Finite Group Databasebeta

← #6379032

Subgroups — #6379032

Order 512.

4585 conjugacy classes of subgroups.

The subgroup lattice has 677 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3497C211yesmaximal, characteristic
2#60144C212yesmaximal, complemented
3C2×#2889C212yesmaximal, complemented
4#22412C214yesmaximal, complemented
5#22410C218yesmaximal, complemented
6D4×#101C214yesmaximal, complemented
7#21218C212yesmaximal, complemented
8#20992C214yesmaximal, complemented
9C2×#1801C211yesmaximal, characteristic, complemented
10#20593C212yesmaximal, complemented
11C2×C4×#101C211yesmaximal, characteristic, complemented
12C22×2−1+4C2212yes—
13C2×#568C2214yes—
14#3497—24no—
15#3497—24no—
16#3497—24no—
17#3497—24no—
18C2×#550C412yes—
19C2×#550C2212yes—
20C2×D4×Q8C412yes—
21C2×D4×Q8C2212yes—
22C2×#548C414yes—
23C2×#548C2214yes—
24C2×#537C414yes—
25C2×#537C2212yes—
26C2×#532C414yes—
27C2×#532C2211yescharacteristic
28C2×#532C2211yescharacteristic
29C22×C4×Q8C2211yescharacteristic
30#2935—24no—
31#2935—24no—
32D4×M16C2218yes—
33D4×M16C2218yes—
34#2932—24no—
35#2931—24no—
36C2×#423C2212yes—
37#2889—216no—
38C2×#413C2214yes—
39C2×#413C2218yes—
40C2×#413C2214yes—
41C22×#101C2211yescharacteristic
42C22×#101C2212yes—
43C22×#101C2218yes—
44C22×#101C2211yescharacteristic
45#2879C2214yes—
46C2×C4×M16C2212yes—
47#1967C2218yes—
48#1966C2214yes—
49#1965C2214yes—
50#1901C22116yes—
51#1900C22116yes—
52#1801C2214yes—
53#1801C2214yes—
54#1785C2218yes—
55#1784C2218yes—
56C4×#101C2214yes—
57C4×#101C2214yes—
58C2×#343C2212yes—
59C2×#343C2212yes—
60C2×2−1+4—28no—
61C2×2−1+4—28no—
62C2×2−1+4—28no—
63C2×2−1+4—28no—
64C2×2−1+4—48no—
65C22×#41C2×C418yes—
66C22×#41C2×C414yes—
67C22×#41C2312yes—
68C22×#41C2314yes—
69C22×#41C2312yes—
70C23×Q8C2×C414yes—
71C23×Q8C2314yes—
72C23×Q8C2311yescharacteristic
73#568—216no—
74#568—48no—
75C2×#131—28no—
76C2×#131—28no—
77C22×M16C2312yes—
78C22×M16C2314yes—
79C22×M16C23116yes—
80C22×M16C2312yes—
81#550—28no—
82#550—28no—
83#550—28no—
84#550—28no—
85D4×Q8—28no—
86D4×Q8—28no—
87D4×Q8—28no—
88D4×Q8—28no—
89#548—216no—
90#548—216no—
91#548—216no—
92#548—216no—
93#537—28no—
94#537—28no—
95#537—216no—
96#537—216no—
97#532—28no—
98#532—28no—
99#532—28no—
100#532—28no—
101#532—22no—
102#532—22no—
103#532—22no—
104#532—22no—
105#532D414yes—
106#532D414yes—
107#532D414yes—
108#532D414yes—
109C2×C4⋊Q8C2×C414yes—
110C2×C4⋊Q8C2×C411yescharacteristic
111C2×C4⋊Q8C2311yescharacteristic
112C2×#127C2×C414yes—
113C2×#127C2×C411yescharacteristic
114C2×#127C2311yescharacteristic
115C2×#124C2×C418yes—
116C2×#124C2×C412yes—
117C2×#124C2312yes—
118C2×C22⋊Q8C2×C418yes—
119C2×C22⋊Q8C2×C418yes—
120C2×C22⋊Q8C2×C414yes—
121C2×C22⋊Q8C2314yes—
