Finite Group Databasebeta

← #6852428

Subgroups — #6852428

Order 512.

1653 conjugacy classes of subgroups.

The subgroup lattice has 689 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36544C211yesmaximal, characteristic
2C2×#2582C211yesmaximal, characteristic, complemented
3#32957C211yesmaximal, characteristic, complemented
4#32657C214yesmaximal, complemented
5#32578C214yesmaximal, complemented
6#29957C212yesmaximal, complemented
7#29298C212yesmaximal, complemented
8#29237C212yesmaximal
9#28264C212yesmaximal, complemented
10#27948C211yesmaximal, characteristic, complemented
11#27677C211yesmaximal, characteristic
12#25471C212yesmaximal, complemented
13#25061C212yesmaximal, complemented
14#24331C212yesmaximal, complemented
15Q8×C22⋊C8C211yesmaximal, characteristic, complemented
16#22041C212yesmaximal
17C4×#448C211yesmaximal, characteristic
18C2×#554C2211yescharacteristic
19C22×C4×Q8C2211yescharacteristic
20C22×C4×Q8C2211yescharacteristic
21C2×#477C2212yes—
22C2×#448C2211yescharacteristic
23C2×C8×Q8C2212yes—
24C2×C4×SD16C2212yes—
25#2857C2211yescharacteristic
26#2851C2212yes—
27#2584C2212yes—
28#2582—21no—
29#2582—21no—
30#2582—22no—
31#2582C2212yes—
32#2582C2214yes—
33#2582C2212yes—
34#2580C2212yes—
35#2567C2211yescharacteristic
36#2538C2211yescharacteristic
37#2465—22no—
38#2449—22no—
39#2423—21no—
40#2409—21no—
41C2×#398—22no—
42#2360—21no—
43#2355C2211yescharacteristic
44D4×C4⋊C4—21no—
45C4×#121C2212yes—
46C4×#121C2211yescharacteristic
47C2×#391—21no—
48C2×#391C2212yes—
49C2×#389—21no—
50C2×#389C2212yes—
51C2×#384C2211yescharacteristic
52#2099C2212yes—
53#2097C2214yes—
54#2093C2212yes—
55#2087C2214yes—
56#2085C2214yes—
57#2084C2214yes—
58#2068C2214yes—
59#2066C2214yes—
60#2065C2214yes—
61#2059C2214yes—
62#2026C2214yes—
63#1990C2212yes—
64#1984C2212yes—
65#1984C2214yes—
66#1968C2214yes—
67#1947C2212yes—
68#1942C2214yes—
69#1941C2214yes—
70#1933C2214yes—
71#1927C2214yes—
72#1919C2214yes—
73#1880C2212yes—
74#1877C2214yes—
75#1874C2212yes—
76#1874C2211yescharacteristic
77#1865C2212yes—
78#1863C2214yes—
79#1862C2214yes—
80#1853C2212yes—
81#1853C2212yes—
82#1853C2211yescharacteristic
83#1852C2212yes—
84#1852C2212yes—
85#1844C2212yes—
86#1803C2212yes—
87#1802C2212yes—
88#1797C2212yes—
89#1797C2211yescharacteristic
90#1787C2211yescharacteristic
91#1787C2212yes—
92C4×#103C2212yes—
93C4×C22⋊C8C2212yes—
94C4×C22⋊C8C2211yescharacteristic
95C23×Q8D411yescharacteristic
96C23×Q8C2311yescharacteristic
97C22×SD16C2312yes—
98#554—44no—
99C2×C4⋊Q8C2311yescharacteristic
100C2×#125C2312yes—
101C2×C22⋊Q8—21no—
102C2×C22⋊Q8—21no—
103C2×C22⋊Q8—21no—
104C2×C22⋊Q8—22no—
105C2×#121C2311yescharacteristic
106C2×C4×Q8—21no—
