Finite Group Databasebeta

← #357173

Subgroups — #357173

Order 512.

1152 conjugacy classes of subgroups.

The subgroup lattice has 686 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1938C211yesmaximal, characteristic
2C2×#1929C211yesmaximal, characteristic
3#20872C211yesmaximal, characteristic
4C2×#1773C211yesmaximal, characteristic, complemented
5#20439C211yesmaximal, characteristic, complemented
6C2×#1759C211yesmaximal, characteristic
7#20378C211yesmaximal, characteristic
8#10934C211yesmaximal, characteristic
9#10929C211yesmaximal, characteristic
10#10911C211yesmaximal, characteristic, complemented
11#10909C211yesmaximal, characteristic, complemented
12#10887C211yesmaximal, characteristic, complemented
13#10879C211yesmaximal, characteristic, complemented
14#10706C211yesmaximal, characteristic, complemented
15#10694C211yesmaximal, characteristic, complemented
16C2×C4×#41C2211yescharacteristic
17C22×#108C2211yescharacteristic
18C2×#427C2211yescharacteristic
19C2×#424C2211yescharacteristic
20C22×#104C2211yescharacteristic
21C22×#104C2211yescharacteristic
22C2×#416C2211yescharacteristic
23C2×#416C2211yescharacteristic
24C2×#408C2211yescharacteristic
25C2×#408C2211yescharacteristic
26C2×#405C2211yescharacteristic
27C2×#405C2211yescharacteristic
28C2×#405C2211yescharacteristic
29C2×#405C2211yescharacteristic
30C2×C4×M16C2211yescharacteristic
31C22×#97C2211yescharacteristic
32C22×C4×C8C2211yescharacteristic
33#2296C2211yescharacteristic
34C2×#376C2211yescharacteristic
35#1938C418yescomplemented
36#1929C418yescomplemented
37#1865C412yescomplemented
38#1864C412yescomplemented
39#1863C412yescomplemented
40#1862C412yescomplemented
41#1773—24no—
42#1771—21no—
43#1771—21no—
44#1770—21no—
45C8×C4⋊C4—21no—
46#1759—24no—
47#1750—21no—
48#1750—21no—
49#1749—21no—
50C4×C22⋊C8—21no—
51C2×#341C2211yescharacteristic
52C2×#341C2211yescharacteristic
53C2×#341C2211yescharacteristic
54#1735C2211yescharacteristic
55C2×#339C2211yescharacteristic
56C2×#339C2211yescharacteristic
57C2×#339C2211yescharacteristic
58C2×#339C2211yescharacteristic
59C2×#339C2211yescharacteristic
60C2×#339C2211yescharacteristic
61#1729C2211yescharacteristic
62#1728C2211yescharacteristic
63C2×#336C2211yescharacteristic
64C2×#336C2211yescharacteristic
65C2×#336C2211yescharacteristic
66C2×#336C2211yescharacteristic
67C22×#41C2311yescharacteristic
68C22×M16C2311yescharacteristic
69C22×M16C2311yescharacteristic
70C22×M16C2311yescharacteristic
71C22×M16C2311yescharacteristic
72C23×C8C2311yescharacteristic
73C23×C8C2311yescharacteristic
74C23×C8C2311yescharacteristic
75C23×C8C2311yescharacteristic
76C4×#41—22no—
77C4×#41—22no—
78C4×#41—41no—
79C4×#41—41no—
80C2×C4×Q8C2×C412yes—
81C2×C4×D4—22no—
82C2×C4×D4C2×C412yes—
83C2×#117—21no—
84C2×#117C2311yescharacteristic
85C2×#117C2311yescharacteristic
86C2×#117C2311yescharacteristic
87C22×C42—21no—
