Finite Group Databasebeta

← #364772

Subgroups — #364772

Order 512.

1528 conjugacy classes of subgroups.

The subgroup lattice has 652 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#477C211yesmaximal, characteristic
2C2×#2369C211yesmaximal, characteristic
3C2×#1990C211yesmaximal, characteristic
4C2×C4×#103C211yesmaximal, characteristic
5#13926C211yesmaximal, characteristic
6#11219C212yesmaximal
7#11067C212yesmaximal
8#10932C212yesmaximal
9#10907C212yesmaximal
10#10849C211yesmaximal, characteristic
11#10834C211yesmaximal, characteristic
12C22×C4⋊Q8C2211yescharacteristic
13C22×C4×Q8C2211yescharacteristic
14C2×#477—24no—
15C2×#477—24no—
16C2×#477C418yescomplemented
17C22×#107C2212yes—
18C22×C4⋊C8C2211yescharacteristic
19C22×#103C2211yescharacteristic
20C22×#103C2212yes—
21C22×#103C2211yescharacteristic
22C22×C4×C8C2211yescharacteristic
23#2369—24no—
24#2369—24no—
25C2×#391—22no—
26C2×#391C2212yes—
27C2×#389—21no—
28C2×#389C2212yes—
29C2×#384C2211yescharacteristic
30C2×C4×C4⋊C4—21no—
31C2×C4×C4⋊C4C2212yes—
32C2×C4×C4⋊C4C2211yescharacteristic
33C2×C4×C4⋊C4C2211yescharacteristic
34#1990—24no—
35C4×#103—24no—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#340C2211yescharacteristic
40C2×#340C2211yescharacteristic
41C2×#340C2212yes—
42C2×#336C2212yes—
43C2×#336C2212yes—
44#1441—22no—
45C4⋊#95—21no—
46C4⋊#95—21no—
47C4⋊#95C2212yes—
48C23×Q8—21no—
49C23×Q8C2311yescharacteristic
50C23×C8C2312yes—
51C23×C8C2311yescharacteristic
52C23×C8C2311yescharacteristic
53C2×C4⋊Q8—22no—
54C2×C4⋊Q8—22no—
55C2×C4⋊Q8—22no—
56C2×C4⋊Q8C2×C414yes—
57C2×C4⋊Q8D414yes—
58C2×C4⋊Q8D414yes—
59C2×C4×Q8—22no—
60C2×C4×Q8—22no—
61C2×C4×Q8—22no—
62C2×C4×Q8—22no—
63C2×C4×Q8—22no—
64C2×C4×Q8C2×C414yes—
65C22×C4⋊C4—21no—
66C22×C4⋊C4—21no—
67C22×C4⋊C4—22no—
68C22×C4⋊C4—21no—
69C22×C4⋊C4—21no—
70C22×C4⋊C4D411yescharacteristic
71C22×C4⋊C4D411yescharacteristic
72C22×C4⋊C4C2312yes—
73C22×C4⋊C4C2311yescharacteristic
74C22×C4⋊C4C2311yescharacteristic
75C22×C4⋊C4C2312yes—
76C22×C4⋊C4C2311yescharacteristic
77C22×C42—21no—
78C22×C42—21no—
79C22×C42—22no—
80C22×C42C2312yes—
81C22×C42C2311yescharacteristic
82#477—232no—
83C2×#107—24no—
84C2×#107—24no—
85C2×#107C2×C418yes—
86C2×C4⋊C8—22no—
87C2×C4⋊C8—22no—
88C2×C4⋊C8C2×C414yes—
89C2×#103—22no—
90C2×#103—22no—
91C2×#103—24no—
92C2×#103—24no—
93C2×#103—22no—
94C2×#103—22no—
95C2×#103—22no—
96C2×#103C2×C418yes—
97C2×#103Q814yescomplemented
98C2×C4×C8—22no—
99C2×C4×C8—22no—
100C2×C4×C8—22no—
101#391—216no—
102#391—24no—
103#391—42no—
104#389—28no—
105#389—24no—
106#389—24no—
