Finite Group Databasebeta

← #10720181

Subgroups — #10720181

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×#895C211yesmaximal, characteristic
2C22×#892C211yesmaximal, characteristic, complemented
3C2×#4988C211yesmaximal, characteristic, complemented
4C2×#4939C211yesmaximal, characteristic, complemented
5C2×C4×(C2×D3)⋊C4C211yesmaximal, characteristic, complemented
6C3×C4⋊#95C211yesmaximal, characteristic, complemented
7#71563C211yesmaximal, characteristic, complemented
8#69426C214yesmaximal, complemented
9#69408C214yesmaximal, complemented
10#14136—31nomaximal, Sylow
11C23×D12C2211yescharacteristic
12C23×D12C2212yes—
13C23×C4×D3C2211yescharacteristic
14C22×C3×C4⋊C4C2211yescharacteristic
15C22×C3×C4⋊C4C2211yescharacteristic
16C22×C3×C42C2211yescharacteristic
17C22×(C2×D3)⋊C4C2212yes—
18C22×(C2×D3)⋊C4C2212yes—
19C22×(C2×D3)⋊C4C2214yes—
20C22×(C2×D3)⋊C4C2214yes—
21C22×(C2×D3)⋊C4C2212yes—
22C22×C4⋊Dic3C2211yescharacteristic
23C22×C4×Dic3C2212yes—
24C22×C4×Dic3C2211yescharacteristic
25C2×#895—24no—
26C2×#895—24no—
27C2×#895C418yescomplemented
28C2×#892—24no—
29C2×#892—24no—
30C2×#892—24no—
31C2×C3×#95C2214yes—
32C2×C3⋊#95C2212yes—
33C2×C3⋊#95C2214yes—
34#4988—24no—
35#4939—24no—
36C4×(C2×D3)⋊C4—24no—
37C22×#121—31no—
38C22×#117—31no—
39C2×#390—31no—
40C2×#390—31no—
41C2×C4×C22⋊C4—31no—
42#1438—34no—
43#1437—34no—
44C4⋊#95—31no—
45C4⋊#95D311yescharacteristic, complemented
46C24×D3C2312yes—
47C24×D3C2311yescharacteristic
48C23×C3×C4C2312yes—
49C23×C3×C4C2312yes—
50C23×C3×C4C2312yes—
51C23×C3×C4C2311yescharacteristic
52C23×Dic3C2312yes—
53C23×Dic3C2312yes—
54C23×Dic3C2311yescharacteristic
55C22×D12—24no—
56C22×D12—24no—
57C22×D12—22no—
58C22×D12—22no—
59C22×D12—22no—
60C22×D12—22no—
61C22×D12—22no—
62C22×D12—22no—
63C22×D12—44no—
64C22×D12—44no—
65C22×D12C2×C414yes—
66C22×D12C2×C418yes—
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⋊C4—22no—
77C2×C3×C4⋊C4—22no—
78C2×C3×C4⋊C4C2×C414yes—
79C2×C3×C4⋊C4D414yescomplemented
80C2×C3×C42—22no—
81C2×C3×C42—22no—
82C2×C3×C42—22no—
83C2×(C2×D3)⋊C4—24no—
84C2×(C2×D3)⋊C4—24no—
85C2×(C2×D3)⋊C4—24no—
86C2×(C2×D3)⋊C4—24no—
87C2×(C2×D3)⋊C4—24no—
88C2×(C2×D3)⋊C4—24no—
89C2×(C2×D3)⋊C4—24no—
90C2×(C2×D3)⋊C4—24no—
91C2×(C2×D3)⋊C4—28no—
92C2×(C2×D3)⋊C4—28no—
93C2×(C2×D3)⋊C4—28no—
94C2×(C2×D3)⋊C4—28no—
95C2×(C2×D3)⋊C4—44no—
96C2×(C2×D3)⋊C4—44no—
97C2×(C2×D3)⋊C4C2×C418yes—
98C2×C4⋊Dic3—22no—
99C2×C4⋊Dic3—22no—
