Finite Group Databasebeta

← #379172

Subgroups — #379172

Order 512.

1266 conjugacy classes of subgroups.

The subgroup lattice has 673 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2096C211yesmaximal, characteristic
2C2×#1941C211yesmaximal, characteristic
3C2×#1920C211yesmaximal, characteristic
4C2×#1865C211yesmaximal, characteristic
5C2×C4×#103C211yesmaximal, characteristic
6#14049C211yesmaximal, characteristic
7#13976C211yesmaximal, characteristic
8#11378C211yesmaximal, characteristic
9#11323C211yesmaximal, characteristic
10#11122C211yesmaximal, characteristic
11#10926C211yesmaximal, characteristic
12#10926C211yesmaximal, characteristic
13#10861C211yesmaximal, characteristic
14#10841C211yesmaximal, characteristic
15C8×#95C211yesmaximal, characteristic
16C22×C4×Q8C2211yescharacteristic
17C22×#106C2211yescharacteristic
18C22×#106C2211yescharacteristic
19C22×#106C2211yescharacteristic
20C22×#106C2211yescharacteristic
21C22×#103C2211yescharacteristic
22C22×#103C2211yescharacteristic
23C22×#103C2211yescharacteristic
24C22×#103C2211yescharacteristic
25C22×C4×C8C2211yescharacteristic
26C22×C4×C8C2211yescharacteristic
27C22×C4×C8C2211yescharacteristic
28C2×#391C2211yescharacteristic
29C2×#391C2211yescharacteristic
30C2×#389C2211yescharacteristic
31C2×#389C2211yescharacteristic
32C2×#387—22no—
33C2×#384C2211yescharacteristic
34C2×#384C2211yescharacteristic
35C2×C4×C4⋊C4C2211yescharacteristic
36#2096C418yescomplemented
37#1941—24no—
38#1920—24no—
39#1865—24no—
40C4×#103—24no—
41C2×#340C2211yescharacteristic
42C2×#340C2211yescharacteristic
43C2×#340C2211yescharacteristic
44C2×#340C2211yescharacteristic
45C2×#340C2211yescharacteristic
46C2×#340C2211yescharacteristic
47C2×#340C2211yescharacteristic
48C2×#340C2211yescharacteristic
49C2×#336C2211yescharacteristic
50C2×#336C2211yescharacteristic
51C2×#336C2211yescharacteristic
52#1445—22no—
53#1444—22no—
54#1436—22no—
55C4⋊#95C2211yescharacteristic
56C4⋊#95C2211yescharacteristic
57C4⋊#95C2211yescharacteristic
58C4⋊#95C2211yescharacteristic
59C4×#95C2211yescharacteristic
60C23×Q8C2311yescharacteristic
61C23×C8C2311yescharacteristic
62C23×C8C2311yescharacteristic
63C23×C8C2311yescharacteristic
64C23×C8C2311yescharacteristic
65C2×C4×Q8—22no—
66C2×C4×Q8—22no—
67C2×C4×Q8—22no—
68C2×C4×Q8—22no—
69C2×C4×Q8—41no—
70C2×C4×Q8—41no—
71C22×C4⋊C4—22no—
72C22×C4⋊C4—21no—
73C22×C4⋊C4—21no—
74C22×C4⋊C4—22no—
75C22×C4⋊C4—21no—
76C22×C4⋊C4C2311yescharacteristic
77C22×C4⋊C4C2311yescharacteristic
78C22×C4⋊C4C2311yescharacteristic
79C22×C4⋊C4C2311yescharacteristic
80C22×C4⋊C4C2311yescharacteristic
81C22×C4⋊C4C2311yescharacteristic