107#384—22no—
108#384—22no—
109C4×C4⋊C4—28no—
110C4×C4⋊C4—24no—
111C4×C4⋊C4—24no—
112C4×C4⋊C4—22no—
113C4×C4⋊C4—22no—
114C4×C4⋊C4—22no—
115C4×C4⋊C4D414yes—
116C2×#95—22no—
117C2×#95—22no—
118C2×#95—21no—
119C2×#95—22no—
120C2×#95—22no—
121C2×#95—22no—
122C2×#95—22no—
123C2×#95D412yes—
124#340—24no—
125#340—24no—
126#340—22no—
127#340—22no—
128#340—24no—
129#340—42no—
130#336—24no—
131#336—42no—
132C22×Q8—24no—
133C22×Q8—24no—
134C22×Q8—24no—
135C22×Q8—21no—
136C22×Q8—21no—
137C22×Q8—21no—
138C22×Q8—21no—
139C22×Q8—21no—
140C22×Q8—21no—
141C22×Q8—21no—
142C22×Q8—21no—
143C22×Q8—21no—
144C22×Q8—21no—
145C22×Q8—21no—
146C22×Q8—81no—
147C22×Q8—81no—
148C22×Q8—81no—
149C22×Q8—81no—
150C22×Q8C22×C412yes—
151C22×Q8C2×Q812yes—
152C22×Q8#4112yes—
153C23×C4—22no—
154C23×C4—21no—
155C23×C4—21no—
156C23×C4—21no—
157C23×C4—21no—
158C23×C4—22no—
159C23×C4C2×D412yes—
160C23×C4C2×D412yes—
161C23×C4C2×D411yescharacteristic
162C23×C4C2×D411yescharacteristic
163C23×C4C2411yescharacteristic
164C22×C8—21no—
165C22×C8—21no—
166C22×C8—21no—
167C22×C8—21no—
168C22×C8—22no—
169C22×C8—22no—
170C22×C8—22no—
171C22×C8—22no—
172C22×C8—21no—
173C22×C8—21no—
174C22×C8—21no—
175C22×C8—21no—
176C22×C8—21no—
177C22×C8—22no—
178C22×C8—22no—
179C22×C8—21no—
180C22×C8—21no—
181C22×C8—21no—
182C22×C8C22×C414yes—
183C22×C8C2×Q812yes—
184C22×C8#4112yes—
185C4⋊Q8—28no—
186C4⋊Q8—28no—
187C4⋊Q8—28no—
188C4⋊Q8C22⋊C4116yes—
189C4×Q8—28no—
190C4×Q8—28no—
191C4×Q8—44no—
192C4×Q8—44no—
193C2×C4⋊C4—24no—
194C2×C4⋊C4—24no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—24no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—24no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—28no—
225C2×C4⋊C4—42no—
226C2×C4⋊C4—42no—
227C2×C4⋊C4—42no—
228C2×C4⋊C4—42no—
229C2×C4⋊C4—42no—
230C2×C4⋊C4—42no—
231C2×C4⋊C4—81no—
232C2×C4⋊C4—81no—
233C2×C4⋊C4—82no—
234C2×C4⋊C4—82no—
235C2×C4⋊C4Q1614yes—
236C2×C4⋊C4C22×C412yes—
237C2×C4⋊C4C22×C414yes—
238C2×C4⋊C4C2×D412yes—
239C2×C4⋊C4C2×D414yes—
240C2×C4⋊C4C2×D414yes—
241C2×C4⋊C4C2×D412yes—
242C2×C4⋊C4C2×Q812yes—
243C2×C4⋊C4#4112yes—
244C2×C42—24no—
245C2×C42—24no—
246C2×C42—22no—
247C2×C42—22no—
248C2×C42—22no—
249C2×C42—22no—
250C2×C42—22no—
251C2×C42—22no—
252C2×C42—28no—
253C2×C42—42no—
254C2×C42—42no—
255C2×C42—42no—
256C2×C42—42no—
257C2×C42—82no—
258C2×C42—82no—
259C2×C42C22×C412yes—
260C2×C42C2×D412yes—
261C2×C42C2×D412yes—
262C2×C42C2×D412yes—
263C2×C42#4112yes—
264C2×C42#4112yes—
265#107—216no—
266C4⋊C8—28no—
267#103—28no—
268#103—216no—
269#103—28no—
270C4×C8—44no—
271#95—28no—
272#95—28no—