100C2×C4⋊Dic3D414yescomplemented
101C2×C4×Dic3—24no—
102C2×C4×Dic3—24no—
103C2×C4×Dic3—24no—
104C2×C4×Dic3—22no—
105C2×C4×Dic3—22no—
106C2×C4×Dic3—22no—
107#895—232no—
108#892—232no—
109C3×#95—44no—
110C3⋊#95—24no—
111C3⋊#95—44no—
112C23×D4—31no—
113C23×D4—32no—
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⋊C4C2×D311yescharacteristic
123C22×C4⋊C4C2×D311yescharacteristic
124C22×C22⋊C4—32no—
125C22×C22⋊C4—32no—
126C22×C22⋊C4—32no—
127C22×C22⋊C4—34no—
128C22×C22⋊C4—34no—
129C22×C42—32no—
130C22×C42—31no—
131C22×C42C2×D311yescharacteristic
132#390—64no—
133#390—64no—
134C4×C22⋊C4—64no—
135C2×#95—32no—
136C2×#95—34no—
137C2×#95C2×D314yes—
138C24×C3C2411yescharacteristic
139C23×D3—21no—
140C23×D3—21no—
141C23×D3—22no—
142C23×D3—22no—
143C23×D3—21no—
144C23×D3—21no—
145C23×D3—22no—
146C23×D3—22no—
147C23×D3—22no—
148C23×D3—22no—
149C23×D3—22no—
150C23×D3—22no—
151C23×D3—24no—
152C23×D3—24no—
153C23×D3—22no—
154C23×D3—22no—
155C23×D3—24no—
156C23×D3—24no—
157C23×D3C22×C414yes—
158C23×D3C22×C412yes—
159C23×D3C2×D412yes—
160C23×D3C2×D412yes—
161C22×C3×C4—22no—
162C22×C3×C4—22no—
163C22×C3×C4—22no—
164C22×C3×C4—22no—
165C22×C3×C4—22no—
166C22×C3×C4—22no—
167C22×C3×C4—22no—
168C22×C3×C4—24no—
169C22×C3×C4—22no—
170C22×C3×C4—24no—
171C22×C3×C4—22no—
172C22×C3×C4—24no—
173C22×C3×C4—24no—
174C22×C3×C4—22no—
175C22×C3×C4C22×C414yes—
176C22×C3×C4C22×C412yes—
177C22×C3×C4C2×D414yes—
178C22×C3×C4C2×D412yes—
179C22×C3×C4C2×D412yes—
180C22×C3×C4C2×D412yes—
181C22×C3×C4C2×D412yes—
182C22×C3×C4#4112yes—
183C22×C3×C4#4114yes—
184C22×C3×C4#4112yes—
185C22×Dic3—21no—
186C22×Dic3—21no—
187C22×Dic3—21no—
188C22×Dic3—21no—
189C22×Dic3—22no—
190C22×Dic3—22no—
191C22×Dic3—22no—
192C22×Dic3—22no—
193C22×Dic3—22no—
194C22×Dic3—22no—
195C22×Dic3—24no—
196C22×Dic3—22no—
197C22×Dic3—24no—
198C22×Dic3—22no—
199C22×Dic3C22×C412yes—
200C22×Dic3C2×D414yes—
201C22×Dic3C2×D412yes—
202C22×Dic3C2×D412yes—
203C22×Dic3#4114yes—
204C2×D12—216no—
205C2×D12—28no—
206C2×D12—28no—
207C2×D12—28no—
208C2×D12—416no—
209C2×D12—48no—
210C2×D12—48no—
211C2×D12—416no—
212C2×D12—44no—
213C2×D12—44no—
214C2×D12—44no—
215C2×D12—44no—
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×C4⋊C4—28no—
225C3×C42—44no—
226(C2×D3)⋊C4—216no—
227(C2×D3)⋊C4—216no—
228(C2×D3)⋊C4—216no—
229(C2×D3)⋊C4—416no—
230(C2×D3)⋊C4—416no—
231C4⋊Dic3—28no—
232C4×Dic3—216no—
233C4×Dic3—44no—
234C25—31no—
235C25—32no—
