Finite Group Databasebeta

← #6373328

Subgroups — #6373328

Order 512.

2582 conjugacy classes of subgroups.

The subgroup lattice has 660 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3109C211yesmaximal, characteristic
2#34222C211yesmaximal, characteristic, complemented
3C2×#2275C211yesmaximal, characteristic, complemented
4#21623C214yesmaximal, complemented
5#21580C212yesmaximal, complemented
6C4×#481C212yesmaximal, complemented
7#21513C214yesmaximal, complemented
8#21452C212yesmaximal, complemented
9C4×#447C212yesmaximal, complemented
10#21386C211yesmaximal, characteristic, complemented
11#21363C211yesmaximal, characteristic, complemented
12#21126C214yesmaximal, complemented
13#20733C211yesmaximal, characteristic, complemented
14#20503C214yesmaximal, complemented
15#20469C211yesmaximal, characteristic, complemented
16C2×#532C2211yescharacteristic
17C22×C4×D4C2212yes—
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#2372—21no—
30#2372—21no—
31#2364—21no—
32#2364—21no—
33#2353—22no—
34#2339—22no—
35#2331C2212yescomplemented
36#2328C2212yescomplemented
37#2319C2212yescomplemented
38C4×#121C2212yes—
39C4×#121C2212yes—
40#2283C2211yescharacteristic
41#2275—22no—
42#2275—21no—
43#2275—21no—
44#2275C2214yes—
45#2275C2212yes—
46#2275C2212yes—
47#2275C2212yescomplemented
48C2×C4×C4⋊C4C2212yes—
49C2×C4×C4⋊C4C2211yescharacteristic
50C2×C4×C22⋊C4—22no—
51C2×C4×C22⋊C4—21no—
52C2×#376C2211yescharacteristic
53C22×#95—21no—
54#1988C2212yescomplemented
55#1982C2212yescomplemented
56#1949C2214yescomplemented
57#1926C2214yescomplemented
58#1925C2214yescomplemented
59#1920C2214yescomplemented
60#1919C2214yescomplemented
61#1879C2214yes—
62#1879C2214yes—
63#1874C2212yes—
64#1874C2212yes—
65#1864C2214yes—
66#1862C2214yes—
67#1853C2212yes—
68#1852C2212yes—
69#1824C2211yescharacteristic
70#1824C2211yescharacteristic
71#1804C2214yes—
72#1786C2211yescharacteristic
73#1786C2211yescharacteristic
74#1776C2214yescomplemented
75#1776C2214yescomplemented
76C4×#106C2214yescomplemented
77#1768C2214yescomplemented
78#1762C2214yes—
79#1762C2214yescomplemented
80C4×#102C2214yes—
81C4×#102C2214yes—
82C4×#102C2214yescomplemented
83C4×#102C2214yescomplemented
84#1749C2212yes—
85C4×C22⋊C8C2212yes—
86#1746C2212yescomplemented
87#1726C2212yescomplemented
88C22×#41—21no—
89C23×D4C2311yescharacteristic
90C24×C4—22no—
91C24×C4—21no—
92C22×D8C2314yes—
93#532—24no—
94#532C2×C418yescomplemented
95C2×C4⋊Q8C2311yescharacteristic
96C2×#127C2311yescharacteristic
97C2×#124—21no—
98C2×#121—21no—
99C2×#121C2312yes—
100C2×C4×D4—216no—
101C2×C4×D4—22no—
102C2×C4×D4—22no—
103C2×C4×D4—21no—
104C2×C4×D4—22no—
105C2×C4×D4—22no—
106C2×C4×D4—21no—
107C2×C4×D4—21no—
108C2×C4×D4—24no—
109C2×C4×D4C2×C418yescomplemented
110C2×C4×D4C2312yes—
111C2×C4×D4C2312yes—
112C2×C4×D4C2314yes—
