Finite Group Databasebeta

← #6355006

Subgroups — #6355006

Order 512.

1602 conjugacy classes of subgroups.

The subgroup lattice has 698 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3249C211yesmaximal, characteristic, complemented
2#60864C211yesmaximal, characteristic
3#60804C211yesmaximal, characteristic
4#60250C211yesmaximal, characteristic, complemented
5#19575C218yesmaximal, complemented
6#19440C212yesmaximal, complemented
7#19439C212yesmaximal, complemented
8#19404C212yesmaximal, complemented
9#19403C212yesmaximal, complemented
10#19242C214yesmaximal, complemented
11#19240C214yesmaximal, complemented
12#17807C211yesmaximal, characteristic, complemented
13#17718C211yesmaximal, characteristic, complemented
14C2×#1540C211yesmaximal, characteristic
15C2×#533C2211yescharacteristic
16C2×#533C2211yescharacteristic
17C2×#532C2211yescharacteristic
18#3249—22no—
19#3249—22no—
20#3249C2214yes—
21#3249C2214yescomplemented
22#3235—22no—
23#3234—22no—
24#3228C2214yes—
25#3227C2212yes—
26#3226C2212yes—
27#3216C2214yes—
28#3215—22no—
29#3210—22no—
30#3209C2212yes—
31#3208C2212yes—
32#3165C2212yes—
33#3152—21no—
34#3141—21no—
35#3138C2212yes—
36#3094—22no—
37#3077—22no—
38C2×#477—21no—
39C2×#477C2212yescomplemented
40C2×#467—22no—
41#3037C2212yes—
42#3037C2212yes—
43C2×#462—21no—
44C2×#462C2212yescomplemented
45#3007—22no—
46#2986C2214yescomplemented
47#2985—22no—
48#2984—22no—
49#2983C2212yescomplemented
50#2982C2212yescomplemented
51#2962C2214yes—
52#2960—22no—
53#2924—22no—
54#2920C2211yescharacteristic
55#2920C2211yescharacteristic
56C2×#431C2211yescharacteristic
57C22×C4⋊C8C2211yescharacteristic
58#1702C2218yescomplemented
59#1698C2218yescomplemented
60#1694C2218yescomplemented
61#1692C2218yescomplemented
62#1682C2218yescomplemented
63#1678C2218yescomplemented
64#1674C2218yescomplemented
65#1672C2218yescomplemented
66#1573C2214yescomplemented
67#1571C2212yescomplemented
68#1570C2212yescomplemented
69#1565C2214yes—
70#1554C2214yescomplemented
71#1548C2211yescharacteristic, complemented
72#1548C2211yescharacteristic, complemented
73#1546C2211yescharacteristic, complemented
74#1546C2211yescharacteristic, complemented
75#1542C2214yes—
76#1540C2214yes—
77#1540C2212yes—
78#1540C2212yes—
79C2×#332C2212yes—
80C2×#331C2212yes—
81C22×#41—21no—
82C2×#137—22no—
83C2×#137—22no—
84C22×Q16—22no—
85C23×C8—21no—
86#533—21no—
87#533—21no—
88#533—21no—
89#533—21no—
90#533—21no—
91#533—21no—
92#533—21no—
93#533—21no—
94#533—21no—
95#533—21no—
96#533D412yes—
97#533D412yes—
98#533C2312yes—
99#533C2312yes—
100#533C2312yes—
101#533C2312yes—
102#532—21no—
103#532—21no—
104#532—21no—
105#532—21no—
106#532—21no—
107#532—21no—
108#532C2312yes—
109#532C2312yes—
110C2×C4⋊Q8D411yescharacteristic
111C2×C4⋊Q8D411yescharacteristic
112C2×C4⋊Q8C2311yescharacteristic
113C2×C4⋊Q8C2311yescharacteristic
114C2×C4⋊Q8C2311yescharacteristic
