Finite Group Databasebeta

← #209308

Subgroups — #209308

Order 512.

964 conjugacy classes of subgroups.

The subgroup lattice has 649 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2962C211yesmaximal, characteristic, complemented
2#23568C211yesmaximal, characteristic
3#21486C211yesmaximal, characteristic
4C2×#1582C211yesmaximal, characteristic, complemented
5C2×C8×SD16C211yesmaximal, characteristic, complemented
6#8952C211yesmaximal, characteristic, complemented
7#8948C211yesmaximal, characteristic, complemented
8#8939C211yesmaximal, characteristic
9#8850C211yesmaximal, characteristic, complemented
10#8846C211yesmaximal, characteristic, complemented
11#8830C211yesmaximal, characteristic, complemented
12#8314C211yesmaximal, characteristic, complemented
13#8207C211yesmaximal, characteristic, complemented
14#8203C211yesmaximal, characteristic, complemented
15#8146C211yesmaximal, characteristic, complemented
16C2×#533C2211yescharacteristic
17#2962—24no—
18#2962—24no—
19C2×#446—21no—
20C2×#446—21no—
21C2×#446C2211yescharacteristic
22C2×#446C2211yescharacteristic
23C2×C8×Q8C2211yescharacteristic
24C2×C8×Q8C2211yescharacteristic
25C2×C4×SD16C2211yescharacteristic
26C2×C4×SD16C2211yescharacteristic
27C2×#435—21no—
28C2×C8×D4C2211yescharacteristic
29C2×#424—21no—
30C22×C4⋊C8C2211yescharacteristic
31C2×#405—21no—
32C2×C4×M16C2211yescharacteristic
33C4×#121C2211yescharacteristic
34#2058C411yescharacteristic
35#2043C411yescharacteristic
36#2035C411yescharacteristic
37#2035C411yescharacteristic
38#2023C411yescharacteristic
39#2013C411yescharacteristic
40#2000C411yescharacteristic
41#2000C411yescharacteristic
42#1954C2211yescharacteristic
43#1939C2211yescharacteristic
44#1924—21no—
45#1870—21no—
46#1797C2211yescharacteristic
47#1770C2211yescharacteristic
48C4×#103C2211yescharacteristic
49C4×#103C2211yescharacteristic
50#1752—21no—
51C4×C22⋊C8C2211yescharacteristic
52C4×C22⋊C8C2211yescharacteristic
53#1582—24no—
54C8×SD16—24no—
55C2×#334C2211yescharacteristic
56C2×#334C2211yescharacteristic
57C2×Q8⋊C8C2211yescharacteristic
58C2×Q8⋊C8C2211yescharacteristic
59C2×Q8⋊C8C2211yescharacteristic
60C2×Q8⋊C8C2211yescharacteristic
61C2×D4⋊C8C2211yescharacteristic
62C2×D4⋊C8C2211yescharacteristic
63C2×#322C2211yescharacteristic
64C2×C82C2211yescharacteristic
65#1279C2211yescharacteristic
66#1279C2211yescharacteristic
67#1277C2211yescharacteristic
68#1277C2211yescharacteristic
69#1273C2211yescharacteristic
70#1273C2211yescharacteristic
71C22×SD16C2×C411yescharacteristic
72C22×SD16C2×C411yescharacteristic
73C22×M16D411yescharacteristic
74C22×M16D411yescharacteristic
75C23×C8—21no—
76C23×C8—21no—
77#533—22no—
78#533—22no—
