Finite Group Databasebeta

← #6382776

Subgroups — #6382776

Order 512.

2742 conjugacy classes of subgroups.

The subgroup lattice has 678 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#497C211yesmaximal, characteristic
2C2×#2950C211yesmaximal, characteristic, complemented
3C2×#2949C211yesmaximal, characteristic
4#34211C211yesmaximal, characteristic
5#34209C211yesmaximal, characteristic, complemented
6#23194C218yesmaximal, complemented
7#23192C218yesmaximal, complemented
8#21261C211yesmaximal, characteristic
9#21259C211yesmaximal, characteristic, complemented
10#21232C211yesmaximal, characteristic, complemented
11#21231C211yesmaximal, characteristic
12#20936C211yesmaximal, characteristic
13#20935C211yesmaximal, characteristic, complemented
14#20935C211yesmaximal, characteristic, complemented
15#20933C211yesmaximal, characteristic
16#20451C211yesmaximal, characteristic, complemented
17#20438C211yesmaximal, characteristic, complemented
18C23×Q16C2211yescharacteristic
19C23×Q16C2211yescharacteristic
20C23×SD16C2211yescharacteristic
21C23×SD16C2211yescharacteristic
22C22×C4⋊Q8C2211yescharacteristic
23C22×#124C2211yescharacteristic
24C2×#519C2211yescharacteristic
25C2×#519C2211yescharacteristic
26C2×#519C2211yescharacteristic
27C2×#518C2211yescharacteristic
28C2×#497C4116yescomplemented
29C2×#497C2214yes—
30C2×#497C2214yes—
31#2950—28no—
32#2949—28no—
33C2×#441C2214yescomplemented
34C2×#440C2214yes—
35C2×C4×Q16C2214yescomplemented
36C2×C4×SD16C2214yes—
37C2×#427C2211yescharacteristic
38C2×#427C2211yescharacteristic
39C2×#423C2211yescharacteristic
40C2×#419C2211yescharacteristic
41C2×#419C2211yescharacteristic
42C2×#419C2211yescharacteristic
43C2×#419C2211yescharacteristic
44C2×#419C2211yescharacteristic
45C2×#419C2211yescharacteristic
46C2×#418C2211yescharacteristic
47C2×#418C2211yescharacteristic
48C2×#405C2211yescharacteristic
49C2×#405C2211yescharacteristic
50C22×#97C2211yescharacteristic
51#2367C2218yes—
52#2365C2218yescomplemented
53#2344—24no—
54#2342—24no—
55#2327—22no—
56#2326—22no—
57#2310C2211yescharacteristic
58#2310C2211yescharacteristic
59#2310C2211yescharacteristic
60#2309C2211yescharacteristic
61C4×C4⋊Q8—22no—
62C4×#124—22no—
63#2285C2211yescharacteristic
64C4×#117C2211yescharacteristic
65C4×#117C2211yescharacteristic
66#1983C2218yes—
67#1981C2218yescomplemented
68#1973C2214yes—
69#1971C2214yescomplemented
70#1969C2214yescomplemented
71#1968C2214yes—
72#1880C2218yes—
73#1878C2218yescomplemented
74#1878C2218yescomplemented
75#1877C2218yes—
76#1776C2214yescomplemented
77#1776C2214yescomplemented
78#1771C2214yescomplemented
79#1771C2214yescomplemented
80#1735C2211yescharacteristic
81#1735C2211yescharacteristic
82#1729C2211yescharacteristic
83#1729C2211yescharacteristic
84C23×Q8C2311yescharacteristic
85C23×Q8C2311yescharacteristic
86C23×Q8C2311yescharacteristic