115C2×#127C2311yescharacteristic
116C2×#125—21no—
117C2×#125—21no—
118C2×#124—21no—
119C2×C22⋊Q8—21no—
120C2×C22⋊Q8—21no—
121C2×C22⋊Q8—21no—
122C2×C22⋊Q8D412yes—
123C2×#121—21no—
124C2×#121—21no—
125C2×C4×Q8—21no—
126C2×C4×Q8—21no—
127C2×C4×Q8—21no—
128C2×C4×Q8C2312yes—
129C2×C4×D4—21no—
130C2×C4×D4—21no—
131C2×#117—21no—
132C2×#117—21no—
133C22×C42C2311yescharacteristic
134#501—24no—
135#500C2314yes—
136#498C2314yes—
137#497—24no—
138#494—22no—
139#492—22no—
140#490—24no—
141#489—24no—
142#487—22no—
143#486—22no—
144#486C2314yes—
145#486C2314yes—
146#482—28no—
147#481—24no—
148#480—24no—
149#478—28no—
150#477—24no—
151#477—24no—
152#477—24no—
153#477C2318yes—
154#476—24no—
155#474—24no—
156#467—28no—
157#467—28no—
158#464—24no—
159#464—24no—
160#462—24no—
161#462—24no—
162#462—24no—
163#462C2314yes—
164#462C2314yes—
165#462C2318yes—
166#460C2314yes—
167#460C2314yes—
168#452—24no—
169#452—24no—
170#451—24no—
171#450—24no—
172#446—24no—
173#446—24no—
174#446—28no—
175#446C2318yes—
176C8×Q8—24no—
177C8×Q8—24no—
178C8×Q8C2318yes—
179#433—28no—
180#432—21no—
181#432—24no—
182#432D412yes—
183#432D412yes—
184#432C2312yes—
185#431—21no—
186#431—21no—
187#431—21no—
188#431—24no—
189#431C2311yescharacteristic
190#431C2311yescharacteristic
191#431C2312yes—
192#431C2312yes—
193#431C2312yes—
194#431C2312yes—
195#428—24no—
196C2×#107—22no—
197#424—24no—
198#423D412yes—
199#423C2311yescharacteristic
200#423C2311yescharacteristic
201C2×C4⋊C8—21no—
202C2×C4⋊C8—21no—
203C2×C4⋊C8—21no—
204C2×C4⋊C8C2312yes—
205C2×C4⋊C8C2312yes—
206C2×C4⋊C8C2312yes—
207C2×C4⋊C8C2312yes—
208#418—24no—
209#417—22no—
210#417—22no—
211C2×#103—22no—
212C2×#103—22no—
213#407—21no—
214#407D412yes—
215C2×C22⋊C8—21no—
216#405—24no—
217C4×M16—21no—
218C4×M16C2312yes—
219C2×C4×C8—21no—
220#335C2318yes—
221#334C2318yes—
222#333C2314yes—
223#333C2314yes—
224#332C2314yes—
225#332C2314yes—
226#332C2314yes—
227#332C2314yes—
228#331C2314yes—
229#331C2314yes—
230C2×#41—22no—
231C2×#41—22no—
232C2×#41—41no—
233C2×#41—41no—
234C2×#41—82no—
235C2×#41—82no—
236C22×Q8—21no—
237C22×Q8—21no—
238C22×Q8—21no—
239C22×Q8—21no—
240C22×Q8C2×D412yes—
241C22×D4—21no—
242C22×D4—21no—
243C22×D4—21no—
244C22×D4—41no—
245C23×C4—41no—
246C23×C4C2×D411yescharacteristic
247C23×C4C2×D411yescharacteristic
248C23×C4#4111yescharacteristic
249#137—84no—
250#137—84no—
251C2×Q16—28no—
252C2×Q16—24no—
253C2×Q16—24no—
254C2×Q16—84no—
255C2×SD16—24no—
256C2×SD16—24no—
257C2×M16—21no—
258C2×M16—21no—
259C2×M16C2×D414yes—
260C22×C8—22no—
261C22×C8—22no—
262C22×C8—22no—
263C22×C8—22no—
264C22×C8—82no—
265C4⋊Q8—21no—
266C4⋊Q8—21no—
267C4⋊Q8—21no—
268C4⋊Q8—21no—
269C4⋊Q8—21no—
270C4⋊Q8—21no—
271C4⋊Q8—21no—
272C4⋊Q8—21no—
273C4⋊Q8—21no—
274C4⋊Q8—21no—
275C4⋊Q8—21no—
276C4⋊Q8—21no—
277C4⋊Q8—21no—
278C4⋊Q8C2×D412yes—
279C4⋊Q8C2×D412yes—
280C4⋊Q8C2×D412yes—
281C4⋊Q8C2×D412yes—