113C2×C4×D4C2312yes—
114C2×C4×D4C2312yes—
115C2×C4×D4C2312yes—
116C2×C4×D4C2314yes—
117C22×C4⋊C4C2312yes—
118C22×C4⋊C4C2311yescharacteristic
119C22×C22⋊C4—22no—
120C22×C22⋊C4—21no—
121C22×C42—21no—
122C22×C42—22no—
123C22×C42C2312yes—
124C22×C42C2311yescharacteristic
125#481C2×C4116yescomplemented
126#476C2×C4116yescomplemented
127#459C2×C4116yescomplemented
128#447C2×C4116yescomplemented
129#442—216no—
130#442C2×C4132yescomplemented
131C4×D8—216no—
132#432C2×C418yescomplemented
133#431—24no—
134C2×#107C2314yes—
135C2×#106C2314yes—
136C2×#106C2314yes—
137C2×C4⋊C8C2312yes—
138C2×C4⋊C8C2312yes—
139C2×#102C2314yes—
140C2×#102C2314yes—
141C2×#102C2314yes—
142C2×#102C2314yes—
143C2×#102C2314yes—
144C2×#102C2314yes—
145C2×C22⋊C8C2311yescharacteristic
146C2×C22⋊C8C2311yescharacteristic
147C2×C22⋊C8C2312yes—
148C2×#97C2314yes—
149C2×#97C2312yes—
150C2×C4×C8C2312yes—
151C2×C4×C8C2314yes—
152#391—22no—
153#391—22no—
154#390C2312yes—
155#390C2312yes—
156#389C2312yes—
157#389C2312yes—
158#388—22no—
159#388—22no—
160#388—22no—
161#388—22no—
162#387—22no—
163#387—22no—
164#386—22no—
165#386—22no—
166#386—22no—
167#386—22no—
168#386—22no—
169#386—22no—
170#383—21no—
171#382C2312yes—
172#381—21no—
173#381—22no—
174#380—22no—
175#380C2311yescharacteristic
176#380C2311yescharacteristic
177C4×C4⋊C4—22no—
178C4×C4⋊C4—22no—
179C4×C4⋊C4—21no—
180C4×C4⋊C4—21no—
181C4×C4⋊C4C2312yes—
182C4×C4⋊C4C2312yes—
183C4×C4⋊C4C2312yes—
184C4×C4⋊C4C2314yes—
185C4×C4⋊C4C2312yes—
186C4×C4⋊C4C2312yes—
187C4×C4⋊C4C2314yes—
188C4×C22⋊C4—21no—
189C4×C22⋊C4—22no—
190C4×C22⋊C4—22no—
191C4×C22⋊C4—22no—
192C4×C22⋊C4—22no—
193C4×C22⋊C4—22no—
194C4×C22⋊C4—24no—
195C4×C22⋊C4—22no—
196C4×C22⋊C4—24no—
197C4×C22⋊C4—24no—
198C4×C22⋊C4—24no—
199C4×C22⋊C4—24no—
200C4×C22⋊C4—22no—
201#376—21no—
202#376D412yes—
203#376D411yescharacteristic
204#376D411yescharacteristic
205#376C2311yescharacteristic
206#376C2312yes—
207#376C2311yescharacteristic
208C2×#95—22no—
209C2×#95—21no—
210C2×#95—22no—
211C2×#95—22no—
212C2×#95—24no—
213C2×#95—22no—
214C2×#95D412yes—
215C2×#95D411yescharacteristic
216C2×#95D411yescharacteristic
217#340C2314yes—
218#340C2314yes—
219#340C2314yes—
220#340C2314yes—
221#340C2314yes—
222#340C2314yes—
223#336C2314yes—
224#336C2314yes—
225C25—21no—
226C2×#41—28no—
227C2×#41—82no—
228C2×#41—82no—
229C22×Q8—21no—
230C22×D4—28no—
231C22×D4—22no—
232C22×D4—21no—
233C22×D4—21no—
234C22×D4—21no—
235C22×D4—22no—
236C22×D4C22×C414yes—
237C22×D4C2412yes—
238C22×D4C2412yes—
239C23×C4—22no—
240C23×C4—21no—
241C23×C4—216no—
242C23×C4—28no—
243C23×C4—24no—
244C23×C4—22no—
245C23×C4—22no—
246C23×C4—24no—
247C23×C4—22no—
248C23×C4—22no—
249C23×C4—22no—
250C23×C4—84no—
251C23×C4—84no—
252C23×C4—84no—
253C23×C4C2×D412yes—
254C23×C4C2×D412yes—
255C23×C4C2×D411yescharacteristic