236C22×D4—34no—
237C22×D4—38no—
238C22×D4—64no—
239C22×D4—64no—
240C22×D4—62no—
241C22×D4—62no—
242C22×D4—62no—
243C22×D4—62no—
244C22×D4—62no—
245C22×D4—62no—
246C22×D4—124no—
247C22×D4—124no—
248C23×C4—31no—
249C23×C4—34no—
250C23×C4—34no—
251C23×C4—34no—
252C23×C4—32no—
253C23×C4—32no—
254C23×C4—62no—
255C23×C4—62no—
256C23×C4—62no—
257C23×C4—62no—
258C23×C4C22×D312yes—
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—22no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4—34no—
269C2×C4⋊C4—62no—
270C2×C4⋊C4—62no—
271C2×C4⋊C4C4×D314yes—
272C2×C4⋊C4D1214yescomplemented
273C2×C22⋊C4—38no—
274C2×C22⋊C4—64no—
275C2×C22⋊C4—64no—
276C2×C22⋊C4—64no—
277C2×C22⋊C4—64no—
278C2×C22⋊C4—64no—
279C2×C22⋊C4—64no—
280C2×C22⋊C4—64no—
281C2×C22⋊C4—64no—
282C2×C22⋊C4—68no—
283C2×C22⋊C4—68no—
284C2×C22⋊C4—68no—
285C2×C22⋊C4—68no—
286C2×C22⋊C4—124no—
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—44no—
298#95—64no—
299#95—124no—
300C23×C3C23×C411yescharacteristic
301C23×C3C22×D411yescharacteristic
302C23×C3C22×D411yescharacteristic
303C23×C3C22×D411yescharacteristic
304C23×C3C22×D411yescharacteristic
305C23×C3C2×#4111yescharacteristic
306C23×C3C2×#4111yescharacteristic
307C23×C3C2×#4112yes—
308C23×C3C2×#4112yes—
309C23×C3H(5,𝔽2)12yes—
310C23×C3H(5,𝔽2)12yes—
311C22×D3—22no—
312C22×D3—22no—
313C22×D3—24no—
314C22×D3—24no—
315C22×D3—22no—
316C22×D3—22no—
317C22×D3—22no—
318C22×D3—22no—
319C22×D3—24no—
320C22×D3—22no—
321C22×D3—22no—
322C22×D3—22no—
323C22×D3—22no—
324C22×D3—24no—
325C22×D3—24no—
326C22×D3—24no—
327C22×D3—24no—
328C22×D3—28no—
329C22×D3—28no—
330C22×D3—42no—
331C22×D3—42no—
332C22×D3—42no—
333C22×D3—42no—
334C22×D3—42no—
335C22×D3—42no—
336C22×D3—42no—
337C22×D3—42no—
338C22×D3—42no—
339C22×D3—42no—
340C22×D3—42no—
341C22×D3—42no—
342C22×D3—44no—
343C22×D3—42no—
344C22×D3—42no—
345C22×D3—42no—
346C22×D3—42no—
347C22×D3—44no—
348C22×D3—44no—
349C22×D3—44no—
350C22×D3—44no—
351C22×D3—44no—
352C22×D3—44no—
353C22×D3—44no—
354C22×D3—44no—
355C22×D3—44no—
356C22×D3C2×C22⋊C414yes—
357C2×C3×C4—24no—
358C2×C3×C4—28no—
359C2×C3×C4—24no—
360C2×C3×C4—24no—
361C2×C3×C4—24no—
362C2×C3×C4—28no—
363C2×C3×C4—24no—
364C2×C3×C4—42no—
365C2×C3×C4—42no—
366C2×C3×C4—42no—
367C2×C3×C4—42no—
368C2×C3×C4—44no—
369C2×C3×C4—42no—
370C2×C3×C4—42no—
371C2×C3×C4—42no—
372C2×C3×C4—44no—
373C2×C3×C4—42no—
374C2×C3×C4C2×C22⋊C414yes—
375C2×C3×C4C4×D414yes—
376C2×C3×C4C4×D414yes—
