Finite Group Databasebeta

← #6373336

Subgroups — #6373336

Order 512.

2480 conjugacy classes of subgroups.

The subgroup lattice has 680 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3109C211yesmaximal, characteristic
2#34226C211yesmaximal, characteristic, complemented
3C4×#518C211yesmaximal, characteristic, complemented
4#21614C212yesmaximal, complemented
5C4×#476C212yesmaximal, complemented
6#21584C214yesmaximal, complemented
7#21503C212yesmaximal, complemented
8C4×#459C212yesmaximal, complemented
9#21458C214yesmaximal, complemented
10#21390C211yesmaximal, characteristic, complemented
11#21364C211yesmaximal, characteristic, complemented
12#21126C214yesmaximal, complemented
13#20732C211yesmaximal, characteristic, complemented
14#20503C214yesmaximal, complemented
15#20470C211yesmaximal, characteristic, complemented
16C2×#532C2211yescharacteristic
17C2×#518C2212yes—
18C22×C4×D4C2211yescharacteristic
19#3109C4116yescomplemented
20C2×#481C2212yes—
21C2×#476C2212yes—
22C2×#459C2212yes—
23C2×#447C2212yes—
24C2×#442C2214yes—
25C2×#442C2214yes—
26C2×C4×D8C2214yes—
27C2×#432C2211yescharacteristic
28C2×#431C2211yescharacteristic
29#2367—22no—
30#2363—22no—
31#2357—21no—
32#2349—21no—
33#2341—21no—
34#2339—21no—
35#2324C2214yes—
36#2319C2212yescomplemented
37C4×C4⋊Q8C2212yescomplemented
38C4×#127C2212yescomplemented
39#2285C2211yescharacteristic
40#2275—22no—
41#2275C2212yes—
42#2275C2212yes—
43#2275C2212yescomplemented
44#2274C2212yes—
45#2273—21no—
46#2273—21no—
47C42×D4—21no—
48C42×D4—21no—
49C42×D4C2212yes—
50C42×D4C2212yes—
51C4×#117C2211yescharacteristic
52C4×#117C2211yescharacteristic
53C2×C4×C22⋊C4—22no—
54#1987C2211yescharacteristic, complemented
55#1987C2211yescharacteristic, complemented
56#1982C2211yescharacteristic, complemented
57#1982C2211yescharacteristic, complemented
58#1948C2212yescomplemented
59#1947C2212yescomplemented
60#1926C2214yescomplemented
61#1925C2214yescomplemented
62#1920C2214yescomplemented
63#1919C2214yescomplemented
64#1879C2214yes—
65#1879C2214yes—
66#1875C2214yes—
67#1864C2214yes—
68#1862C2214yes—
69#1854C2214yes—
70#1823C2212yes—
71#1802C2212yes—
72#1802C2212yes—
73#1786C2212yes—
74#1776C2214yescomplemented
75#1776C2214yescomplemented
76C4×#106C2214yescomplemented
77#1767C2212yescomplemented
78C4×C4⋊C8C2212yescomplemented
79#1762C2214yes—
80#1762C2214yescomplemented
81C4×#102C2214yes—
82C4×#102C2214yes—
83C4×#102C2214yescomplemented
84C4×#102C2214yescomplemented
85#1750C2214yes—
86#1746C2212yescomplemented
87#1726C2212yescomplemented
88C22×#41—21no—
89C23×D4C2311yescharacteristic
90C24×C4—21no—
91C22×D8C2314yes—
92#532—24no—
93#532C2×C418yescomplemented
94C2×C4⋊Q8C2311yescharacteristic
95C2×#127C2311yescharacteristic
96C2×#124—21no—
97C2×#121—21no—
98C2×#121C2312yes—
99#518—28no—
100#518—28no—
101C2×C4×Q8—22no—
102C2×C4×D4—22no—
103C2×C4×D4—22no—
104C2×C4×D4—21no—
105C2×C4×D4—21no—
106C2×C4×D4—21no—
