Finite Group Databasebeta

← #364996

Subgroups — #364996

Order 512.

1958 conjugacy classes of subgroups.

The subgroup lattice has 729 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2536C211yesmaximal, characteristic
2C2×#2354C211yesmaximal, characteristic
3C2×#1955C211yesmaximal, characteristic
4C2×#1873C212yesmaximal, complemented
5C2×#1803C211yesmaximal, characteristic, complemented
6C2×#1763C211yesmaximal, characteristic, complemented
7#11200C212yesmaximal, complemented
8#10970C212yesmaximal, complemented
9#10850C211yesmaximal, characteristic, complemented
10#10843C211yesmaximal, characteristic, complemented
11#10731C212yesmaximal, complemented
12C22×C22⋊Q8C2212yes—
13C22×C4×Q8C2211yescharacteristic
14C22×C4×D4C2211yescharacteristic
15C22×#117C2212yes—
16C22×C4⋊C8C2211yescharacteristic
17C22×#103C2211yescharacteristic
18C22×#103C2212yes—
19C22×#103C2211yescharacteristic
20C22×C22⋊C8C2212yes—
21C22×C22⋊C8C2212yes—
22C22×#97C2211yescharacteristic
23#2536—24no—
24#2536C418yescomplemented
25C2×#398—21no—
26C2×#394—21no—
27#2354—24no—
28#2354—24no—
29C2×#390—21no—
30C2×#388—22no—
31C2×#388—22no—
32C2×#384C2211yescharacteristic
33C2×#382C2211yescharacteristic
34C2×#380C2212yes—
35C2×#380C2212yes—
36C2×C4×C4⋊C4—21no—
37C2×C4×C4⋊C4C2211yescharacteristic
38#1955—24no—
39#1873—28no—
40#1803—24no—
41#1763—24no—
42C2×#340C2212yes—
43C2×#340C2212yes—
44C2×#340C2212yes—
45C2×#340C2211yescharacteristic
46C2×#340C2211yescharacteristic
47C2×#340C2212yes—
48C2×#340C2212yes—
49C23×Q8C2311yescharacteristic
50C23×D4D411yescharacteristic
51C24×C4C2312yes—
52C23×C8C2312yes—
53C23×C8C2311yescharacteristic
54C23×C8C2311yescharacteristic
55C2×C22⋊Q8—24no—
56C2×C22⋊Q8—24no—
57C2×C22⋊Q8—24no—
58C2×C22⋊Q8—42no—
59C2×C22⋊Q8—42no—
60C2×C22⋊Q8C2×C418yes—
61C2×C4×Q8—22no—
62C2×C4×Q8—22no—
63C2×C4×Q8—22no—
64C2×C4×Q8—22no—
65C2×C4×Q8—22no—
66C2×C4×Q8C2×C414yes—
67C2×C4×D4—22no—
68C2×C4×D4—22no—
69C2×C4×D4—22no—
70C2×C4×D4—22no—
71C2×C4×D4C2×C414yes—
72C2×C4×D4D414yes—
73C2×#117—24no—
74C2×#117—24no—
75C2×#117—24no—
76C2×#117—24no—
77C2×#117—42no—
78C2×#117—42no—
79C22×C4⋊C4—21no—
80C22×C4⋊C4—22no—
81C22×C4⋊C4—21no—
82C22×C4⋊C4—21no—
83C22×C4⋊C4D411yescharacteristic
84C22×C4⋊C4C2311yescharacteristic
85C22×C4⋊C4C2312yes—
86C22×C4⋊C4C2312yes—
87C22×C4⋊C4C2311yescharacteristic
88C22×C4⋊C4C2311yescharacteristic
89C22×C22⋊C4—22no—
90C22×C22⋊C4—22no—
91C22×C22⋊C4—22no—
92C22×C22⋊C4—22no—
93C22×C22⋊C4D412yes—
94C22×C42—22no—
95C22×C42—21no—
96C22×C42—21no—
97C22×C42C2311yescharacteristic
98C2×C4⋊C8—22no—
99C2×C4⋊C8—22no—
100C2×C4⋊C8—22no—
101C2×#103—22no—
102C2×#103—22no—
103C2×#103—22no—
104C2×#103—22no—
105C2×#103—22no—
106C2×#103—24no—
107C2×#103—24no—
108C2×#103—42no—
109C2×#103D414yescomplemented
110C2×C22⋊C8—24no—
