Finite Group Databasebeta

← #6683895

Subgroups — #6683895

Order 512.

1834 conjugacy classes of subgroups.

The subgroup lattice has 699 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3381C211yesmaximal, characteristic, complemented
2C2×#3368C211yesmaximal, characteristic
3#38952C211yesmaximal, characteristic
4#36593C211yesmaximal, characteristic, complemented
5#33031C211yesmaximal, characteristic, complemented
6#30951C214yesmaximal, complemented
7#30835C214yesmaximal, complemented
8#30277C211yesmaximal, characteristic, complemented
9#29864C211yesmaximal, characteristic, complemented
10#28684C211yesmaximal, characteristic, complemented
11#28637C211yesmaximal, characteristic, complemented
12#26387C214yesmaximal, complemented
13#26163C214yesmaximal, complemented
14#25935C211yesmaximal, characteristic, complemented
15#25836C211yesmaximal, characteristic, complemented
16#25835C211yesmaximal, characteristic, complemented
17#25645C211yesmaximal, characteristic
18#22240C211yesmaximal, characteristic, complemented
19#21120C211yesmaximal, characteristic, complemented
20C2×#557C2211yescharacteristic
21C2×#557C2211yescharacteristic
22C2×#554C2211yescharacteristic
23C2×#550C2211yescharacteristic
24#3381—28no—
25#3368—28no—
26C2×#499C2212yes—
27C2×#498C2211yescharacteristic
28C2×#498C2211yescharacteristic
29C2×#490C2212yes—
30C2×#486C2211yescharacteristic
31C2×#486C2211yescharacteristic
32C2×#479C2212yes—
33C2×#477C2211yescharacteristic
34C2×#478C2212yes—
35C2×#478C2212yes—
36C2×#474C2211yescharacteristic
37C2×#474C2211yescharacteristic
38C2×#460C2211yescharacteristic
39C2×#460C2212yes—
40C2×#446C2211yescharacteristic
41C2×C8×Q8C2211yescharacteristic
42C2×#440C2212yes—
43C2×C4×SD16C2212yes—
44C2×C4×SD16C2211yescharacteristic
45#2868—21no—
46#2868C2211yescharacteristic
47#2858C2211yescharacteristic
48#2672—24no—
49#2620—21no—
50#2567C2211yescharacteristic
51#2552—24no—
52#2552C2214yes—
53#2548C2211yescharacteristic
54#2546—21no—
55#2546—21no—
56#2546C2211yescharacteristic
57#2546C2211yescharacteristic
58#2501—21no—
59#2454C2214yes—
60#2445C2214yes—
61#2432—21no—
62Q8×C4⋊C4C2214yes—
63#2311—21no—
64C4×C4⋊Q8—21no—
65C4×C4⋊Q8C2211yescharacteristic
66#2293C2211yescharacteristic
67C42×Q8C2211yescharacteristic
68#2094C2214yes—
69#2094C2214yes—
70#2088C2214yes—
71#2083C2214yes—
72#2071C2214yes—
73#2067C2214yes—
74#2059C2214yes—
75#2059C2214yes—
76#2041C2214yes—
77#2025C2214yes—
78#1988C2211yescharacteristic
79#1988C2211yescharacteristic
80#1987C2211yescharacteristic
81#1984C2211yescharacteristic
82#1984C2211yescharacteristic
83#1982C2211yescharacteristic
84#1943C2214yes—
85#1934C2214yes—
86#1925C2214yes—
87#1925C2212yes—
88#1920C2212yes—
89#1917C2214yes—
90#1916C2214yes—
91#1880C2214yes—
92#1880C2214yes—
93#1880C2212yes—
94#1865C2214yes—
95#1864C2214yes—