87C23×D4C2311yescharacteristic
88C22×Q16—24no—
89C22×Q16—24no—
90C22×Q16—42no—
91C22×Q16—42no—
92C22×Q16C2×C418yescomplemented
93C22×Q16C2×C418yescomplemented
94C22×Q16C2312yes—
95C22×Q16C2312yes—
96C22×Q16C2314yes—
97C22×SD16—24no—
98C22×SD16—24no—
99C22×SD16—42no—
100C22×SD16—42no—
101C22×SD16C2×C418yes—
102C22×SD16C2×C418yes—
103C22×SD16C2312yes—
104C22×SD16C2312yes—
105C22×SD16C2314yes—
106C22×M16C2311yescharacteristic
107C22×M16C2311yescharacteristic
108C23×C8C2311yescharacteristic
109C23×C8C2311yescharacteristic
110C2×C4⋊Q8—21no—
111C2×C4⋊Q8—21no—
112C2×C4⋊Q8—24no—
113C2×C4⋊Q8C2×C418yescomplemented
114C2×C4⋊Q8C2314yes—
115C2×#124—21no—
116C2×#124—21no—
117C2×#124—24no—
118C2×#124C2×C418yes—
119C2×#124C2314yes—
120#519—24no—
121#519—24no—
122#519—24no—
123#519—42no—
124#519—42no—
125#519—42no—
126#518—24no—
127#518—42no—
128C2×C4×Q8—22no—
129C2×C4×Q8—22no—
130C2×C4×Q8—22no—
131C2×C4×Q8C2314yes—
132C2×C4×Q8C2314yes—
133C2×C4×Q8C2314yes—
134C2×C4×D4—22no—
135C2×C4×D4C2314yes—
136C2×#117—22no—
137C2×#117—22no—
138C2×#117—22no—
139C2×#117C2311yescharacteristic
140C2×#117C2311yescharacteristic
141C2×#117C2311yescharacteristic
142C2×#117C2311yescharacteristic
143C2×#117C2311yescharacteristic
144C2×#117C2311yescharacteristic
145C22×C4⋊C4—22no—
146C22×C22⋊C4—22no—
147C22×C22⋊C4—22no—
148C22×C42C2311yescharacteristic
149#497C2×C4132yescomplemented
150#497C2×C4132yescomplemented
151#441—216no—
152#440—216no—
153C4×Q16—216no—
154C4×SD16—216no—
155#427—24no—
156#427D418yes—
157C2×#107C2312yes—
158C2×#107C2312yes—
159C2×#106C2312yes—
160C2×#106C2312yes—
161#423—24no—
162C2×C4⋊C8C2314yes—
163#419—24no—
164#419—24no—
165#419—24no—
166#419—42no—
167#419—42no—
168#419—42no—
169#418—24no—
170#418—42no—
171C2×#103C2314yes—
172C2×#103C2314yes—
173C2×#103C2314yes—
174C2×#103C2314yes—
175C2×#103C2314yes—
176C2×#103C2314yes—
177C2×#102C2314yes—
178C2×#102C2314yes—
179#405—24no—
180#405—24no—
181C2×#97C2×C418yescomplemented
182C2×#97C2312yes—
183C2×#97C2312yes—
184C2×#97C2312yes—
185C2×#97C2312yes—
186C2×C4×C8C2312yes—
187C2×C4×C8C2312yes—
188#391—22no—
189#391—22no—
190#391—22no—
191#391C2314yes—
192#391C2314yes—
193#391C2314yes—
194#390—22no—
195#390C2314yes—
196#389—28no—
197#388—28no—
198#388—28no—
199#387—28no—
200#387—28no—
201#385—28no—
202#384D412yes—
203#382D412yes—
204#381—22no—
205#381—22no—
206#381—22no—
207#380D414yes—
208#379—22no—
209C4×C4⋊C4—28no—
210C4×C4⋊C4C2314yes—
211C4×C4⋊C4C2314yes—
212C4×C4⋊C4C2314yes—
213C4×C22⋊C4—22no—
214C4×C22⋊C4—22no—
215C4×C22⋊C4—28no—
216#376—21no—
217#376—21no—
218C43—21no—
219C43—21no—
220#340C2314yes—
221#340C2314yes—
222#336C2314yes—
223#336C2314yes—
224C25—21no—
225C22×Q8—21no—
226C22×Q8—21no—
227C22×Q8—21no—
228C22×Q8—22no—
229C22×Q8—22no—
230C22×Q8—22no—
231C22×Q8—22no—
232C22×Q8—22no—
233C22×Q8—22no—
234C22×Q8—22no—
235C22×Q8—22no—
236C22×Q8—22no—