79#533—22no—
80#533—22no—
81C2×C4⋊Q8C2×C411yescharacteristic
82C2×C4⋊Q8C2×C411yescharacteristic
83C2×#125—21no—
84C2×C22⋊Q8—21no—
85C2×C22⋊Q8C2×C411yescharacteristic
86C2×C22⋊Q8C2×C411yescharacteristic
87C2×#121C2×C411yescharacteristic
88C2×C4×Q8—21no—
89C2×C4×Q8C2311yescharacteristic
90C2×C4×Q8C2311yescharacteristic
91C2×C4×D4—21no—
92C2×C4×D4C2311yescharacteristic
93C2×#117—21no—
94C22×C42C2311yescharacteristic
95#446—28no—
96#446—28no—
97#446—22no—
98#446—22no—
99#446—22no—
100#446D414yescomplemented
101C8×Q8—22no—
102C8×Q8—22no—
103C8×Q8—22no—
104C8×Q8D414yescomplemented
105C4×SD16—22no—
106C4×SD16—22no—
107C4×SD16—22no—
108C4×SD16—22no—
109#435—44no—
110C8×D4—22no—
111C8×D4—22no—
112C2×#106C2×C411yescharacteristic
113C2×#106C2×C411yescharacteristic
114#424—28no—
115C2×C4⋊C8—21no—
116C2×C4⋊C8—21no—
117C2×C4⋊C8—21no—
118C2×C4⋊C8—21no—
119C2×C4⋊C8—21no—
120C2×C4⋊C8—22no—
121C2×C4⋊C8—22no—
122C2×C4⋊C8—21no—
123C2×C4⋊C8C2311yescharacteristic
124C2×C4⋊C8C2311yescharacteristic
125C2×C4⋊C8C2311yescharacteristic
126C2×C4⋊C8C2311yescharacteristic
127C2×#103C2×C411yescharacteristic
128C2×#103C2×C411yescharacteristic
129C2×#103C2×C411yescharacteristic
130C2×#103C2×C411yescharacteristic
131C2×#103C2×C411yescharacteristic
132C2×#103C2×C411yescharacteristic
133C2×#102C2×C411yescharacteristic
134C2×#102C2×C411yescharacteristic
135C2×C22⋊C8—21no—
136C2×C22⋊C8—21no—
137C2×C22⋊C8—21no—
138C2×C22⋊C8—21no—
139C2×C22⋊C8—21no—
140C2×C22⋊C8C2×C411yescharacteristic
141C2×C22⋊C8C2×C411yescharacteristic
142#405—28no—
143C4×M16—22no—
144C4×M16—22no—
145C2×#97—21no—
146C2×#97—21no—
147C2×#97—21no—
148C2×#97D411yescharacteristic
149C2×#97D411yescharacteristic
150C2×C4×C8—21no—
151C2×C4×C8—21no—
152C2×C4×C8C2311yescharacteristic
153C2×C4×C8C2311yescharacteristic
154C2×C4×C8C2311yescharacteristic
155C2×C4×C8C2311yescharacteristic
156C2×C4×C8C2311yescharacteristic
157C2×C4×C8C2311yescharacteristic
158#390C2×C411yescharacteristic
159#389C2×C411yescharacteristic
160#388—21no—
161#386—21no—
162#380C2×C411yescharacteristic
163C4×C4⋊C4C2311yescharacteristic
164C4×C22⋊C4—21no—
165#340C2×C411yescharacteristic
166#340C2×C411yescharacteristic
167#340C2×C411yescharacteristic
168#340C2×C411yescharacteristic
169#336—21no—
170#336—21no—
171#336—21no—
172#336—21no—
173#336C2×C411yescharacteristic
174#336C2×C411yescharacteristic
175#334—22no—
176#334—22no—
177Q8⋊C8—22no—
178Q8⋊C8—22no—
179Q8⋊C8—22no—
180Q8⋊C8—22no—
181D4⋊C8—22no—
182D4⋊C8—22no—
183#322—22no—
184C82—22no—
185C22×Q8—21no—
186C22×Q8C22×C411yescharacteristic
187C22×Q8C22×C411yescharacteristic
188C22×D4—21no—
189C22×D4C22×C411yescharacteristic
190C23×C4—21no—
