Finite Group Databasebeta

← #362748

Subgroups — #362748

Order 512.

2570 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
1C22×#544C211yesmaximal, characteristic
2C2×#1993C211yesmaximal, characteristic, complemented
3C2×#1992C211yesmaximal, characteristic, complemented
4C2×#1991C211yesmaximal, characteristic, complemented
5C2×#1990C211yesmaximal, characteristic, complemented
6C2×#1750C211yesmaximal, characteristic
7#14340C211yesmaximal, characteristic
8#11385C212yesmaximal, complemented
9#11384C212yesmaximal, complemented
10#10731C212yesmaximal
11#10730C212yesmaximal
12C2×#544—24no—
13C2×#544—24no—
14C2×#544—24no—
15C2×#544—24no—
16C2×#544—24no—
17C2×#544C418yescomplemented
18C22×C4⋊Q8—21no—
19C22×C4⋊Q8C2211yescharacteristic
20C22×C4⋊Q8C2211yescharacteristic
21C22×#125—21no—
22C22×#125C2211yescharacteristic
23C22×#125C2211yescharacteristic
24C22×C22⋊Q8—21no—
25C22×C22⋊Q8—21no—
26C22×#117C2211yescharacteristic
27C22×#117C2211yescharacteristic
28C23×C4⋊C4C2211yescharacteristic
29C22×C22⋊C8C2212yes—
30C22×#97C2211yescharacteristic
31C22×C4×C8C2211yescharacteristic
32C2×#384C2211yescharacteristic
33C2×#382C2211yescharacteristic
34C2×#380C2212yes—
35#1993—24no—
36#1992—24no—
37#1991—24no—
38#1990—24no—
39#1750—24no—
40C2×#340C2212yes—
41C2×#340C2212yes—
42C2×#340C2212yes—
43C2×#340C2212yes—
44C2×#340C2212yes—
45C2×#340C2212yes—
46C2×#340C2212yes—
47C2×#340C2212yes—
48C2×#336C2212yes—
49#1441—21no—
50#1441—21no—
51#1439—21no—
52#1439—21no—
53C4⋊#95C2212yes—
54C23×Q8—21no—
55C23×Q8—21no—
56C24×C4—21no—
57C24×C4C2311yescharacteristic
58C23×C8C2312yes—
59C23×C8C2312yes—
60#544—232no—
61#544—232no—
62#544—232no—
63#544—232no—
64C2×C4⋊Q8—28no—
65C2×C4⋊Q8—28no—
66C2×C4⋊Q8—28no—
67C2×C4⋊Q8—22no—
68C2×C4⋊Q8—22no—
69C2×C4⋊Q8—22no—
70C2×C4⋊Q8—22no—
71C2×C4⋊Q8—22no—
72C2×C4⋊Q8—22no—
73C2×C4⋊Q8—22no—
74C2×C4⋊Q8—22no—
75C2×C4⋊Q8C2×C414yes—
76C2×C4⋊Q8C2×C414yes—
77C2×C4⋊Q8D414yes—
78C2×C4⋊Q8D414yes—
79C2×#125—28no—
80C2×#125—28no—
81C2×#125—28no—
82C2×#125—22no—
83C2×#125—22no—
84C2×#125—22no—
85C2×#125—22no—
86C2×#125—22no—
87C2×#125—22no—
88C2×#125—22no—
89C2×#125—22no—
90C2×#125C2×C414yes—
91C2×#125C2×C414yes—
92C2×#125D414yes—
93C2×#125D414yes—
94C2×C22⋊Q8—28no—
95C2×C22⋊Q8—28no—
96C2×C22⋊Q8—28no—
97C2×C22⋊Q8—28no—
98C2×C22⋊Q8—28no—
99C2×C22⋊Q8—28no—
100C2×#117—22no—
101C2×#117—22no—
102C2×#117—22no—
103C2×#117—22no—
104C2×#117—22no—
105C2×#117—22no—
106C2×#117—22no—
107C2×#117—22no—
108C2×#117—22no—
109C2×#117C2×C414yes—
110C2×#117C2×C414yes—
111C2×#117D414yes—
112C22×C4⋊C4—21no—
113C22×C4⋊C4—22no—
114C22×C4⋊C4—21no—
115C22×C4⋊C4—21no—
116C22×C4⋊C4—21no—
117C22×C4⋊C4—21no—
118C22×C4⋊C4—21no—
