Finite Group Databasebeta

← #360773

Subgroups — #360773

Order 512.

1408 conjugacy classes of subgroups.

The subgroup lattice has 733 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2913C211yesmaximal, characteristic
2#20882C211yesmaximal, characteristic
3#20881C211yesmaximal, characteristic
4C2×#1860C211yesmaximal, characteristic
5C2×#1859C211yesmaximal, characteristic
6#20802C211yesmaximal, characteristic, complemented
7C2×#1847C211yesmaximal, characteristic, complemented
8C2×#1815C211yesmaximal, characteristic
9C2×#1765C211yesmaximal, characteristic
10#11340C211yesmaximal, characteristic, complemented
11#11339C211yesmaximal, characteristic, complemented
12#11331C211yesmaximal, characteristic, complemented
13#11330C211yesmaximal, characteristic, complemented
14#10870C211yesmaximal, characteristic, complemented
15#10860C211yesmaximal, characteristic, complemented
16C22×#131C2211yescharacteristic
17C2×#520C2211yescharacteristic
18C2×#520C2211yescharacteristic
19C22×#108C2211yescharacteristic
20#2913—24no—
21#2913—24no—
22C2×#428—21no—
23C2×#428—21no—
24C2×#428C2211yescharacteristic
25C2×#428C2211yescharacteristic
26C2×#428C2211yescharacteristic
27C2×#428C2211yescharacteristic
28C2×#427—21no—
29C2×#427C2211yescharacteristic
30C22×#107—21no—
31C22×#107C2211yescharacteristic
32C22×#107C2211yescharacteristic
33C22×#106C2211yescharacteristic
34C22×#106C2211yescharacteristic
35C22×#104C2211yescharacteristic
36C2×#417C2211yescharacteristic
37C2×#417C2211yescharacteristic
38C2×#416C2211yescharacteristic
39C2×#405C2211yescharacteristic
40C22×#97C2211yescharacteristic
41#2317C2211yescharacteristic
42#2317C2211yescharacteristic
43C2×#384C2211yescharacteristic
44C2×#382C2211yescharacteristic
45#1865—22no—
46#1864—22no—
47#1863—22no—
48#1862—22no—
49#1860—24no—
50#1859—24no—
51#1847—24no—
52#1847—21no—
53#1847—21no—
54#1846—21no—
55#1844—21no—
56#1815—24no—
57#1765C418yescomplemented
58C2×#343C2211yescharacteristic
59C2×#343C2211yescharacteristic
60C2×#343C2211yescharacteristic
61C2×#343C2211yescharacteristic
62C2×#341C2211yescharacteristic
63#1736C2211yescharacteristic
64#1736C2211yescharacteristic
65C2×#339C2211yescharacteristic
66C2×#339C2211yescharacteristic
67C2×#339C2211yescharacteristic
68#1729C2211yescharacteristic
69#1729C2211yescharacteristic
70C22×#41C2311yescharacteristic
71C2×#131—22no—
72C2×#131—22no—
73C2×#131—22no—
74C2×#131C2×C414yes—
75C2×#131D414yes—
76C2×#131Q814yes—
77C22×M16—21no—
78C22×M16C2311yescharacteristic
79C22×M16C2311yescharacteristic
80C22×M16C2311yescharacteristic
81C22×M16C2311yescharacteristic
82C23×C8—21no—
83C23×C8C2311yescharacteristic
84C23×C8C2311yescharacteristic
85#520—22no—
86#520—22no—
87#520—22no—
88#520—22no—
89#520—22no—
90#520—22no—
91#520—41no—
92#520—41no—
93C2×C4×Q8—21no—
94C2×C4×Q8—21no—
95C2×C4×D4—21no—
96C2×C4×D4—21no—
97C2×C4×D4—41no—
98C2×C4×D4—41no—
99C2×#117—21no—
100C2×#117—21no—
101C2×#117C2311yescharacteristic
102C2×#117C2311yescharacteristic
103C2×#117C2311yescharacteristic