377C2×C3×C4#12014yes—
378C2×C3×C4#12114yes—
379C2×C3×C4#12114yes—
380C2×C3×C4#12314yes—
381C2×Dic3—22no—
382C2×Dic3—22no—
383C2×Dic3—22no—
384C2×Dic3—22no—
385C2×Dic3—22no—
386C2×Dic3—22no—
387C2×Dic3—24no—
388C2×Dic3—24no—
389C2×Dic3—24no—
390C2×Dic3—28no—
391C2×Dic3—24no—
392C2×Dic3—42no—
393C2×Dic3—42no—
394C2×Dic3—42no—
395C2×Dic3—42no—
396C2×Dic3—42no—
397C2×Dic3—44no—
398C2×Dic3—42no—
399C2×Dic3C2×C22⋊C414yes—
400D12—432no—
401D12—416no—
402C4×D3—232no—
403C24—32no—
404C24—32no—
405C24—32no—
406C24—34no—
407C24—61no—
408C24—61no—
409C24—61no—
410C24—62no—
411C24—62no—
412C24—61no—
413C24—62no—
414C24—62no—
415C24—62no—
416C24—62no—
417C24—62no—
418C24—62no—
419C24—62no—
420C24—64no—
421C24—64no—
422C24—62no—
423C24—64no—
424C24—64no—
425C24C23×D311yescharacteristic
426C2×D4—616no—
427C2×D4—68no—
428C2×D4—68no—
429C2×D4—68no—
430C2×D4—1216no—
431C2×D4—128no—
432C2×D4—128no—
433C2×D4—1216no—
434C2×D4—124no—
435C2×D4—124no—
436C2×D4—124no—
437C2×D4—124no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—24no—
447C22×C4—24no—
448C22×C4—24no—
449C22×C4—22no—
450C22×C4—24no—
451C22×C4—22no—
452C22×C4—32no—
453C22×C4—32no—
454C22×C4—32no—
455C22×C4—316no—
456C22×C4—34no—
457C22×C4—34no—
458C22×C4—61no—
459C22×C4—61no—
460C22×C4—61no—
461C22×C4—61no—
462C22×C4—68no—
463C22×C4—68no—
464C22×C4—68no—
465C22×C4—68no—
466C22×C4—68no—
467C22×C4—68no—
468C22×C4—62no—
469C22×C4—62no—
470C22×C4—62no—
471C22×C4—62no—
472C22×C4—62no—
473C22×C4—62no—
474C22×C4—62no—
475C22×C4—64no—
476C22×C4—62no—
477C22×C4—64no—
478C22×C4C2×C4×D314yes—
479C22×C4C2×C4×D312yes—
480C22×C4C2×D1214yes—
481C22×C4C2×D1212yes—
482C22×C4#23512yes—
483C22×C4D3×D412yes—
484C22×C4D3×D412yes—
485C22×C4D3×D412yes—
486C22×C4#23914yes—
487C22×C4#23912yes—
488C4⋊C4—28no—
489C4⋊C4—28no—
490C4⋊C4—68no—
491C22⋊C4—616no—
492C22⋊C4—616no—
493C22⋊C4—616no—
494C22⋊C4—1216no—
495C22⋊C4—1216no—
496C42—44no—
497C42—616no—
498C42—124no—
499C22×C3C22×C22⋊C411yescharacteristic
500C22×C3C2×C4×D411yescharacteristic
501C22×C3C2×C4×D411yescharacteristic
502C22×C3C4×#4112yes—
503C22×C3#51812yes—
504C22×C3C2×#12011yescharacteristic
505C22×C3C2×#12111yescharacteristic
506C22×C3C2×#12111yescharacteristic
507C22×C3C2×#12311yescharacteristic
508C22×C3#52512yes—
509C22×C3#53211yescharacteristic
510C22×C3#53211yescharacteristic
511C22×C3#53511yescharacteristic
512C22×C3#53511yescharacteristic
513C22×C3#53512yes—