107C2×C4×D4—22no—
108C2×C4×D4—22no—
109C2×C4×D4—24no—
110C2×C4×D4C2×C418yescomplemented
111C2×C4×D4C2314yes—
112C2×C4×D4C2312yes—
113C2×C4×D4C2312yes—
114C2×C4×D4C2312yes—
115C2×C4×D4C2312yes—
116C2×C4×D4C2312yes—
117C2×C4×D4C2312yes—
118C2×C4×D4C2312yes—
119C2×#117—21no—
120C2×#117C2312yes—
121C2×#117C2312yes—
122C22×C4⋊C4C2311yescharacteristic
123C22×C22⋊C4—21no—
124C22×C22⋊C4—22no—
125C22×C22⋊C4—22no—
126C22×C42—22no—
127C22×C42—22no—
128C22×C42C2311yescharacteristic
129#481C2×C4116yescomplemented
130#476C2×C4116yescomplemented
131#459C2×C4116yescomplemented
132#447C2×C4116yescomplemented
133#442—216no—
134#442C2×C4132yescomplemented
135C4×D8—216no—
136#432C2×C418yescomplemented
137#431—24no—
138C2×#107C2314yes—
139C2×#106C2314yes—
140C2×#106C2314yes—
141C2×C4⋊C8C2311yescharacteristic
142C2×C4⋊C8C2311yescharacteristic
143C2×C4⋊C8C2312yes—
144C2×#102C2314yes—
145C2×#102C2314yes—
146C2×#102C2314yes—
147C2×#102C2314yes—
148C2×#102C2314yes—
149C2×#102C2314yes—
150C2×C22⋊C8C2312yes—
151C2×C22⋊C8C2312yes—
152C2×#97C2314yes—
153C2×#97C2312yes—
154C2×C4×C8C2312yes—
155C2×C4×C8C2314yes—
156#391—22no—
157#390—24no—
158#390—22no—
159#390—22no—
160#390C2314yes—
161#389—22no—
162#389C2314yes—
163#388—24no—
164#388—24no—
165#386—24no—
166#385—24no—
167#384C2312yes—
168#382—21no—
169#381—22no—
170#380—21no—
171#380—22no—
172#380C2312yes—
173C4×C4⋊C4—22no—
174C4×C4⋊C4—22no—
175C4×C4⋊C4—22no—
176C4×C4⋊C4—21no—
177C4×C4⋊C4D411yescharacteristic
178C4×C4⋊C4D411yescharacteristic
179C4×C4⋊C4D412yes—
180C4×C4⋊C4C2314yes—
181C4×C4⋊C4C2312yes—
182C4×C4⋊C4C2312yes—
183C4×C4⋊C4C2312yes—
184C4×C4⋊C4C2311yescharacteristic
185C4×C4⋊C4C2311yescharacteristic
186C4×C4⋊C4C2312yes—
187C4×C4⋊C4C2312yes—
188C4×C4⋊C4C2312yes—
189C4×C22⋊C4—21no—
190C4×C22⋊C4—21no—
191C4×C22⋊C4—22no—
192C4×C22⋊C4—22no—
193C4×C22⋊C4—22no—
194C4×C22⋊C4—22no—
195C4×C22⋊C4—22no—
196C4×C22⋊C4—22no—
197C4×C22⋊C4—22no—
198C4×C22⋊C4—22no—
199C4×C22⋊C4—22no—
200C4×C22⋊C4—24no—
201C4×C22⋊C4—22no—
202C4×C22⋊C4—22no—
203C4×C22⋊C4—22no—
204C4×C22⋊C4—22no—
205C4×C22⋊C4—22no—
206C4×C22⋊C4—22no—
207C4×C22⋊C4D411yescharacteristic
208C4×C22⋊C4D411yescharacteristic
209C4×C22⋊C4D412yes—
210#376—21no—
211#376C2311yescharacteristic
212#376C2311yescharacteristic
213C2×#95—22no—
214C2×#95—22no—
215C2×#95—22no—
216C43—21no—
217C43C2312yes—
218#340C2314yes—
219#340C2314yes—
220#340C2314yes—
221#340C2314yes—
222#340C2314yes—
223#340C2314yes—
224#336C2314yes—
225#336C2314yes—
226C25—21no—
227C2×#41—28no—
228C2×#41—84no—
229C22×Q8—21no—
230C22×D4—24no—
231C22×D4—24no—
232C22×D4—22no—
233C22×D4—21no—
234C22×D4—21no—
235C22×D4—21no—
236C22×D4—22no—
237C22×D4C22×C414yes—
238C22×D4C2412yes—
239C22×D4C2412yes—
240C23×C4—21no—
241C23×C4—22no—
242C23×C4—28no—
243C23×C4—22no—
244C23×C4—22no—
245C23×C4—22no—
246C23×C4—22no—
247C23×C4—22no—
248C23×C4—22no—
249C23×C4—24no—
250C23×C4—82no—
251C23×C4—82no—
