Finite Group Databasebeta

← #362763

Subgroups — #362763

Order 512.

2506 conjugacy classes of subgroups.

The subgroup lattice has 667 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#557C211yesmaximal, characteristic
2C2×#1991C211yesmaximal, characteristic
3C2×#1989C211yesmaximal, characteristic
4C2×#1984C211yesmaximal, characteristic
5C2×#1805C211yesmaximal, characteristic
6C2×C4×#103C211yesmaximal, characteristic
7#13925C211yesmaximal, characteristic
8#11385C212yesmaximal
9#11384C212yesmaximal
10#10789C212yesmaximal
11#10789C212yesmaximal
12C2×#557—24no—
13C2×#557—24no—
14C2×#557—24no—
15C2×#557—24no—
16C2×#557—24no—
17C2×#557C418yescomplemented
18C22×C4⋊Q8—21no—
19C22×C4⋊Q8C2211yescharacteristic
20C22×#125—21no—
21C22×#125—21no—
22C22×#125C2211yescharacteristic
23C22×#125C2211yescharacteristic
24C22×C4×Q8—21no—
25C22×C4×Q8C2211yescharacteristic
26C22×C4×Q8C2211yescharacteristic
27C22×C4×Q8C2211yescharacteristic
28C22×C4×Q8C2211yescharacteristic
29C22×C4⋊C8C2211yescharacteristic
30C22×#103C2212yes—
31C22×#103C2212yes—
32C22×C4×C8C2211yescharacteristic
33C2×#382C2211yescharacteristic
34C2×C4×C4⋊C4C2211yescharacteristic
35#1991—24no—
36#1989—24no—
37#1984—24no—
38#1805—24no—
39C4×#103—24no—
40C2×#340C2212yes—
41C2×#340C2212yes—
42C2×#340C2212yes—
43C2×#340C2212yes—
44C2×#340C2212yes—
45C2×#340C2212yes—
46C2×#336C2212yes—
47C2×#336C2212yes—
48#1446—21no—
49#1445—21no—
50#1436—21no—
51#1436—21no—
52C4⋊#95C2212yes—
53C4⋊#95C2212yes—
54C23×Q8—21no—
55C23×Q8D411yescharacteristic
56C23×Q8C2311yescharacteristic
57C23×C8C2312yes—
58C23×C8C2312yes—
59#557—232no—
60#557—232no—
61#557—232no—
62#557—232no—
63C2×C4⋊Q8—28no—
64C2×C4⋊Q8—28no—
65C2×C4⋊Q8—28no—
66C2×C4⋊Q8—22no—
67C2×C4⋊Q8—22no—
68C2×C4⋊Q8—22no—
69C2×C4⋊Q8—22no—
70C2×C4⋊Q8—22no—
71C2×C4⋊Q8C2×C414yes—
72C2×#125—28no—
73C2×#125—28no—
74C2×#125—28no—
75C2×#125—28no—
76C2×#125—28no—
77C2×#125—28no—
78C2×#125—22no—
79C2×#125—22no—
80C2×#125—22no—
81C2×#125—22no—
82C2×#125—22no—
83C2×#125—22no—
84C2×#125—22no—
85C2×#125—22no—
86C2×#125C2×C414yes—
87C2×#125C2×C414yes—
88C2×#125D414yes—
89C2×#125D414yes—
90C2×C4×Q8—28no—
91C2×C4×Q8—28no—
92C2×C4×Q8—22no—
93C2×C4×Q8—22no—
94C2×C4×Q8—22no—
95C2×C4×Q8—22no—
96C2×C4×Q8—28no—
97C2×C4×Q8—22no—
98C2×C4×Q8—22no—
99C2×C4×Q8—22no—
100C2×C4×Q8—22no—
101C2×C4×Q8—22no—
102C2×C4×Q8—22no—
103C2×C4×Q8—22no—
104C2×C4×Q8—22no—
105C2×C4×Q8—22no—
106C2×C4×Q8—22no—
107C2×C4×Q8—22no—
108C2×C4×Q8—22no—
109C2×C4×Q8—22no—
110C2×C4×Q8C2×C414yes—
111C2×C4×Q8C2×C414yes—
112C2×C4×Q8C2×C414yes—
113C2×C4×Q8C2×C414yes—
114C2×C4×Q8D414yes—
115C2×C4×Q8D414yes—
116C2×C4×Q8D414yes—
117C22×C4⋊C4—21no—
118C22×C4⋊C4—21no—
119C22×C4⋊C4—21no—
120C22×C4⋊C4—21no—
121C22×C4⋊C4—21no—
122C22×C4⋊C4—21no—
123C22×C4⋊C4—21no—
124C22×C4⋊C4—21no—
125C22×C4⋊C4D411yescharacteristic
126C22×C4⋊C4D411yescharacteristic
127C22×C4⋊C4D411yescharacteristic
128C22×C4⋊C4C2312yes—
129C22×C4⋊C4C2312yes—
130C22×C4⋊C4C2311yescharacteristic