273#95—24no—
274#95—24no—
275#95—44no—
276#95—44no—
277#95—82no—
278#95—82no—
279C24C22×D411yescharacteristic
280C2×Q8—22no—
281C2×Q8—22no—
282C2×Q8—22no—
283C2×Q8—216no—
284C2×Q8—22no—
285C2×Q8—22no—
286C2×Q8—22no—
287C2×Q8—22no—
288C2×Q8—22no—
289C2×Q8—22no—
290C2×Q8—22no—
291C2×Q8—22no—
292C2×Q8—22no—
293C2×Q8—22no—
294C2×Q8—22no—
295C2×Q8—22no—
296C2×Q8—41no—
297C2×Q8—41no—
298C2×Q8—41no—
299C2×Q8—41no—
300C2×Q8—41no—
301C2×Q8—41no—
302C2×Q8—41no—
303C2×Q8—41no—
304C2×Q8—41no—
305C2×Q8—41no—
306C2×Q8—41no—
307C2×Q8—41no—
308C2×Q8—84no—
309C2×Q8—84no—
310C2×Q8—84no—
311C2×Q8—84no—
312C2×Q8—84no—
313C2×Q8—84no—
314C2×Q8C4×Q814yes—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—21no—
318C22×C4—21no—
319C22×C4—22no—
320C22×C4—24no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—24no—
324C22×C4—24no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—41no—
344C22×C4—41no—
345C22×C4—41no—
346C22×C4—41no—
347C22×C4—41no—
348C22×C4—41no—
349C22×C4—41no—
350C22×C4—42no—
351C22×C4—42no—
352C22×C4—41no—
353C22×C4—42no—
354C22×C4—42no—
355C22×C4—41no—
356C22×C4—41no—
357C22×C4—41no—
358C22×C4—41no—
359C22×C4—41no—
360C22×C4—41no—
361C22×C4—41no—
362C22×C4—41no—
363C22×C4—41no—
364C22×C4—41no—
365C22×C4—41no—
366C22×C4—41no—
367C22×C4—42no—
368C22×C4—42no—
369C22×C4—42no—
370C22×C4—42no—
371C22×C4—42no—
372C22×C4—42no—
373C22×C4—42no—
374C22×C4C4×D412yes—
375C22×C4C4×D412yes—
376C22×C4#12012yes—
377C22×C4#12012yes—
378C22×C4#12112yes—
379C22×C4#12112yes—
380C22×C4C2×Q1614yes—
381C22×C4C2×Q1612yes—
382C22×C4#13512yes—
383C22×C4#13712yes—
384C22×C4#13712yes—
385C22×C4C23×C411yescharacteristic
386C22×C4C22×D411yescharacteristic
387C22×C4C22×D411yescharacteristic
388C22×C4C22×D411yescharacteristic
389C22×C4C22×Q811yescharacteristic
390C22×C4C2×#4111yescharacteristic
391C22×C4C2×#4111yescharacteristic
392C22×C4C2×#4111yescharacteristic
393C22×C4C2×#4111yescharacteristic
394C22×C4C2×#4111yescharacteristic
395C22×C4H(5,𝔽2)11yescharacteristic
396C22×C42−1+411yescharacteristic
397C22×C42−1+411yescharacteristic
398C22×C42−1+411yescharacteristic
399C2×C8—24no—
400C2×C8—24no—
401C2×C8—24no—
402C2×C8—22no—
403C2×C8—22no—
404C2×C8—22no—
405C2×C8—22no—
406C2×C8—22no—
407C2×C8—22no—
408C2×C8—41no—
409C2×C8—41no—
410C2×C8—41no—
411C2×C8—41no—
412C2×C8—42no—
413C2×C8—42no—
414C2×C8—41no—
415C2×C8—42no—
416C2×C8—42no—
417C2×C8—41no—
418C2×C8—41no—
419C2×C8—41no—
420C4⋊C4—24no—
421C4⋊C4—24no—
422C4⋊C4—24no—
423C4⋊C4—22no—
424C4⋊C4—22no—
425C4⋊C4—22no—
426C4⋊C4—216no—
427C4⋊C4—28no—
428C4⋊C4—22no—