122C2×#121C2×C412yes—
123C2×#121C2×C418yes—
124C2×#121C2×C412yes—
125C2×#121C2×C418yes—
126C2×#121C2×C418yes—
127C2×#121C2×C414yes—
128C2×#121C2×C412yes—
129C2×#121C2×C412yes—
130C2×#121C2314yes—
131C2×C4×Q8—22no—
132C2×C4×Q8—22no—
133C2×C4×Q8—22no—
134C2×C4×Q8—22no—
135C2×C4×Q8C2×C412yes—
136C2×C4×Q8C2×C412yes—
137C2×C4×Q8C2312yes—
138C2×C4×Q8C2312yes—
139C2×C4×D4C2×C418yes—
140C2×C4×D4C2×C414yes—
141C2×C4×D4C2×C414yes—
142C2×C4×D4C2×C412yes—
143C2×C4×D4C2×C412yes—
144C2×C4×D4C2312yes—
145C2×C4×D4C2312yes—
146C22×C4⋊C4C2×C412yes—
147C22×C4⋊C4C2311yescharacteristic
148C22×C42C2×C412yes—
149C22×C42C2311yescharacteristic
150#436—28no—
151#436—28no—
152#435—216no—
153#435—216no—
154C8×D4—28no—
155C8×D4—28no—
156#423—22no—
157#423—22no—
158#423C2314yes—
159#423C2314yes—
160C2×C4⋊C8—22no—
161C2×C4⋊C8—22no—
162#413—216no—
163#413—232no—
164#413—48no—
165C2×#101—28no—
166C2×#101—24no—
167C2×#101—232no—
168C2×#101D418yescomplemented
169C2×#101C2318yes—
170C2×#101C2318yes—
171C2×#101C2314yes—
172C2×#101C2318yes—
173C2×#101C2314yes—
174C2×#100C2318yes—
175C2×#100C2318yes—
176#408—28no—
177#408—28no—
178#407C23116yes—
179C4×M16—22no—
180C4×M16—22no—
181C4×M16C2314yes—
182C4×M16C2314yes—
183C2×#97—22no—
184C2×C4×C8—22no—
185#343C2318yes—
186#343C2318yes—
1872−1+4—432no—
1882−1+4—432no—
1892−1+4—432no—
1902−1+4—432no—
191C2×#41—216no—
192C2×#41—28no—
193C2×#41—28no—
194C2×#41—216no—
195C2×#41—216no—
196C2×#41—216no—
197C2×#41—232no—
198C2×#41—24no—
199C2×#41—24no—
200C2×#41—24no—
201C2×#41—216no—
202C2×#41—24no—
203C2×#41—24no—
204C2×#41—24no—
205C2×#41—28no—
206C2×#41—28no—
207C2×#41—416no—
208C2×#41—48no—
209C2×#41—48no—
210C2×#41C2×D418yes—
211C2×#41C2×D418yes—
212C22×Q8—216no—
213C22×Q8—216no—
214C22×Q8—28no—
215C22×Q8—28no—
216C22×Q8—22no—
217C22×Q8—22no—
218C22×Q8—216no—
219C22×Q8—24no—
220C22×Q8—48no—
221C22×Q8—48no—
222C22×Q8C22×C418yes—
223C22×Q8C22×C412yes—
224C22×Q8C22×C412yes—
225C22×Q8C2×D414yes—
226C22×Q8#4114yes—
227C22×Q8C2412yes—
228C22×Q8C2412yes—
229C22×D4C22×C414yes—
230C22×D4C22×C418yes—
231C22×D4C22×C414yes—
232C22×D4C22×C412yes—
233C22×D4C22×C412yes—
234C22×D4C22×C412yes—
235C22×D4C22×C418yes—
236C22×D4C22×C412yes—
237C22×D4C22×C412yes—
238C22×D4C2412yes—
239C23×C4C22×C414yes—
240C23×C4C22×C418yes—
241C23×C4C22×C418yes—
242C23×C4C22×C412yes—
243C23×C4C22×C412yes—
244C23×C4C22×C411yescharacteristic
245C23×C4C22×C411yescharacteristic
246C23×C4C2414yes—
247C23×C4C2411yescharacteristic
248#131—432no—
249#131—432no—
250C2×M16—22no—
251C2×M16—24no—
252C2×M16—24no—
253C2×M16—216no—
254C2×M16—22no—
255C2×M16—28no—
256C2×M16—24no—
257C2×M16—28no—
258C2×M16—24no—
259C2×M16—232no—
260C2×M16—416no—
261C2×M16C2×D418yes—