107C2×C4×Q8—21no—
108C2×C4×Q8—44no—
109C2×C4×Q8—44no—
110C2×C4×Q8D411yescharacteristic
111C2×C4×Q8D411yescharacteristic
112C2×C4×Q8D412yes—
113C2×C4×Q8C2312yes—
114C2×C4×Q8C2312yes—
115C2×C4×Q8C2311yescharacteristic
116C2×C4×Q8C2311yescharacteristic
117C2×C4×Q8C2312yes—
118C2×C4×D4—21no—
119C2×C4×D4—22no—
120C2×C4×D4C2311yescharacteristic
121C2×C4×D4C2312yes—
122C22×C4⋊C4—21no—
123C22×C4⋊C4—21no—
124C22×C4⋊C4—21no—
125C22×C4⋊C4—22no—
126C22×C4⋊C4—21no—
127C22×C4⋊C4D411yescharacteristic
128C22×C4⋊C4D412yes—
129C22×C4⋊C4C2311yescharacteristic
130C22×C4⋊C4C2312yes—
131C22×C42—21no—
132C22×C42C2311yescharacteristic
133C22×C42C2312yes—
134#477—44no—
135#448Q818yescomplemented
136C8×Q8—44no—
137C4×SD16—44no—
138C2×#107C2314yes—
139C2×#107C2314yes—
140C2×#106C2312yes—
141C2×#106C2314yes—
142C2×#106C2314yes—
143C2×C4⋊C8C2312yes—
144C2×C4⋊C8C2314yes—
145C2×#103C2314yes—
146C2×#103C2314yes—
147C2×#103C2312yes—
148C2×#103C2312yes—
149C2×#103C2312yes—
150C2×#102C2314yes—
151C2×#102C2312yes—
152C2×C22⋊C8C2312yes—
153C2×C22⋊C8C2311yescharacteristic
154C2×C22⋊C8C2311yescharacteristic
155C2×C4×C8C2312yes—
156C2×C4×C8C2314yes—
157#399—22no—
158#398—24no—
159#398—24no—
160#398—24no—
161#398—24no—
162#398—22no—
163#396—22no—
164#395—22no—
165#395—22no—
166#393—22no—
167#391—21no—
168#391—22no—
169#391—24no—
170#391—22no—
171#391—21no—
172#391—22no—
173#391—22no—
174#391—21no—
175#391C2314yes—
176#391C2314yes—
177#390C2312yes—
178#390C2311yescharacteristic
179#389—22no—
180#389—22no—
181#389—24no—
182#389—22no—
183#389—22no—
184#389—22no—
185#389—22no—
186#389—21no—
187#389C2312yes—
188#389C2312yes—
189#389C2314yes—
190#389C2314yes—
191#389C2312yes—
192#389C2312yes—
193#389C2311yescharacteristic
194#388—22no—
195#388—21no—
196#387—22no—
197#387—22no—
198#387—22no—
199#386—22no—
200#386—21no—
201#385—22no—
202#385—22no—
203#385—22no—
204#385—22no—
205#384—21no—
206#384—21no—
207#384—21no—
208#384C2312yes—
209#384C2311yescharacteristic
210#384C2312yes—
211#382C2312yes—
212#380—21no—
213#380—21no—
214#380C2311yescharacteristic
215#380C2312yes—
216C4×C4⋊C4—21no—
217C4×C4⋊C4C2311yescharacteristic
218C4×C4⋊C4C2312yes—
219C4×C22⋊C4—21no—
220C4×C22⋊C4—22no—
221C2×#95—22no—
222C2×#95—22no—
223C2×#95—22no—
224#340C2314yes—
225#340C2314yes—
226#340C2314yes—
227#340C2314yes—
228#340C2312yes—
229#340C2312yes—
230#340C2314yes—
231#336C2312yes—
232#336C2314yes—
233#336C2314yes—
234#336C2312yes—
235C22×Q8—24no—
236C22×Q8—24no—
237C22×Q8—21no—
238C22×Q8—21no—
239C22×Q8—22no—
240C22×Q8—42no—
241C22×Q8—42no—
242C22×Q8—42no—
243C22×Q8C2×D411yescharacteristic
244C22×Q8C2×D412yes—
245C22×Q8C2×D411yescharacteristic
246C22×Q8C2×Q814yes—