237C22×Q8—22no—
238C22×Q8—22no—
239C22×Q8—22no—
240C22×Q8—22no—
241C22×Q8—22no—
242C22×Q8—22no—
243C22×Q8C22×C414yes—
244C22×Q8C22×C414yes—
245C22×Q8C22×C414yes—
246C22×Q8C2412yes—
247C22×Q8C2412yes—
248C22×Q8C2412yes—
249C22×D4—22no—
250C22×D4—21no—
251C22×D4—22no—
252C22×D4—22no—
253C22×D4—22no—
254C22×D4—22no—
255C22×D4C22×C414yes—
256C22×D4C2412yes—
257C23×C4—21no—
258C23×C4—21no—
259C23×C4—21no—
260C23×C4—24no—
261C23×C4C2×D412yes—
262C23×C4C2411yescharacteristic
263C2×Q16—232no—
264C2×Q16—24no—
265C2×Q16—24no—
266C2×Q16—24no—
267C2×Q16—24no—
268C2×Q16—416no—
269C2×Q16—44no—
270C2×Q16—44no—
271C2×Q16C22×C418yes—
272C2×Q16C22×C418yes—
273C2×Q16C22×C4116yes—
274C2×SD16—232no—
275C2×SD16—24no—
276C2×SD16—24no—
277C2×SD16—24no—
278C2×SD16—24no—
279C2×SD16—416no—
280C2×SD16—44no—
281C2×SD16—44no—
282C2×SD16C22×C418yes—
283C2×SD16C22×C418yes—
284C2×SD16C22×C4116yes—
285C2×M16—22no—
286C2×M16—22no—
287C2×M16—22no—
288C2×M16—22no—
289C2×M16—22no—
290C2×M16—22no—
291C22×C8—22no—
292C22×C8—22no—
293C22×C8C22×C414yes—
294C22×C8C22×C414yes—
295C22×C8C2×D414yes—
296C22×C8#4114yes—
297C22×C8C2412yes—
298C22×C8C2412yes—
299C22×C8C2412yes—
300C22×C8C2411yescharacteristic
301C22×C8C2411yescharacteristic
302C4⋊Q8—28no—
303C4⋊Q8—28no—
304C4⋊Q8—28no—
305C4⋊Q8C22×C4116yes—
306#124—28no—
307#124—28no—
308#124—28no—
309#124C22×C4116yes—
310C4×Q8—28no—
311C4×Q8—28no—
312C4×Q8—28no—
313C4×Q8—44no—
314C4×Q8—44no—
315C4×Q8—44no—
316C4×Q8—48no—
317C4×Q8—48no—
318C4×Q8—48no—
319C4×D4—28no—
320C4×D4—44no—
321C4×D4—48no—
322#117—22no—
323#117—22no—
324#117—22no—
325#117—22no—
326#117—22no—
327#117—22no—
328#117—22no—
329#117—22no—
330#117—22no—
331#117—22no—
332#117—48no—
333#117—48no—
334#117—48no—
335#117C2×D414yes—
336#117C2×D414yes—
337C2×C4⋊C4—216no—
338C2×C4⋊C4—24no—
339C2×C4⋊C4—24no—
340C2×C4⋊C4—24no—
341C2×C4⋊C4—24no—
342C2×C4⋊C4—24no—
343C2×C4⋊C4C2×D412yes—
344C2×C4⋊C4C2×D412yes—
345C2×C4⋊C4C2×D418yes—
346C2×C4⋊C4C2412yes—
347C2×C4⋊C4C2412yes—
348C2×C4⋊C4C2412yes—
349C2×C4⋊C4C2412yes—
350C2×C22⋊C4—21no—
351C2×C22⋊C4—21no—
352C2×C22⋊C4—216no—
353C2×C22⋊C4—24no—
354C2×C22⋊C4—24no—
355C2×C22⋊C4—21no—
356C2×C22⋊C4—21no—
357C2×C22⋊C4—24no—
358C2×C22⋊C4—28no—
359C2×C22⋊C4—48no—
360C2×C22⋊C4C2×D412yes—
361C2×C22⋊C4C2×D412yes—
362C2×C42—21no—
363C2×C42—21no—
364C2×C42—24no—
365C2×C42—24no—
366C2×C42—24no—
367C2×C42—21no—
368C2×C42—21no—
369C2×C42—24no—
370C2×C42—24no—
371C2×C42C22×C414yes—
372C2×C42C2×D411yescharacteristic
373C2×C42C2×D411yescharacteristic
374C2×C42#4114yes—
375C2×C42C2412yes—
376C2×C42C2412yes—
377C2×C42C2412yes—
378#107—24no—
379#107C2×D418yes—
380#106—24no—
381#106C2×D418yes—
382C4⋊C8—28no—
383#103—28no—
384#103—28no—
385#103—28no—
386#103—44no—
387#103—44no—
388#103—44no—
389#102—28no—
390#102—44no—
391#97—24no—
392#97—24no—
393#97C22×C418yes—
394#97C22×C418yes—
395C4×C8—24no—
396C4×C8—24no—