514C22×C3#54612yes—
515C22×C3#54612yes—
516C22×C3#55012yes—
517C22×C3#55012yes—
518C22×C3#55312yes—
519C22×C3#55312yes—
520C22×C3#55512yes—
521C22×C3#55512yes—
522C2×D3—24no—
523C2×D3—24no—
524C2×D3—24no—
525C2×D3—44no—
526C2×D3—44no—
527C2×D3—44no—
528C2×D3—44no—
529C2×D3—42no—
530C2×D3—42no—
531C2×D3—42no—
532C2×D3—42no—
533C2×D3—44no—
534C2×D3—48no—
535C2×D3—48no—
536C2×D3—44no—
537C2×D3—44no—
538C2×D3—84no—
539C2×D3—84no—
540C2×D3—82no—
541C2×D3—82no—
542C2×D3—82no—
543C2×D3—82no—
544C3×C4—44no—
545C3×C4—44no—
546C3×C4—44no—
547C3×C4#38618yes—
548Dic3—24no—
549Dic3—44no—
550Dic3—44no—
551C23—34no—
552C23—62no—
553C23—62no—
554C23—62no—
555C23—62no—
556C23—62no—
557C23—62no—
558C23—62no—
559C23—62no—
560C23—62no—
561C23—62no—
562C23—64no—
563C23—64no—
564C23—64no—
565C23—64no—
566C23—64no—
567C23—64no—
568C23—64no—
569C23—68no—
570C23—68no—
571C23—122no—
572C23—122no—
573C23—122no—
574C23—122no—
575C23—122no—
576C23—122no—
577C23—122no—
578C23—122no—
579C23—122no—
580C23—122no—
581C23—124no—
582C23—122no—
583C23—122no—
584C23—124no—
585C23—122no—
586C23—122no—
587C23—122no—
588C23—122no—
589C23—124no—
590C23—124no—
591C23—124no—
592C23—124no—
593C23—124no—
594C23—124no—
595C23—124no—
596C23—124no—
597C23C22×C4×D311yescharacteristic
598C23C22×D1211yescharacteristic
599C23C2×#23511yescharacteristic
600C23C2×D3×D411yescharacteristic
601C23C2×D3×D411yescharacteristic
602C23C2×D3×D411yescharacteristic
603C23#100412yes—
604C23C2×#23911yescharacteristic
605C23C2×#23912yes—
606C23D3×#4112yes—
607C23#100912yes—
608D4—1232no—
609D4—1216no—
610C2×C4—24no—
611C2×C4—24no—
612C2×C4—24no—
613C2×C4—28no—
614C2×C4—28no—
615C2×C4—24no—
616C2×C4—24no—
617C2×C4—34no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—44no—
625C2×C4—42no—
626C2×C4—44no—
627C2×C4—42no—
628C2×C4—62no—
629C2×C4—62no—
630C2×C4—62no—
631C2×C4—62no—
632C2×C4—62no—
633C2×C4—62no—
634C2×C4—632no—
635C2×C4—64no—
636C2×C4—64no—
637C2×C4—64no—
638C2×C4—64no—
639C2×C4—68no—
640C2×C4—122no—
641C2×C4—122no—
642C2×C4—122no—
643C2×C4—122no—
644C2×C4—124no—
645C2×C4—122no—
646C2×C4—122no—
647C2×C4C4×D1214yes—
648C2×C4D3×C22⋊C414yes—
649C2×C4#88214yes—
650C2×C4#88414yes—
651C2×C4#89314yes—
652C2×C4#89414yes—
653C2×C4#89514yes—
654C2×C3C2×#38611yescharacteristic
655C2×C3#229711yescharacteristic
656C2×C3#230911yescharacteristic
657C2×C3#236312yes—
658C2×C3#236312yes—
659C2×C3#241412yes—
660C2×C3#241412yes—