429C4⋊C4—22no—
430C4⋊C4—22no—
431C4⋊C4—22no—
432C4⋊C4—41no—
433C4⋊C4—41no—
434C4⋊C4—41no—
435C4⋊C4—41no—
436C4⋊C4—48no—
437C4⋊C4—44no—
438C4⋊C4—42no—
439C4⋊C4—42no—
440C4⋊C4—42no—
441C4⋊C4—42no—
442C4⋊C4—84no—
443C4⋊C4—88no—
444C4⋊C4—84no—
445C4⋊C4C2×C22⋊C418yes—
446C4⋊C4C22⋊Q814yes—
447C42—42no—
448C42—42no—
449C42—42no—
450C42—42no—
451C42—48no—
452C42—48no—
453C42—88no—
454C42C2×C22⋊C414yes—
455C42C4×D414yes—
456C42#12114yes—
457C42#12314yes—
458C23C2×C4×D411yescharacteristic
459C23C2×#12011yescharacteristic
460C23C2×#12111yescharacteristic
461C23#54611yescharacteristic
462C23#54711yescharacteristic
463C23#54811yescharacteristic
464C23D4×Q811yescharacteristic
465C23C22×Q1611yescharacteristic
466C23C2×#13511yescharacteristic
467C23C2×#13711yescharacteristic
468C23C2×#13711yescharacteristic
469C23#57611yescharacteristic
470C23#57611yescharacteristic
471C23#57711yescharacteristic
472C23#57711yescharacteristic
473Q8—24no—
474Q8—24no—
475Q8—42no—
476Q8—42no—
477Q8—42no—
478Q8—42no—
479Q8—42no—
480Q8—42no—
481Q8—816no—
482C2×C4—22no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—22no—
486C2×C4—22no—
487C2×C4—22no—
488C2×C4—24no—
489C2×C4—24no—
490C2×C4—24no—
491C2×C4—24no—
492C2×C4—24no—
493C2×C4—28no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—44no—
507C2×C4—44no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—44no—
512C2×C4—44no—
513C2×C4—44no—
514C2×C4—44no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—81no—
530C2×C4—81no—
531C2×C4—81no—
532C2×C4—81no—
533C2×C4—81no—
534C2×C4—81no—
535C2×C4—81no—
536C2×C4—81no—
537C2×C4—82no—
538C2×C4—82no—
539C2×C4—82no—
540C2×C4—82no—
541C2×C4—82no—
542C2×C4—82no—
543C2×C4—82no—
544C2×C4#38614yes—
545C2×C4#38614yes—
546C2×C4C4×Q1614yes—
547C2×C4#44114yes—
548C2×C4#45114yes—
549C2×C4#45214yes—
550C2×C4#46214yes—
551C2×C4#46414yes—
552C2×C4C22×C22⋊C411yescharacteristic
553C2×C4C2×C4×D411yescharacteristic
554C2×C4C2×C4×Q811yescharacteristic
555C2×C4C4×#4111yescharacteristic
556C2×C4#51911yescharacteristic
557C2×C4#52011yescharacteristic
558C2×C4C2×#12111yescharacteristic
559C2×C4C2×C22⋊Q811yescharacteristic
560C2×C4C2×#12311yescharacteristic
561C2×C4#52511yescharacteristic
562C2×C4#52911yescharacteristic
563C2×C4#53311yescharacteristic
564C2×C4#53611yescharacteristic
565C2×C4#53711yescharacteristic
566C2×C4#54111yescharacteristic
567C2×C4#54211yescharacteristic
568C2×C4#54811yescharacteristic
569C2×C4#54811yescharacteristic
570C2×C4#54811yescharacteristic
571C2×C4#54811yescharacteristic
572C2×C4D4×Q811yescharacteristic