88C22×C42C2311yescharacteristic
89C22×C42C2311yescharacteristic
90C22×C42C2311yescharacteristic
91C2×#108—22no—
92C2×#108—22no—
93C2×#108C2×C414yescomplemented
94#427—22no—
95#427—22no—
96C2×#107C2×C412yes—
97C2×#106C2×C412yes—
98#424—22no—
99#424C2×C414yescomplemented
100C2×C4⋊C8—21no—
101C2×C4⋊C8—21no—
102C2×#104—22no—
103C2×#104—22no—
104C2×#104—22no—
105C2×#104—22no—
106C2×#104C2×C414yes—
107C2×#104C2×C414yes—
108#416—41no—
109#416—41no—
110#416C2×C414yes—
111#416C2×C414yes—
112C2×#103C2×C412yes—
113C2×#103C2×C411yescharacteristic
114C2×#103C2×C411yescharacteristic
115C2×#102C2×C412yes—
116C2×#102C2×C411yescharacteristic
117C2×#102C2×C411yescharacteristic
118#408—22no—
119#408—22no—
120#408C2×C414yes—
121#408C2×C414yes—
122C2×C22⋊C8—21no—
123C2×C22⋊C8—21no—
124C2×C22⋊C8—21no—
125C2×C22⋊C8—21no—
126#405—22no—
127#405—22no—
128#405—22no—
129#405—22no—
130#405—41no—
131#405—41no—
132#405—41no—
133#405C2×C414yescomplemented
134C4×M16—22no—
135C4×M16—41no—
136C2×#97—21no—
137C2×#97—22no—
138C2×#97—22no—
139C2×#97—21no—
140C2×#97—21no—
141C2×#97—21no—
142C2×#97—21no—
143C2×#97C2×C414yescomplemented
144C2×C4×C8—21no—
145C2×C4×C8—21no—
146C2×C4×C8—21no—
147C2×C4×C8—21no—
148C2×C4×C8—22no—
149C2×C4×C8—21no—
150C2×C4×C8—22no—
151C2×C4×C8—41no—
152#389C2×C412yes—
153#389C2×C412yes—
154#385—22no—
155C4×C4⋊C4—21no—
156C4×C4⋊C4—21no—
157#376—22no—
158#376—41no—
159C2×#95—22no—
160#341—22no—
161#341—41no—
162#341C2×C414yescomplemented
163#340C2×C412yes—
164#340C2×C412yes—
165#339—22no—
166#339—22no—
167#339—41no—
168#339C2×C414yescomplemented
169#339C2×C414yescomplemented
170#339D414yes—
171#336—21no—
172#336—21no—
173#336—22no—
174#336—21no—
175#336—21no—
176#336C2×C412yes—
177#336C2×C412yes—
178#336C2×C414yescomplemented
179#336D411yescharacteristic
180#336D411yescharacteristic
181#336D411yescharacteristic
182#336D411yescharacteristic
183C2×#41—21no—
184C2×#41—21no—
185C2×#41—21no—
186C2×#41—21no—
187C2×#41—21no—
188C2×#41—21no—
189C2×#41—21no—
190C2×#41—21no—
191C2×#41—21no—
192C2×#41—21no—
193C2×#41C22×C412yes—
194C2×#41C22×C412yes—
195C22×Q8C22×C411yescharacteristic
196C22×D4—21no—
197C22×D4C22×C411yescharacteristic
198C23×C4—21no—
199C23×C4—21no—
200C23×C4—21no—
201C23×C4—22no—
202C23×C4#4112yes—
203C23×C4C2411yescharacteristic
204C2×M16—21no—
205C2×M16—21no—
206C2×M16—21no—
207C2×M16—21no—
208C2×M16—21no—
209C2×M16—21no—
210C2×M16—21no—
211C2×M16—21no—
212C2×M16—21no—
213C2×M16—21no—
214C2×M16—21no—
215C2×M16—21no—
216C2×M16—21no—
217C2×M16—21no—
218C2×M16—21no—
219C2×M16—21no—
220C2×M16—21no—
221C2×M16—21no—
222C2×M16C22×C412yes—
223C2×M16C22×C412yes—
224C2×M16C22×C412yes—
225C2×M16C22×C412yes—
226C2×M16C2×D412yes—
227C2×M16#4112yes—
228C22×C8—21no—
229C22×C8—21no—
230C22×C8—21no—
231C22×C8—21no—
232C22×C8—21no—
233C22×C8—21no—
234C22×C8—21no—
235C22×C8—21no—
236C22×C8—21no—
237C22×C8—21no—
238C22×C8—21no—
239C22×C8—21no—
