Finite Group Databasebeta

← #377233

Subgroups — #377233

Order 512.

845 conjugacy classes of subgroups.

The subgroup lattice has 680 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23656C211yesmaximal, characteristic
2#21147C211yesmaximal, characteristic
3C2×#1922C211yesmaximal, characteristic
4C2×#1773C211yesmaximal, characteristic
5#20441C211yesmaximal, characteristic
6#11370C211yesmaximal, characteristic
7#11317C211yesmaximal, characteristic
8#11221C211yesmaximal, characteristic
9#10911C211yesmaximal, characteristic
10#10910C211yesmaximal, characteristic
11#10909C211yesmaximal, characteristic
12#10887C211yesmaximal, characteristic
13#10886C211yesmaximal, characteristic
14#10883C211yesmaximal, characteristic
15#10877C211yesmaximal, characteristic
16C22×#108C2211yescharacteristic
17C2×#427C2211yescharacteristic
18C2×#427C2211yescharacteristic
19C2×#424C2211yescharacteristic
20C2×#424C2211yescharacteristic
21C22×C4⋊C8C2211yescharacteristic
22C2×#405C2211yescharacteristic
23C2×#405C2211yescharacteristic
24C2×#405C2211yescharacteristic
25C2×C4×M16C2211yescharacteristic
26C22×#97C2211yescharacteristic
27#2296C2211yescharacteristic
28#2285C2211yescharacteristic
29#2085C411yescharacteristic, complemented
30#2085C411yescharacteristic, complemented
31#2085C411yescharacteristic, complemented
32#2085C411yescharacteristic, complemented
33#2084C411yescharacteristic, complemented
34#2084C411yescharacteristic, complemented
35#2083C411yescharacteristic, complemented
36#2083C411yescharacteristic, complemented
37#1940—21no—
38#1940—21no—
39#1939—21no—
40#1939—21no—
41#1922—24no—
42#1773—24no—
43#1771—21no—
44#1771—21no—
45#1770—21no—
46C8×C4⋊C4—21no—
47C2×#341C2211yescharacteristic
48C2×#341C2211yescharacteristic
49C2×#341C2211yescharacteristic
50#1735C2211yescharacteristic
51#1735C2211yescharacteristic
52C2×#339C2211yescharacteristic
53C2×#339C2211yescharacteristic
54C2×#339C2211yescharacteristic
55C2×#339C2211yescharacteristic
56C2×#339C2211yescharacteristic
57C2×#339C2211yescharacteristic
58C2×#339C2211yescharacteristic
59C2×#339C2211yescharacteristic
60#1729C2211yescharacteristic
61#1729C2211yescharacteristic
62#1729C2211yescharacteristic
63#1728C2211yescharacteristic
64C2×#336C2211yescharacteristic
65C2×#336C2211yescharacteristic
66C2×#336C2211yescharacteristic
67C2×#336C2211yescharacteristic
68C4×#95C2211yescharacteristic
69C22×M16C2311yescharacteristic
70C22×M16C2311yescharacteristic
71C22×M16C2311yescharacteristic
72C22×M16C2311yescharacteristic
73C23×C8C2311yescharacteristic
74C23×C8C2311yescharacteristic
75C23×C8C2311yescharacteristic
76C23×C8C2311yescharacteristic
77C2×#117C2311yescharacteristic
78C2×#117C2311yescharacteristic
79C2×#117C2311yescharacteristic
80C2×#117C2311yescharacteristic
81C22×C42C2311yescharacteristic
82C22×C42C2311yescharacteristic
83C22×C42C2311yescharacteristic