96#1863C2212yes—
97#1862C2214yes—
98#1831C2211yescharacteristic
99C4×#107C2211yescharacteristic
100C4×#103C2211yescharacteristic
101C4×#102C2211yescharacteristic
102#1744C2211yescharacteristic
103C22×#41—21no—
104C22×SD16C2312yes—
105C22×SD16C2311yescharacteristic
106#557—24no—
107#557—24no—
108#557—42no—
109#557—42no—
110#554—24no—
111#554—42no—
112#550—24no—
113#550—42no—
114C2×C4⋊Q8—21no—
115C2×C4⋊Q8—21no—
116C2×C4⋊Q8—21no—
117C2×C4⋊Q8—21no—
118C2×C4⋊Q8—21no—
119C2×C4⋊Q8—21no—
120C2×C4⋊Q8D411yescharacteristic
121C2×C4⋊Q8C2311yescharacteristic
122C2×C4⋊Q8C2311yescharacteristic
123C2×C4⋊Q8C2311yescharacteristic
124C2×#127C2311yescharacteristic
125C2×#127C2312yes—
126C2×#125—22no—
127C2×#125—22no—
128C2×#125C2312yes—
129C2×#125C2312yes—
130C2×#125C2312yes—
131C2×#124—21no—
132C2×#124—22no—
133C2×#123—24no—
134C2×C22⋊Q8—21no—
135C2×C22⋊Q8—22no—
136C2×#121—21no—
137C2×#121—22no—
138C2×C4×Q8—21no—
139C2×C4×Q8—21no—
140C2×C4×Q8C2312yes—
141C2×C4×Q8C2312yes—
142C2×C4×Q8C2312yes—
143C2×C4×Q8C2311yescharacteristic
144C2×C4×Q8C2311yescharacteristic
145C2×C4×Q8C2311yescharacteristic
146C2×C4×D4C2312yes—
147C2×C4×D4C2312yes—
148C2×C4×D4C2311yescharacteristic
149C2×#117—21no—
150C22×C4⋊C4—21no—
151#499—28no—
152#498—24no—
153#498—42no—
154#490—28no—
155#486—24no—
156#486—42no—
157#479—28no—
158#478—28no—
159#478—28no—
160#477—24no—
161#474—24no—
162#474—24no—
163#460—28no—
164#460—24no—
165#446—24no—
166C8×Q8—24no—
167#440—28no—
168C4×SD16—28no—
169C4×SD16D418yes—
170C2×#107C2312yes—
171C2×#107C2312yes—
172C2×#107C2312yes—
173C2×#107C2312yes—
174C2×#106C2312yes—
175C2×#106C2312yes—
176C2×#106C2314yes—
177C2×#106C2311yescharacteristic
178C2×C4⋊C8C2312yes—
179C2×C4⋊C8C2312yes—
180C2×C4⋊C8C2311yescharacteristic
181C2×C4⋊C8C2311yescharacteristic
182C2×#103C2312yes—
183C2×#103C2312yes—
184C2×#103C2311yescharacteristic
185C2×#102C2312yes—
186C2×#102C2312yes—
187C2×#102C2312yes—
188C2×#102C2312yes—
189C2×#102C2314yes—
190C2×#102C2312yes—
191C2×#102C2312yes—
192C2×#102C2314yes—
193C2×#102C2311yescharacteristic
194C2×#97C2312yes—
195C2×C4×C8C2311yescharacteristic
196C2×C4×C8C2312yes—
197C2×C4×C8C2311yescharacteristic
198#399—24no—
199#398—22no—
200#398—22no—
201#398—24no—
202#398—24no—
203#396—24no—
204#395—22no—
205#394—22no—
206#394—24no—
207#394—24no—
208#391—21no—
209#391—21no—
210#391—21no—
211#391D411yescharacteristic
212#391D411yescharacteristic
213#390—21no—
214#390—21no—
215#390—21no—
216#390C2314yes—
217#390C2312yes—
218#390C2314yes—
219#389—24no—
220#389—24no—
221#389—24no—
222#389D414yes—
223#389C2314yes—
224#389C2314yes—
225#389C2312yes—
226#389C2314yes—
227#388—22no—
228#386—24no—
229#385—24no—
230#384—21no—
231#384—24no—
232#384—24no—
233#384—22no—
234#384D411yescharacteristic
235#384C2314yes—
236#380—24no—
237C4×C4⋊C4—24no—
238C4×C4⋊C4—21no—