104C22×C4⋊C4—21no—
105C22×C4⋊C4—21no—
106C22×C4⋊C4—41no—
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2311yescharacteristic
109C22×C4⋊C4C2311yescharacteristic
110C22×C4⋊C4C2311yescharacteristic
111C22×C42C2311yescharacteristic
112C2×#108—22no—
113C2×#108—22no—
114C2×#108—22no—
115#428—28no—
116#428—28no—
117#428—22no—
118#428—22no—
119#428—22no—
120#428—22no—
121#428—22no—
122#428—22no—
123#428—41no—
124#428—41no—
125#427—28no—
126#427—22no—
127#427—22no—
128C2×#107—28no—
129C2×#107—21no—
130C2×#107—21no—
131C2×#107—22no—
132C2×#107—21no—
133C2×#107—21no—
134C2×#107—22no—
135C2×#107—21no—
136C2×#107—21no—
137C2×#107—22no—
138C2×#107—41no—
139C2×#107D411yescharacteristic
140C2×#107D414yes—
141C2×#107D411yescharacteristic
142C2×#106—21no—
143C2×#106—21no—
144C2×#106—22no—
145C2×#106—21no—
146C2×#106—22no—
147C2×#106—21no—
148C2×#106—22no—
149C2×#106—21no—
150C2×#106—21no—
151C2×#106—22no—
152C2×#106—41no—
153C2×#106D411yescharacteristic
154C2×#106D411yescharacteristic
155C2×#104C2×C414yes—
156C2×#104D414yes—
157C2×#104Q814yes—
158#417—22no—
159#417—22no—
160#417—22no—
161#417—22no—
162#416C2×C414yes—
163#416Q814yes—
164C2×#103—21no—
165C2×#103—21no—
166C2×#103D412yes—
167C2×#102—21no—
168C2×#102—21no—
169C2×#102D412yes—
170#405—22no—
171#405C2×C414yescomplemented
172C2×#97—22no—
173C2×#97—21no—
174C2×#97—22no—
175C2×#97C2×C414yescomplemented
176C2×C4×C8—21no—
177#389—22no—
178#389—22no—
179#385—41no—
180#385—41no—
181#384—21no—
182#384—21no—
183#384—22no—
184#384—22no—
185#382—21no—
186#382—21no—
187#382—22no—
188#382—22no—
189C2×#95—41no—
190#343—22no—
191#343—22no—
192#343—22no—
193#343—22no—
194#341C2×C414yescomplemented
195#340—22no—
196#340—22no—
197#339—22no—
198#339—22no—
199#339C2×C414yescomplemented
200#336—22no—
201#336—22no—
202C2×#41—21no—
203C2×#41—21no—
204C2×#41—21no—
205C2×#41—21no—
206C2×#41—21no—
207C2×#41—21no—
208C2×#41C22×C412yes—
209C2×#41C2×D412yes—
210C2×#41C2×D412yes—
211C2×#41C2×Q812yes—
212C2×#41C2×Q812yes—
213C2×#41C2×Q812yes—
214C22×Q8C2×D411yescharacteristic
215C22×D4—21no—
216C22×D4C2×D411yescharacteristic
217C23×C4—21no—
218C23×C4—21no—
219C23×C4—21no—
220C23×C4—21no—
221C23×C4—21no—
222C23×C4—21no—
223C23×C4—41no—
224C23×C4—41no—
225C23×C4C2411yescharacteristic
226#131—28no—
227#131—28no—
228#131—28no—
229#131C4⋊C4116yescomplemented
230C2×M16—21no—
231C2×M16—21no—
232C2×M16—21no—
233C2×M16—21no—
234C2×M16—21no—
235C2×M16—21no—
236C2×M16—24no—
237C2×M16—24no—
238C2×M16—24no—
239C2×M16—21no—
240C2×M16—21no—
241C2×M16—21no—
242C2×M16—21no—
243C2×M16—21no—
244C2×M16—21no—
245C2×M16—21no—
246C2×M16—21no—
247C2×M16—21no—
248C2×M16—21no—
249C2×M16—21no—
250C2×M16—21no—
251C2×M16C22×C412yes—
252C2×M16C22×C412yes—
253C2×M16C2×D412yes—
254C2×M16C2×D412yes—
255C2×M16C2×Q812yes—
256C2×M16C2×Q812yes—
257C22×C8—21no—
258C22×C8—21no—
259C22×C8—21no—
260C22×C8—21no—
261C22×C8—21no—
262C22×C8—21no—
263C22×C8—24no—
264C22×C8—24no—
265C22×C8—24no—
266C22×C8—21no—
267C22×C8—21no—
268C22×C8C22×C412yes—