131C22×C4⋊C4C2311yescharacteristic
132C22×C4⋊C4C2311yescharacteristic
133C22×C42—21no—
134C22×C42—21no—
135C22×C42—21no—
136C22×C42—21no—
137C22×C42C2311yescharacteristic
138C22×C42C2311yescharacteristic
139C22×C42C2311yescharacteristic
140C2×C4⋊C8—22no—
141C2×C4⋊C8—22no—
142C2×C4⋊C8—22no—
143C2×#103—24no—
144C2×#103—24no—
145C2×#103—24no—
146C2×#103—24no—
147C2×#103—24no—
148C2×#103Q818yescomplemented
149C2×C4×C8—22no—
150C2×C4×C8—22no—
151C2×C4×C8—22no—
152#382—22no—
153#382—22no—
154C4×C4⋊C4—22no—
155C4×C4⋊C4—22no—
156C2×#95—22no—
157C2×#95—22no—
158C2×#95—21no—
159C2×#95—21no—
160C2×#95—21no—
161C2×#95—22no—
162C2×#95—22no—
163#340—24no—
164#340—24no—
165#340—24no—
166#340—24no—
167#340—24no—
168#340—24no—
169#336—24no—
170#336—24no—
171C22×Q8—24no—
172C22×Q8—24no—
173C22×Q8—24no—
174C22×Q8—22no—
175C22×Q8—22no—
176C22×Q8—24no—
177C22×Q8—21no—
178C22×Q8—21no—
179C22×Q8—21no—
180C22×Q8—22no—
181C22×Q8—22no—
182C22×Q8—42no—
183C22×Q8—44no—
184C22×Q8—44no—
185C22×Q8C22⋊C414yes—
186C22×Q8C22×C412yes—
187C22×Q8C2×D412yes—
188C22×Q8C2×D412yes—
189C22×Q8C2×Q814yes—
190C22×Q8#4114yes—
191C23×C4—21no—
192C23×C4—21no—
193C23×C4—21no—
194C23×C4—21no—
195C23×C4—22no—
196C23×C4—22no—
197C23×C4C2×D411yescharacteristic
198C23×C4C2×D411yescharacteristic
199C23×C4C2×D411yescharacteristic
200C23×C4C2×D411yescharacteristic
201C23×C4C2×D411yescharacteristic
202C23×C4C2×D411yescharacteristic
203C23×C4C2411yescharacteristic
204C22×C8—22no—
205C22×C8—22no—
206C22×C8—22no—
207C22×C8—22no—
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×C8C2×Q814yes—
218C4⋊Q8—264no—
219C4⋊Q8—28no—
220C4⋊Q8—28no—
221C4⋊Q8—28no—
222C4⋊Q8—28no—
223#125—264no—
224#125—28no—
225#125—28no—
226#125—264no—
227#125—28no—
228#125—28no—
229#125—28no—
230#125—28no—
231#125—28no—
232#125C22⋊C4116yes—
233C4×Q8—28no—
234C4×Q8—28no—
235C4×Q8—264no—
236C4×Q8—28no—
237C4×Q8—28no—
238C4×Q8—28no—
239C4×Q8—28no—
240C4×Q8—28no—
241C4×Q8—28no—
242C4×Q8—28no—
243C4×Q8—28no—
244C4×Q8—28no—
245C4×Q8—28no—
246C4×Q8—28no—
247C4×Q8—28no—
248C4×Q8—28no—
249C4×Q8C22⋊C4116yes—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—24no—
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—24no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—22no—
279C2×C4⋊C4—22no—
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—22no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4C22⋊C414yes—
301C2×C4⋊C4C22⋊C414yes—
302C2×C4⋊C4C22⋊C414yes—
303C2×C4⋊C4C22×C412yes—
304C2×C4⋊C4C22×C412yes—
305C2×C4⋊C4C22×C412yes—
306C2×C4⋊C4C2×D412yes—
307C2×C4⋊C4C2×D412yes—
308C2×C4⋊C4C2×D412yes—
309C2×C4⋊C4C2×D412yes—
310C2×C4⋊C4C2×D412yes—
311C2×C4⋊C4C2×D412yes—
312C2×C4⋊C4C2×Q814yes—
313C2×C42—24no—
314C2×C42—24no—
315C2×C42—24no—
316C2×C42—24no—
317C2×C42—24no—
318C2×C42—24no—
319C2×C42—24no—
320C2×C42—24no—
321C2×C42—24no—
322C2×C42—21no—
323C2×C42—21no—
324C2×C42—21no—
325C2×C42—21no—
326C2×C42—21no—
327C2×C42—21no—
328C2×C42—21no—