111C2×C22⋊C8—24no—
112C2×C22⋊C8—24no—
113C2×C22⋊C8—24no—
114C2×C22⋊C8—42no—
115C2×C22⋊C8—42no—
116C2×#97—22no—
117C2×#97—22no—
118C2×#97—22no—
119#398—28no—
120#394—28no—
121#390—28no—
122#388—216no—
123#388—216no—
124#384—22no—
125#384D414yes—
126#382—22no—
127#382C2×C414yes—
128#380—24no—
129#380—24no—
130#380—24no—
131#380C2×C418yes—
132C4×C4⋊C4—28no—
133C4×C4⋊C4—22no—
134C4×C4⋊C4D414yes—
135C2×#95—21no—
136C2×#95—22no—
137C2×#95—22no—
138C2×#95—21no—
139C2×#95—21no—
140#340—24no—
141#340—24no—
142#340—24no—
143#340—22no—
144#340—22no—
145#340—42no—
146#340—42no—
147C25C2×D412yes—
148C22×Q8—21no—
149C22×Q8—21no—
150C22×Q8—21no—
151C22×Q8—21no—
152C22×Q8—21no—
153C22×Q8—21no—
154C22×Q8—21no—
155C22×Q8—21no—
156C22×Q8—21no—
157C22×Q8—21no—
158C22×Q8—21no—
159C22×Q8C22×C412yes—
160C22×Q8C2×D412yes—
161C22×Q8#4112yes—
162C22×D4—22no—
163C22×D4—22no—
164C22×D4—42no—
165C22×D4—42no—
166C22×D4C22⋊C414yes—
167C23×C4—21no—
168C23×C4—21no—
169C23×C4—22no—
170C23×C4—22no—
171C23×C4—22no—
172C23×C4—22no—
173C23×C4—22no—
174C23×C4—22no—
175C23×C4—22no—
176C23×C4—22no—
177C23×C4—22no—
178C23×C4—22no—
179C23×C4—22no—
180C23×C4—22no—
181C23×C4—22no—
182C23×C4—22no—
183C23×C4—21no—
184C23×C4—21no—
185C23×C4C22×C414yes—
186C23×C4C2×D412yes—
187C23×C4C2×D414yes—
188C23×C4C2×D411yescharacteristic
189C23×C4C2×D411yescharacteristic
190C23×C4C2411yescharacteristic
191C22×C8—21no—
192C22×C8—21no—
193C22×C8—21no—
194C22×C8—21no—
195C22×C8—21no—
196C22×C8—21no—
197C22×C8—21no—
198C22×C8—21no—
199C22×C8—21no—
200C22×C8—21no—
201C22×C8—21no—
202C22×C8—21no—
203C22×C8—22no—
204C22×C8—22no—
205C22×C8—22no—
206C22×C8—22no—
207C22×C8—22no—
208C22×C8—22no—
209C22×C8—22no—
210C22×C8C2×D412yes—
211C22×C8#4112yes—
212C22⋊Q8—216no—
213C22⋊Q8—48no—
214C22⋊Q8—48no—
215C22⋊Q8—48no—
216C4×Q8—28no—
217C4×Q8—28no—
218C4×Q8—44no—
219C4×Q8—44no—
220C4×D4—28no—
221C4×D4—28no—
222C4×D4—28no—
223C4×D4—28no—
224#117—48no—
225#117—48no—
226#117—48no—
227#117—48no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4—21no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—21no—
239C2×C4⋊C4—21no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—21no—
242C2×C4⋊C4—21no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—28no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—21no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—44no—
258C2×C4⋊C4—44no—
259C2×C4⋊C4—44no—
260C2×C4⋊C4—42no—
261C2×C4⋊C4—42no—
262C2×C4⋊C4—82no—
263C2×C4⋊C4C22⋊C414yes—
264C2×C4⋊C4C22×C412yes—
265C2×C4⋊C4C22×C414yes—
266C2×C4⋊C4C2×D414yes—
267C2×C4⋊C4C2×D412yes—
268C2×C4⋊C4C2×D412yes—
269C2×C4⋊C4C2×D412yes—
270C2×C4⋊C4#4112yes—
271C2×C22⋊C4—24no—
272C2×C22⋊C4—24no—
273C2×C22⋊C4—28no—
274C2×C22⋊C4—28no—
275C2×C22⋊C4—28no—
276C2×C22⋊C4—28no—