269C22×C8C22×C412yes—
270C22×C8C2×D412yes—
271C22×C8C2×D412yes—
272C22×C8C2×D411yescharacteristic
273C22×C8C2×D411yescharacteristic
274C22×C8C2×D412yes—
275C22×C8C2×Q812yes—
276C22×C8C2×Q812yes—
277C22×C8#4112yes—
278C22×C8#4112yes—
279C4×Q8—42no—
280C4×Q8—42no—
281C4×Q8—42no—
282C4×Q8—42no—
283C4×D4—42no—
284C4×D4—42no—
285C4×D4—44no—
286C4×D4—44no—
287C4×D4—44no—
288C4×D4—44no—
289C4×D4—42no—
290C4×D4—42no—
291#117—21no—
292#117—21no—
293#117—24no—
294#117—24no—
295#117—21no—
296#117—21no—
297#117—21no—
298#117—21no—
299#117—21no—
300#117—21no—
301#117—21no—
302#117—21no—
303#117—42no—
304#117—42no—
305#117C22×C412yes—
306#117C2×Q812yes—
307C2×C4⋊C4—24no—
308C2×C4⋊C4—24no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—21no—
315C2×C4⋊C4—21no—
316C2×C4⋊C4—21no—
317C2×C4⋊C4—21no—
318C2×C4⋊C4—21no—
319C2×C4⋊C4—21no—
320C2×C4⋊C4—21no—
321C2×C4⋊C4—21no—
322C2×C4⋊C4—21no—
323C2×C4⋊C4—21no—
324C2×C4⋊C4—21no—
325C2×C4⋊C4—21no—
326C2×C4⋊C4—21no—
327C2×C4⋊C4—21no—
328C2×C4⋊C4—21no—
329C2×C4⋊C4—21no—
330C2×C4⋊C4—21no—
331C2×C4⋊C4—42no—
332C2×C4⋊C4—42no—
333C2×C4⋊C4—41no—
334C2×C4⋊C4—41no—
335C2×C4⋊C4—44no—
336C2×C4⋊C4—42no—
337C2×C4⋊C4—42no—
338C2×C4⋊C4—84no—
339C2×C4⋊C4C2×D411yescharacteristic
340C2×C4⋊C4C2×D411yescharacteristic
341C2×C4⋊C4C2×D412yes—
342C2×C4⋊C4C2×D412yes—
343C2×C4⋊C4C2×D411yescharacteristic
344C2×C4⋊C4C2×D411yescharacteristic
345C2×C4⋊C4C2×D412yes—
346C2×C22⋊C4—21no—
347C2×C22⋊C4—21no—
348C2×C22⋊C4—21no—
349C2×C22⋊C4—41no—
350C2×C22⋊C4—41no—
351C2×C42—21no—
352C2×C42—21no—
353C2×C42—21no—
354C2×C42—21no—
355C2×C42—21no—
356C2×C42—21no—
357C2×C42—41no—
358C2×C42—41no—
359C2×C42C22×C412yes—
360C2×C42C2×D412yes—
361C2×C42C2×Q812yes—
362#108—44no—
363#107—24no—
364#107—24no—
365#107—22no—
366#107—24no—
367#107—24no—
368#107—22no—
369#107—44no—
370#107—42no—
371#107—42no—
372#106—24no—
373#106—24no—
374#106—22no—
375#106—22no—
376#106—44no—
377#106—42no—
378#106—42no—
379#106—42no—
380#106—42no—
381#104C4⋊C4116yescomplemented
382#103—42no—
383#103—42no—
384#103C4⋊C418yescomplemented
385#102—42no—
386#102—42no—
387#102C4⋊C418yescomplemented
388#97—28no—
389#97—24no—
390C4×C8—24no—
391#95—44no—
392#95—42no—
393#95—42no—
394C24—21no—
395C24H(5,𝔽2)11yescharacteristic
396#41—22no—
397#41—22no—
398#41—22no—
399#41—22no—
400#41—22no—
401#41—22no—
402#41—22no—
403#41—22no—
404#41—22no—
405#41—22no—
406#41—22no—
407#41—22no—
408#41C2×C4⋊C414yes—
409#41C2×C4⋊C414yes—
410#41C4⋊Q814yes—
411#41C4⋊Q814yes—
412C2×Q8—21no—
413C2×Q8—21no—
414C2×Q8—21no—
415C2×Q8C2×C4⋊C412yes—
416C2×Q8#12712yes—
417C2×Q8C4⋊Q812yes—
418C2×D4—22no—
419C2×D4—22no—
420C2×D4—22no—
421C2×D4—22no—
422C2×D4—22no—
423C2×D4—22no—
424C2×D4—21no—
425C2×D4—21no—
426C2×D4—21no—
427C2×D4C2×C4⋊C412yes—
428C2×D4#12712yes—
429C2×D4C4⋊Q812yes—
430C22×C4—21no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—42no—
448C22×C4—42no—
449C22×C4—41no—
450C22×C4—41no—
451C22×C4—44no—
452C22×C4—44no—
453C22×C4—41no—
454C22×C4—41no—
455C22×C4—41no—
456C22×C4—41no—