240C22×C8—21no—
241C22×C8—21no—
242C22×C8—21no—
243C22×C8—21no—
244C22×C8—21no—
245C22×C8—21no—
246C22×C8—21no—
247C22×C8—21no—
248C22×C8—21no—
249C22×C8—21no—
250C22×C8C22×C412yes—
251C22×C8C22×C411yescharacteristic
252C22×C8C22×C411yescharacteristic
253C22×C8C22×C412yes—
254C22×C8C22×C412yes—
255C22×C8C22×C412yes—
256C22×C8C22×C412yes—
257C22×C8C2×D411yescharacteristic
258C22×C8C2×D411yescharacteristic
259C22×C8C2×D411yescharacteristic
260C22×C8C2×D411yescharacteristic
261C22×C8#4112yes—
262C22×C8#4112yes—
263C4×Q8—24no—
264C4×Q8—42no—
265C4×D4—24no—
266C4×D4—42no—
267C4×D4—44no—
268C4×D4—44no—
269#117—21no—
270#117—21no—
271#117—21no—
272#117—21no—
273#117—21no—
274#117—21no—
275#117—21no—
276#117—21no—
277#117—42no—
278#117—42no—
279#117C22×C412yes—
280#117C22×C412yes—
281#117C2×D412yes—
282#117#4112yes—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4C22×C412yes—
287C2×C4⋊C4C22×C411yescharacteristic
288C2×C4⋊C4C22×C411yescharacteristic
289C2×C4⋊C4C22×C412yes—
290C2×C22⋊C4—21no—
291C2×C22⋊C4—21no—
292C2×C22⋊C4—21no—
293C2×C22⋊C4—22no—
294C2×C42—21no—
295C2×C42—21no—
296C2×C42—21no—
297C2×C42—21no—
298C2×C42—21no—
299C2×C42—21no—
300C2×C42—21no—
301C2×C42—21no—
302C2×C42—21no—
303C2×C42—21no—
304C2×C42—21no—
305C2×C42—21no—
306C2×C42—21no—
307C2×C42—22no—
308C2×C42—21no—
309C2×C42—21no—
310C2×C42—42no—
311C2×C42—42no—
312C2×C42C22×C412yes—
313C2×C42C22×C412yes—
314C2×C42C22×C412yes—
315C2×C42C2×D411yescharacteristic
316C2×C42C2×D411yescharacteristic
317C2×C42C2×D412yes—
318C2×C42#4112yes—
319#108—28no—
320#107—24no—
321#106—24no—
322C4⋊C8—24no—
323C4⋊C8—24no—
324#104—28no—
325#104—28no—
326#103—42no—
327#103C4214yes—
328#103C4214yes—
329#102—42no—
330#102C4214yes—
331#102C4214yes—
332C22⋊C8—24no—
333C22⋊C8—24no—
334C22⋊C8—24no—
335C22⋊C8—24no—
336#97—24no—
337#97—28no—
338#97—42no—
339#97—42no—
340#97—42no—
341#97—42no—
342C4×C8—24no—
343C4×C8—44no—
344C4×C8—42no—
345C4×C8—42no—
346C4×C8—42no—
347C4×C8—42no—
348#95—22no—
349#95—22no—
350#95—44no—
351C24—21no—
352C24C2×#4111yescharacteristic
353#41—22no—
354#41—22no—
355#41—22no—
356#41—22no—
357#41—22no—
358#41—22no—
359#41—22no—
360#41—22no—
361#41—41no—
362#41—41no—
363#41—41no—
364#41—41no—
365#41—41no—
366#41—41no—
367#41—41no—
368#41—41no—
369C2×Q8—21no—
370C2×Q8—21no—
371C2×Q8—21no—
372C2×Q8—21no—
373C2×Q8—21no—
374C2×Q8C2×C4212yes—
375C2×D4—22no—
376C2×D4—22no—
377C2×D4—21no—
378C2×D4—21no—
379C2×D4—21no—
380C2×D4—21no—
381C2×D4—21no—
382C2×D4—41no—
383C2×D4—41no—
384C2×D4—41no—
385C2×D4—41no—
386C2×D4C2×C4212yes—
387C22×C4—21no—
388C22×C4—21no—
389C22×C4—22no—
390C22×C4—21no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—42no—
401C22×C4—42no—
402C22×C4—44no—
403C22×C4—44no—
404C22×C4—42no—
405C22×C4—42no—
406C22×C4—41no—
407C22×C4—41no—
408C22×C4—41no—
409C22×C4—41no—
410C22×C4—42no—
411C22×C4—42no—
412C22×C4—41no—
413C22×C4—41no—
414C22×C4—82no—