277C2×C22⋊C4—44no—
278C2×C22⋊C4—44no—
279C2×C22⋊C4—44no—
280C2×C22⋊C4—44no—
281C2×C22⋊C4—44no—
282C2×C22⋊C4—44no—
283C2×C22⋊C4—82no—
284C2×C22⋊C4—82no—
285C2×C22⋊C4C22⋊C418yes—
286C2×C42—28no—
287C2×C42—24no—
288C2×C42—24no—
289C2×C42—44no—
290C2×C42—44no—
291C2×C42—44no—
292C2×C42—42no—
293C2×C42—42no—
294C2×C42C22×C412yes—
295C2×C42C2×D412yes—
296C2×C42C2×D412yes—
297C2×C42C2×D412yes—
298C2×C42#4112yes—
299C2×C42#4112yes—
300C4⋊C8—44no—
301#103—28no—
302#103—28no—
303#103—48no—
304C22⋊C8—48no—
305C22⋊C8—48no—
306#97—44no—
307#95—28no—
308#95—28no—
309#95—24no—
310#95—24no—
311#95—24no—
312C24—22no—
313C24—22no—
314C24—22no—
315C24—22no—
316C24—22no—
317C24—22no—
318C24—22no—
319C24—22no—
320C24—22no—
321C24—22no—
322C24—22no—
323C24—22no—
324C24—22no—
325C24—22no—
326C24C2×C22⋊C414yes—
327C24C22×D411yescharacteristic
328C2×Q8—22no—
329C2×Q8—22no—
330C2×Q8—22no—
331C2×Q8—22no—
332C2×Q8—22no—
333C2×Q8—22no—
334C2×Q8—22no—
335C2×Q8—22no—
336C2×Q8—22no—
337C2×Q8—22no—
338C2×Q8—22no—
339C2×Q8—22no—
340C2×Q8—22no—
341C2×Q8—22no—
342C2×Q8—22no—
343C2×Q8—41no—
344C2×Q8—41no—
345C2×Q8—41no—
346C2×Q8—41no—
347C2×Q8—41no—
348C2×Q8—41no—
349C2×Q8—41no—
350C2×Q8—41no—
351C2×Q8—41no—
352C2×Q8—41no—
353C2×Q8—41no—
354C2×Q8—41no—
355C2×Q8C4×D414yes—
356C2×D4—28no—
357C2×D4—48no—
358C2×D4—48no—
359C2×D4—44no—
360C2×D4—44no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—24no—
378C22×C4—24no—
379C22×C4—24no—
380C22×C4—24no—
381C22×C4—24no—
382C22×C4—24no—
383C22×C4—24no—
384C22×C4—24no—
385C22×C4—24no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—24no—
393C22×C4—41no—
394C22×C4—41no—
395C22×C4—41no—
396C22×C4—41no—
397C22×C4—42no—
398C22×C4—42no—
399C22×C4—42no—
400C22×C4—42no—
401C22×C4—42no—
402C22×C4—42no—
403C22×C4—42no—
404C22×C4—42no—
405C22×C4—42no—
406C22×C4—42no—
407C22×C4—42no—
408C22×C4—42no—
409C22×C4—42no—
410C22×C4—42no—
411C22×C4—42no—
412C22×C4—42no—
413C22×C4—42no—
414C22×C4—42no—
415C22×C4—41no—
416C22×C4—41no—
417C22×C4—42no—
418C22×C4—42no—
419C22×C4—42no—
420C22×C4—42no—
421C22×C4—41no—
422C22×C4—41no—
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⋊C414yes—
436C22×C4C2×C22⋊C412yes—
437C22×C4C2×C22⋊C412yes—
438C22×C4#12012yes—
439C22×C4#12012yes—
440C22×C4#12312yes—
441C22×C4#12312yes—
442C22×C4C23×C411yescharacteristic
443C22×C4C22×D411yescharacteristic
444C22×C4C22×D411yescharacteristic
445C22×C4C22×D411yescharacteristic
446C22×C4C22×D411yescharacteristic
447C22×C4C2×#4111yescharacteristic
448C22×C4C2×#4111yescharacteristic
449C22×C4C2×#4111yescharacteristic
450C22×C4C2×#4111yescharacteristic
451C22×C4C2×#4111yescharacteristic
452C22×C4H(5,𝔽2)11yescharacteristic
453C22×C4H(5,𝔽2)11yescharacteristic
454C22×C4H(5,𝔽2)11yescharacteristic
455C22×C42−1+411yescharacteristic