84C2×#108—22no—
85C2×#108—22no—
86C2×#108Q814yes—
87#427—22no—
88#427—22no—
89#427—22no—
90#427—22no—
91C2×#107C2×C411yescharacteristic
92C2×#107C2×C411yescharacteristic
93C2×#107C2×C411yescharacteristic
94C2×#107C2×C411yescharacteristic
95C2×#106C2×C411yescharacteristic
96C2×#106C2×C411yescharacteristic
97C2×#106C2×C411yescharacteristic
98C2×#106C2×C411yescharacteristic
99#424—22no—
100#424—22no—
101#424—22no—
102#424—41no—
103C2×C4⋊C8—22no—
104C2×C4⋊C8—22no—
105C2×C4⋊C8—21no—
106C2×C4⋊C8—21no—
107C2×C4⋊C8—21no—
108C2×C4⋊C8—21no—
109C2×C4⋊C8D411yescharacteristic
110C2×C4⋊C8D411yescharacteristic
111C2×C4⋊C8D411yescharacteristic
112C2×C4⋊C8D411yescharacteristic
113#405—22no—
114#405—22no—
115#405—22no—
116#405—41no—
117#405—41no—
118#405—41no—
119C4×M16—22no—
120C4×M16—22no—
121C2×#97—22no—
122C2×#97—22no—
123C2×#97—22no—
124C2×#97—21no—
125C2×#97—21no—
126C2×#97—21no—
127C2×#97—21no—
128C2×C4×C8—21no—
129C2×C4×C8—21no—
130C2×C4×C8—21no—
131C2×C4×C8—21no—
132#389C2×C411yescharacteristic
133#389C2×C411yescharacteristic
134#389C2×C411yescharacteristic
135#389C2×C411yescharacteristic
136#385—21no—
137#385—21no—
138#384C2×C411yescharacteristic
139#384C2×C411yescharacteristic
140#382C2×C411yescharacteristic
141#382C2×C411yescharacteristic
142#380—21no—
143#380—21no—
144C4×C4⋊C4—21no—
145C4×C4⋊C4—21no—
146C4×C4⋊C4—21no—
147C4×C4⋊C4—21no—
148C2×#95—21no—
149C2×#95—21no—
150#341—22no—
151#341—22no—
152#341—22no—
153#340C2×C411yescharacteristic
154#340C2×C411yescharacteristic
155#340C2×C411yescharacteristic
156#340C2×C411yescharacteristic
157#340C2×C411yescharacteristic
158#340C2×C411yescharacteristic
159#340C2×C411yescharacteristic
160#340C2×C411yescharacteristic
161#339—22no—
162#339—22no—
163#339—22no—
164#339—22no—
165#339—22no—
166#339—41no—
167#339—41no—
168#339D414yes—
169#336—21no—
170#336—21no—
171#336—21no—
172#336—21no—
173#336—21no—
174#336—21no—
175#336—22no—
176#336—22no—
177#336—22no—
178#336—21no—
179#336—21no—
180#336C2×C411yescharacteristic
181#336C2×C411yescharacteristic
182#336C2×C411yescharacteristic
183#336C2×C411yescharacteristic
184C23×C4—21no—
185C23×C4—21no—
186C23×C4#4111yescharacteristic
187C23×C4#4111yescharacteristic
188C23×C4C2411yescharacteristic
189C2×M16—21no—
190C2×M16—21no—
191C2×M16—21no—
192C2×M16—21no—
193C2×M16—21no—
194C2×M16—21no—
195C2×M16—21no—
196C2×M16—21no—
197C2×M16—21no—
198C2×M16—21no—
199C2×M16—21no—
200C2×M16—21no—
201C2×M16—21no—
202C2×M16—21no—
203C2×M16—21no—
204C2×M16—21no—
205C2×M16—21no—
206C2×M16—21no—
207C2×M16—21no—
208C2×M16C2×D412yes—
209C2×M16C2×Q812yes—
210C2×M16C2×Q812yes—
211C2×M16#4112yes—
212C2×M16#4112yes—
213C22×C8—21no—
214C22×C8—21no—
215C22×C8—21no—
216C22×C8—21no—
217C22×C8—21no—
218C22×C8—21no—
219C22×C8—21no—
220C22×C8—21no—
221C22×C8—21no—
222C22×C8—21no—
223C22×C8—21no—
224C22×C8—21no—
225C22×C8—21no—
226C22×C8—21no—
227C22×C8—21no—