661C2×C3#243111yescharacteristic
662C2×C3#243111yescharacteristic
663C2×C3#250011yescharacteristic
664C2×C3#250011yescharacteristic
665D3—44no—
666D3—84no—
667C22—64no—
668C22—64no—
669C22—64no—
670C22—122no—
671C22—124no—
672C22—122no—
673C22—122no—
674C22—124no—
675C22—124no—
676C22—122no—
677C22—124no—
678C22—124no—
679C22—124no—
680C22—128no—
681C22—124no—
682C22—128no—
683C22—242no—
684C22—244no—
685C22—242no—
686C22—242no—
687C22—244no—
688C22—242no—
689C22C2×C4×D1211yescharacteristic
690C22C2×D3×C22⋊C411yescharacteristic
691C22C2×#88211yescharacteristic
692C22C2×#88411yescharacteristic
693C22C2×#89311yescharacteristic
694C22C2×#89411yescharacteristic
695C22C2×#89511yescharacteristic
696C22#551412yes—
697C22#553212yes—
698C22#554512yes—
699C22#555412yes—
700C22C4×#23912yes—
701C22#557712yes—
702C22#559912yes—
703C22#560712yes—
704C22#562712yes—
705C22#563012yes—
706C22#566811yescharacteristic
707C22#567411yescharacteristic
708C22#569512yes—
709C22#571511yescharacteristic
710C22#572511yescharacteristic
711C22#573412yes—
712C4—44no—
713C4—44no—
714C4—44no—
715C4—64no—
716C4—124no—
717C4—124no—
718C4#466518yes—
719C3#1413611yescharacteristic, complemented, Sylow
720C2—124no—
721C2—244no—
722C2C2×#466511yescharacteristic
723C2#7364912yes—
724C2#7401912yes—
725C2#7416611yescharacteristic
726C2#7436711yescharacteristic
727C2#7598512yes—
728C2#7620912yes—
729C2#7631311yescharacteristic
730C2#7642811yescharacteristic
731C2#7647811yescharacteristic
732C2#7662611yescharacteristic
subgroupG/H
center, FrattiniC24C23×D3
commutatorC23×C3C23×C4
FittingC3×C4⋊#95C2
socle, cosocleC24×C3C24
prime radicalC22×#895C2
hypercenterC4⋊#95D3
hypocenterC3#14136
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C24 ◁ C4⋊#95

Complements

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

NH ≅ G/Ntype
2. C22×#892C2semidirect
3. C2×#4988C2semidirect
4. C2×#4939C2semidirect
5. C2×C4×(C2×D3)⋊C4C2semidirect
6. C3×C4⋊#95C2semidirect
7. #71563C2semidirect
8. #69426C2semidirect
9. #69408C2semidirect
27. C2×#895C4semidirect
45. C4⋊#95D3semidirect
79. C2×C3×C4⋊C4D4semidirect
100. C2×C4⋊Dic3D4semidirect
222. C2×C4×D3C22⋊C4semidirect
272. C2×C4⋊C4D12semidirect
719. C3#14136semidirect

Maximal subgroups

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

subgroupindex
C22×#8952
C22×#8922
C2×#49882
C2×#49392
C2×C4×(C2×D3)⋊C42
C3×C4⋊#952
#715632
#694262
#694082
#141363