239C4×C4⋊C4—22no—
240C4×C4⋊C4C2311yescharacteristic
241C4×C4⋊C4C2312yes—
242C4×C4⋊C4C2311yescharacteristic
243C4×C4⋊C4C2314yes—
244C4×C22⋊C4—21no—
245C43—21no—
246C43C2311yescharacteristic
247#340C2312yes—
248#340C2312yes—
249#340C2314yes—
250#340C2314yes—
251#340C2314yes—
252#340C2312yes—
253#336C2314yes—
254#336C2314yes—
255#336C2314yes—
256#336C2312yes—
257#336C2312yes—
258C2×#41—44no—
259C2×#41—44no—
260C2×#41—82no—
261C22×Q8—21no—
262C22×Q8—21no—
263C22×Q8—21no—
264C22×Q8C2×D411yescharacteristic
265C22×Q8C2×D411yescharacteristic
266C22×Q8C2411yescharacteristic
267C22×D4—21no—
268C22×D4—22no—
269C22×D4—22no—
270C22×D4—21no—
271C22×D4C2412yes—
272C22×D4C2411yescharacteristic
273C23×C4—21no—
274C23×C4—22no—
275C23×C4—22no—
276C2×SD16—24no—
277C2×SD16—24no—
278C2×SD16—28no—
279C2×SD16C2×D414yes—
280C22×C8C2412yes—
281C22×C8C2412yes—
282C22×C8C2412yes—
283C22×C8C2411yescharacteristic
284C22×C8C2411yescharacteristic
285C4⋊Q8—22no—
286C4⋊Q8—22no—
287C4⋊Q8—22no—
288C4⋊Q8—22no—
289C4⋊Q8—22no—
290C4⋊Q8—22no—
291C4⋊Q8—42no—
292C4⋊Q8—44no—
293C4⋊Q8—44no—
294C4⋊Q8—44no—
295C4⋊Q8—82no—
296C4⋊Q8—82no—
297C4⋊Q8—82no—
298#127—24no—
299#127—22no—
300#127—22no—
301#127—42no—
302#125—24no—
303#125—24no—
304#125—24no—
305#125—42no—
306#125—42no—
307#125—42no—
308#125—48no—
309#125—48no—
310#124—84no—
311#124—82no—
312#123—88no—
313C22⋊Q8—48no—
314C22⋊Q8—44no—
315#121—48no—
316#121—44no—
317C4×Q8—24no—
318C4×Q8—24no—
319C4×Q8—22no—
320C4×Q8—22no—
321C4×Q8—22no—
322C4×Q8—22no—
323C4×Q8—22no—
324C4×Q8—24no—
325C4×Q8—24no—
326C4×Q8—42no—
327C4×Q8—42no—
328C4×Q8—44no—
329C4×Q8—44no—
330C4×Q8C2×D414yes—
331C4×D4—24no—
332C4×D4—24no—
333C4×D4—22no—
334C4×D4—42no—
335C4×D4—42no—
336C4×D4C2×D414yes—
337#117—82no—
338C2×C4⋊C4—21no—
339C2×C4⋊C4—22no—
340C2×C4⋊C4—22no—
341C2×C4⋊C4—21no—
342C2×C4⋊C4—22no—
343C2×C4⋊C4—22no—
344C2×C4⋊C4—22no—
345C2×C4⋊C4—22no—
346C2×C4⋊C4—22no—
347C2×C4⋊C4—21no—
348C2×C4⋊C4—21no—
349C2×C4⋊C4—22no—
350C2×C4⋊C4—22no—
351C2×C4⋊C4—24no—
352C2×C4⋊C4—22no—
353C2×C4⋊C4—24no—
354C2×C4⋊C4—24no—
355C2×C4⋊C4—24no—
356C2×C4⋊C4—24no—
357C2×C4⋊C4—44no—
358C2×C4⋊C4C2×D411yescharacteristic
359C2×C4⋊C4C2×D411yescharacteristic
360C2×C4⋊C4C2×D412yes—
361C2×C4⋊C4C2×D412yes—
362C2×C4⋊C4C2×D414yes—
363C2×C4⋊C4C2×D414yes—
364C2×C4⋊C4C2412yes—
365C2×C4⋊C4C2412yes—
366C2×C4⋊C4C2412yes—
367C2×C4⋊C4C2412yes—
368C2×C4⋊C4C2411yescharacteristic
369C2×C4⋊C4C2411yescharacteristic
370C2×C4⋊C4C2412yes—
371C2×C22⋊C4—21no—
372C2×C22⋊C4—22no—
373C2×C22⋊C4—22no—
374C2×C22⋊C4—22no—
375C2×C22⋊C4—24no—
376C2×C22⋊C4—22no—
377C2×C22⋊C4—24no—
378C2×C22⋊C4—24no—
379C2×C22⋊C4—24no—
380C2×C42—21no—
381C2×C42—22no—
382C2×C42—21no—
383C2×C42—22no—
384C2×C42—22no—
385C2×C42—21no—
386C2×C42—22no—
387C2×C42C2×D411yescharacteristic
388C2×C42C2×D412yes—
389C2×C42C2×D411yescharacteristic
390C2×C42C2412yes—