252C23×C4C2×D411yescharacteristic
253C23×C4C2×D411yescharacteristic
254C23×C4C2411yescharacteristic
255C2×D8—28no—
256C2×D8—28no—
257C2×D8C22×C4116yes—
258C22×C8C2412yes—
259C22×C8C2412yes—
260C22×C8C2414yes—
261C4⋊Q8—22no—
262C4⋊Q8C22×C414yes—
263#127—22no—
264#127C22×C414yes—
265#124—28no—
266#121—28no—
267#121—24no—
268#121C22×C418yes—
269C4×Q8—84no—
270C4×D4—216no—
271C4×D4—216no—
272C4×D4—24no—
273C4×D4—24no—
274C4×D4—24no—
275C4×D4—24no—
276C4×D4—28no—
277C4×D4—28no—
278C4×D4—28no—
279C4×D4—24no—
280C4×D4—24no—
281C4×D4—24no—
282C4×D4—24no—
283C4×D4—24no—
284C4×D4—28no—
285C4×D4—28no—
286C4×D4—84no—
287C4×D4—84no—
288C4×D4C22×C418yes—
289C4×D4C22×C418yes—
290C4×D4C22×C418yes—
291C4×D4#4118yescomplemented
292C4×D4#4118yescomplemented
293#117—24no—
294#117—24no—
295#117—24no—
296#117—24no—
297#117—82no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—24no—
304C2×C4⋊C4C22×C414yes—
305C2×C4⋊C4C2×D412yes—
306C2×C4⋊C4C2×D412yes—
307C2×C4⋊C4C2414yes—
308C2×C4⋊C4C2412yes—
309C2×C4⋊C4C2412yes—
310C2×C4⋊C4C2412yes—
311C2×C4⋊C4C2412yes—
312C2×C22⋊C4—216no—
313C2×C22⋊C4—216no—
314C2×C22⋊C4—22no—
315C2×C22⋊C4—28no—
316C2×C22⋊C4—22no—
317C2×C22⋊C4—22no—
318C2×C22⋊C4—22no—
319C2×C22⋊C4—22no—
320C2×C22⋊C4—22no—
321C2×C22⋊C4—22no—
322C2×C22⋊C4—22no—
323C2×C22⋊C4—22no—
324C2×C22⋊C4—24no—
325C2×C22⋊C4—22no—
326C2×C22⋊C4—24no—
327C2×C22⋊C4—24no—
328C2×C22⋊C4C2×D412yes—
329C2×C22⋊C4C2×D412yes—
330C2×C42—22no—
331C2×C42—21no—
332C2×C42—21no—
333C2×C42—22no—
334C2×C42—24no—
335C2×C42—22no—
336C2×C42—22no—
337C2×C42—22no—
338C2×C42—22no—
339C2×C42—22no—
340C2×C42—84no—
341C2×C42—84no—
342C2×C42C22×C414yes—
343C2×C42C2×D412yes—
344C2×C42C2×D412yes—
345C2×C42C2×D412yes—
346C2×C42C2×D412yes—
347C2×C42C2×D411yescharacteristic
348C2×C42C2×D411yescharacteristic
349C2×C42#4114yes—
350C2×C42C2412yes—
351C2×C42C2411yescharacteristic
352C2×C42C2411yescharacteristic
353C2×C42C2412yes—
354#107—28no—
355#106—28no—
356#106C22×C4116yes—
357C4⋊C8—22no—
358C4⋊C8—22no—
359C4⋊C8C22×C418yes—
360#102—28no—
361#102—28no—
362#102—28no—
363#102—28no—
364#102C22×C4116yes—
365#102C22×C4116yes—
366C22⋊C8—24no—
367C22⋊C8C22×C418yes—
368#97—28no—
369#97C22×C418yes—
370C4×C8—24no—
371C4×C8—28no—
372#95—22no—
373#95—22no—
374#95—22no—
375#95—22no—
376#95—22no—
377#95—22no—
378#95—24no—
379#95—21no—
380#95—21no—
381#95—22no—
382#95—24no—
383#95C2×D411yescharacteristic
384#95C2×D412yes—
385#95C2×D412yes—
386#95C2×D411yescharacteristic
387#95C2×D411yescharacteristic
388#95C2×D411yescharacteristic
389C24—28no—
390C24—22no—
391C24—24no—
392C24—22no—
393C24—81no—
394C24—81no—
395C24—82no—
396C24C22×D411yescharacteristic
397#41—816no—
398C2×Q8—24no—
399C2×Q8—44no—
400C2×D4—28no—
401C2×D4—28no—
402C2×D4—28no—
403C2×D4—216no—
404C2×D4—28no—
405C2×D4—24no—
406C2×D4—24no—
407C2×D4—24no—
408C2×D4—22no—
409C2×D4—22no—
410C2×D4—22no—
411C2×D4—24no—
412C2×D4—24no—
413C2×D4—22no—
414C2×D4—44no—