191C23×C4C22×C411yescharacteristic
192C23×C4C2×D411yescharacteristic
193C23×C4#4111yescharacteristic
194C2×SD16—22no—
195C2×SD16—22no—
196C2×SD16—22no—
197C2×SD16—22no—
198C2×SD16—22no—
199C2×SD16—41no—
200C2×M16—22no—
201C2×M16—22no—
202C2×M16—22no—
203C2×M16—22no—
204C2×M16—41no—
205C2×M16SD1614yescomplemented
206C22×C8—24no—
207C22×C8—21no—
208C22×C8—21no—
209C22×C8—21no—
210C22×C8—21no—
211C22×C8—21no—
212C22×C8—21no—
213C22×C8—42no—
214C22×C8—42no—
215C22×C8—42no—
216C22×C8—42no—
217C22×C8—42no—
218C22×C8C22×C411yescharacteristic
219C22×C8C22×C411yescharacteristic
220C22×C8C22×C411yescharacteristic
221C22×C8C22×C411yescharacteristic
222C22×C8C2×D411yescharacteristic
223C22×C8C2×D411yescharacteristic
224C22×C8C2×D411yescharacteristic
225C22×C8C2×D411yescharacteristic
226C4⋊Q8—22no—
227C4⋊Q8—22no—
228C4⋊Q8—22no—
229C4⋊Q8—22no—
230#125—24no—
231#125—24no—
232C22⋊Q8—24no—
233C22⋊Q8—24no—
234C22⋊Q8—22no—
235C22⋊Q8—22no—
236C22⋊Q8—22no—
237C22⋊Q8—22no—
238#121—22no—
239#121—41no—
240C4×Q8—24no—
241C4×Q8—24no—
242C4×Q8—21no—
243C4×Q8—21no—
244C4×Q8—21no—
245C4×Q8—21no—
246C4×Q8C2×D412yes—
247C4×Q8C2×D412yes—
248C4×Q8#4112yes—
249C4×Q8#4112yes—
250C4×D4—21no—
251C4×D4—21no—
252C4×D4—21no—
253C4×D4—21no—
254C4×D4—42no—
255C4×D4—42no—
256#117—24no—
257#117—24no—
258C2×C4⋊C4—21no—
259C2×C4⋊C4—21no—
260C2×C4⋊C4—21no—
261C2×C4⋊C4—21no—
262C2×C4⋊C4—21no—
263C2×C4⋊C4—21no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—21no—
266C2×C4⋊C4C22×C411yescharacteristic
267C2×C4⋊C4C22×C411yescharacteristic
268C2×C4⋊C4C22×C411yescharacteristic
269C2×C4⋊C4C22×C411yescharacteristic
270C2×C4⋊C4C22×C411yescharacteristic
271C2×C22⋊C4—21no—
272C2×C22⋊C4—21no—
273C2×C22⋊C4—21no—
274C2×C22⋊C4—21no—
275C2×C22⋊C4—21no—
276C2×C22⋊C4—21no—
277C2×C22⋊C4—21no—
278C2×C42—21no—
279C2×C42—21no—
280C2×C42—21no—
281C2×C42—21no—
282C2×C42—21no—
283C2×C42—21no—
284C2×C42—21no—
285C2×C42C22×C411yescharacteristic
286C2×C42C2×D411yescharacteristic
287C2×C42#4111yescharacteristic
288C2×C42C2411yescharacteristic
289#106—22no—
290#106—22no—
291C4⋊C8—24no—
292C4⋊C8—24no—
293C4⋊C8—24no—
294C4⋊C8—24no—
295C4⋊C8—24no—
296C4⋊C8—21no—
297C4⋊C8—21no—
298C4⋊C8—21no—
299C4⋊C8—21no—
300C4⋊C8—21no—
301C4⋊C8—21no—
302C4⋊C8—42no—
303C4⋊C8C2×D412yes—
304C4⋊C8C2×D412yes—
305#103—22no—
306#103—22no—
307#103—22no—
308#103—41no—
309#103—41no—
310#103—41no—
311#102—22no—
312#102—22no—
313C22⋊C8—22no—
314C22⋊C8—41no—
315C22⋊C8—42no—
316C22⋊C8—42no—
317C22⋊C8—42no—
318C22⋊C8—42no—
319C22⋊C8—42no—
320#97—24no—
321#97—24no—
322#97—22no—
323#97—22no—
324#97—42no—
325C4×C8—24no—
326C4×C8—24no—
327C4×C8—21no—
328C4×C8—21no—
329C4×C8—21no—
330C4×C8—21no—
331C4×C8—21no—
332C4×C8—21no—