456C2×C8—22no—
457C2×C8—22no—
458C2×C8—22no—
459C2×C8—22no—
460C2×C8—22no—
461C2×C8—22no—
462C2×C8—41no—
463C2×C8—41no—
464C2×C8—41no—
465C2×C8—41no—
466C2×C8—41no—
467C2×C8—41no—
468C2×C8—41no—
469C2×C8—41no—
470C2×C8—42no—
471C2×C8—42no—
472C2×C8—42no—
473C2×C8—42no—
474C2×C8—42no—
475C2×C8—42no—
476C2×C8—42no—
477C4⋊C4—22no—
478C4⋊C4—22no—
479C4⋊C4—22no—
480C4⋊C4—28no—
481C4⋊C4—22no—
482C4⋊C4—22no—
483C4⋊C4—24no—
484C4⋊C4—24no—
485C4⋊C4—22no—
486C4⋊C4—22no—
487C4⋊C4—24no—
488C4⋊C4—24no—
489C4⋊C4—416no—
490C4⋊C4—48no—
491C4⋊C4—42no—
492C4⋊C4—41no—
493C4⋊C4—41no—
494C4⋊C4—42no—
495C4⋊C4—42no—
496C4⋊C4—42no—
497C4⋊C4—41no—
498C4⋊C4—41no—
499C4⋊C4—48no—
500C4⋊C4—84no—
501C4⋊C4#12114yes—
502C22⋊C4—216no—
503C22⋊C4—416no—
504C22⋊C4—416no—
505C22⋊C4—416no—
506C22⋊C4—88no—
507C42—48no—
508C42—416no—
509C42—48no—
510C42C4×D414yes—
511C42C4×D414yes—
512C42#12114yes—
513C42#12114yes—
514C23—24no—
515C23—24no—
516C23—24no—
517C23—24no—
518C23—24no—
519C23—24no—
520C23—24no—
521C23—42no—
522C23—42no—
523C23—42no—
524C23—42no—
525C23—42no—
526C23—42no—
527C23—42no—
528C23—42no—
529C23—42no—
530C23—42no—
531C23—42no—
532C23—42no—
533C23—42no—
534C23—42no—
535C23—42no—
536C23—42no—
537C23—42no—
538C23—42no—
539C23—42no—
540C23—42no—
541C23—42no—
542C23—42no—
543C23—42no—
544C23—42no—
545C23—42no—
546C23—42no—
547C23—42no—
548C23—42no—
549C23C22×C22⋊C411yescharacteristic
550C23C2×#12011yescharacteristic
551C23C2×#12311yescharacteristic
552C23D4211yescharacteristic
553C23#54611yescharacteristic
554C23#54611yescharacteristic
555C23#54711yescharacteristic
556C23#57512yes—
557C23#57512yes—
558C23#57712yes—
559C23#57812yes—
560Q8—24no—
561Q8—24no—
562Q8—42no—
563Q8—42no—
564Q8—42no—
565Q8—42no—
566Q8—42no—
567Q8—42no—
568D4—416no—
569C2×C4—24no—
570C2×C4—24no—
571C2×C4—24no—
572C2×C4—24no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—24no—
578C2×C4—24no—
579C2×C4—24no—
580C2×C4—24no—
581C2×C4—24no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—44no—
594C2×C4—42no—
595C2×C4—44no—
596C2×C4—44no—
597C2×C4—44no—
598C2×C4—44no—
599C2×C4—44no—
600C2×C4—44no—
601C2×C4—44no—
602C2×C4—44no—
603C2×C4—44no—
604C2×C4—44no—
605C2×C4—44no—
606C2×C4—44no—
607C2×C4—44no—
608C2×C4—42no—
609C2×C4—44no—
610C2×C4—42no—
611C2×C4—44no—
612C2×C4—42no—
613C2×C4—44no—
614C2×C4—42no—
615C2×C4—82no—
616C2×C4—82no—
617C2×C4—81no—
618C2×C4—81no—
619C2×C4—82no—
620C2×C4—82no—
621C2×C4—82no—
622C2×C4—82no—
623C2×C4—82no—
624C2×C4—82no—
625C2×C4—82no—
626C2×C4—82no—
627C2×C4—82no—
628C2×C4#38614yes—
629C2×C4#38614yes—
630C2×C4C2×C4×D411yescharacteristic
631C2×C4C2×C4×D411yescharacteristic
632C2×C4C2×C4×D411yescharacteristic
633C2×C4C4×#4111yescharacteristic
634C2×C4#51811yescharacteristic
635C2×C4#52011yescharacteristic