119C22×C4⋊C4—21no—
120C22×C4⋊C4—22no—
121C22×C4⋊C4—22no—
122C22×C4⋊C4—22no—
123C22×C4⋊C4—22no—
124C22×C4⋊C4—21no—
125C22×C4⋊C4—21no—
126C22×C4⋊C4C2×C414yes—
127C22×C4⋊C4D411yescharacteristic
128C22×C4⋊C4D411yescharacteristic
129C22×C4⋊C4C2312yes—
130C22×C4⋊C4C2312yes—
131C22×C4⋊C4C2311yescharacteristic
132C22×C4⋊C4C2311yescharacteristic
133C22×C4⋊C4C2311yescharacteristic
134C22×C4⋊C4C2311yescharacteristic
135C22×C22⋊C4—21no—
136C22×C22⋊C4—22no—
137C22×C22⋊C4D411yescharacteristic
138C22×C22⋊C4D411yescharacteristic
139C22×C42—21no—
140C22×C42C2311yescharacteristic
141C22×C42C2311yescharacteristic
142C2×C22⋊C8—24no—
143C2×C22⋊C8—24no—
144C2×C22⋊C8Q818yescomplemented
145C2×#97—22no—
146C2×#97—22no—
147C2×#97—22no—
148C2×C4×C8—22no—
149C2×C4×C8—22no—
150C2×C4×C8—22no—
151#384—22no—
152#384—22no—
153#382—22no—
154#382—22no—
155#380—28no—
156C2×#95—21no—
157C2×#95—21no—
158C2×#95—22no—
159C2×#95—22no—
160C2×#95—22no—
161#340—24no—
162#340—24no—
163#340—24no—
164#340—24no—
165#340—24no—
166#340—24no—
167#340—24no—
168#340—24no—
169#336—24no—
170C25C2×D411yescharacteristic
171C22×Q8—24no—
172C22×Q8—24no—
173C22×Q8—24no—
174C22×Q8—24no—
175C22×Q8—24no—
176C22×Q8—24no—
177C22×Q8—82no—
178C22×Q8—82no—
179C22×Q8—82no—
180C22×Q8—82no—
181C23×C4—21no—
182C23×C4—21no—
183C23×C4—21no—
184C23×C4—21no—
185C23×C4—22no—
186C23×C4—22no—
187C23×C4—21no—
188C23×C4—21no—
189C23×C4—21no—
190C23×C4—21no—
191C23×C4—24no—
192C23×C4—24no—
193C23×C4—24no—
194C23×C4—22no—
195C23×C4—22no—
196C23×C4—82no—
197C23×C4—82no—
198C23×C4C22×C412yes—
199C23×C4C2×D411yescharacteristic
200C23×C4C2×D411yescharacteristic
201C23×C4C2×D411yescharacteristic
202C23×C4C2×D411yescharacteristic
203C23×C4C2×D411yescharacteristic
204C23×C4C2×D412yes—
205C23×C4C2×Q814yes—
206C23×C4#4114yes—
207C23×C4C2411yescharacteristic
208C22×C8—22no—
209C22×C8—22no—
210C22×C8—22no—
211C22×C8—22no—
212C22×C8—22no—
213C22×C8—22no—
214C22×C8—22no—
215C22×C8—22no—
216C22×C8—22no—
217C22×C8—22no—
218C22×C8—22no—
219C22×C8—22no—
220C22×C8C2×Q814yes—
221C22×C8C2×Q814yes—
222C4⋊Q8—264no—
223C4⋊Q8—28no—
224C4⋊Q8—28no—
225C4⋊Q8—28no—
226C4⋊Q8—28no—
227C4⋊Q8—28no—
228C4⋊Q8—28no—
229C4⋊Q8—28no—
230C4⋊Q8C22⋊C4116yescomplemented
231#125—264no—
232#125—28no—
233#125—28no—
234#125—28no—
235#125—28no—
236#125—28no—
237#125—28no—
238#125—28no—
239#125C22⋊C4116yescomplemented
240C22⋊Q8—264no—
241C22⋊Q8—264no—
242#117—28no—
243#117—28no—
244#117—28no—
245#117—28no—
246#117—28no—
247#117—28no—
248#117—28no—
249#117—28no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—28no—
253C2×C4⋊C4—28no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—24no—
262C2×C4⋊C4—24no—