256C23×C4C2×D411yescharacteristic
257C23×C4C2411yescharacteristic
258C2×D8—28no—
259C2×D8—28no—
260C2×D8C22×C4116yes—
261C22×C8C2412yes—
262C22×C8C2412yes—
263C22×C8C2414yes—
264C4⋊Q8—22no—
265C4⋊Q8C22×C414yescomplemented
266#127—22no—
267#127C22×C414yescomplemented
268#124—28no—
269#121—28no—
270#121—24no—
271#121C22×C418yes—
272C4×D4—216no—
273C4×D4—216no—
274C4×D4—28no—
275C4×D4—28no—
276C4×D4—28no—
277C4×D4—28no—
278C4×D4—28no—
279C4×D4—24no—
280C4×D4—24no—
281C4×D4—28no—
282C4×D4—24no—
283C4×D4—28no—
284C4×D4—28no—
285C4×D4—84no—
286C4×D4—84no—
287C4×D4C22×C418yes—
288C4×D4C22×C418yes—
289C4×D4C22×C418yescomplemented
290C4×D4#41116yescomplemented
291C2×C4⋊C4—22no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—28no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4C22×C414yes—
299C2×C4⋊C4C2412yes—
300C2×C4⋊C4C2412yes—
301C2×C4⋊C4C2412yes—
302C2×C4⋊C4C2412yes—
303C2×C4⋊C4C2414yes—
304C2×C22⋊C4—216no—
305C2×C22⋊C4—28no—
306C2×C22⋊C4—24no—
307C2×C22⋊C4—22no—
308C2×C22⋊C4—22no—
309C2×C22⋊C4—24no—
310C2×C22⋊C4—22no—
311C2×C22⋊C4—22no—
312C2×C22⋊C4—21no—
313C2×C22⋊C4—21no—
314C2×C22⋊C4—22no—
315C2×C22⋊C4—22no—
316C2×C22⋊C4—22no—
317C2×C22⋊C4—22no—
318C2×C22⋊C4—22no—
319C2×C22⋊C4—22no—
320C2×C42—28no—
321C2×C42—22no—
322C2×C42—22no—
323C2×C42—22no—
324C2×C42—24no—
325C2×C42—22no—
326C2×C42—22no—
327C2×C42—24no—
328C2×C42—82no—
329C2×C42—84no—
330C2×C42C22×C414yes—
331C2×C42C2×D412yes—
332C2×C42C2×D412yes—
333C2×C42C2×D411yescharacteristic
334C2×C42C2×D411yescharacteristic
335C2×C42#4114yes—
336C2×C42C2412yes—
337C2×C42C2411yescharacteristic
338C2×C42C2411yescharacteristic
339C2×C42C2412yes—
340#107—28no—
341#106—28no—
342#106C22×C4116yescomplemented
343C4⋊C8—24no—
344C4⋊C8C22×C418yescomplemented
345#102—28no—
346#102—28no—
347#102—28no—
348#102—28no—
349#102C22×C4116yes—
350#102C22×C4116yescomplemented
351C22⋊C8—22no—
352C22⋊C8—22no—
353C22⋊C8C22×C418yes—
354#97—28no—
355#97C22×C418yescomplemented
356C4×C8—24no—
357C4×C8—28no—
358#95—24no—
359#95—22no—
360#95—22no—
361#95—24no—
362#95—22no—
363#95—22no—
364#95—22no—
365#95—21no—
366#95—21no—
367#95—22no—
368#95—22no—
369#95—22no—
370#95—22no—
371#95C2×D411yescharacteristic
372#95C2×D412yes—
373#95C2×D411yescharacteristic
374#95C2×D412yes—
375#95C2×D411yescharacteristic
376#95C2×D412yes—
377#95C2×D412yes—
378#95C2×D412yes—
379#95C2×D412yes—
380#95C2×D411yescharacteristic
381C24—28no—
382C24—24no—
383C24—22no—
384C24—22no—
385C24—44no—
386C24—42no—
387C24—42no—
388C24C22×D411yescharacteristic
389#41—816no—
390C2×Q8—24no—
391C2×Q8—81no—
392C2×Q8—81no—
393C2×D4—28no—
394C2×D4—216no—
395C2×D4—216no—
396C2×D4—28no—
397C2×D4—24no—
398C2×D4—24no—
399C2×D4—24no—
400C2×D4—24no—
401C2×D4—22no—
402C2×D4—24no—
403C2×D4—24no—
404C2×D4—22no—
405C2×D4—44no—
406C2×D4—44no—
407C2×D4—82no—
408C2×D4—81no—