247C22×Q8C2412yes—
248C22×D4—21no—
249C22×D4C2411yescharacteristic
250C23×C4—21no—
251C23×C4—22no—
252C23×C4—21no—
253C23×C4—21no—
254C23×C4—22no—
255C23×C4C2×D411yescharacteristic
256C23×C4C2×D412yes—
257C23×C4C2×D411yescharacteristic
258C23×C4C2×D412yes—
259C23×C4C2411yescharacteristic
260C2×SD16—42no—
261C2×SD16—42no—
262C2×SD16C2×Q818yescomplemented
263C22×C8C2412yes—
264C22×C8C2414yes—
265C22×C8C2412yes—
266C4⋊Q8—24no—
267#125—44no—
268C22⋊Q8—48no—
269C22⋊Q8—44no—
270C22⋊Q8—44no—
271C22⋊Q8—82no—
272#121—22no—
273#121C2×Q814yes—
274C4×Q8—24no—
275C4×Q8—24no—
276C4×Q8—24no—
277C4×Q8—24no—
278C4×Q8—44no—
279C4×Q8—44no—
280C4×Q8—42no—
281C4×Q8—42no—
282C4×Q8—42no—
283C4×Q8—42no—
284C4×Q8—44no—
285C4×Q8—44no—
286C4×D4—24no—
287C4×D4—44no—
288C4×D4—44no—
289C4×D4—84no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—24no—
305C2×C4⋊C4—24no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—42no—
311C2×C4⋊C4—44no—
312C2×C4⋊C4—44no—
313C2×C4⋊C4—42no—
314C2×C4⋊C4—44no—
315C2×C4⋊C4—44no—
316C2×C4⋊C4—44no—
317C2×C4⋊C4—82no—
318C2×C4⋊C4—84no—
319C2×C4⋊C4C2×D411yescharacteristic
320C2×C4⋊C4C2×D412yes—
321C2×C4⋊C4C2×D411yescharacteristic
322C2×C4⋊C4C2×D412yes—
323C2×C4⋊C4C2×D412yes—
324C2×C4⋊C4C2×D412yes—
325C2×C4⋊C4C2×D412yes—
326C2×C4⋊C4C2412yes—
327C2×C4⋊C4C2414yes—
328C2×C4⋊C4C2411yescharacteristic
329C2×C4⋊C4C2412yes—
330C2×C4⋊C4C2411yescharacteristic
331C2×C4⋊C4C2412yes—
332C2×C4⋊C4C2411yescharacteristic
333C2×C22⋊C4—21no—
334C2×C22⋊C4—22no—
335C2×C22⋊C4—21no—
336C2×C22⋊C4—21no—
337C2×C22⋊C4—22no—
338C2×C22⋊C4—21no—
339C2×C22⋊C4—22no—
340C2×C22⋊C4—22no—
341C2×C22⋊C4—21no—
342C2×C42—24no—
343C2×C42—21no—
344C2×C42—22no—
345C2×C42—22no—
346C2×C42—22no—
347C2×C42—44no—
348C2×C42—44no—
349C2×C42C2×D411yescharacteristic
350C2×C42C2×D411yescharacteristic
351C2×C42C2×D412yes—
352C2×C42C2×D412yes—
353C2×C42C2412yes—
354C2×C42C2411yescharacteristic
355C2×C42C2412yes—
356C2×C42C2411yescharacteristic
357#107—44no—
358#107—44no—
359#106—44no—
360#106—44no—
361#106—42no—
362C4⋊C8—24no—
363C4⋊C8—44no—
364#103—44no—
365#103—42no—
366#103—42no—
367#103—44no—
368#103C2×Q818yescomplemented
369#102—44no—
370#102—42no—
371C22⋊C8—24no—
372C22⋊C8—22no—
373C22⋊C8C2×Q814yes—
374C4×C8—24no—
375C4×C8—44no—
376#95—24no—
377#95—24no—
378#95—24no—
379#95—24no—
380#95—24no—
381#95—24no—
382#95—21no—
383#95—22no—
384#95—21no—
385#95—22no—
386#95—22no—
387#95—21no—
388#95—22no—
389#95—22no—
390#95—22no—
391#95—22no—
392#95—22no—
393C24—21no—
394C24C22×D411yescharacteristic
395C2×Q8—22no—
396C2×Q8—22no—
397C2×Q8—24no—
398C2×Q8—22no—
399C2×Q8—24no—
400C2×Q8—22no—
401C2×Q8—24no—
402C2×Q8—22no—
403C2×Q8—22no—
404C2×Q8—22no—
405C2×Q8—22no—