457C22×C4—41no—
458C22×C4—41no—
459C22×C4—41no—
460C22×C4—41no—
461C22×C4—82no—
462C22×C4—82no—
463C22×C4—82no—
464C22×C4—82no—
465C22×C4—162no—
466C22×C4—162no—
467C22×C4C23×C411yescharacteristic
468C22×C4C22×D411yescharacteristic
469C22×C4C22×D411yescharacteristic
470C22×C4C22×D411yescharacteristic
471C22×C4C22×D411yescharacteristic
472C22×C4C22×D411yescharacteristic
473C22×C4C22×Q811yescharacteristic
474C22×C4C22×Q811yescharacteristic
475C22×C4C22×Q811yescharacteristic
476C22×C4C2×#4111yescharacteristic
477C22×C4C2×#4111yescharacteristic
478C22×C4H(5,𝔽2)11yescharacteristic
479C22×C42−1+411yescharacteristic
480C22×C42−1+411yescharacteristic
481M16—22no—
482M16—22no—
483M16—22no—
484M16—22no—
485M16—22no—
486M16—22no—
487M16—216no—
488M16—41no—
489M16—41no—
490M16—41no—
491M16—41no—
492M16—41no—
493M16—41no—
494M16—41no—
495M16—41no—
496M16C2×C4⋊C414yes—
497M16C2×C4⋊C414yes—
498C2×C8—22no—
499C2×C8—22no—
500C2×C8—22no—
501C2×C8—22no—
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×C8—216no—
511C2×C8—41no—
512C2×C8—41no—
513C2×C8—41no—
514C2×C8—41no—
515C2×C8—41no—
516C2×C8—41no—
517C2×C8C2×C4⋊C414yes—
518C2×C8C2×C4⋊C414yes—
519C2×C8C4×D412yes—
520C2×C8C4×D412yes—
521C2×C8C4×D414yes—
522C2×C8#12012yes—
523C2×C8#12012yes—
524C2×C8C22⋊Q812yes—
525C2×C8C22⋊Q812yes—
526C2×C8C22⋊Q814yes—
527C2×C8#12314yes—
528C4⋊C4—22no—
529C4⋊C4—21no—
530C4⋊C4—21no—
531C4⋊C4—22no—
532C4⋊C4—22no—
533C4⋊C4—21no—
534C4⋊C4—21no—
535C4⋊C4—22no—
536C4⋊C4—21no—
537C4⋊C4—21no—
538C4⋊C4—22no—
539C4⋊C4—21no—
540C4⋊C4—21no—
541C4⋊C4—44no—
542C4⋊C4—44no—
543C4⋊C4—41no—
544C4⋊C4—41no—
545C4⋊C4—41no—
546C4⋊C4—41no—
547C4⋊C4—41no—
548C4⋊C4—41no—
549C4⋊C4—41no—
550C4⋊C4—41no—
551C4⋊C4—41no—
552C4⋊C4—41no—
553C4⋊C4—41no—
554C4⋊C4—41no—
555C4⋊C4—88no—
556C4⋊C4—162no—
557C4⋊C4—162no—
558C4⋊C4C2×C4⋊C414yes—
559C22⋊C4—22no—
560C22⋊C4—22no—
561C22⋊C4—44no—
562C22⋊C4—44no—
563C22⋊C4—41no—
564C22⋊C4—41no—
565C22⋊C4—41no—
566C22⋊C4—41no—
567C42—22no—
568C42—22no—
569C42—22no—
570C42—22no—
571C42—22no—
572C42—44no—
573C42—44no—
574C42—41no—
575C42—41no—
576C42—41no—
577C42—41no—
578C42C2×C4⋊C414yes—
579C23—22no—
580C23—22no—
581C23—22no—
582C23—22no—
583C23—22no—
584C23—42no—
585C23—42no—
586C23#52011yescharacteristic
587C23#53511yescharacteristic
588C23#54411yescharacteristic
589C23D4211yescharacteristic
590C23D4211yescharacteristic
591C23#55411yescharacteristic
592C23#55411yescharacteristic
593Q8—22no—
594Q8—22no—
595Q8—22no—
596Q8#38414yes—
597D4—24no—
598D4—24no—
599D4—24no—
600D4—24no—
601D4—22no—
602D4—22no—
603D4—22no—
604D4#38414yes—
605C2×C4—22no—
606C2×C4—22no—
607C2×C4—22no—
608C2×C4—24no—
609C2×C4—24no—
610C2×C4—24no—
611C2×C4—24no—
612C2×C4—24no—
613C2×C4—24no—
614C2×C4—42no—
615C2×C4—44no—
616C2×C4—42no—
617C2×C4—44no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—41no—
621C2×C4—41no—
622C2×C4—41no—
623C2×C4—41no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—82no—
631C2×C4—82no—
632C2×C4—82no—
633C2×C4—82no—
634C2×C4—82no—
635C2×C4—82no—