636C2×C4C2×#12111yescharacteristic
637C2×C4C2×#12111yescharacteristic
638C2×C4C2×#12111yescharacteristic
639C2×C4#52511yescharacteristic
640C2×C4#52911yescharacteristic
641C2×C4#53211yescharacteristic
642C2×C4#53511yescharacteristic
643C2×C4#53711yescharacteristic
644C2×C4#54011yescharacteristic
645C2×C4#54211yescharacteristic
646C2×C4D4211yescharacteristic
647C2×C4D4211yescharacteristic
648C2×C4#54611yescharacteristic
649C2×C4#54611yescharacteristic
650C2×C4#54611yescharacteristic
651C2×C4#54711yescharacteristic
652C2×C4#54711yescharacteristic
653C2×C4#54811yescharacteristic
654C2×C4#55011yescharacteristic
655C2×C4#55311yescharacteristic
656C2×C4#55311yescharacteristic
657C2×C4#55511yescharacteristic
658C8—42no—
659C8—42no—
660C8—82no—
661C22—44no—
662C22—44no—
663C22—44no—
664C22—44no—
665C22—44no—
666C22—44no—
667C22—44no—
668C22—82no—
669C22—82no—
670C22—82no—
671C22—82no—
672C22—82no—
673C22—82no—
674C22—82no—
675C22—82no—
676C22C2×#38611yescharacteristic
677C22D4×C22⋊C411yescharacteristic
678C22#233911yescharacteristic
679C22#240711yescharacteristic
680C22#240811yescharacteristic
681C22#242011yescharacteristic
682C22#242211yescharacteristic
683C22#289612yes—
684C22#289812yes—
685C22#300511yescharacteristic
686C22#300511yescharacteristic
687C22#300911yescharacteristic
688C22#301111yescharacteristic
689C22#309312yes—
690C22#309712yes—
691C22#326912yes—
692C22#327112yes—
693C22#331312yes—
694C22#331512yes—
695C22#331712yes—
696C22#332112yes—
697C22#334812yes—
698C22#334912yes—
699C4—24no—
700C4—24no—
701C4—44no—
702C4—44no—
703C4—44no—
704C4—82no—
705C4—84no—
706C4—82no—
707C4—84no—
708C4C4×#12111yescharacteristic
709C4#232411yescharacteristic
710C4#236111yescharacteristic
711C4#236311yescharacteristic
712C4#242711yescharacteristic
713C4#242811yescharacteristic
714C4#253511yescharacteristic
715C4#253711yescharacteristic
716C2—84no—
717C2#1397011yescharacteristic
718C2#2091011yescharacteristic
719C2#2091311yescharacteristic
720C2#2242612yes—
721C2#2249612yes—
722C2#2413311yescharacteristic
723C2#2413511yescharacteristic
724C2#2425311yescharacteristic
725C2#2425411yescharacteristic
726C2#2498012yes—
727C2#2502512yes—
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×D4D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×C4×D4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. C2×#1873C2semidirect
5. C2×#1803C2semidirect
6. C2×#1763C2semidirect
7. #11200C2semidirect
8. #10970C2semidirect
9. #10850C2semidirect
10. #10843C2semidirect
11. #10731C2semidirect
24. #2536C4semidirect
109. C2×#103D4semidirect

Maximal subgroups

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

subgroupindex
C2×#25362
C2×#23542
C2×#19552
C2×#18732
C2×#18032
C2×#17632
#112002
#109702
#108502
#108432
#107312