391C2×C42C2412yes—
392C2×C42C2412yes—
393C2×C42C2411yescharacteristic
394#107—24no—
395#107—24no—
396#107—24no—
397#107—24no—
398#106—24no—
399#106—24no—
400#106—44no—
401#106C2×D414yes—
402C4⋊C8—24no—
403C4⋊C8—24no—
404C4⋊C8—22no—
405C4⋊C8—22no—
406#103—24no—
407#103—24no—
408#103C2×D414yes—
409#102—24no—
410#102—24no—
411#102—24no—
412#102—24no—
413#102—24no—
414#102—24no—
415#102—28no—
416#102—44no—
417#102C2×D414yes—
418#97—24no—
419C4×C8—24no—
420C4×C8—22no—
421C4×C8C2×D414yes—
422#95—22no—
423#95—21no—
424#95—22no—
425#95—24no—
426#95—24no—
427#95—21no—
428#95—24no—
429#95C2×D414yes—
430#95C2×D414yes—
431C24—21no—
432C24—22no—
433#41—816no—
434C2×Q8—22no—
435C2×Q8—22no—
436C2×Q8—22no—
437C2×Q8—22no—
438C2×Q8—22no—
439C2×Q8—22no—
440C2×Q8—21no—
441C2×Q8—22no—
442C2×Q8—42no—
443C2×Q8—42no—
444C2×Q8—42no—
445C2×Q8—42no—
446C2×Q8—42no—
447C2×Q8—42no—
448C2×Q8—42no—
449C2×Q8—82no—
450C2×Q8C22×D412yes—
451C2×Q82−1+414yes—
452C2×D4—22no—
453C2×D4—21no—
454C2×D4—22no—
455C2×D4—22no—
456C2×D4—22no—
457C2×D4—24no—
458C2×D4—24no—
459C2×D4—44no—
460C2×D4—44no—
461C2×D4—44no—
462C2×D4—42no—
463C2×D4—42no—
464C2×D4—42no—
465C2×D4—42no—
466C2×D4—82no—
467C2×D4—84no—
468C2×D4—82no—
469C2×D4C22×D412yes—
470C22×C4—21no—
471C22×C4—22no—
472C22×C4—21no—
473C22×C4—21no—
474C22×C4—24no—
475C22×C4—22no—
476C22×C4—22no—
477C22×C4—22no—
478C22×C4—22no—
479C22×C4—44no—
480C22×C4—44no—
481C22×C4—44no—
482C22×C4—42no—
483C22×C4—42no—
484C22×C4—82no—
485C22×C4—84no—
486C22×C4C22×D412yes—
487C22×C4C22×D412yes—
488C22×C4C22×D412yes—
489C22×C4C22×D412yes—
490C22×C4C22×D412yes—
491C22×C4C22×D411yescharacteristic
492C22×C4C22×D412yes—
493C22×C4C22×D411yescharacteristic
494C22×C4C2511yescharacteristic
495SD16—28no—
496SD16—48no—
497C2×C8—22no—
498C2×C8—22no—
499C2×C8—22no—
500C2×C8—22no—
501C2×C8—22no—
502C2×C8—24no—
503C2×C8—22no—
504C2×C8—22no—
505C2×C8—24no—
506C2×C8C22×D412yes—
507C2×C8C22×D412yes—
508C4⋊C4—28no—
509C4⋊C4—24no—
510C4⋊C4—22no—
511C4⋊C4—22no—
512C4⋊C4—24no—
513C4⋊C4—24no—
514C4⋊C4—28no—
515C4⋊C4—21no—
516C4⋊C4—21no—
517C4⋊C4—22no—
518C4⋊C4—22no—
519C4⋊C4—22no—
520C4⋊C4—22no—
521C4⋊C4—22no—
522C4⋊C4—22no—
523C4⋊C4—22no—
524C4⋊C4—22no—
525C4⋊C4—22no—
526C4⋊C4—44no—
527C4⋊C4—44no—
528C4⋊C4—44no—
529C4⋊C4—44no—
530C4⋊C4—44no—
531C4⋊C4—44no—
532C4⋊C4—44no—
533C4⋊C4—44no—
534C4⋊C4—44no—
535C4⋊C4—48no—
536C4⋊C4—42no—
537C4⋊C4—42no—
538C4⋊C4—42no—
539C4⋊C4—82no—
540C4⋊C4—84no—
541C4⋊C4—84no—
542C4⋊C4—84no—
543C4⋊C4—84no—
544C4⋊C4C22×D412yes—
545C4⋊C4C22×D412yes—
546C4⋊C42−1+414yes—
547C22⋊C4—24no—
548C22⋊C4—44no—
549C22⋊C4—44no—
550C22⋊C4—44no—
551C22⋊C4—48no—
552C22⋊C4—82no—
553C22⋊C4—84no—
554C22⋊C4—84no—
555C22⋊C4—84no—
556C42—28no—
557C42—24no—
558C42—22no—
559C42—22no—
560C42—24no—
561C42—22no—
562C42—22no—
563C42—22no—
564C42—22no—
565C42—22no—
566C42—22no—