228C22×C8—21no—
229C22×C8—21no—
230C22×C8—21no—
231C22×C8—21no—
232C22×C8—21no—
233C22×C8—21no—
234C22×C8—21no—
235C22×C8—21no—
236C22×C8—21no—
237C22×C8C22×C411yescharacteristic
238C22×C8C22×C411yescharacteristic
239C22×C8C22×C411yescharacteristic
240C22×C8C22×C411yescharacteristic
241C22×C8C2×D411yescharacteristic
242C22×C8C2×D411yescharacteristic
243C22×C8C2×D411yescharacteristic
244C22×C8C2×D411yescharacteristic
245C22×C8C2×Q812yes—
246C22×C8#4112yes—
247C22×C8#4112yes—
248C22×C8#4112yes—
249#117—21no—
250#117—21no—
251#117—21no—
252#117—21no—
253#117—21no—
254#117—21no—
255#117—21no—
256#117—21no—
257#117—21no—
258#117—21no—
259#117—21no—
260#117—21no—
261#117—21no—
262#117—21no—
263#117C2×D412yes—
264#117#4112yes—
265C2×C4⋊C4—21no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—21no—
268C2×C4⋊C4—21no—
269C2×C4⋊C4—21no—
270C2×C4⋊C4—21no—
271C2×C4⋊C4—21no—
272C2×C4⋊C4—21no—
273C2×C4⋊C4C22×C411yescharacteristic
274C2×C4⋊C4C22×C411yescharacteristic
275C2×C4⋊C4C22×C411yescharacteristic
276C2×C4⋊C4C22×C411yescharacteristic
277C2×C4⋊C4C22×C411yescharacteristic
278C2×C4⋊C4C22×C411yescharacteristic
279C2×C4⋊C4C22×C411yescharacteristic
280C2×C4⋊C4C22×C411yescharacteristic
281C2×C22⋊C4—21no—
282C2×C22⋊C4—21no—
283C2×C22⋊C4—21no—
284C2×C22⋊C4—21no—
285C2×C42—21no—
286C2×C42—21no—
287C2×C42—21no—
288C2×C42—21no—
289C2×C42—21no—
290C2×C42—21no—
291C2×C42—21no—
292C2×C42—21no—
293C2×C42—21no—
294C2×C42—21no—
295C2×C42—21no—
296C2×C42—21no—
297C2×C42—21no—
298C2×C42—21no—
299C2×C42—21no—
300C2×C42—21no—
301C2×C42—21no—
302C2×C42—21no—
303C2×C42—21no—
304C2×C42C22×C411yescharacteristic
305C2×C42C22×C411yescharacteristic
306C2×C42C2×D412yes—
307C2×C42C2×D411yescharacteristic
308C2×C42C2×D411yescharacteristic
309#108—28no—
310#107—22no—
311#107—22no—
312#107—22no—
313#107—22no—
314#106—22no—
315#106—22no—
316#106—22no—
317#106—22no—
318C4⋊C8—22no—
319C4⋊C8—22no—
320C4⋊C8—22no—
321C4⋊C8—22no—
322C4⋊C8—42no—
323C4⋊C8—42no—
324C4⋊C8—42no—
325C4⋊C8—42no—
326#97—28no—
327#97—24no—
328#97—42no—
329#97—42no—
330#97—42no—
331C4×C8—24no—
332C4×C8—42no—
333C4×C8—42no—
334C4×C8—42no—
335#95—21no—
336#95—21no—
337#95—21no—
338#95—21no—
339#95—21no—
340#95—21no—
341#95—21no—
342#95—21no—
343#95—42no—
344#95—42no—
345C24C2×#4111yescharacteristic
346C22×C4—21no—
347C22×C4—21no—
348C22×C4—21no—
349C22×C4—21no—
350C22×C4—21no—
351C22×C4—21no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—21no—
355C22×C4—21no—
356C22×C4—21no—
357C22×C4—21no—
358C22×C4—21no—
359C22×C4—21no—
360C22×C4—22no—
361C22×C4—41no—
362C22×C4—41no—
363C22×C4—41no—
364C22×C4—41no—
365C22×C4—41no—
366C22×C4—41no—
367C22×C4—41no—
368C22×C4—41no—
369C22×C4—41no—
370C22×C4—41no—
371C22×C4—41no—
372C22×C4—41no—
373C22×C4—41no—
374C22×C4—41no—
375C22×C4—41no—
376C22×C4C4×D411yescharacteristic
377C22×C4C4×D411yescharacteristic
378C22×C4C4×D411yescharacteristic