82C22×C4⋊C4C2311yescharacteristic
83C22×C42—21no—
84C22×C42—21no—
85C22×C42C2311yescharacteristic
86C22×C42C2311yescharacteristic
87C22×C42C2311yescharacteristic
88C2×#106—22no—
89C2×#106—22no—
90C2×#106—22no—
91C2×#106—22no—
92C2×#106—22no—
93C2×#106—22no—
94C2×#106—22no—
95C2×#106—41no—
96C2×#106—41no—
97C2×#106—41no—
98C2×#106C2×C414yes—
99C2×#106C2×C414yes—
100C2×#103—22no—
101C2×#103—22no—
102C2×#103—22no—
103C2×#103—22no—
104C2×#103—22no—
105C2×#103—22no—
106C2×#103—22no—
107C2×#103—22no—
108C2×#103—41no—
109C2×#103C2×C414yes—
110C2×#103C2×C414yes—
111C2×#103Q814yes—
112C2×C4×C8—22no—
113C2×C4×C8—22no—
114C2×C4×C8—22no—
115C2×C4×C8—22no—
116C2×C4×C8—41no—
117C2×C4×C8—41no—
118C2×C4×C8—41no—
119C2×C4×C8—41no—
120C2×C4×C8D414yes—
121#391—22no—
122#391—22no—
123#391—41no—
124#391C2×C414yes—
125#389—22no—
126#389—22no—
127#389—22no—
128#389—22no—
129#387—48no—
130#384—22no—
131#384—22no—
132#384—41no—
133#384C2×C414yes—
134C4×C4⋊C4—22no—
135C4×C4⋊C4—22no—
136C2×#95—22no—
137C2×#95—22no—
138C2×#95—22no—
139C2×#95—22no—
140C2×#95—21no—
141C2×#95—22no—
142C2×#95—21no—
143C2×#95—21no—
144C2×#95—22no—
145C2×#95D412yes—
146C2×#95D412yes—
147#340—22no—
148#340—22no—
149#340—22no—
150#340—22no—
151#340—22no—
152#340—22no—
153#340—22no—
154#340—22no—
155#336—22no—
156#336C2×C414yes—
157#336D414yes—
158C22×Q8—21no—
159C22×Q8—21no—
160C22×Q8—21no—
161C22×Q8—21no—
162C22×Q8—21no—
163C22×Q8—21no—
164C22×Q8—21no—
165C22×Q8—21no—
166C22×Q8—21no—
167C22×Q8—21no—
168C22×Q8—21no—
169C22×Q8C22×C412yes—
170C22×Q8C2×Q812yes—
171C22×Q8#4112yes—
172C23×C4—21no—
173C23×C4—21no—
174C23×C4—21no—
175C23×C4—21no—
176C23×C4—21no—
177C23×C4—21no—
178C23×C4—21no—
179C23×C4—21no—
180C23×C4C2×D411yescharacteristic
181C23×C4C2×D412yes—
182C23×C4C2×D412yes—
183C23×C4C2×D411yescharacteristic
184C23×C4C2411yescharacteristic
185C22×C8—21no—
186C22×C8—21no—
187C22×C8—21no—
188C22×C8—21no—
189C22×C8—21no—
190C22×C8—21no—
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×C8C22×C412yes—
204C22×C8C22×C412yes—
205C22×C8C2×D412yes—
206C22×C8C2×D412yes—
207C22×C8C2×D412yes—
208C22×C8C2×D412yes—
209C22×C8C2×Q812yes—
210C22×C8#4112yes—
211C22×C8#4112yes—
212C22×C8#4112yes—
213C4×Q8—28no—
214C4×Q8—44no—
215C4×Q8—44no—
216C4×Q8—44no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4—21no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—28no—
238C2×C4⋊C4—21no—
239C2×C4⋊C4—21no—
240C2×C4⋊C4—21no—
241C2×C4⋊C4—21no—
242C2×C4⋊C4—21no—
243C2×C4⋊C4—21no—
244C2×C4⋊C4—21no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4—21no—
253C2×C4⋊C4—21no—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—42no—