415C22×C4C4×D412yes—
416C22×C4C4×D412yes—
417C22×C4C23×C411yescharacteristic
418C22×C4C23×C411yescharacteristic
419C22×C4C23×C411yescharacteristic
420C22×C4C22×D411yescharacteristic
421C22×C4C22×D411yescharacteristic
422C22×C4C2×#4111yescharacteristic
423C22×C4C2×#4111yescharacteristic
424C22×C4C2×#4111yescharacteristic
425C22×C4C2×#4111yescharacteristic
426C22×C4C2×#4111yescharacteristic
427C22×C4H(5,𝔽2)11yescharacteristic
428C22×C4H(5,𝔽2)11yescharacteristic
429C22×C42−1+411yescharacteristic
430C22×C42−1+411yescharacteristic
431M16—22no—
432M16—22no—
433M16—22no—
434M16—22no—
435M16—22no—
436M16—22no—
437M16—22no—
438M16—22no—
439M16—22no—
440M16—41no—
441M16—41no—
442M16—41no—
443M16—41no—
444M16—41no—
445M16—41no—
446M16C4×D414yes—
447C2×C8—22no—
448C2×C8—22no—
449C2×C8—22no—
450C2×C8—21no—
451C2×C8—21no—
452C2×C8—22no—
453C2×C8—22no—
454C2×C8—22no—
455C2×C8—22no—
456C2×C8—21no—
457C2×C8—21no—
458C2×C8—22no—
459C2×C8—22no—
460C2×C8—21no—
461C2×C8—21no—
462C2×C8—22no—
463C2×C8—21no—
464C2×C8—21no—
465C2×C8—21no—
466C2×C8—21no—
467C2×C8—21no—
468C2×C8—21no—
469C2×C8—22no—
470C2×C8—22no—
471C2×C8—22no—
472C2×C8—22no—
473C2×C8—22no—
474C2×C8—22no—
475C2×C8—22no—
476C2×C8—22no—
477C2×C8—22no—
478C2×C8—22no—
479C2×C8—22no—
480C2×C8—41no—
481C2×C8—41no—
482C2×C8—41no—
483C2×C8—41no—
484C2×C8—41no—
485C2×C8—41no—
486C2×C8—41no—
487C2×C8—41no—
488C2×C8—41no—
489C2×C8—41no—
490C2×C8C2×C4212yes—
491C2×C8C2×C4212yes—
492C2×C8C4×D412yes—
493C2×C8C4×D412yes—
494C2×C8C4×D412yes—
495C2×C8C4×D412yes—
496C4⋊C4—21no—
497C4⋊C4—21no—
498C4⋊C4—22no—
499C4⋊C4—22no—
500C4⋊C4—22no—
501C4⋊C4—44no—
502C4⋊C4—44no—
503C4⋊C4—82no—
504C4⋊C4C2×C4212yes—
505C4⋊C4C2×C4212yes—
506C4⋊C4C4×D414yes—
507C22⋊C4—22no—
508C22⋊C4—22no—
509C22⋊C4—22no—
510C22⋊C4—22no—
511C22⋊C4—44no—
512C22⋊C4—41no—
513C22⋊C4—41no—
514C42—22no—
515C42—22no—
516C42—22no—
517C42—22no—
518C42—22no—
519C42—21no—
520C42—22no—
521C42—21no—
522C42—22no—
523C42—21no—
524C42—21no—
525C42—22no—
526C42—22no—
527C42—22no—
528C42—22no—
529C42—44no—
530C42—41no—
531C42—41no—
532C42—41no—
533C42—41no—
534C42C4×D414yes—
535C23—22no—
536C23—22no—
537C23—22no—
538C23—41no—
539C23—41no—
540C23—42no—
541C23—42no—
542C23C2×C4×D411yescharacteristic
543C23C4×#4111yescharacteristic
544C23C4×#4111yescharacteristic
545C23#54611yescharacteristic
546C23#55111yescharacteristic
547C23#55211yescharacteristic
548C23#55611yescharacteristic
549Q8—22no—
550Q8—22no—
551Q8—41no—
552Q8—41no—
553D4—22no—
554D4—22no—
555D4—42no—
556D4—42no—
557D4—42no—
558D4—42no—
559D4—41no—
560D4—41no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—44no—
575C2×C4—44no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
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—82no—
593C2×C4—82no—
594C2×C4—88no—
595C2×C4—82no—
596C2×C4—82no—
597C2×C4C22×C4211yescharacteristic
598C2×C4C2×C4×D411yescharacteristic