379C22×C4C4×D411yescharacteristic
380C22×C4C23×C411yescharacteristic
381C22×C4C22×D411yescharacteristic
382C22×C4C22×D411yescharacteristic
383C22×C4C22×Q811yescharacteristic
384C22×C4C2×#4111yescharacteristic
385C22×C4C2×#4111yescharacteristic
386C22×C4C2×#4111yescharacteristic
387C22×C4C2×#4111yescharacteristic
388C22×C4C2×#4111yescharacteristic
389C22×C4C2×#4111yescharacteristic
390C22×C4H(5,𝔽2)11yescharacteristic
391C22×C42−1+411yescharacteristic
392C22×C42−1+411yescharacteristic
393C22×C42−1+411yescharacteristic
394M16—22no—
395M16—22no—
396M16—22no—
397M16—22no—
398M16—22no—
399M16—22no—
400M16—22no—
401M16—41no—
402M16—41no—
403M16—41no—
404M16—41no—
405M16—41no—
406M16—41no—
407M16—41no—
408M16—41no—
409M16C22⋊Q814yes—
410C2×C8—21no—
411C2×C8—21no—
412C2×C8—21no—
413C2×C8—21no—
414C2×C8—22no—
415C2×C8—22no—
416C2×C8—22no—
417C2×C8—22no—
418C2×C8—22no—
419C2×C8—22no—
420C2×C8—22no—
421C2×C8—21no—
422C2×C8—21no—
423C2×C8—22no—
424C2×C8—21no—
425C2×C8—21no—
426C2×C8—22no—
427C2×C8—22no—
428C2×C8—22no—
429C2×C8—21no—
430C2×C8—21no—
431C2×C8—21no—
432C2×C8—21no—
433C2×C8—21no—
434C2×C8—21no—
435C2×C8—21no—
436C2×C8—21no—
437C2×C8—22no—
438C2×C8—21no—
439C2×C8—21no—
440C2×C8—21no—
441C2×C8—21no—
442C2×C8—22no—
443C2×C8—41no—
444C2×C8—41no—
445C2×C8—41no—
446C2×C8—41no—
447C2×C8—41no—
448C2×C8—41no—
449C2×C8—41no—
450C2×C8—41no—
451C2×C8—41no—
452C2×C8—41no—
453C2×C8—41no—
454C2×C8—41no—
455C2×C8—41no—
456C2×C8—41no—
457C2×C8—41no—
458C2×C8C4×Q812yes—
459C2×C8C4×Q812yes—
460C2×C8#12312yes—
461C2×C8#12312yes—
462C4⋊C4—21no—
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—21no—
468C4⋊C4—21no—
469C4⋊C4—21no—
470C4⋊C4—21no—
471C4⋊C4—21no—
472C4⋊C4—21no—
473C4⋊C4—21no—
474C4⋊C4—21no—
475C4⋊C4—21no—
476C4⋊C4—42no—
477C4⋊C4—42no—
478C4⋊C4—42no—
479C4⋊C4—42no—
480C4⋊C4—81no—
481C4⋊C4—81no—
482C4⋊C4—81no—
483C4⋊C4—81no—
484C4⋊C4C4×D412yes—
485C4⋊C4C4×D412yes—
486C22⋊C4—22no—
487C22⋊C4—22no—
488C22⋊C4—41no—
489C22⋊C4—41no—
490C22⋊C4—41no—
491C22⋊C4—41no—
492C22⋊C4—41no—
493C22⋊C4—41no—
494C42—21no—
495C42—21no—
496C42—21no—
497C42—21no—
498C42—21no—
499C42—21no—
500C42—21no—
501C42—21no—
502C42—22no—
503C42—22no—
504C42—22no—
505C42—22no—
506C42—41no—
507C42—41no—
508C42—41no—
509C42—41no—
510C42—41no—
511C42—41no—
512C42—41no—
513C42—41no—
514C42—41no—
515C42—41no—
516C42—41no—
517C42—41no—
518C23—21no—
519C23—21no—
520C23—21no—
521C23—21no—
522C23C2×C4×D411yescharacteristic
523C23#52911yescharacteristic
524C23#52911yescharacteristic
525C23#54711yescharacteristic
526C23#55311yescharacteristic
527C23#55611yescharacteristic
528C23#55711yescharacteristic
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—42no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—81no—
573C2×C4—81no—
574C2×C4—81no—
575C2×C4—81no—
576C2×C4—81no—
577C2×C4—81no—
578C2×C4—81no—
579C2×C4—81no—
580C2×C4—81no—