397#95—24no—
398#95—24no—
399#95—28no—
400#95—28no—
401#95—28no—
402#95C2×D418yes—
403C24—22no—
404C24—24no—
405C24—42no—
406C24—42no—
407C24—84no—
408C24C22×D411yescharacteristic
409C2×Q8—24no—
410C2×Q8—24no—
411C2×Q8—24no—
412C2×Q8—22no—
413C2×Q8—22no—
414C2×Q8—22no—
415C2×Q8—28no—
416C2×Q8—28no—
417C2×Q8—28no—
418C2×Q8—28no—
419C2×Q8—28no—
420C2×Q8—28no—
421C2×Q8—22no—
422C2×Q8—22no—
423C2×Q8—22no—
424C2×Q8—22no—
425C2×Q8—22no—
426C2×Q8—22no—
427C2×Q8—22no—
428C2×Q8—22no—
429C2×Q8—22no—
430C2×Q8—22no—
431C2×Q8—22no—
432C2×Q8—22no—
433C2×Q8—42no—
434C2×Q8—42no—
435C2×Q8—42no—
436C2×Q8—44no—
437C2×Q8—44no—
438C2×Q8—44no—
439C2×Q8—42no—
440C2×Q8—42no—
441C2×Q8—42no—
442C2×Q8—44no—
443C2×Q8—44no—
444C2×Q8—44no—
445C2×Q8C23×C414yes—
446C2×Q8C23×C414yes—
447C2×Q8C23×C414yes—
448C2×D4—28no—
449C2×D4—24no—
450C2×D4—28no—
451C2×D4—22no—
452C2×D4—22no—
453C2×D4—22no—
454C2×D4—22no—
455C2×D4—22no—
456C2×D4—42no—
457C2×D4—42no—
458C2×D4—44no—
459C2×D4—44no—
460C2×D4C23×C414yes—
461C22×C4—24no—
462C22×C4—24no—
463C22×C4—24no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—44no—
472C22×C4—48no—
473C22×C4—42no—
474C22×C4—42no—
475C22×C4—42no—
476C22×C4—42no—
477C22×C4—42no—
478C22×C4—42no—
479C22×C4—88no—
480C22×C4C4×D418yes—
481C22×C4C23×C412yes—
482C22×C4C22×D414yes—
483C22×C4C22×D414yes—
484C22×C4C22×D412yes—
485C22×C4C22×D412yes—
486C22×C4C22×D411yescharacteristic
487C22×C4C22×D412yes—
488C22×C4C2×#4112yes—
489C22×C4C2×#4112yes—
490C22×C4H(5,𝔽2)12yes—
491C22×C4H(5,𝔽2)12yes—
492C22×C4C2511yescharacteristic
493Q16—216no—
494Q16—216no—
495Q16—416no—
496SD16—216no—
497SD16—216no—
498SD16—416no—
499M16—44no—
500M16—44no—
501C2×C8—28no—
502C2×C8—22no—
503C2×C8—22no—
504C2×C8—22no—
505C2×C8—22no—
506C2×C8—22no—
507C2×C8—22no—
508C2×C8—22no—
509C2×C8—22no—
510C2×C8C4×D4116yes—
511C2×C8C23×C414yes—
512C2×C8C23×C412yes—
513C2×C8C23×C412yes—
514C2×C8C22×D412yes—
515C2×C8C22×D412yes—
516C2×C8C2×#4112yes—
517C2×C8C2×#4112yes—
518C4⋊C4—216no—
519C4⋊C4—216no—
520C4⋊C4—24no—
521C4⋊C4—24no—
522C4⋊C4—22no—
523C4⋊C4—22no—
524C4⋊C4—22no—
525C4⋊C4—22no—
526C4⋊C4—22no—
527C4⋊C4—22no—
528C4⋊C4—48no—
529C4⋊C4—48no—
530C4⋊C4—48no—
531C4⋊C4—48no—
532C4⋊C4C22×D414yes—
533C4⋊C4C22×D414yes—
534C22⋊C4—24no—
535C22⋊C4—24no—
536C22⋊C4—24no—
537C22⋊C4—24no—
538C22⋊C4—216no—
539C22⋊C4—216no—
540C22⋊C4—48no—
541C22⋊C4—42no—
542C22⋊C4—42no—
543C22⋊C4—88no—
544C42—24no—
545C42—24no—
546C42—22no—
547C42—22no—
548C42—22no—
549C42—22no—
550C42—48no—
551C42—48no—
552C42—48no—
553C42—48no—
554C42—42no—
555C42—42no—
556C42—84no—
557C42C4×D414yes—
558C42C4×D414yes—
559C42C23×C414yes—
560C42C2×#4114yes—
561C23—24no—
562C23—24no—
563C23—48no—
564C23—42no—
565C23—42no—
566C23—48no—
567C23—88no—
568C23—164no—
569C23—164no—
570C23C2×C4×D412yes—
571C23D4212yes—
572C23#55012yes—
573C23C23×D411yescharacteristic
574Q8—24no—
575Q8—24no—