262C2×M16C2418yes—
263C2×M16C2414yes—
264C2×M16C2414yes—
265C22×C8—22no—
266C22×C8—24no—
267C22×C8—216no—
268C22×C8—22no—
269C4⋊Q8—28no—
270C4⋊Q8—28no—
271C4⋊Q8—22no—
272C4⋊Q8—22no—
273C4⋊Q8C22⋊C414yes—
274C4⋊Q8C22⋊C414yes—
275#127—28no—
276#127—28no—
277#127—22no—
278#127—22no—
279#127C22⋊C414yes—
280#127C22⋊C414yes—
281#124—216no—
282#124—216no—
283#124—24no—
284#124—24no—
285#124C22⋊C418yes—
286#124C22⋊C418yes—
287C22⋊Q8—216no—
288C22⋊Q8—216no—
289C22⋊Q8—232no—
290C22⋊Q8—232no—
291#121—216no—
292#121—216no—
293#121—216no—
294#121—232no—
295#121—22no—
296#121—22no—
297#121—22no—
298#121—22no—
299#121—24no—
300#121—24no—
301#121—216no—
302#121—232no—
303#121C22⋊C414yes—
304#121C22⋊C414yes—
305#121C22⋊C414yes—
306#121C22⋊C414yes—
307#121C22⋊C418yes—
308#121C22⋊C418yes—
309C4×Q8—24no—
310C4×Q8—24no—
311C4×Q8—24no—
312C4×Q8—24no—
313C4×Q8—24no—
314C4×Q8—24no—
315C4×Q8—24no—
316C4×Q8—24no—
317C4×D4—28no—
318C4×D4—28no—
319C4×D4—28no—
320C4×D4—28no—
321C4×D4—216no—
322C4×D4—216no—
323C4×D4—24no—
324C4×D4—24no—
325C4×D4—24no—
326C4×D4—24no—
327C4×D4C2×D418yes—
328C4×D4C2×D418yes—
329C4×D4C2×D418yes—
330C4×D4C2×D418yes—
331C2×C4⋊C4—24no—
332C2×C4⋊C4—24no—
333C2×C4⋊C4—22no—
334C2×C4⋊C4—22no—
335C2×C4⋊C4C22×C414yes—
336C2×C4⋊C4C22×C414yes—
337C2×C4⋊C4C22×C412yes—
338C2×C4⋊C4C22×C412yes—
339C2×C4⋊C4C22×C412yes—
340C2×C4⋊C4C22×C418yes—
341C2×C4⋊C4C22×C412yes—
342C2×C4⋊C4C22×C412yes—
343C2×C4⋊C4C2412yes—
344C2×C22⋊C4C22×C414yes—
345C2×C22⋊C4C22×C414yes—
346C2×C22⋊C4C22×C418yes—
347C2×C22⋊C4C22×C418yes—
348C2×C22⋊C4C22×C414yes—
349C2×C22⋊C4C22×C414yes—
350C2×C42—22no—
351C2×C42—22no—
352C2×C42—22no—
353C2×C42—22no—
354C2×C42C22×C414yes—
355C2×C42C22×C414yes—
356C2×C42C22×C411yescharacteristic
357C2×C42C22×C411yescharacteristic
358C2×C42C2×D412yes—
359C2×C42C2×D412yes—
360C2×C42C2×D412yes—
361C2×C42C2×D412yes—
362C2×C42C2411yescharacteristic
363C2×C42C2411yescharacteristic
364C4⋊C8—24no—
365C4⋊C8—24no—
366C4⋊C8—24no—
367C4⋊C8—24no—
368#101—216no—
369#101—28no—
370#101—216no—
371#101—48no—
372#101C2×D4116yes—
373#100—216no—
374#100—48no—
375C22⋊C8—216no—
376C22⋊C8—216no—
377#97—24no—
378#97—24no—
379C4×C8—24no—
380C4×C8—24no—
381C24C23×C414yes—
382C24C23×C414yes—
383C24C23×C411yescharacteristic
384#41—216no—
385#41—216no—
386#41—216no—
387#41—216no—
388#41—464no—
389#41—464no—
390#41—432no—
391#41—432no—
392#41—464no—
393#41—48no—
394#41—48no—
395#41—48no—
396#41—48no—
397C2×Q8—24no—
398C2×Q8—216no—
399C2×Q8—24no—
400C2×Q8—216no—
401C2×Q8—24no—
402C2×Q8—24no—
403C2×Q8—28no—
404C2×Q8—28no—
405C2×Q8—28no—
406C2×Q8—28no—
407C2×Q8—24no—
408C2×Q8—22no—
409C2×Q8—24no—
410C2×Q8—22no—
411C2×Q8—24no—
412C2×Q8—24no—
413C2×Q8—48no—
414C2×Q8—432no—
415C2×Q8—432no—