581C2×C4—81no—
582C2×C4#44412yes—
583C2×C4C2×C4×D411yescharacteristic
584C2×C4C2×C4×Q811yescharacteristic
585C2×C4C4×#4111yescharacteristic
586C2×C4C4×#4111yescharacteristic
587C2×C4#51911yescharacteristic
588C2×C4#52011yescharacteristic
589C2×C4C2×C22⋊Q811yescharacteristic
590C2×C4C2×#12311yescharacteristic
591C2×C4#52511yescharacteristic
592C2×C4#52511yescharacteristic
593C2×C4#52911yescharacteristic
594C2×C4#53311yescharacteristic
595C2×C4#53911yescharacteristic
596C2×C4#53911yescharacteristic
597C2×C4#54111yescharacteristic
598C2×C4#54111yescharacteristic
599C2×C4D4211yescharacteristic
600C2×C4#54711yescharacteristic
601C2×C4#54711yescharacteristic
602C2×C4#54811yescharacteristic
603C2×C4#54811yescharacteristic
604C2×C4D4×Q811yescharacteristic
605C2×C4#55211yescharacteristic
606C2×C4#55211yescharacteristic
607C2×C4#55311yescharacteristic
608C2×C4#55411yescharacteristic
609C2×C4#55611yescharacteristic
610C2×C4#55711yescharacteristic
611C8—22no—
612C8—22no—
613C8—24no—
614C8—22no—
615C8—22no—
616C8—41no—
617C8—41no—
618C8—41no—
619C8—41no—
620C8—42no—
621C8—42no—
622C8—81no—
623C22—21no—
624C22—21no—
625C22—21no—
626C22—21no—
627C22—21no—
628C22—21no—
629C22—21no—
630C22—21no—
631C22—21no—
632C22—21no—
633C22—21no—
634C22—21no—
635C22C4×#12311yescharacteristic
636C22#236111yescharacteristic
637C22#236511yescharacteristic
638C22#245311yescharacteristic
639C22#247811yescharacteristic
640C22#259011yescharacteristic
641C22#259411yescharacteristic
642C22C2×#44411yescharacteristic
643C22#295511yescharacteristic
644C22#295611yescharacteristic
645C22#295711yescharacteristic
646C4—22no—
647C4—41no—
648C4—41no—
649C4—41no—
650C4—42no—
651C4—42no—
652C4—82no—
653C4—82no—
654C4—81no—
655C4—81no—
656C4—81no—
657C4—81no—
658C4—81no—
659C4—81no—
660C4C4×C22⋊Q811yescharacteristic
661C4#233611yescharacteristic
662C4#236211yescharacteristic
663C4#237011yescharacteristic
664C4#244311yescharacteristic
665C4#244611yescharacteristic
666C4#244911yescharacteristic
667C4#247511yescharacteristic
668C2—21no—
669C2—21no—
670C2—21no—
671C2—21no—
672C2#1419111yescharacteristic
673C2#2159311yescharacteristic
674C2#2159411yescharacteristic
675C2#2301611yescharacteristic
676C2#2301711yescharacteristic
677C2#2317111yescharacteristic
678C2#2317211yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
socleC23C2×C4×D4
prime radicalC24C2×#41
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C22×C4⋊C8 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
29. #2085C4semidirect
30. #2085C4semidirect
31. #2085C4semidirect
32. #2085C4semidirect
33. #2084C4semidirect
34. #2084C4semidirect
35. #2083C4semidirect
36. #2083C4semidirect

Maximal subgroups

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

subgroupindex
#236562
#211472
C2×#19222
C2×#17732
#204412
#113702
#113172
#112212
#109112
#109102
#109092
#108872
#108862
#108832
#108772