329C2×C42—21no—
330C2×C42—42no—
331C2×C42—82no—
332C2×C42C22×C412yes—
333C2×C42C22×C412yes—
334C2×C42C22×C412yes—
335C2×C42C2×D412yes—
336C2×C42C2×D412yes—
337C2×C42C2×D412yes—
338C2×C42C2×D412yes—
339C2×C42C2×D412yes—
340C2×C42C2×D412yes—
341C2×C42C2×D412yes—
342C4⋊C8—44no—
343#103—216no—
344#103—48no—
345C4×C8—44no—
346#95—44no—
347#95—42no—
348#95—42no—
349#95—42no—
350#95—44no—
351#95—82no—
352#95—82no—
353C24C22×D411yescharacteristic
354C2×Q8—216no—
355C2×Q8—22no—
356C2×Q8—24no—
357C2×Q8—22no—
358C2×Q8—22no—
359C2×Q8—24no—
360C2×Q8—22no—
361C2×Q8—22no—
362C2×Q8—24no—
363C2×Q8—24no—
364C2×Q8—24no—
365C2×Q8—24no—
366C2×Q8—24no—
367C2×Q8—24no—
368C2×Q8—28no—
369C2×Q8—28no—
370C2×Q8—44no—
371C2×Q8—48no—
372C2×Q8—48no—
373C2×Q8—48no—
374C2×Q8—416no—
375C2×Q8—416no—
376C2×Q8—48no—
377C2×Q8C2×C22⋊C414yes—
378C2×Q8C4×Q818yes—
379C2×Q8C22⋊Q818yes—
380C2×Q8#12414yes—
381C2×Q8C4⋊Q814yes—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—21no—
386C22×C4—21no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—21no—
398C22×C4—21no—
399C22×C4—21no—
400C22×C4—21no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—21no—
404C22×C4—21no—
405C22×C4—22no—
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—24no—
416C22×C4—42no—
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×C4C2×C22⋊C412yes—
436C22×C4C2×C22⋊C412yes—
437C22×C4C2×C22⋊C412yes—
438C22×C4C2×C22⋊C412yes—
439C22×C4C2×C22⋊C412yes—
440C22×C4C2×C22⋊C412yes—
441C22×C4#12012yes—
442C22×C4#12012yes—
443C22×C4#12012yes—
444C22×C4#12012yes—
445C22×C4#13514yes—
446C22×C4C23×C411yescharacteristic
447C22×C4C22×D411yescharacteristic
448C22×C4C22×D411yescharacteristic
449C22×C4C22×D411yescharacteristic
450C22×C4C22×D411yescharacteristic
451C22×C4C22×D411yescharacteristic
452C22×C4C22×Q812yes—
453C22×C4C2×#4112yes—
454C22×C42−1+412yes—
455C22×C42−1+412yes—
456C2×C8—24no—
457C2×C8—24no—
458C2×C8—24no—
459C2×C8—42no—
460C2×C8—42no—
461C2×C8—42no—
462C2×C8—42no—
463C2×C8—42no—
464C2×C8—42no—
465C2×C8—42no—
466C2×C8—42no—
467C2×C8—42no—
468C2×C8—42no—
469C2×C8—42no—
470C4⋊C4—216no—
471C4⋊C4—216no—
472C4⋊C4—216no—
473C4⋊C4—216no—
474C4⋊C4—216no—
475C4⋊C4—22no—
476C4⋊C4—22no—
477C4⋊C4—22no—
478C4⋊C4—22no—
479C4⋊C4—22no—
480C4⋊C4—22no—
481C4⋊C4—22no—
482C4⋊C4—22no—
483C4⋊C4—22no—
484C4⋊C4—216no—
485C4⋊C4—216no—
486C4⋊C4—216no—
487C4⋊C4—28no—
488C4⋊C4—28no—
489C4⋊C4—28no—
490C4⋊C4—24no—
491C4⋊C4—24no—
492C4⋊C4—42no—
493C4⋊C4—42no—
494C4⋊C4—42no—
495C4⋊C4—42no—
496C4⋊C4—42no—
497C4⋊C4—42no—
498C4⋊C4C2×C22⋊C414yes—
499C4⋊C4C2×C22⋊C414yes—
500C4⋊C4C2×C22⋊C414yes—
501C42—216no—
502C42—216no—
503C42—22no—
504C42—22no—
505C42—22no—
506C42—22no—
507C42—22no—
508C42—22no—
509C42—22no—
510C42—22no—
511C42—216no—
512C42—84no—
513C42C2×C22⋊C414yes—
514C42C2×C22⋊C414yes—
515C42#12014yes—
516C42#12014yes—
517C23C22×C22⋊C411yescharacteristic
518C23C2×#12011yescharacteristic
519C23C2×#12011yescharacteristic
520C23#54812yes—
521C23D4×Q812yes—