406C2×Q8—22no—
407C2×Q8—22no—
408C2×Q8—42no—
409C2×Q8—48no—
410C2×Q8—416no—
411C2×Q8—42no—
412C2×Q8—42no—
413C2×Q8—48no—
414C2×Q8—42no—
415C2×Q8—81no—
416C2×Q8—81no—
417C2×Q8#13714yes—
418C2×Q8C22×Q814yes—
419C2×D4—24no—
420C2×D4—22no—
421C2×D4—21no—
422C2×D4—22no—
423C2×D4—42no—
424C2×D4—42no—
425C2×D4C22×Q812yes—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—24no—
431C22×C4—24no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—21no—
435C22×C4—24no—
436C22×C4—22no—
437C22×C4—24no—
438C22×C4—21no—
439C22×C4—22no—
440C22×C4—24no—
441C22×C4—22no—
442C22×C4—21no—
443C22×C4—22no—
444C22×C4—42no—
445C22×C4—41no—
446C22×C4—42no—
447C22×C4—42no—
448C22×C4—44no—
449C22×C4—42no—
450C22×C4—42no—
451C22×C4—42no—
452C22×C4—44no—
453C22×C4—44no—
454C22×C4—42no—
455C22×C4—44no—
456C22×C4—42no—
457C22×C4—42no—
458C22×C4—42no—
459C22×C4—42no—
460C22×C4—82no—
461C22×C4—82no—
462C22×C4#13514yes—
463C22×C4C22×D412yes—
464C22×C4C22×D412yes—
465C22×C4C22×D412yes—
466C22×C4C22×D411yescharacteristic
467C22×C4C22×D411yescharacteristic
468C22×C4C22×D411yescharacteristic
469C22×C4C22×D411yescharacteristic
470C22×C4C22×D412yes—
471C22×C4C22×D411yescharacteristic
472C22×C4C22×Q812yes—
473C22×C42−1+412yes—
474C22×C4C2511yescharacteristic
475SD16—48no—
476C2×C8—22no—
477C2×C8—22no—
478C2×C8—22no—
479C2×C8—24no—
480C2×C8—22no—
481C2×C8—24no—
482C2×C8—24no—
483C2×C8—22no—
484C2×C8C22×Q814yes—
485C4⋊C4—22no—
486C4⋊C4—24no—
487C4⋊C4—22no—
488C4⋊C4—24no—
489C4⋊C4—21no—
490C4⋊C4—22no—
491C4⋊C4—22no—
492C4⋊C4—24no—
493C4⋊C4—22no—
494C4⋊C4—24no—
495C4⋊C4—24no—
496C4⋊C4—22no—
497C4⋊C4—24no—
498C4⋊C4—42no—
499C4⋊C4—42no—
500C4⋊C4—42no—
501C4⋊C4—44no—
502C4⋊C4—42no—
503C4⋊C4—44no—
504C4⋊C4—44no—
505C4⋊C4—44no—
506C4⋊C4—44no—
507C4⋊C4—42no—
508C4⋊C4—44no—
509C4⋊C4—44no—
510C4⋊C4—42no—
511C4⋊C4—42no—
512C4⋊C4—44no—
513C4⋊C4—82no—
514C4⋊C4—82no—
515C4⋊C4—82no—
516C4⋊C4—84no—
517C4⋊C4—84no—
518C4⋊C4—82no—
519C4⋊C4—81no—
520C4⋊C4—82no—
521C4⋊C4C22×Q812yes—
522C22⋊C4—24no—
523C22⋊C4—44no—
524C22⋊C4—42no—
525C22⋊C4—44no—
526C22⋊C4—42no—
527C22⋊C4—42no—
528C22⋊C4—44no—
529C22⋊C4—81no—
530C22⋊C4—82no—
531C42—22no—
532C42—22no—
533C42—24no—
534C42—22no—
535C42—24no—
536C42—42no—
537C42—42no—
538C42—44no—
539C42—44no—
540C42—42no—
541C42—82no—
542C42H(5,𝔽2)14yes—
543C23—21no—
544C23—21no—
545C23—21no—
546C23—21no—
547C23—22no—
548C23—41no—
549C23—41no—
550C23—41no—
551C23—41no—
552C23—41no—
553C23—41no—
554C23D4×Q812yes—
555C23C2×#13512yes—
556C23#57712yes—
557C23C23×D411yescharacteristic
558Q8—28no—
559Q8—24no—
560Q8—24no—
561Q8—48no—
562Q8—48no—
563Q8—44no—
564Q8—42no—
565Q8—42no—
566Q8—162no—
567D4—24no—
568D4—48no—