409C2×D4—81no—
410C2×D4C23×C414yes—
411C2×D4C23×C414yes—
412C2×D4C2×#4118yes—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—24no—
416C22×C4—28no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—22no—
421C22×C4—24no—
422C22×C4—22no—
423C22×C4—216no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—28no—
427C22×C4—28no—
428C22×C4—28no—
429C22×C4—216no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—42no—
433C22×C4—42no—
434C22×C4—42no—
435C22×C4—44no—
436C22×C4—44no—
437C22×C4—44no—
438C22×C4—44no—
439C22×C4—44no—
440C22×C4—84no—
441C22×C4—84no—
442C22×C4—84no—
443C22×C4—84no—
444C22×C4—84no—
445C22×C4—84no—
446C22×C4—82no—
447C22×C4—82no—
448C22×C4—82no—
449C22×C4—82no—
450C22×C4—82no—
451C22×C4—82no—
452C22×C4—816no—
453C22×C4—832no—
454C22×C4C4×D414yes—
455C22×C4C4×D414yes—
456C22×C4C23×C412yes—
457C22×C4C22×D412yes—
458C22×C4C22×D412yes—
459C22×C4C22×D411yescharacteristic
460C22×C4C22×D411yescharacteristic
461C22×C4C22×D411yescharacteristic
462C22×C4C22×D411yescharacteristic
463C22×C4C22×D411yescharacteristic
464C22×C4C22×D412yes—
465C22×C4C22×D412yes—
466C22×C4C2×#4112yes—
467C22×C4C2511yescharacteristic
468D8—232no—
469C2×C8—24no—
470C2×C8—24no—
471C2×C8—22no—
472C2×C8—22no—
473C2×C8—24no—
474C2×C8—24no—
475C2×C8C23×C418yes—
476C4⋊C4—28no—
477C4⋊C4—24no—
478C4⋊C4—24no—
479C4⋊C4—22no—
480C4⋊C4—24no—
481C4⋊C4—22no—
482C4⋊C4—24no—
483C4⋊C4—24no—
484C4⋊C4—82no—
485C4⋊C4—82no—
486C4⋊C4C23×C414yes—
487C4⋊C4C23×C414yes—
488C4⋊C4C2×#4118yes—
489C22⋊C4—216no—
490C22⋊C4—28no—
491C22⋊C4—28no—
492C22⋊C4—28no—
493C22⋊C4—28no—
494C22⋊C4—216no—
495C22⋊C4—42no—
496C22⋊C4—42no—
497C22⋊C4—44no—
498C22⋊C4—44no—
499C22⋊C4—82no—
500C22⋊C4—82no—
501C22⋊C4—82no—
502C22⋊C4—82no—
503C42—24no—
504C42—24no—
505C42—24no—
506C42—82no—
507C42—84no—
508C42—82no—
509C42—82no—
510C42—84no—
511C42—82no—
512C42—82no—
513C42C4×D414yes—
514C42C4×D414yes—
515C42C23×C412yes—
516C42C23×C412yes—
517C42C2×#4112yes—
518C42C2×#4112yes—
519C42C2×#4118yes—
520C23—22no—
521C23—22no—
522C23—216no—
523C23—24no—
524C23—28no—
525C23—24no—
526C23—48no—
527C23—416no—
528C23—416no—
529C23—44no—
530C23—48no—
531C23—42no—
532C23—42no—
533C23—42no—
534C23—42no—
535C23—42no—
536C23—42no—
537C23—42no—
538C23—42no—
539C23C2×C4×D412yes—
540C23#57612yes—
541C23#57712yes—
542C23C23×D411yescharacteristic
543Q8—84no—
544D4—216no—
545D4—24no—
546D4—28no—
547D4—28no—
548D4—416no—
549D4—44no—
550D4—84no—
551D4—84no—
552D4C4×#4118yes—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—28no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—24no—
570C2×C4—24no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—28no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—44no—
577C2×C4—48no—
578C2×C4—48no—
579C2×C4—48no—
580C2×C4—416no—
581C2×C4—44no—
582C2×C4—44no—