263C2×C4⋊C4—24no—
264C2×C4⋊C4—24no—
265C2×C4⋊C4—24no—
266C2×C4⋊C4—24no—
267C2×C4⋊C4—24no—
268C2×C4⋊C4—24no—
269C2×C4⋊C4—24no—
270C2×C4⋊C4—24no—
271C2×C4⋊C4—24no—
272C2×C4⋊C4—24no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—21no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—28no—
290C2×C4⋊C4—28no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—24no—
296C2×C4⋊C4—24no—
297C2×C4⋊C4—24no—
298C2×C4⋊C4—24no—
299C2×C4⋊C4—42no—
300C2×C4⋊C4—42no—
301C2×C4⋊C4—82no—
302C2×C4⋊C4—82no—
303C2×C4⋊C4C22⋊C414yes—
304C2×C4⋊C4C22⋊C414yes—
305C2×C4⋊C4C22×C412yes—
306C2×C4⋊C4C22×C412yes—
307C2×C4⋊C4C22×C412yes—
308C2×C4⋊C4C22×C412yes—
309C2×C4⋊C4C2×D412yes—
310C2×C4⋊C4C2×D412yes—
311C2×C4⋊C4C2×D412yes—
312C2×C4⋊C4C2×D412yes—
313C2×C4⋊C4C2×D412yes—
314C2×C4⋊C4C2×D412yes—
315C2×C4⋊C4C2×D412yes—
316C2×C4⋊C4C2×D412yes—
317C2×C22⋊C4—24no—
318C2×C22⋊C4—24no—
319C2×C22⋊C4—24no—
320C2×C22⋊C4—22no—
321C2×C22⋊C4—22no—
322C2×C22⋊C4—22no—
323C2×C22⋊C4—22no—
324C2×C22⋊C4—44no—
325C2×C22⋊C4—82no—
326C2×C22⋊C4—82no—
327C2×C22⋊C4C22⋊C414yes—
328C2×C22⋊C4C22⋊C414yes—
329C2×C42—24no—
330C2×C42—24no—
331C2×C42—24no—
332C2×C42—21no—
333C2×C42—21no—
334C2×C42—21no—
335C2×C42—21no—
336C2×C42C22×C412yes—
337C2×C42C22×C412yes—
338C2×C42C2×D412yes—
339C2×C42C2×D412yes—
340C2×C42C2×D412yes—
341C2×C42C2×D412yes—
342C2×C42C2×D412yes—
343C2×C42C2×D412yes—
344C22⋊C8—216no—
345#97—44no—
346C4×C8—44no—
347#95—42no—
348#95—42no—
349#95—44no—
350#95—82no—
351#95—82no—
352C24—21no—
353C24—21no—
354C24—21no—
355C24—21no—
356C24—22no—
357C24—22no—
358C24—22no—
359C24—22no—
360C24C2×C22⋊C412yes—
361C24#12412yes—
362C24C4⋊Q812yes—
363C24C22×D411yescharacteristic
364C2×Q8—216no—
365C2×Q8—216no—
366C2×Q8—88no—
367C2×Q8—84no—
368C2×Q8—84no—
369C2×Q8—88no—
370C2×Q8—88no—
371C2×Q8—84no—
372C2×Q8—84no—
373C2×Q8—88no—
374C22×C4—21no—
375C22×C4—21no—
376C22×C4—21no—
377C22×C4—21no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—24no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—24no—
392C22×C4—24no—
393C22×C4—24no—
394C22×C4—24no—
395C22×C4—24no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—21no—
401C22×C4—21no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—21no—
405C22×C4—21no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—216no—
417C22×C4—42no—
418C22×C4—42no—
419C22×C4—42no—
420C22×C4—42no—
421C22×C4—42no—
422C22×C4—42no—
423C22×C4—42no—
424C22×C4—42no—
425C22×C4—42no—
426C22×C4—42no—
427C22×C4—42no—
428C22×C4—42no—
429C22×C4—42no—
430C22×C4—42no—
431C22×C4—42no—
432C22×C4—42no—
433C22×C4—42no—
434C22×C4—42no—
435C22×C4—42no—
436C22×C4—42no—
437C22×C4—42no—
438C22×C4—42no—
439C22×C4—88no—
440C22×C4—84no—