576Q8—24no—
577Q8—24no—
578Q8—24no—
579Q8—24no—
580Q8—48no—
581Q8—48no—
582Q8—48no—
583Q8—42no—
584Q8—42no—
585Q8—42no—
586Q8—42no—
587Q8—42no—
588Q8—42no—
589D4—24no—
590D4—24no—
591D4—48no—
592D4—42no—
593D4—42no—
594C2×C4—24no—
595C2×C4—24no—
596C2×C4—28no—
597C2×C4—28no—
598C2×C4—24no—
599C2×C4—24no—
600C2×C4—24no—
601C2×C4—24no—
602C2×C4—24no—
603C2×C4—24no—
604C2×C4—24no—
605C2×C4—48no—
606C2×C4—44no—
607C2×C4—44no—
608C2×C4—48no—
609C2×C4—48no—
610C2×C4—48no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—44no—
618C2×C4—44no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—168no—
622C2×C4C2×C4×D414yes—
623C2×C4C2×C4×D414yes—
624C2×C4C2×C4×D412yes—
625C2×C4C2×C4×D414yes—
626C2×C4#51814yes—
627C2×C4D4212yes—
628C2×C4#55012yes—
629C2×C4#55014yes—
630C2×C4#55914yes—
631C2×C4C24×C411yescharacteristic
632C2×C4C23×D411yescharacteristic
633C2×C4C22×#4111yescharacteristic
634C2×C4C22×#4111yescharacteristic
635C2×C4C2×H(5,𝔽2)11yescharacteristic
636C2×C4C2×H(5,𝔽2)11yescharacteristic
637C8—24no—
638C8—42no—
639C8—42no—
640C8C2×C4×D414yes—
641C8C2×C4×D414yes—
642C22—24no—
643C22—24no—
644C22—24no—
645C22—48no—
646C22—44no—
647C22—84no—
648C22—84no—
649C22—168no—
650C22#236614yes—
651C22C22×C4×D411yescharacteristic
652C22C2×D4211yescharacteristic
653C22C2×#55011yescharacteristic
654C22#358614yes—
655C4—24no—
656C4—24no—
657C4—44no—
658C4—44no—
659C4—44no—
660C4—44no—
661C4—44no—
662C4—44no—
663C4—44no—
664C4—82no—
665C4—82no—
666C4#236614yes—
667C4C22×C4×D411yescharacteristic
668C4C2×#51811yescharacteristic
669C4C2×#55011yescharacteristic
670C4C2×#55911yescharacteristic
671C2—24no—
672C2—84no—
673C2C2×#236611yescharacteristic
674C2#6016912yes—
675C2#6108312yes—
676C2#6141812yes—
subgroupG/H
center, socleC23C23×D4
commutatorC2×C4C24×C4
Frattini, cosocleC22×C4C25
prime radicalC23×D4C23
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ #2285 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2950C2semidirect
5. #34209C2semidirect
6. #23194C2semidirect
7. #23192C2semidirect
9. #21259C2semidirect
10. #21232C2semidirect
13. #20935C2semidirect
14. #20935C2semidirect
16. #20451C2semidirect
17. #20438C2semidirect
28. C2×#497C4semidirect
33. C2×#441C22semidirect
35. C2×C4×Q16C22semidirect
52. #2365C22semidirect
67. #1981C22semidirect
69. #1971C22semidirect
70. #1969C22semidirect
73. #1878C22semidirect
74. #1878C22semidirect
76. #1776C22semidirect
77. #1776C22semidirect
78. #1771C22semidirect
79. #1771C22semidirect
92. C22×Q16C2×C4semidirect
93. C22×Q16C2×C4semidirect
113. C2×C4⋊Q8C2×C4semidirect
149. #497C2×C4semidirect
150. #497C2×C4semidirect
181. C2×#97C2×C4semidirect

Maximal subgroups

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

subgroupindex
C22×#4972
C2×#29502
C2×#29492
#342112
#342092
#231942
#231922
#212612
#212592
#212322
#212312
#209362
#209352
#209332
#204512
#204382