415C2×D4—42no—
416C2×D4—42no—
417C2×D4—84no—
418C2×D4C23×C414yes—
419C2×D4C23×C414yes—
420C2×D4C2×#4114yes—
421C2×D4C2×#4114yes—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—28no—
431C22×C4—28no—
432C22×C4—24no—
433C22×C4—28no—
434C22×C4—28no—
435C22×C4—28no—
436C22×C4—28no—
437C22×C4—28no—
438C22×C4—24no—
439C22×C4—42no—
440C22×C4—44no—
441C22×C4—44no—
442C22×C4—48no—
443C22×C4—816no—
444C22×C4—82no—
445C22×C4—82no—
446C22×C4—82no—
447C22×C4—82no—
448C22×C4—82no—
449C22×C4—82no—
450C22×C4—82no—
451C22×C4—82no—
452C22×C4—82no—
453C22×C4—82no—
454C22×C4—84no—
455C22×C4—84no—
456C22×C4—84no—
457C22×C4—81no—
458C22×C4—81no—
459C22×C4C4×D414yes—
460C22×C4C4×D414yes—
461C22×C4C23×C412yes—
462C22×C4C22×D412yes—
463C22×C4C22×D412yes—
464C22×C4C22×D411yescharacteristic
465C22×C4C22×D411yescharacteristic
466C22×C4C22×D411yescharacteristic
467C22×C4C22×D411yescharacteristic
468C22×C4C22×D412yes—
469C22×C4C22×D412yes—
470C22×C4C22×D411yescharacteristic
471C22×C4C2×#4112yes—
472C22×C4C2511yescharacteristic
473D8—232no—
474C2×C8—24no—
475C2×C8—24no—
476C2×C8—22no—
477C2×C8—22no—
478C2×C8—24no—
479C2×C8—24no—
480C2×C8C23×C418yes—
481C4⋊C4—24no—
482C4⋊C4—24no—
483C4⋊C4—24no—
484C4⋊C4—24no—
485C4⋊C4—22no—
486C4⋊C4—22no—
487C4⋊C4—22no—
488C4⋊C4—22no—
489C4⋊C4—24no—
490C4⋊C4—24no—
491C4⋊C4—44no—
492C4⋊C4—44no—
493C4⋊C4—84no—
494C4⋊C4C23×C414yes—
495C4⋊C4C23×C414yes—
496C4⋊C4C2×#4114yes—
497C4⋊C4C2×#4114yes—
498C22⋊C4—28no—
499C22⋊C4—28no—
500C22⋊C4—28no—
501C22⋊C4—28no—
502C22⋊C4—28no—
503C22⋊C4—28no—
504C22⋊C4—28no—
505C22⋊C4—24no—
506C22⋊C4—24no—
507C22⋊C4—28no—
508C22⋊C4—28no—
509C22⋊C4—48no—
510C22⋊C4—84no—
511C22⋊C4—84no—
512C22⋊C4—82no—
513C42—28no—
514C42—24no—
515C42—24no—
516C42—22no—
517C42—22no—
518C42—42no—
519C42—42no—
520C42—42no—
521C42—42no—
522C42—82no—
523C42—82no—
524C42—82no—
525C42—82no—
526C42—82no—
527C42—82no—
528C42—84no—
529C42C4×D414yes—
530C42C4×D414yes—
531C42C23×C412yes—
532C42C23×C412yes—
533C42C2×#4112yes—
534C42C2×#4112yes—
535C42C2×#4114yes—
536C42C2×#4114yes—
537C23—216no—
538C23—22no—
539C23—24no—
540C23—22no—
541C23—24no—
542C23—24no—
543C23—21no—
544C23—21no—
545C23—24no—
546C23—42no—
547C23—42no—
548C23—42no—
549C23—42no—
550C23—44no—
551C23—44no—
552C23—42no—
553C23—42no—
554C23—84no—
555C23—84no—
556C23—88no—
557C23—88no—
558C23C2×C4×D412yes—
559C23C23×D411yescharacteristic
560Q8—48no—
561D4—216no—
562D4—24no—
563D4—28no—
564D4—24no—
565D4—24no—
566D4—48no—
567D4—42no—
568D4—42no—
569D4—48no—
570D4—88no—
571D4C4×#4118yes—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—21no—
579C2×C4—22no—
580C2×C4—21no—
581C2×C4—24no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—28no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—24no—
588C2×C4—24no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—24no—
592C2×C4—24no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—44no—