599C2×C4C2×C4×D411yescharacteristic
600C2×C4C2×C4×D411yescharacteristic
601C2×C4C2×C4×D411yescharacteristic
602C2×C4C2×C4×D411yescharacteristic
603C2×C4C4×#4111yescharacteristic
604C2×C4C4×#4111yescharacteristic
605C2×C4C4×#4111yescharacteristic
606C2×C4C4×#4111yescharacteristic
607C2×C4#51811yescharacteristic
608C2×C4#51911yescharacteristic
609C2×C4#52011yescharacteristic
610C2×C4#52011yescharacteristic
611C2×C4#52011yescharacteristic
612C2×C4#52011yescharacteristic
613C2×C4D4211yescharacteristic
614C2×C4#54611yescharacteristic
615C2×C4#54711yescharacteristic
616C2×C4#54711yescharacteristic
617C2×C4#54811yescharacteristic
618C2×C4#54811yescharacteristic
619C2×C4#55111yescharacteristic
620C2×C4#55211yescharacteristic
621C2×C4#55311yescharacteristic
622C2×C4#55311yescharacteristic
623C2×C4#55611yescharacteristic
624C2×C4#55911yescharacteristic
625C8—22no—
626C8—22no—
627C8—22no—
628C8—22no—
629C8—24no—
630C8—22no—
631C8—22no—
632C8—42no—
633C8—42no—
634C8—42no—
635C8—42no—
636C22—21no—
637C22—21no—
638C22—21no—
639C22—21no—
640C22—22no—
641C22—22no—
642C22—22no—
643C22—22no—
644C22—42no—
645C22—42no—
646C22—42no—
647C22—82no—
648C22—82no—
649C22C42×D411yescharacteristic
650C22#235811yescharacteristic
651C22#235911yescharacteristic
652C22#236211yescharacteristic
653C22#236711yescharacteristic
654C22#237311yescharacteristic
655C22#237311yescharacteristic
656C22#295512yes—
657C22#295712yes—
658C4—44no—
659C4—42no—
660C4—42no—
661C4—42no—
662C4—42no—
663C4—44no—
664C4—42no—
665C4—82no—
666C4—82no—
667C4C42×D411yescharacteristic
668C4#227411yescharacteristic
669C4#235911yescharacteristic
670C4#236111yescharacteristic
671C4#236111yescharacteristic
672C4#236211yescharacteristic
673C4#236711yescharacteristic
674C4#237411yescharacteristic
675C2—21no—
676C2—21no—
677C2—21no—
678C2—21no—
679C2—82no—
680C2#1382311yescharacteristic
681C2#2051011yescharacteristic
682C2#2051311yescharacteristic
683C2#2288712yes—
684C2#2308212yes—
subgroupG/H
centerC22×C4C22×D4
commutatorC2×C4C22×C42
Frattini, cosocleC23×C4C24
socleC23C2×C4×D4
prime radicalC22×#41C23
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C2×#336 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. C2×#1773C2semidirect
5. #20439C2semidirect
10. #10911C2semidirect
11. #10909C2semidirect
12. #10887C2semidirect
13. #10879C2semidirect
14. #10706C2semidirect
15. #10694C2semidirect
35. #1938C4semidirect
36. #1929C4semidirect
37. #1865C4semidirect
38. #1864C4semidirect
39. #1863C4semidirect
40. #1862C4semidirect
93. C2×#108C2×C4semidirect
99. #424C2×C4semidirect
133. #405C2×C4semidirect
143. C2×#97C2×C4semidirect
162. #341C2×C4semidirect
168. #339C2×C4semidirect
169. #339C2×C4semidirect
178. #336C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#19382
C2×#19292
#208722
C2×#17732
#204392
C2×#17592
#203782
#109342
#109292
#109112
#109092
#108872
#108792
#107062
#106942