256C2×C4⋊C4—42no—
257C2×C4⋊C4—42no—
258C2×C4⋊C4—44no—
259C2×C4⋊C4—44no—
260C2×C4⋊C4—44no—
261C2×C4⋊C4—42no—
262C2×C4⋊C4—42no—
263C2×C4⋊C4—44no—
264C2×C4⋊C4—44no—
265C2×C4⋊C4—82no—
266C2×C4⋊C4—82no—
267C2×C4⋊C4C22×C412yes—
268C2×C4⋊C4C22×C412yes—
269C2×C4⋊C4C22×C412yes—
270C2×C4⋊C4C2×Q812yes—
271C2×C4⋊C4#4112yes—
272C2×C42—21no—
273C2×C42—21no—
274C2×C42—21no—
275C2×C42—21no—
276C2×C42—21no—
277C2×C42—21no—
278C2×C42—21no—
279C2×C42—21no—
280C2×C42—21no—
281C2×C42—21no—
282C2×C42—21no—
283C2×C42—21no—
284C2×C42—21no—
285C2×C42—21no—
286C2×C42—42no—
287C2×C42—42no—
288C2×C42—42no—
289C2×C42—42no—
290C2×C42—44no—
291C2×C42C22×C412yes—
292C2×C42C2×D412yes—
293C2×C42C2×D412yes—
294C2×C42#4112yes—
295#106—28no—
296#106—28no—
297#106—44no—
298#106—44no—
299#103—28no—
300#103—28no—
301#103—28no—
302#103—44no—
303C4×C8—44no—
304C4×C8—44no—
305C4×C8—44no—
306#95—28no—
307#95—24no—
308#95—44no—
309#95—42no—
310#95—42no—
311#95—42no—
312#95—44no—
313#95—44no—
314#95—44no—
315#95—82no—
316#95SD1618yes—
317C24C22×D411yescharacteristic
318C2×Q8—22no—
319C2×Q8—22no—
320C2×Q8—22no—
321C2×Q8—22no—
322C2×Q8—22no—
323C2×Q8—22no—
324C2×Q8—22no—
325C2×Q8—22no—
326C2×Q8—22no—
327C2×Q8—22no—
328C2×Q8—22no—
329C2×Q8—22no—
330C2×Q8—22no—
331C2×Q8—22no—
332C2×Q8—22no—
333C2×Q8—41no—
334C2×Q8—41no—
335C2×Q8—41no—
336C2×Q8—41no—
337C2×Q8—41no—
338C2×Q8—41no—
339C2×Q8—41no—
340C2×Q8—41no—
341C2×Q8—41no—
342C2×Q8—41no—
343C2×Q8—41no—
344C2×Q8—41no—
345C2×Q8C4×Q814yes—
346C22×C4—21no—
347C22×C4—21no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—21no—
358C22×C4—21no—
359C22×C4—22no—
360C22×C4—22no—
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—41no—
372C22×C4—41no—
373C22×C4—41no—
374C22×C4—41no—
375C22×C4—41no—
376C22×C4—41no—
377C22×C4—41no—
378C22×C4—41no—
379C22×C4—41no—
380C22×C4—41no—
381C22×C4—41no—
382C22×C4—41no—
383C22×C4—41no—
384C22×C4—41no—
385C22×C4—42no—
386C22×C4—42no—
387C22×C4—42no—
388C22×C4—42no—
389C22×C4—42no—
390C22×C4—42no—
391C22×C4—42no—
392C22×C4—42no—
393C22×C4—42no—
394C22×C4—42no—
395C22×C4—41no—
396C22×C4—41no—
397C22×C4—41no—
398C22×C4—41no—
399C22×C4—41no—
400C22×C4—41no—
401C22×C4—42no—
402C22×C4—42no—
403C22×C4—42no—
404C22×C4—42no—
405C22×C4C4×D414yes—
406C22×C4#12712yes—
407C22×C4#12712yes—
408C22×C4C2×SD1614yes—
409C22×C4C2×SD1614yes—
410C22×C4C2×SD1614yes—
411C22×C4#13514yes—
412C22×C4C23×C411yescharacteristic
413C22×C4C22×D411yescharacteristic
414C22×C4C22×D411yescharacteristic
415C22×C4C22×Q811yescharacteristic
416C22×C4C2×#4111yescharacteristic
417C22×C4C2×#4111yescharacteristic
418C22×C4C2×#4111yescharacteristic