567C42—44no—
568C42—44no—
569C42—42no—
570C42—42no—
571C42—44no—
572C42C22×D412yes—
573C42H(5,𝔽2)12yes—
574C23—22no—
575C23—42no—
576C23—41no—
577C23—41no—
578C23—42no—
579C23—42no—
580C23—44no—
581C23—42no—
582C23—42no—
583C23—44no—
584C23C23×D411yescharacteristic
585Q8—22no—
586Q8—24no—
587Q8—44no—
588Q8—44no—
589Q8—84no—
590Q8—84no—
591Q8—162no—
592Q8#53614yes—
593D4—22no—
594D4—22no—
595D4—42no—
596D4—42no—
597D4—42no—
598D4—44no—
599D4—88no—
600D4—84no—
601D4—162no—
602D4—164no—
603C2×C4—22no—
604C2×C4—22no—
605C2×C4—24no—
606C2×C4—22no—
607C2×C4—22no—
608C2×C4—22no—
609C2×C4—22no—
610C2×C4—22no—
611C2×C4—22no—
612C2×C4—22no—
613C2×C4—22no—
614C2×C4—22no—
615C2×C4—24no—
616C2×C4—22no—
617C2×C4—22no—
618C2×C4—22no—
619C2×C4—22no—
620C2×C4—22no—
621C2×C4—22no—
622C2×C4—24no—
623C2×C4—24no—
624C2×C4—41no—
625C2×C4—42no—
626C2×C4—41no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—48no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—42no—
637C2×C4—42no—
638C2×C4—42no—
639C2×C4—42no—
640C2×C4—44no—
641C2×C4—88no—
642C2×C4—164no—
643C2×C4D4212yes—
644C2×C4D4212yes—
645C2×C4#57512yes—
646C2×C4#57612yes—
647C2×C4#57714yes—
648C2×C4C23×D411yescharacteristic
649C2×C4C2×H(5,𝔽2)11yescharacteristic
650C2×C4C2×2−1+412yes—
651C2×C4C2×2−1+412yes—
652C8—22no—
653C8—22no—
654C8—42no—
655C8—42no—
656C8—42no—
657C22—42no—
658C22—42no—
659C22—42no—
660C22—82no—
661C22—82no—
662C22—82no—
663C22—82no—
664C22—84no—
665C22—82no—
666C22—84no—
667C22C2×D4211yescharacteristic
668C22#348612yes—
669C22C2×#57511yescharacteristic
670C22C2×#57611yescharacteristic
671C22C2×#57712yes—
672C4—24no—
673C4—24no—
674C4—22no—
675C4—42no—
676C4—42no—
677C4—42no—
678C4—42no—
679C4—42no—
680C4—42no—
681C4—42no—
682C4—44no—
683C4—82no—
684C4—82no—
685C4—82no—
686C4—82no—
687C4#326814yes—
688C4C2×#53611yescharacteristic
689C4#349911yescharacteristic
690C4#355212yes—
691C2—82no—
692C2—162no—
693C2#3767511yescharacteristic
694C2C2×#326811yescharacteristic
695C2#6108711yescharacteristic
696C2#6183612yes—
697C2#6186812yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#550C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×C4⋊Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#3381C2semidirect
4. #36593C2semidirect
5. #33031C2semidirect
6. #30951C2semidirect
7. #30835C2semidirect
8. #30277C2semidirect
9. #29864C2semidirect
10. #28684C2semidirect
11. #28637C2semidirect
12. #26387C2semidirect
13. #26163C2semidirect
14. #25935C2semidirect
15. #25836C2semidirect
16. #25835C2semidirect
18. #22240C2semidirect
19. #21120C2semidirect

Maximal subgroups

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

subgroupindex
C2×#33812
C2×#33682
#389522
#365932
#330312
#309512
#308352
#302772
#298642
#286842
#286372
#263872
#261632
#259352
#258362
#258352
#256452
#222402
#211202