333C4×C8—21no—
334C4×C8—21no—
335C4×C8—21no—
336C4×C8—21no—
337C4×C8C2×D412yes—
338C4×C8C2×D412yes—
339#95—21no—
340#95—21no—
341#95—21no—
342C24—21no—
343C24C4×D411yescharacteristic
344C2×Q8—24no—
345C2×Q8—21no—
346C2×Q8—21no—
347C2×Q8—21no—
348C2×Q8—21no—
349C2×Q8—21no—
350C2×Q8—21no—
351C2×Q8—21no—
352C2×Q8—21no—
353C2×Q8—21no—
354C2×Q8—21no—
355C2×Q8—81no—
356C2×Q8—81no—
357C2×Q8C4×D412yes—
358C2×Q8C4×D412yes—
359C2×D4—21no—
360C2×D4—21no—
361C2×D4—21no—
362C2×D4—21no—
363C2×D4—21no—
364C2×D4—42no—
365C2×D4—42no—
366C2×D4—81no—
367C2×D4#13112yes—
368C22×C4—21no—
369C22×C4—21no—
370C22×C4—21no—
371C22×C4—21no—
372C22×C4—21no—
373C22×C4—21no—
374C22×C4—21no—
375C22×C4—21no—
376C22×C4—21no—
377C22×C4—21no—
378C22×C4—21no—
379C22×C4—21no—
380C22×C4—21no—
381C22×C4—21no—
382C22×C4—21no—
383C22×C4—21no—
384C22×C4—21no—
385C22×C4—21no—
386C22×C4—21no—
387C22×C4—41no—
388C22×C4—41no—
389C22×C4—41no—
390C22×C4—41no—
391C22×C4—41no—
392C22×C4—41no—
393C22×C4—41no—
394C22×C4—41no—
395C22×C4C4×D411yescharacteristic
396C22×C4C4×D411yescharacteristic
397C22×C4C4×D411yescharacteristic
398C22×C4C2×SD1612yes—
399C22×C4#13512yes—
400C22×C4C23×C411yescharacteristic
401C22×C4C22×D411yescharacteristic
402C22×C4C2×#4111yescharacteristic
403SD16—28no—
404SD16—44no—
405M16—28no—
406M16—44no—
407C2×C8—21no—
408C2×C8—21no—
409C2×C8—21no—
410C2×C8—21no—
411C2×C8—21no—
412C2×C8—21no—
413C2×C8—22no—
414C2×C8—21no—
415C2×C8—21no—
416C2×C8—21no—
417C2×C8—22no—
418C2×C8—21no—
419C2×C8—22no—
420C2×C8—21no—
421C2×C8—21no—
422C2×C8—21no—
423C2×C8—21no—
424C2×C8—21no—
425C2×C8—21no—
426C2×C8—21no—
427C2×C8—21no—
428C2×C8—41no—
429C2×C8—48no—
430C2×C8—41no—
431C2×C8—41no—
432C2×C8—41no—
433C2×C8—41no—
434C2×C8—41no—
435C2×C8—41no—
436C2×C8—41no—
437C2×C8—41no—
438C2×C8—41no—
439C2×C8—41no—
440C2×C8—41no—
441C2×C8—41no—
442C2×C8—41no—
443C2×C8—41no—
444C2×C8—84no—
445C2×C8#12012yes—
446C2×C8#12012yes—
447C2×C8#13112yes—
448C2×C8C2×SD1612yes—
449C2×C8C2×SD1612yes—
450C2×C8#13712yes—
451C4⋊C4—24no—
452C4⋊C4—24no—
453C4⋊C4—24no—
454C4⋊C4—24no—
455C4⋊C4—24no—
456C4⋊C4—21no—
457C4⋊C4—21no—
458C4⋊C4—22no—
459C4⋊C4—22no—
460C4⋊C4—22no—
461C4⋊C4—22no—
462C4⋊C4—21no—
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—21no—
468C4⋊C4—42no—
469C4⋊C4C4×D412yes—
470C4⋊C4C4×D412yes—
471C22⋊C4—24no—
472C22⋊C4—24no—
473C22⋊C4—42no—
474C22⋊C4—42no—
475C22⋊C4—41no—
476C22⋊C4—41no—
477C22⋊C4—42no—
478C22⋊C4—81no—
479C42—21no—
480C42—21no—
481C42—22no—
482C42—22no—
483C42—21no—
484C42—21no—
485C42—21no—
486C42—21no—
487C42—22no—
488C42—22no—
489C42—41no—
490C42—41no—
491C42C22×D411yescharacteristic
492C42C22×D411yescharacteristic