419C22×C4C2×#4111yescharacteristic
420C22×C4C2×#4111yescharacteristic
421C22×C4C2×#4111yescharacteristic
422C22×C4H(5,𝔽2)11yescharacteristic
423C22×C4H(5,𝔽2)11yescharacteristic
424C22×C42−1+411yescharacteristic
425C22×C42−1+411yescharacteristic
426C2×C8—22no—
427C2×C8—22no—
428C2×C8—22no—
429C2×C8—22no—
430C2×C8—22no—
431C2×C8—22no—
432C2×C8—22no—
433C2×C8—22no—
434C2×C8—22no—
435C2×C8—22no—
436C2×C8—22no—
437C2×C8—22no—
438C2×C8—22no—
439C2×C8—22no—
440C2×C8—22no—
441C2×C8—22no—
442C2×C8—22no—
443C2×C8—22no—
444C2×C8—22no—
445C2×C8—22no—
446C2×C8—22no—
447C2×C8—41no—
448C2×C8—41no—
449C2×C8—41no—
450C2×C8—41no—
451C2×C8C4×D414yes—
452C2×C8C4×D414yes—
453C2×C8C22⋊Q814yes—
454C4⋊C4—22no—
455C4⋊C4—22no—
456C4⋊C4—22no—
457C4⋊C4—22no—
458C4⋊C4—22no—
459C4⋊C4—22no—
460C4⋊C4—22no—
461C4⋊C4—22no—
462C4⋊C4—22no—
463C4⋊C4—22no—
464C4⋊C4—48no—
465C4⋊C4—41no—
466C4⋊C4—41no—
467C4⋊C4—41no—
468C4⋊C4—41no—
469C4⋊C4—41no—
470C4⋊C4—41no—
471C4⋊C4—41no—
472C4⋊C4—41no—
473C4⋊C4—41no—
474C4⋊C4—41no—
475C4⋊C4—416no—
476C4⋊C4—41no—
477C4⋊C4—41no—
478C4⋊C4—41no—
479C4⋊C4—41no—
480C4⋊C4—41no—
481C4⋊C4—41no—
482C4⋊C4—41no—
483C4⋊C4—41no—
484C4⋊C4—88no—
485C4⋊C4—84no—
486C4⋊C4—84no—
487C42—22no—
488C42—22no—
489C42—22no—
490C42—22no—
491C42—22no—
492C42—22no—
493C42—22no—
494C42—22no—
495C42—48no—
496C42—41no—
497C42—41no—
498C42—41no—
499C42—41no—
500C42—84no—
501C23C2×C4×D411yescharacteristic
502C23C2×#12711yescharacteristic
503C23#53211yescharacteristic
504C23#54611yescharacteristic
505C23#54611yescharacteristic
506C23#54711yescharacteristic
507C23D4×Q811yescharacteristic
508C23C22×SD1612yes—
509C23C2×#13512yes—
510C23#57712yes—
511C23#57812yes—
512Q8—24no—
513Q8—24no—
514Q8—42no—
515Q8—42no—
516Q8—42no—
517Q8—42no—
518Q8—42no—
519Q8—42no—
520C2×C4—24no—
521C2×C4—24no—
522C2×C4—24no—
523C2×C4—22no—
524C2×C4—24no—
525C2×C4—24no—
526C2×C4—24no—
527C2×C4—24no—
528C2×C4—24no—
529C2×C4—22no—
530C2×C4—24no—
531C2×C4—24no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—24no—
535C2×C4—24no—
536C2×C4—22no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—44no—
553C2×C4—44no—
554C2×C4—44no—
555C2×C4—44no—
556C2×C4—44no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—82no—
560C2×C4—81no—
561C2×C4—81no—
562C2×C4—81no—
563C2×C4—81no—
564C2×C4—81no—
565C2×C4—81no—
566C2×C4—81no—
567C2×C4—81no—
568C2×C4—81no—
569C2×C4—81no—
570C2×C4—82no—
571C2×C4—82no—
572C2×C4—82no—
573C2×C4—82no—
574C2×C4—81no—
575C2×C4—81no—
576C2×C4—81no—
577C2×C4—81no—
578C2×C4—82no—
579C2×C4—82no—
580C2×C4—82no—
581C2×C4—82no—
582C2×C4—81no—
583C2×C4—81no—
584C2×C4C4×SD1618yes—
585C2×C4#49214yes—