282C4⋊Q8C2412yes—
283C4⋊Q8C2412yes—
284C4⋊Q8C2412yes—
285#127—21no—
286#127—21no—
287#127—21no—
288#127C2412yes—
289#125—22no—
290#125—22no—
291#125—22no—
292#125—22no—
293#125—22no—
294#125—22no—
295#125—41no—
296#125—41no—
297#124—22no—
298#124—22no—
299#124—41no—
300#124—41no—
301C22⋊Q8—22no—
302C22⋊Q8—22no—
303C22⋊Q8—22no—
304C22⋊Q8—22no—
305C22⋊Q8—22no—
306C22⋊Q8—22no—
307C22⋊Q8—22no—
308C22⋊Q8—22no—
309C22⋊Q8—22no—
310C22⋊Q8—22no—
311C22⋊Q8—41no—
312C22⋊Q8—41no—
313C22⋊Q8—41no—
314C22⋊Q8—41no—
315C22⋊Q8C2×D414yes—
316C22⋊Q8C2×D414yes—
317#121—22no—
318#121—22no—
319#121—22no—
320#121—22no—
321#121—41no—
322#121—41no—
323#121—41no—
324#121—41no—
325C4×Q8—22no—
326C4×Q8—22no—
327C4×Q8—22no—
328C4×Q8—22no—
329C4×Q8—22no—
330C4×Q8—22no—
331C4×Q8—22no—
332C4×Q8—22no—
333C4×Q8—22no—
334C4×Q8—22no—
335C4×Q8—41no—
336C4×Q8—41no—
337C4×Q8—41no—
338C4×Q8—41no—
339C4×Q8C2×D414yes—
340C4×Q8C2414yes—
341C4×D4—22no—
342C4×D4—22no—
343C4×D4—22no—
344C4×D4—22no—
345C4×D4—41no—
346C4×D4—41no—
347C4×D4—41no—
348C4×D4—41no—
349#117—22no—
350#117—22no—
351#117—22no—
352#117—22no—
353#117—22no—
354#117—22no—
355#117—41no—
356#117—41no—
357C2×C4⋊C4—21no—
358C2×C4⋊C4—21no—
359C2×C4⋊C4—21no—
360C2×C4⋊C4—22no—
361C2×C4⋊C4—22no—
362C2×C4⋊C4—21no—
363C2×C4⋊C4—21no—
364C2×C4⋊C4—21no—
365C2×C4⋊C4—21no—
366C2×C4⋊C4—21no—
367C2×C4⋊C4—21no—
368C2×C4⋊C4—41no—
369C2×C4⋊C4—41no—
370C2×C4⋊C4C2×D412yes—
371C2×C22⋊C4—22no—
372C2×C22⋊C4—41no—
373C2×C22⋊C4—41no—
374C2×C22⋊C4—41no—
375C2×C42—22no—
376C2×C42—41no—
377C2×C42C2×D411yescharacteristic
378C2×C42C2×D411yescharacteristic
379C2×C42C2×D411yescharacteristic
380C2×C42C2×D411yescharacteristic
381C2×C42C2×D411yescharacteristic
382C2×C42C2×D411yescharacteristic
383C2×C42#4111yescharacteristic
384C2×C42#4111yescharacteristic
385C2×C42#4111yescharacteristic
386C2×C42C2411yescharacteristic
387C2×C42C2411yescharacteristic
388C2×C42C2411yescharacteristic
389#107—28no—
390#107—28no—
391#106—28no—
392C4⋊C8—28no—
393C4⋊C8—28no—
394C4⋊C8—22no—
395C4⋊C8—22no—
396C4⋊C8—22no—
397C4⋊C8—22no—
398C4⋊C8—21no—
399C4⋊C8—21no—
400C4⋊C8C2×D414yes—
401C4⋊C8C2×D414yes—
402C4⋊C8C2412yes—
403C4⋊C8C2412yes—
404C4⋊C8C2414yes—
405C4⋊C8C2412yes—
406C4⋊C8C2412yes—
407#103—28no—
408#103—28no—
409#103—24no—
410#103—24no—
411#102—28no—
412#102—28no—
413#102—24no—
414#102—24no—
415C22⋊C8—21no—
416C22⋊C8—21no—
417C22⋊C8—22no—
418C22⋊C8—22no—
419C22⋊C8—22no—
420C22⋊C8C2×D414yes—
421C22⋊C8C2×D414yes—
422#97—28no—
423#97—22no—
424#97C2×D414yes—
425C4×C8—28no—
426C4×C8—21no—
427C4×C8—22no—
428C4×C8—21no—
429C4×C8—22no—
430C4×C8C2414yes—
431C24—41no—
432C24#12111yescharacteristic
433#41—44no—
434#41—44no—
435#41—82no—
436#41—82no—
437#41—84no—
438#41—84no—
439C2×Q8—22no—
440C2×Q8—22no—
441C2×Q8—22no—
442C2×Q8—24no—
443C2×Q8—24no—
444C2×Q8—22no—
445C2×Q8—22no—
446C2×Q8—22no—
447C2×Q8—41no—
448C2×Q8—41no—
449C2×Q8—41no—