569D4—42no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—22no—
574C2×C4—21no—
575C2×C4—22no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—28no—
579C2×C4—24no—
580C2×C4—24no—
581C2×C4—21no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—21no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—24no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—21no—
595C2×C4—44no—
596C2×C4—48no—
597C2×C4—48no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—42no—
604C2×C4—44no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—41no—
608C2×C4—42no—
609C2×C4—44no—
610C2×C4—41no—
611C2×C4—42no—
612C2×C4—41no—
613C2×C4—41no—
614C2×C4—44no—
615C2×C4—42no—
616C2×C4—41no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—84no—
621C2×C4—84no—
622C2×C4—88no—
623C2×C4—84no—
624C2×C4—164no—
625C2×C4#53412yes—
626C2×C4#53412yes—
627C2×C4D4×Q812yes—
628C2×C4C2×#13512yes—
629C2×C4C2×#13512yes—
630C2×C4C2×#13714yes—
631C2×C4C2×#13712yes—
632C2×C4C23×Q811yescharacteristic
633C2×C4C2×H(5,𝔽2)12yes—
634C2×C4C2×2−1+411yescharacteristic
635C2×C4#58512yes—
636C8—24no—
637C8—42no—
638C8—44no—
639C22—22no—
640C22—22no—
641C22—22no—
642C22—22no—
643C22—22no—
644C22—22no—
645C22—42no—
646C22—42no—
647C22—42no—
648C22—81no—
649C22—81no—
650C22—81no—
651C22—81no—
652C22#338114yes—
653C22C2×#53411yescharacteristic
654C22C2×D4×Q811yescharacteristic
655C22#348711yescharacteristic
656C22C22×#13511yescharacteristic
657C22C22×#13711yescharacteristic
658C22C2×#57711yescharacteristic
659C22#358511yescharacteristic
660C4—22no—
661C4—24no—
662C4—22no—
663C4—22no—
664C4—42no—
665C4—41no—
666C4—42no—
667C4—42no—
668C4—42no—
669C4—44no—
670C4—81no—
671C4—82no—
672C4—82no—
673C4—81no—
674C4—84no—
675C4—82no—
676C4#319014yes—
677C4#350112yes—
678C4#354112yes—
679C2—24no—
680C2—82no—
681C2#3959511yescharacteristic
682C2C2×#319011yescharacteristic
683C2#6076511yescharacteristic
684C2C2×#338111yescharacteristic
685C2Q8×#13711yescharacteristic
686C2#6188511yescharacteristic
687C2#6190611yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#121C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2582C2semidirect
3. #32957C2semidirect
4. #32657C2semidirect
5. #32578C2semidirect
6. #29957C2semidirect
7. #29298C2semidirect
9. #28264C2semidirect
10. #27948C2semidirect
12. #25471C2semidirect
13. #25061C2semidirect
14. #24331C2semidirect
15. Q8×C22⋊C8C2semidirect
135. #448Q8semidirect
262. C2×SD16C2×Q8semidirect
368. #103C2×Q8semidirect

Maximal subgroups

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

subgroupindex
#365442
C2×#25822
#329572
#326572
#325782
#299572
#292982
#292372
#282642
#279482
#276772
#254712
#250612
#243312
Q8×C22⋊C82
#220412
C4×#4482