573C2×C4D4×Q811yescharacteristic
574C2×C4#55011yescharacteristic
575C2×C4#55011yescharacteristic
576C2×C4#55211yescharacteristic
577C2×C4#55211yescharacteristic
578C2×C4#55411yescharacteristic
579C2×C4#55611yescharacteristic
580C8—44no—
581C8—42no—
582C8—42no—
583C22C2×#38611yescharacteristic
584C22#236011yescharacteristic
585C22#236211yescharacteristic
586C22#240711yescharacteristic
587C22#240911yescharacteristic
588C22#242611yescharacteristic
589C22#242911yescharacteristic
590C22C2×C4×Q1611yescharacteristic
591C22C2×#44111yescharacteristic
592C22#294611yescharacteristic
593C22#294611yescharacteristic
594C22C2×#45111yescharacteristic
595C22C2×#45211yescharacteristic
596C22#301211yescharacteristic
597C22#301211yescharacteristic
598C22C2×#46211yescharacteristic
599C22C2×#46411yescharacteristic
600C22#304111yescharacteristic
601C22#304211yescharacteristic
602C22#329911yescharacteristic
603C22#330511yescharacteristic
604C22#332911yescharacteristic
605C22#333311yescharacteristic
606C22#333811yescharacteristic
607C22#334011yescharacteristic
608C22#335611yescharacteristic
609C22#335711yescharacteristic
610C22#336111yescharacteristic
611C22#336311yescharacteristic
612C22#337211yescharacteristic
613C22#337511yescharacteristic
614C22#338111yescharacteristic
615C22Q8×Q1611yescharacteristic
616C22#338511yescharacteristic
617C22#338711yescharacteristic
618C4—24no—
619C4—44no—
620C4—44no—
621C4—44no—
622C4—48no—
623C4—82no—
624C4—82no—
625C4—82no—
626C4—162no—
627C4#187818yes—
628C4C4×C22⋊Q811yescharacteristic
629C4#232311yescharacteristic
630C4Q8×C22⋊C411yescharacteristic
631C4#234111yescharacteristic
632C4#242111yescharacteristic
633C4#242311yescharacteristic
634C4#253611yescharacteristic
635C4#253811yescharacteristic
636C2#1396911yescharacteristic
637C2C2×#187811yescharacteristic
638C2#2092911yescharacteristic
639C2#2296511yescharacteristic
640C2#2297111yescharacteristic
641C2#2306311yescharacteristic
642C2#2306911yescharacteristic
643C2#2412111yescharacteristic
644C2#2412211yescharacteristic
645C2#2433111yescharacteristic
646C2#2433311yescharacteristic
647C2#2527711yescharacteristic
648C2#2527811yescharacteristic
649C2#2545411yescharacteristic
650C2#2545511yescharacteristic
subgroupG/H
center, socle, prime radicalC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
16. C2×#477C4semidirect
97. C2×#103Q8semidirect

Maximal subgroups

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

subgroupindex
C22×#4772
C2×#23692
C2×#19902
C2×C4×#1032
#139262
#112192
#110672
#109322
#109072
#108492
#108342