
436C2×D4—128no—
437C2×D4—1216no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—24no—
443C22×C4—24no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—32no—
449C22×C4—32no—
450C22×C4—32no—
451C22×C4—316no—
452C22×C4—34no—
453C22×C4—34no—
454C22×C4—34no—
455C22×C4—61no—
456C22×C4—61no—
457C22×C4—61no—
458C22×C4—61no—
459C22×C4—68no—
460C22×C4—68no—
461C22×C4—68no—
462C22×C4—68no—
463C22×C4—68no—
464C22×C4—68no—
465C22×C4—62no—
466C22×C4—62no—
467C22×C4—62no—
468C22×C4—62no—
469C22×C4—62no—
470C22×C4—62no—
471C22×C4—62no—
472C22×C4—62no—
473C22×C4—64no—
474C22×C4—64no—
475C22×C4—64no—
476C22×C4—64no—
477C22×C4—62no—
478C22×C4—62no—
479C22×C4C2×C4×D312yes—
480C22×C4#23512yes—
481C22×C4D3×D412yes—
482C22×C4D3×D412yes—
483C22×C4D3×D412yes—
484C22×C4#23714yes—
485C22×C4#23712yes—
486C22×C4C2×#6714yes—
487C22×C4C2×#6712yes—
488C4⋊C4—28no—
489C4⋊C4—68no—
490C4⋊C4—68no—
491C22⋊C4—416no—
492C22⋊C4—616no—
493C22⋊C4—616no—
494C22⋊C4—616no—
495C22⋊C4—1216no—
496C42—44no—
497C42—616no—
498C42—124no—
499C22×C3—44no—
500C22×C3—48no—
501C22×C3—48no—
502C22×C3—44no—
503C22×C3—44no—
504C22×C3—84no—
505C22×C3—84no—
506C22×C3—82no—
507C22×C3—82no—
508C22×C3—82no—
509C22×C3—82no—
510C22×C3C22×C22⋊C411yescharacteristic
511C22×C3C2×C4×D411yescharacteristic
512C22×C3C2×C4×D411yescharacteristic
513C22×C3C4×#4112yes—
514C22×C3#51812yes—
515C22×C3C2×#12011yescharacteristic
516C22×C3C2×#12111yescharacteristic
517C22×C3C2×#12111yescharacteristic
518C22×C3C2×#12311yescharacteristic
519C22×C3#52512yes—
520C22×C3#53211yescharacteristic
521C22×C3#53211yescharacteristic
522C22×C3#53511yescharacteristic
523C22×C3#53511yescharacteristic
524C22×C3#53512yes—
525C22×C3#54612yes—
526C22×C3#54612yes—
527C22×C3#55012yes—
528C22×C3#55012yes—
529C22×C3#55312yes—
530C22×C3#55312yes—
531C22×C3#55512yes—
532C22×C3#55512yes—
533C2×D3—24no—
534C2×D3—24no—
535C2×D3—24no—
536C2×D3—44no—
537C2×D3—44no—
538C2×D3—44no—
539C2×D3—44no—
540C2×D3—42no—
541C2×D3—42no—
542C2×D3—42no—
543C2×D3—42no—
544C3×C4—44no—
545C3×C4—44no—
546C3×C4#38618yes—
547Dic3—24no—
548Dic3—44no—
549Dic3—44no—
550Dic3—44no—
551C23—24no—
552C23—24no—
553C23—24no—
554C23—28no—
555C23—28no—
556C23—34no—
557C23—42no—
558C23—42no—
559C23—44no—
560C23—42no—
561C23—42no—
562C23—44no—
563C23—42no—
564C23—42no—
565C23—42no—
566C23—42no—
567C23—44no—
568C23—44no—
569C23—44no—
570C23—44no—
571C23—44no—
572C23—44no—
573C23—44no—
574C23—44no—
575C23—62no—
576C23—62no—
577C23—62no—
578C23—62no—
579C23—62no—
580C23—62no—
581C23—62no—
582C23—62no—
583C23—64no—
584C23—62no—
585C23—62no—
586C23—64no—
587C23—64no—
588C23—64no—
589C23—122no—
590C23—122no—
591C23—122no—
592C23—122no—
593C23—122no—
594C23—122no—
595C23—122no—
596C23—122no—
597C23C22×C4×D311yescharacteristic
598C23C2×#23511yescharacteristic
599C23C2×D3×D411yescharacteristic
600C23C2×D3×D411yescharacteristic
601C23C2×D3×D411yescharacteristic
602C23C2×#23712yes—
603C23C2×#23711yescharacteristic
604C23#100412yes—
605C23D3×#4112yes—
606C23#100912yes—
607C23C22×#6711yescharacteristic
608D4—416no—
609D4—1232no—
610C2×C4—24no—
611C2×C4—24no—
612C2×C4—28no—
613C2×C4—24no—
614C2×C4—24no—