450C2×Q8—42no—
451C2×Q8—42no—
452C2×Q8—81no—
453C2×Q8—81no—
454C2×Q8—82no—
455C2×Q8—82no—
456C2×Q8C22×D414yes—
457C2×D4—22no—
458C2×D4—22no—
459C2×D4—22no—
460C2×D4—42no—
461C2×D4—42no—
462C2×D4—41no—
463C2×D4—41no—
464C2×D4—41no—
465C2×D4—44no—
466C2×D4—81no—
467C2×D4—81no—
468C2×D4—82no—
469C2×D4—84no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—24no—
473C22×C4—22no—
474C22×C4—22no—
475C22×C4—41no—
476C22×C4—41no—
477C22×C4—41no—
478C22×C4—42no—
479C22×C4—42no—
480C22×C4—84no—
481C22×C4—81no—
482C22×C4#12111yescharacteristic
483C22×C4#12111yescharacteristic
484C22×C4#12111yescharacteristic
485C22×C4#12111yescharacteristic
486C22×C4#12111yescharacteristic
487C22×C4#12111yescharacteristic
488C22×C4#12111yescharacteristic
489C22×C4#12111yescharacteristic
490C22×C4#12111yescharacteristic
491C22×C4C22×D411yescharacteristic
492C22×C4C22×D411yescharacteristic
493C22×C4C22×D411yescharacteristic
494C22×C4C22×D411yescharacteristic
495C22×C4C22×D411yescharacteristic
496C22×C4C22×D411yescharacteristic
497C22×C4C2×#4111yescharacteristic
498C22×C4C2×#4111yescharacteristic
499C22×C4C2×#4111yescharacteristic
500Q16—88no—
501Q16—84no—
502Q16—84no—
503SD16—84no—
504SD16—84no—
505M16—28no—
506M16—42no—
507M16—42no—
508C2×C8—22no—
509C2×C8—22no—
510C2×C8—22no—
511C2×C8—22no—
512C2×C8—24no—
513C2×C8—44no—
514C2×C8—44no—
515C2×C8—82no—
516C2×C8—82no—
517C2×C8C22×D418yes—
518C4⋊C4—22no—
519C4⋊C4—22no—
520C4⋊C4—22no—
521C4⋊C4—22no—
522C4⋊C4—22no—
523C4⋊C4—22no—
524C4⋊C4—24no—
525C4⋊C4—24no—
526C4⋊C4—24no—
527C4⋊C4—22no—
528C4⋊C4—24no—
529C4⋊C4—22no—
530C4⋊C4—22no—
531C4⋊C4—22no—
532C4⋊C4—22no—
533C4⋊C4—22no—
534C4⋊C4—22no—
535C4⋊C4—41no—
536C4⋊C4—41no—
537C4⋊C4—41no—
538C4⋊C4—41no—
539C4⋊C4—41no—
540C4⋊C4—41no—
541C4⋊C4—42no—
542C4⋊C4—42no—
543C4⋊C4—42no—
544C4⋊C4—42no—
545C4⋊C4C22×D414yes—
546C22⋊C4—24no—
547C22⋊C4—24no—
548C22⋊C4—42no—
549C22⋊C4—42no—
550C22⋊C4—42no—
551C22⋊C4—42no—
552C22⋊C4—42no—
553C22⋊C4—42no—
554C42—24no—
555C42—24no—
556C42—42no—
557C42—42no—
558C42#12111yescharacteristic
559C42#12111yescharacteristic
560C42#12111yescharacteristic
561C42#12111yescharacteristic
562C42#12111yescharacteristic
563C42#12111yescharacteristic
564C42C22×D411yescharacteristic
565C42C22×D411yescharacteristic
566C42C22×D411yescharacteristic
567C42C22×D411yescharacteristic
568C42C22×D411yescharacteristic
569C42C22×D411yescharacteristic
570C42C2×#4111yescharacteristic
571C42C2×#4111yescharacteristic
572C42C2×#4111yescharacteristic
573C42C2511yescharacteristic
574C23—22no—
575C23—22no—
576C23—41no—
577C23—42no—
578C23—82no—
579C23—82no—
580C23—82no—
581C23C2×#12111yescharacteristic
582C23C2×#12111yescharacteristic
583C23C2×#12111yescharacteristic
584Q8—24no—
585Q8—24no—
586Q8—44no—
587Q8—44no—
588Q8—42no—
589Q8—42no—
590Q8—42no—
591Q8—42no—
592Q8—82no—
593Q8—82no—
594Q8—82no—
595Q8—82no—
596Q8—82no—
597D4—42no—