522C23C2×#13512yes—
523C23#57512yes—
524C23#57712yes—
525C23#57812yes—
526Q8—24no—
527Q8—28no—
528Q8—28no—
529Q8—24no—
530Q8—24no—
531Q8—416no—
532Q8—432no—
533Q8#39118yes—
534C2×C4—28no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—24no—
539C2×C4—28no—
540C2×C4—24no—
541C2×C4—24no—
542C2×C4—24no—
543C2×C4—24no—
544C2×C4—24no—
545C2×C4—24no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—24no—
549C2×C4—24no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—24no—
554C2×C4—24no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—24no—
558C2×C4—22no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—44no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—44no—
569C2×C4—44no—
570C2×C4—44no—
571C2×C4—44no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—42no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—44no—
581C2×C4—44no—
582C2×C4—44no—
583C2×C4—44no—
584C2×C4—82no—
585C2×C4—82no—
586C2×C4—82no—
587C2×C4—82no—
588C2×C4—82no—
589C2×C4—82no—
590C2×C4—82no—
591C2×C4—82no—
592C2×C4—82no—
593C2×C4—82no—
594C2×C4—82no—
595C2×C4#37914yes—
596C2×C4#37914yes—
597C2×C4#41614yes—
598C2×C4C22×C22⋊C411yescharacteristic
599C2×C4C22×C22⋊C411yescharacteristic
600C2×C4C2×C4×Q812yes—
601C2×C4#51912yes—
602C2×C4C2×#12011yescharacteristic
603C2×C4C2×#12011yescharacteristic
604C2×C4C2×C22⋊Q812yes—
605C2×C4C2×#12411yescharacteristic
606C2×C4C2×C4⋊Q811yescharacteristic
607C2×C4#53612yes—
608C2×C4#53611yescharacteristic
609C2×C4#53611yescharacteristic
610C2×C4#54812yes—
611C2×C4#54812yes—
612C2×C4#54812yes—
613C2×C4D4×Q812yes—
614C2×C4D4×Q812yes—
615C2×C4D4×Q812yes—
616C8—44no—
617C8—82no—
618C22C2×#37911yescharacteristic
619C22Q8×C22⋊C412yes—
620C22#240912yes—
621C22#243411yescharacteristic
622C22#243611yescharacteristic
623C22C2×#41611yescharacteristic
624C22#289611yescharacteristic
625C22#289812yes—
626C22#300212yes—
627C22#300212yes—
628C22#301011yescharacteristic
629C22#301111yescharacteristic
630C22#301312yes—
631C22#336412yes—
632C22#336512yes—
633C22#337112yes—
634C22#337712yes—
635C22#338112yes—
636C22#338312yes—
637C22#338812yes—
638C22#339012yes—
639C4—28no—
640C4—24no—
641C4—24no—
642C4—24no—
643C4—42no—
644C4—44no—
645C4—44no—
646C4—42no—
647C4—42no—
648C4—44no—
649C4—84no—
650C4—162no—
651C4C2×#39111yescharacteristic
652C4#231011yescharacteristic
653C4Q8×C22⋊C412yes—
654C4#240912yes—
655C4#243411yescharacteristic
656C4#243611yescharacteristic
657C2#1392211yescharacteristic
658C2#2059711yescharacteristic
659C2#2060311yescharacteristic
660C2#2252112yes—
661C2#2252712yes—
662C2#2433412yes—
663C2#2433612yes—
664C2#2584212yes—
665C2#2593712yes—
subgroupG/H
center, socle, prime radicalC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
17. C2×#557C4semidirect
148. C2×#103Q8semidirect

Maximal subgroups

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

subgroupindex
C22×#5572
C2×#19912
C2×#19892
C2×#19842
C2×#18052
C2×C4×#1032
#139252
#113852
#113842
#107892