586C2×C4C2×C4×D411yescharacteristic
587C2×C4C2×C4×D411yescharacteristic
588C2×C4C2×C4×Q811yescharacteristic
589C2×C4C4×#4111yescharacteristic
590C2×C4#52011yescharacteristic
591C2×C4#52011yescharacteristic
592C2×C4C2×C22⋊Q811yescharacteristic
593C2×C4#52511yescharacteristic
594C2×C4#52911yescharacteristic
595C2×C4#53211yescharacteristic
596C2×C4#53211yescharacteristic
597C2×C4#53311yescharacteristic
598C2×C4#53411yescharacteristic
599C2×C4#53611yescharacteristic
600C2×C4#53911yescharacteristic
601C2×C4#53911yescharacteristic
602C2×C4#54611yescharacteristic
603C2×C4#54611yescharacteristic
604C2×C4#54711yescharacteristic
605C2×C4D4×Q811yescharacteristic
606C2×C4#55111yescharacteristic
607C2×C4#55111yescharacteristic
608C2×C4#55211yescharacteristic
609C2×C4#55211yescharacteristic
610C2×C4#55211yescharacteristic
611C2×C4#55211yescharacteristic
612C2×C4#55411yescharacteristic
613C2×C4#55411yescharacteristic
614C8—24no—
615C8—24no—
616C8—24no—
617C8—42no—
618C22C4×#12711yescharacteristic
619C22#235911yescharacteristic
620C22#236011yescharacteristic
621C22#253911yescharacteristic
622C22#254211yescharacteristic
623C22#255311yescharacteristic
624C22#255611yescharacteristic
625C22C2×C4×SD1612yes—
626C22#294712yes—
627C22C2×#49211yescharacteristic
628C22#314711yescharacteristic
629C22#315411yescharacteristic
630C22#315611yescharacteristic
631C22#316712yes—
632C22#317812yes—
633C22#329612yes—
634C22#329812yes—
635C22#329812yes—
636C22#330412yes—
637C22#333512yes—
638C22#333612yes—
639C22Q8×SD1612yes—
640C22#338312yes—
641C4—28no—
642C4—24no—
643C4—44no—
644C4—44no—
645C4—42no—
646C4—82no—
647C4—82no—
648C4—82no—
649C4—82no—
650C4—82no—
651C4—161no—
652C4—161no—
653C4C4×C22⋊Q811yescharacteristic
654C4#232211yescharacteristic
655C4#235911yescharacteristic
656C4#236011yescharacteristic
657C4#255311yescharacteristic
658C4#255611yescharacteristic
659C4#256411yescharacteristic
660C4#261111yescharacteristic
661C2#1423011yescharacteristic
662C2C4×#49211yescharacteristic
663C2#2170311yescharacteristic
664C2#2292012yes—
665C2#2296412yes—
666C2#2833611yescharacteristic
667C2#2834011yescharacteristic
668C2#2853011yescharacteristic
669C2#2853511yescharacteristic
670C2#2884712yes—
671C2#2894712yes—
subgroupG/H
center, socle, prime radicalC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C4×#95 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
36. #2096C4semidirect

Maximal subgroups

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

subgroupindex
C2×#20962
C2×#19412
C2×#19202
C2×#18652
C2×C4×#1032
#140492
#139762
#113782
#113232
#111222
#109262
#108612
#108412
C8×#952