596C2×C4—44no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—416no—
604C2×C4—44no—
605C2×C4—44no—
606C2×C4—44no—
607C2×C4—88no—
608C2×C4—88no—
609C2×C4—88no—
610C2×C4—88no—
611C2×C4—84no—
612C2×C4—88no—
613C2×C4—88no—
614C2×C4—168no—
615C2×C4C2×C4×D412yes—
616C2×C4C2×C4×D412yes—
617C2×C4C2×C4×D412yes—
618C2×C4C2×C4×D412yes—
619C2×C4C2×C4×D412yes—
620C2×C4#52514yes—
621C2×C4#52514yes—
622C2×C4#57612yes—
623C2×C4#57712yes—
624C2×C4C24×C411yescharacteristic
625C2×C4C22×#4112yes—
626C2×C4C22×#4112yes—
627C2×C4C22×#4111yescharacteristic
628C8—28no—
629C8—42no—
630C8—42no—
631C22—22no—
632C22—22no—
633C22—28no—
634C22—22no—
635C22—44no—
636C22—44no—
637C22—44no—
638C22—42no—
639C22—44no—
640C22—44no—
641C22—42no—
642C22—44no—
643C22—44no—
644C22—44no—
645C22—44no—
646C22—816no—
647C22—84no—
648C22—84no—
649C22C22×C4×D411yescharacteristic
650C22C2×#52512yes—
651C22C2×#57611yescharacteristic
652C22C2×#57711yescharacteristic
653C22#358512yes—
654C4—22no—
655C4—22no—
656C4—24no—
657C4—24no—
658C4—22no—
659C4—22no—
660C4—44no—
661C4—42no—
662C4—44no—
663C4—42no—
664C4—44no—
665C4—44no—
666C4—84no—
667C4#294614yes—
668C4C2×C4×#4112yes—
669C4#344411yescharacteristic
670C4#344411yescharacteristic
671C2—42no—
672C2—48no—
673C2—84no—
674C2C4×#52511yescharacteristic
675C2C2×#294611yescharacteristic
676C2#6016811yescharacteristic
677C2#6059012yes—
678C2#6059712yes—
subgroupG/H
center, socleC23C23×D4
commutatorC2×C4C24×C4
Frattini, cosocleC22×C4C25
prime radicalC2×#3109C2
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #34226C2semidirect
3. C4×#518C2semidirect
4. #21614C2semidirect
5. C4×#476C2semidirect
6. #21584C2semidirect
7. #21503C2semidirect
8. C4×#459C2semidirect
9. #21458C2semidirect
10. #21390C2semidirect
11. #21364C2semidirect
12. #21126C2semidirect
13. #20732C2semidirect
14. #20503C2semidirect
15. #20470C2semidirect
19. #3109C4semidirect
36. #2319C22semidirect
37. C4×C4⋊Q8C22semidirect
38. C4×#127C22semidirect
43. #2275C22semidirect
54. #1987C22semidirect
55. #1987C22semidirect
56. #1982C22semidirect
57. #1982C22semidirect
58. #1948C22semidirect
59. #1947C22semidirect
60. #1926C22semidirect
61. #1925C22semidirect
62. #1920C22semidirect
63. #1919C22semidirect
74. #1776C22semidirect
75. #1776C22semidirect
76. C4×#106C22semidirect
77. #1767C22semidirect
78. C4×C4⋊C8C22semidirect
80. #1762C22semidirect
83. C4×#102C22semidirect
84. C4×#102C22semidirect
86. #1746C22semidirect
87. #1726C22semidirect
93. #532C2×C4semidirect
110. C2×C4×D4C2×C4semidirect
129. #481C2×C4semidirect
130. #476C2×C4semidirect
131. #459C2×C4semidirect
132. #447C2×C4semidirect
134. #442C2×C4semidirect
136. #432C2×C4semidirect
291. C4×D4#41semidirect
292. C4×D4#41semidirect

Maximal subgroups

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

subgroupindex
C2×#31092
#342262
C4×#5182
#216142
C4×#4762
#215842
#215032
C4×#4592
#214582
#213902
#213642
#211262
#207322
#205032
#204702