416C2×Q8—464no—
417C2×Q8C2×C22⋊C418yes—
418C2×Q8C4×D418yes—
419C2×Q8C22×D414yes—
420C2×Q8C2×#4114yes—
421C2×D4—28no—
422C2×D4—28no—
423C2×D4—24no—
424C2×D4—24no—
425C2×D4—216no—
426C2×D4—28no—
427C2×D4—28no—
428C2×D4—216no—
429C2×D4—22no—
430C2×D4—22no—
431C2×D4—24no—
432C2×D4—24no—
433C2×D4—216no—
434C2×D4—24no—
435C2×D4—22no—
436C2×D4—24no—
437C2×D4—22no—
438C2×D4—24no—
439C2×D4—44no—
440C2×D4—48no—
441C2×D4—44no—
442C2×D4—82no—
443C2×D4—82no—
444C2×D4—88no—
445C2×D4C2×C22⋊C414yes—
446C2×D4C2×C22⋊C418yes—
447C2×D4C2×C22⋊C418yes—
448C2×D4C2×C22⋊C418yes—
449C2×D4C2×C22⋊C414yes—
450C2×D4C2×C22⋊C418yes—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—24no—
454C22×C4—24no—
455C22×C4—24no—
456C22×C4—28no—
457C22×C4—216no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—216no—
461C22×C4—28no—
462C22×C4—216no—
463C22×C4—216no—
464C22×C4—28no—
465C22×C4—28no—
466C22×C4—42no—
467C22×C4—48no—
468C22×C4—48no—
469C22×C4—81no—
470C22×C4—82no—
471C22×C4C2×C22⋊C411yescharacteristic
472C22×C4C2×C22⋊C411yescharacteristic
473C22×C4C2×C22⋊C411yescharacteristic
474C22×C4C2×C22⋊C411yescharacteristic
475C22×C4C2×C22⋊C412yes—
476C22×C4C2×C22⋊C412yes—
477C22×C4C4×D418yes—
478C22×C4C23×C414yes—
479C22×C4C23×C411yescharacteristic
480C22×C4C23×C414yes—
481C22×C4C23×C414yes—
482C22×C4C23×C414yes—
483C22×C4C23×C411yescharacteristic
484C22×C4C23×C411yescharacteristic
485C22×C4C23×C411yescharacteristic
486C22×C4C23×C411yescharacteristic
487C22×C4C22×D4116yes—
488C22×C4C22×D412yes—
489C22×C4C22×D412yes—
490C22×C4C22×D412yes—
491C22×C4C2×#4112yes—
492C22×C4H(5,𝔽2)14yes—
493C22×C4C2511yescharacteristic
494M16—28no—
495M16—24no—
496M16—28no—
497M16—24no—
498M16—24no—
499M16—28no—
500M16—432no—
501M16—48no—
502M16—44no—
503M16C22×D418yes—
504C2×C8—28no—
505C2×C8—24no—
506C2×C8—24no—
507C2×C8—28no—
508C2×C8—432no—
509C2×C8—48no—
510C2×C8—44no—
511C4⋊C4—28no—
512C4⋊C4—28no—
513C4⋊C4—24no—
514C4⋊C4—22no—
515C4⋊C4—22no—
516C4⋊C4—216no—
517C4⋊C4—24no—
518C4⋊C4—24no—
519C4⋊C4C2×C22⋊C414yes—
520C4⋊C4C2×C22⋊C414yes—
521C4⋊C4C2×C22⋊C418yes—
522C22⋊C4—216no—
523C22⋊C4—24no—
524C22⋊C4—24no—
525C22⋊C4—216no—
526C22⋊C4—28no—
527C22⋊C4C2×C22⋊C418yes—
528C22⋊C4C2×C22⋊C418yes—
529C22⋊C4C2×C22⋊C4116yes—
530C42—24no—
531C42—24no—
532C42—24no—
533C42—24no—
534C42C2×C22⋊C412yes—
535C42C2×C22⋊C412yes—
536C42C2×C22⋊C412yes—
537C42C2×C22⋊C412yes—
538C42C22×D412yes—
539C42C22×D412yes—
540C42C22×D412yes—
541C42C22×D412yes—
542C23—24no—
543C23—24no—
544C23—24no—
545C23—24no—
546C23—24no—
547C23—41no—
548C23—41no—
549C23—44no—
550C23—44no—
551C23—44no—
552C23—44no—
553C23C22×C22⋊C418yes—
554C23C22×C22⋊C411yescharacteristic
555C23C22×C22⋊C411yescharacteristic
556C23C2×C4×D412yes—