583C2×C4—44no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—88no—
593C2×C4—88no—
594C2×C4—816no—
595C2×C4—88no—
596C2×C4—816no—
597C2×C4—164no—
598C2×C4—164no—
599C2×C4C2×C4×D412yes—
600C2×C4C2×C4×D412yes—
601C2×C4C2×C4×D412yes—
602C2×C4C2×C4×D412yes—
603C2×C4C2×C4×D412yes—
604C2×C4#52514yes—
605C2×C4#52514yes—
606C2×C4C24×C411yescharacteristic
607C2×C4C22×#4114yes—
608C2×C4C22×#4111yescharacteristic
609C8—28no—
610C8—42no—
611C8—42no—
612C22—22no—
613C22—24no—
614C22—24no—
615C22—22no—
616C22—28no—
617C22—432no—
618C22—44no—
619C22—48no—
620C22—48no—
621C22—44no—
622C22—44no—
623C22—44no—
624C22—44no—
625C22—82no—
626C22—82no—
627C22—82no—
628C22—82no—
629C22#294614yes—
630C22C22×C4×D411yescharacteristic
631C22C2×#52512yes—
632C22C2×#57611yescharacteristic
633C22C2×#57711yescharacteristic
634C22#358512yes—
635C4—22no—
636C4—22no—
637C4—24no—
638C4—24no—
639C4—22no—
640C4—22no—
641C4—22no—
642C4—44no—
643C4—42no—
644C4—48no—
645C4—42no—
646C4—44no—
647C4—82no—
648C4—82no—
649C4C2×C4×#4112yes—
650C4#344412yes—
651C2—24no—
652C2—48no—
653C2—84no—
654C2C4×#52511yescharacteristic
655C2C2×#294611yescharacteristic
656C2#6016811yescharacteristic
657C2#6057812yes—
658C2#6058812yes—
subgroupG/H
center, socleC23C23×D4
commutatorC2×C4C24×C4
Frattini, cosocleC22×C4C25
prime radicalC2×#3109C2
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×#376 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #34222C2semidirect
3. C2×#2275C2semidirect
4. #21623C2semidirect
5. #21580C2semidirect
6. C4×#481C2semidirect
7. #21513C2semidirect
8. #21452C2semidirect
9. C4×#447C2semidirect
10. #21386C2semidirect
11. #21363C2semidirect
12. #21126C2semidirect
13. #20733C2semidirect
14. #20503C2semidirect
15. #20469C2semidirect
19. #3109C4semidirect
35. #2331C22semidirect
36. #2328C22semidirect
37. #2319C22semidirect
47. #2275C22semidirect
54. #1988C22semidirect
55. #1982C22semidirect
56. #1949C22semidirect
57. #1926C22semidirect
58. #1925C22semidirect
59. #1920C22semidirect
60. #1919C22semidirect
74. #1776C22semidirect
75. #1776C22semidirect
76. C4×#106C22semidirect
77. #1768C22semidirect
79. #1762C22semidirect
82. C4×#102C22semidirect
83. C4×#102C22semidirect
86. #1746C22semidirect
87. #1726C22semidirect
94. #532C2×C4semidirect
109. C2×C4×D4C2×C4semidirect
125. #481C2×C4semidirect
126. #476C2×C4semidirect
127. #459C2×C4semidirect
128. #447C2×C4semidirect
130. #442C2×C4semidirect
132. #432C2×C4semidirect
265. C4⋊Q8C22×C4semidirect
267. #127C22×C4semidirect
289. C4×D4C22×C4semidirect
290. C4×D4#41semidirect
342. #106C22×C4semidirect
344. C4⋊C8C22×C4semidirect
350. #102C22×C4semidirect
355. #97C22×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#31092
#342222
C2×#22752
#216232
#215802
C4×#4812
#215132
#214522
C4×#4472
#213862
#213632
#211262
#207332
#205032
#204692