441C22×C4—84no—
442C22×C4—88no—
443C22×C4C2×C22⋊C412yes—
444C22×C4C2×C22⋊C412yes—
445C22×C4C2×C22⋊C412yes—
446C22×C4C2×C22⋊C412yes—
447C22×C4C2×C22⋊C412yes—
448C22×C4C4×Q818yes—
449C22×C4#12012yes—
450C22×C4#12012yes—
451C22×C4#12012yes—
452C22×C4#12012yes—
453C22×C4C22⋊Q818yes—
454C22×C4#12412yes—
455C22×C4C4⋊Q812yes—
456C22×C4C23×C411yescharacteristic
457C22×C4C22×D411yescharacteristic
458C22×C4C22×D411yescharacteristic
459C22×C4C22×D411yescharacteristic
460C22×C4C22×D411yescharacteristic
461C22×C4C22×D411yescharacteristic
462C22×C4C22×Q812yes—
463C22×C4C2×#4112yes—
464C22×C42−1+412yes—
465C22×C42−1+412yes—
466C2×C8—24no—
467C2×C8—24no—
468C2×C8—24no—
469C2×C8—24no—
470C2×C8—24no—
471C2×C8—24no—
472C2×C8—42no—
473C2×C8—42no—
474C2×C8—42no—
475C2×C8—42no—
476C2×C8—42no—
477C2×C8—42no—
478C2×C8—42no—
479C2×C8—42no—
480C4⋊C4—216no—
481C4⋊C4—216no—
482C4⋊C4—216no—
483C4⋊C4—216no—
484C4⋊C4—22no—
485C4⋊C4—22no—
486C4⋊C4—22no—
487C4⋊C4—22no—
488C4⋊C4—22no—
489C4⋊C4—22no—
490C4⋊C4—22no—
491C4⋊C4—22no—
492C4⋊C4—22no—
493C4⋊C4—22no—
494C4⋊C4—22no—
495C4⋊C4—22no—
496C4⋊C4—216no—
497C4⋊C4—216no—
498C4⋊C4—216no—
499C4⋊C4—28no—
500C4⋊C4—28no—
501C4⋊C4—44no—
502C4⋊C4—44no—
503C4⋊C4—44no—
504C4⋊C4—44no—
505C4⋊C4—84no—
506C4⋊C4—84no—
507C4⋊C4C2×C22⋊C414yes—
508C4⋊C4C2×C22⋊C414yes—
509C4⋊C4C2×C22⋊C414yes—
510C4⋊C4C2×C22⋊C414yes—
511C22⋊C4—216no—
512C22⋊C4—28no—
513C22⋊C4—28no—
514C22⋊C4—88no—
515C42—216no—
516C42—22no—
517C42—22no—
518C42—22no—
519C42—22no—
520C42C2×C22⋊C414yes—
521C42C2×C22⋊C414yes—
522C42#12014yes—
523C42#12014yes—
524C23—22no—
525C23—24no—
526C23—24no—
527C23—24no—
528C23—24no—
529C23—24no—
530C23—24no—
531C23—22no—
532C23—22no—
533C23—22no—
534C23—22no—
535C23—22no—
536C23—42no—
537C23—42no—
538C23—42no—
539C23—42no—
540C23—41no—
541C23—41no—
542C23—41no—
543C23—41no—
544C23—42no—
545C23—42no—
546C23#39114yes—
547C23C22×C22⋊C411yescharacteristic
548C23C2×#12011yescharacteristic
549C23C2×#12011yescharacteristic
550C23C2×#12411yescharacteristic
551C23C2×C4⋊Q811yescharacteristic
552C23#53611yescharacteristic
553C23#53611yescharacteristic
554C23#57612yes—
555C23#57712yes—
556C23#57712yes—
557C23#57812yes—
558Q8—816no—
559Q8—816no—
560C2×C4—22no—
561C2×C4—24no—
562C2×C4—24no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—24no—
572C2×C4—24no—
573C2×C4—22no—
574C2×C4—28no—
575C2×C4—28no—
576C2×C4—28no—
577C2×C4—28no—
578C2×C4—24no—
579C2×C4—24no—
580C2×C4—22no—
581C2×C4—24no—
582C2×C4—22no—
583C2×C4—24no—
584C2×C4—24no—
585C2×C4—24no—
586C2×C4—44no—
587C2×C4—44no—
588C2×C4—44no—
589C2×C4—44no—
590C2×C4—41no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—41no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—41no—