557C23#51812yes—
558C23C24×C411yescharacteristic
559Q8—216no—
560Q8—216no—
561Q8—28no—
562Q8—28no—
563Q8—432no—
564Q8—44no—
565Q8—44no—
566D4—216no—
567D4—216no—
568D4—216no—
569D4—28no—
570D4—44no—
571D4—48no—
572D4—432no—
573D4—44no—
574D4—48no—
575D4—48no—
576D4—44no—
577D4—44no—
578D4—44no—
579D4—84no—
580D4—816no—
581C2×C4—28no—
582C2×C4—28no—
583C2×C4—24no—
584C2×C4—24no—
585C2×C4—24no—
586C2×C4—24no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—216no—
591C2×C4—216no—
592C2×C4—24no—
593C2×C4—24no—
594C2×C4—24no—
595C2×C4—24no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—432no—
604C2×C4—432no—
605C2×C4—416no—
606C2×C4—44no—
607C2×C4—82no—
608C2×C4—82no—
609C2×C4C22×C22⋊C411yescharacteristic
610C2×C4C22×C22⋊C411yescharacteristic
611C2×C4C22×C22⋊C411yescharacteristic
612C2×C4C22×C22⋊C412yes—
613C2×C4C22×C22⋊C412yes—
614C2×C4C22×C22⋊C418yes—
615C2×C4C22×C22⋊C412yes—
616C2×C4C22×C22⋊C411yescharacteristic
617C2×C4C2×C4×D418yes—
618C2×C4C2×C4×D412yes—
619C2×C4#51812yes—
620C2×C4D4214yes—
621C2×C4D4214yes—
622C2×C4#54614yes—
623C2×C4#54614yes—
624C2×C4C23×D411yescharacteristic
625C2×C4C23×D412yes—
626C2×C4C22×#4111yescharacteristic
627C2×C4C2×H(5,𝔽2)12yes—
628C8—28no—
629C8—48no—
630C8—44no—
631C22—21no—
632C22—21no—
633C22—21no—
634C22—21no—
635C22—21no—
636C22—21no—
637C22—28no—
638C22—28no—
639C22—28no—
640C22—42no—
641C22—42no—
642C22—44no—
643C22—44no—
644C22—44no—
645C22—84no—
646C22—84no—
647C22—84no—
648C22—84no—
649C22D4×C22⋊C412yes—
650C22D4×C22⋊C412yes—
651C22C23×C22⋊C411yescharacteristic
652C22C22×C4×D411yescharacteristic
653C22C2×#51811yescharacteristic
654C4—28no—
655C4—28no—
656C4—21no—
657C4—21no—
658C4—21no—
659C4—22no—
660C4—22no—
661C4—44no—
662C4—44no—
663C4D4×C22⋊C414yes—
664C4#288912yescomplemented
665C4C2×D4212yes—
666C4C2×#54612yes—
667C2—21no—
668C2—21no—
669C2—22no—
670C2—22no—
671C2—48no—
672C2—84no—
673C2C2×D4×C22⋊C411yescharacteristic
674C2C2×#288911yescharacteristic
675C2#6007611yescharacteristic
subgroupG/H
center, socleC22C23×C22⋊C4
commutatorC23C24×C4
Frattini, cosocleC22×C4C25
prime radicalC2×#3497C2
LCSG ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×#532 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #60144C2semidirect
3. C2×#2889C2semidirect
4. #22412C2semidirect
5. #22410C2semidirect
6. D4×#101C2semidirect
7. #21218C2semidirect
8. #20992C2semidirect
9. C2×#1801C2semidirect
10. #20593C2semidirect
11. C2×C4×#101C2semidirect
168. C2×#101D4semidirect
664. C4#2889semidirect

Maximal subgroups

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

subgroupindex
C2×#34972
#601442
C2×#28892
#224122
#224102
D4×#1012
#212182
#209922
C2×#18012
#205932
C2×C4×#1012