615C2×C4—34no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—44no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—62no—
624C2×C4—62no—
625C2×C4—62no—
626C2×C4—62no—
627C2×C4—62no—
628C2×C4—62no—
629C2×C4—632no—
630C2×C4—64no—
631C2×C4—68no—
632C2×C4—64no—
633C2×C4—64no—
634C2×C4—68no—
635C2×C4—64no—
636C2×C4—64no—
637C2×C4—122no—
638C2×C4—122no—
639C2×C4—122no—
640C2×C4—122no—
641C2×C4—122no—
642C2×C4—122no—
643C2×C4—122no—
644C2×C4—124no—
645C2×C4—124no—
646C2×C4—122no—
647C2×C4D3×C22⋊C414yes—
648C2×C4#67⋊C414yes—
649C2×C4#88314yes—
650C2×C4#88414yes—
651C2×C4C4×#6714yes—
652C2×C4#93714yes—
653C2×C4#93814yes—
654C2×C3—84no—
655C2×C3C2×#38611yescharacteristic
656C2×C3#229711yescharacteristic
657C2×C3#230911yescharacteristic
658C2×C3#236312yes—
659C2×C3#236312yes—
660C2×C3#241412yes—
661C2×C3#241412yes—
662C2×C3#243111yescharacteristic
663C2×C3#243111yescharacteristic
664C2×C3#250011yescharacteristic
665C2×C3#250011yescharacteristic
666D3—44no—
667C22—44no—
668C22—48no—
669C22—44no—
670C22—48no—
671C22—44no—
672C22—64no—
673C22—64no—
674C22—64no—
675C22—82no—
676C22—84no—
677C22—82no—
678C22—84no—
679C22—82no—
680C22—82no—
681C22—122no—
682C22—124no—
683C22—124no—
684C22—122no—
685C22—124no—
686C22—122no—
687C22—124no—
688C22—122no—
689C22C2×D3×C22⋊C411yescharacteristic
690C22C2×#67⋊C411yescharacteristic
691C22C2×#88311yescharacteristic
692C22C2×#88411yescharacteristic
693C22C4×#23712yes—
694C22#554512yes—
695C22#555812yes—
696C22#556312yes—
697C22#560612yes—
698C22#560712yes—
699C22#561612yes—
700C22#561812yes—
701C22#566811yescharacteristic
702C22#567411yescharacteristic
703C22#567612yes—
704C22#571511yescharacteristic
705C22#571611yescharacteristic
706C22#571812yes—
707C22C2×C4×#6711yescharacteristic
708C22C2×#93711yescharacteristic
709C22C2×#93811yescharacteristic
710C22#580612yes—
711C22#583412yes—
712C4—44no—
713C4—44no—
714C4—64no—
715C4—124no—
716C4—124no—
717C4—124no—
718C4#495618yes—
719C3#1413611yescharacteristic, complemented, Sylow
720C2—84no—
721C2—124no—
722C2#7629011yescharacteristic
723C2#7630412yes—
724C2#7633411yescharacteristic
725C2#7642811yescharacteristic
726C2#7643812yes—
727C2#7645311yescharacteristic
728C2C2×#495611yescharacteristic
729C2#8234112yes—
730C2#8270312yes—
731C2#8295811yescharacteristic
732C2#8318711yescharacteristic
subgroupG/H
center, FrattiniC24C23×D3
commutatorC23×C3C23×C4
FittingC2×C3×#390C2
socle, cosocleC24×C3C24
prime radicalC22×#938C2
hypercenterC2×#390D3
hypocenterC3#14136
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C24 ◁ C2×#390

Complements

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

NH ≅ G/Ntype
2. C22×#892C2semidirect
3. C2×C3×#390C2semidirect
4. C2×#5221C2semidirect
5. C2×C4×(C2×D3)⋊C4C2semidirect
6. #71563C2semidirect
7. #71550C2semidirect
8. #71545C2semidirect
9. #71535C2semidirect
21. C2×#938C4semidirect
40. C2×#390D3semidirect
76. C2×C3×C4⋊C4D4semidirect
101. C2×C4⋊Dic3D4semidirect
222. C2×C4×D3C22⋊C4semidirect
272. C2×C4⋊C4#67semidirect
719. C3#14136semidirect

Maximal subgroups

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

subgroupindex
C22×#9382
C22×#8922
C2×C3×#3902
C2×#52212
C2×C4×(C2×D3)⋊C42
#715632
#715502
#715452
#715352
#141363