636C2×C4—81no—
637C2×C4—81no—
638C2×C4—82no—
639C2×C4—82no—
640C2×C4—161no—
641C2×C4—161no—
642C2×C4—164no—
643C2×C4—164no—
644C2×C4—161no—
645C2×C4—161no—
646C2×C4C22×C4⋊C411yescharacteristic
647C2×C4C22×C4⋊C411yescharacteristic
648C2×C4C22×C4⋊C411yescharacteristic
649C2×C4C2×C4×D411yescharacteristic
650C2×C4C2×C4×D411yescharacteristic
651C2×C4#52011yescharacteristic
652C2×C4C2×#12011yescharacteristic
653C2×C4C2×C22⋊Q811yescharacteristic
654C2×C4C2×C22⋊Q811yescharacteristic
655C2×C4C2×#12311yescharacteristic
656C2×C4C2×#12711yescharacteristic
657C2×C4C2×C4⋊Q811yescharacteristic
658C2×C4C2×C4⋊Q811yescharacteristic
659C2×C4C2×C4⋊Q811yescharacteristic
660C2×C4#53611yescharacteristic
661C2×C4#54411yescharacteristic
662C2×C4D4211yescharacteristic
663C2×C4D4211yescharacteristic
664C2×C4#54711yescharacteristic
665C2×C4#54711yescharacteristic
666C2×C4#54711yescharacteristic
667C2×C4#54711yescharacteristic
668C2×C4D4×Q811yescharacteristic
669C2×C4D4×Q811yescharacteristic
670C2×C4D4×Q811yescharacteristic
671C2×C4D4×Q811yescharacteristic
672C2×C4#55411yescharacteristic
673C2×C4#55411yescharacteristic
674C8—24no—
675C8—24no—
676C8—24no—
677C8—81no—
678C8—81no—
679C8#38514yes—
680C8#38514yes—
681C22—22no—
682C22—22no—
683C22—21no—
684C22—21no—
685C22—21no—
686C22—21no—
687C22—22no—
688C22—22no—
689C22—24no—
690C22—44no—
691C22—42no—
692C22—42no—
693C22—44no—
694C22#231711yescharacteristic
695C22D4×C4⋊C411yescharacteristic
696C22D4×C4⋊C411yescharacteristic
697C22#243311yescharacteristic
698C22#244011yescharacteristic
699C22#244011yescharacteristic
700C22#250211yescharacteristic
701C22#328812yes—
702C22#328912yes—
703C4—24no—
704C4—44no—
705C4—44no—
706C4—81no—
707C4—81no—
708C4—81no—
709C4—81no—
710C4—82no—
711C4—82no—
712C4—161no—
713C4—161no—
714C4C2×#38411yescharacteristic
715C4C2×#38511yescharacteristic
716C4D4×C4⋊C411yescharacteristic
717C4D4×C4⋊C411yescharacteristic
718C4#243411yescharacteristic
719C4#244011yescharacteristic
720C4#244011yescharacteristic
721C4#250111yescharacteristic
722C2—21no—
723C2—21no—
724C2—21no—
725C2—21no—
726C2—44no—
727C2#1388011yescharacteristic
728C2#2269312yes—
729C2#2580911yescharacteristic
730C2#2581011yescharacteristic
731C2#2608512yes—
subgroupG/H
center, socleC23D42
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC22×#41C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
6. #20802C2semidirect
7. C2×#1847C2semidirect
10. #11340C2semidirect
11. #11339C2semidirect
12. #11331C2semidirect
13. #11330C2semidirect
14. #10870C2semidirect
15. #10860C2semidirect
57. #1765C4semidirect
171. #405C2×C4semidirect
175. C2×#97C2×C4semidirect
194. #341C2×C4semidirect
199. #339C2×C4semidirect
229. #131C4⋊C4semidirect
381. #104C4⋊C4semidirect
384. #103C4⋊C4semidirect
387. #102C4⋊C4semidirect

Maximal subgroups

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

subgroupindex
C2×#29132
#208822
#208812
C2×#18602
C2×#18592
#208022
C2×#18472
C2×#18152
C2×#17652
#113402
#113392
#113312
#113302
#108702
#108602