598D4—42no—
599D4—44no—
600D4—82no—
601D4—82no—
602D4—82no—
603D4—84no—
604D4—84no—
605D4—82no—
606C2×C4—22no—
607C2×C4—22no—
608C2×C4—22no—
609C2×C4—22no—
610C2×C4—22no—
611C2×C4—22no—
612C2×C4—21no—
613C2×C4—21no—
614C2×C4—21no—
615C2×C4—21no—
616C2×C4—22no—
617C2×C4—22no—
618C2×C4—24no—
619C2×C4—24no—
620C2×C4—41no—
621C2×C4—41no—
622C2×C4—41no—
623C2×C4—41no—
624C2×C4—41no—
625C2×C4—41no—
626C2×C4—44no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—84no—
631C2×C4—82no—
632C2×C4—82no—
633C2×C4—82no—
634C2×C4—82no—
635C2×C4—84no—
636C2×C4C2×#12111yescharacteristic
637C2×C4C2×#12111yescharacteristic
638C2×C4C2×#12111yescharacteristic
639C2×C4C2×#12111yescharacteristic
640C2×C4C2×#12111yescharacteristic
641C2×C4C2×#12111yescharacteristic
642C2×C4C2×#12111yescharacteristic
643C2×C4C2×#12111yescharacteristic
644C2×C4C2×#12111yescharacteristic
645C2×C4#57612yes—
646C2×C4#57712yes—
647C2×C4#57814yes—
648C2×C4C23×D411yescharacteristic
649C2×C4C23×D411yescharacteristic
650C2×C4C22×#4111yescharacteristic
651C8—28no—
652C8—44no—
653C8—42no—
654C8—42no—
655C8—42no—
656C8—42no—
657C22—21no—
658C22—21no—
659C22—21no—
660C22—21no—
661C22—21no—
662C22—21no—
663C22—21no—
664C22—44no—
665C22—82no—
666C22—82no—
667C22—82no—
668C22—164no—
669C22#304412yes—
670C22#306212yes—
671C22C22×#12111yescharacteristic
672C4—21no—
673C4—21no—
674C4—21no—
675C4—21no—
676C4—21no—
677C4—21no—
678C4—21no—
679C4—21no—
680C4—44no—
681C4—44no—
682C4—82no—
683C4—82no—
684C4—82no—
685C4#304412yes—
686C4#306212yes—
687C4C2×#57611yescharacteristic
688C4C2×#57711yescharacteristic
689C4C2×#57812yes—
690C2—22no—
691C2—22no—
692C2—41no—
693C2—162no—
694C2C2×#304411yescharacteristic
695C2C2×#306211yescharacteristic
696C2#6046911yescharacteristic
subgroupG/H
center, socleC22C22×#121
commutator, Frattini, cosocleC42C25
prime radicalC2×#532C22
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#3249C2semidirect
4. #60250C2semidirect
5. #19575C2semidirect
6. #19440C2semidirect
7. #19439C2semidirect
8. #19404C2semidirect
9. #19403C2semidirect
10. #19242C2semidirect
11. #19240C2semidirect
12. #17807C2semidirect
13. #17718C2semidirect
21. #3249C22semidirect
39. C2×#477C22semidirect
44. C2×#462C22semidirect
46. #2986C22semidirect
49. #2983C22semidirect
50. #2982C22semidirect
58. #1702C22semidirect
59. #1698C22semidirect
60. #1694C22semidirect
61. #1692C22semidirect
62. #1682C22semidirect
63. #1678C22semidirect
64. #1674C22semidirect
65. #1672C22semidirect
66. #1573C22semidirect
67. #1571C22semidirect
68. #1570C22semidirect
70. #1554C22semidirect
71. #1548C22semidirect
72. #1548C22semidirect
73. #1546C22semidirect
74. #1546C22semidirect

Maximal subgroups

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

subgroupindex
C2×#32492
#608642
#608042
#602502
#195752
#194402
#194392
#194042
#194032
#192422
#192402
#178072
#177182
C2×#15402