599C2×C4—42no—
600C2×C4—41no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—44no—
604C2×C4—44no—
605C2×C4—44no—
606C2×C4—42no—
607C2×C4—44no—
608C2×C4—816no—
609C2×C4—82no—
610C2×C4—82no—
611C2×C4—82no—
612C2×C4—82no—
613C2×C4—82no—
614C2×C4—82no—
615C2×C4—82no—
616C2×C4—82no—
617C2×C4—82no—
618C2×C4—82no—
619C2×C4—82no—
620C2×C4—82no—
621C2×C4—82no—
622C2×C4—82no—
623C2×C4#37914yes—
624C2×C4#37914yes—
625C2×C4#39114yes—
626C2×C4C22×C22⋊C411yescharacteristic
627C2×C4C22×C22⋊C411yescharacteristic
628C2×C4C2×C4×Q812yes—
629C2×C4#51912yes—
630C2×C4C2×#12011yescharacteristic
631C2×C4C2×#12011yescharacteristic
632C2×C4C2×C22⋊Q812yes—
633C2×C4#53612yes—
634C2×C4#54812yes—
635C2×C4#54812yes—
636C2×C4#54812yes—
637C2×C4#54812yes—
638C2×C4D4×Q812yes—
639C2×C4D4×Q812yes—
640C2×C4D4×Q812yes—
641C2×C4D4×Q812yes—
642C8—44no—
643C8—44no—
644C22—28no—
645C22—24no—
646C22—42no—
647C22—42no—
648C22—44no—
649C22—44no—
650C22—42no—
651C22—42no—
652C22—42no—
653C22—42no—
654C22—44no—
655C22C2×#37911yescharacteristic
656C22C2×#39111yescharacteristic
657C22#231011yescharacteristic
658C22#243411yescharacteristic
659C22#243411yescharacteristic
660C22#243611yescharacteristic
661C22#243611yescharacteristic
662C22#289712yes—
663C22#289812yes—
664C22#300811yescharacteristic
665C22#300911yescharacteristic
666C22#301011yescharacteristic
667C22#301111yescharacteristic
668C22#301212yes—
669C22#301312yes—
670C22#313612yes—
671C22#313712yes—
672C22#316312yes—
673C22#316412yes—
674C22#324112yes—
675C22#324212yes—
676C22#324312yes—
677C22#324412yes—
678C4—24no—
679C4—24no—
680C4—24no—
681C4—42no—
682C4—44no—
683C4—44no—
684C4—44no—
685C4—42no—
686C4—44no—
687C4—162no—
688C4—162no—
689C4Q8×C22⋊C412yes—
690C4Q8×C22⋊C412yes—
691C4#240912yes—
692C4#240912yes—
693C2—44no—
694C2#1392211yescharacteristic
695C2#2060211yescharacteristic
696C2#2060311yescharacteristic
697C2#2132612yes—
698C2#2184312yes—
699C2#2582011yescharacteristic
700C2#2582111yescharacteristic
701C2#2582212yes—
702C2#2591411yescharacteristic
703C2#2591511yescharacteristic
704C2#2591612yes—
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×#117 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#1993C2semidirect
3. C2×#1992C2semidirect
4. C2×#1991C2semidirect
5. C2×#1990C2semidirect
8. #11385C2semidirect
9. #11384C2semidirect
17. C2×#544C4semidirect
144. C2×C22⋊C8Q8semidirect
230. C4⋊Q8C22⋊C4semidirect
239. #125C22⋊C4semidirect

Maximal subgroups

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

subgroupindex
C22×#5442
C2×#19932
C2×#19922
C2×#19912
C2×#19902
C2×#17502
#143402
#113852
#113842
#107312
#107302