493C42C2×#4111yescharacteristic
494C42C2×#4111yescharacteristic
495C23—21no—
496C23—21no—
497C23—21no—
498C23—21no—
499C23—21no—
500C23—21no—
501C23—22no—
502C23—41no—
503C23—41no—
504C23—41no—
505C23—41no—
506C23—41no—
507C23—41no—
508C23C4×SD1612yes—
509C23C2×C4×D411yescharacteristic
510Q8—22no—
511Q8—22no—
512Q8—22no—
513Q8—22no—
514Q8—41no—
515Q8—41no—
516Q8—41no—
517Q8—41no—
518Q8—84no—
519D4—22no—
520D4—22no—
521D4—41no—
522D4—41no—
523D4—84no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—81no—
564C2×C4—81no—
565C2×C4—81no—
566C2×C4—81no—
567C2×C4—81no—
568C2×C4—81no—
569C2×C4C4×SD1612yes—
570C2×C4C2×C4×D411yescharacteristic
571C2×C4C2×C4×D411yescharacteristic
572C2×C4C2×#12011yescharacteristic
573C2×C4#52511yescharacteristic
574C2×C4#52511yescharacteristic
575C2×C4#52511yescharacteristic
576C2×C4C2×#13111yescharacteristic
577C2×C4C2×#13111yescharacteristic
578C2×C4#56811yescharacteristic
579C2×C4#56811yescharacteristic
580C2×C4C22×SD1611yescharacteristic
581C2×C4C2×#13511yescharacteristic
582C2×C4C2×#13711yescharacteristic
583C2×C4#57511yescharacteristic
584C8—22no—
585C8—22no—
586C8—22no—
587C8—42no—
588C8—41no—
589C8—41no—
590C8—41no—
591C8—41no—
592C8—42no—
593C8—42no—
594C8—81no—
595C8—81no—
596C8—81no—
597C8#44814yes—
598C22—22no—
599C22—22no—
600C22—22no—
601C22—41no—
602C22—41no—
603C22—41no—
604C22—41no—
605C22—42no—
606C22—42no—
607C22—42no—
608C22—81no—
609C22—81no—
610C22—81no—
611C22—81no—
612C22C4×#12011yescharacteristic
613C22#293311yescharacteristic
614C22#293511yescharacteristic
615C22C2×C4×SD1611yescharacteristic
616C22C4×#13711yescharacteristic
617C22#295611yescharacteristic
618C22#295711yescharacteristic
619C4—22no—
620C4—21no—
621C4—21no—
622C4—41no—
623C4—42no—
624C4—42no—
625C4—41no—
626C4—42no—
627C4—42no—
628C4—81no—
629C4—81no—
630C4—81no—
631C4#292911yescharacteristic
632C4#293211yescharacteristic
633C4#293311yescharacteristic
634C4#293511yescharacteristic
635C4C2×#44811yescharacteristic
636C4#300211yescharacteristic
637C4#310311yescharacteristic
638C4#310811yescharacteristic
639C2—42no—
640C2—82no—
641C2#1799511yescharacteristic
642C2#1800911yescharacteristic
643C2#1806711yescharacteristic
644C2#1807711yescharacteristic
645C2#2144311yescharacteristic
646C2C4×#44811yescharacteristic
647C2#2146811yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C2×C4×D4
prime radicalC2×#121C2×C4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C2×C4×M16 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2962C2semidirect
4. C2×#1582C2semidirect
5. C2×C8×SD16C2semidirect
6. #8952C2semidirect
7. #8948C2semidirect
9. #8850C2semidirect
10. #8846C2semidirect
11. #8830C2semidirect
12. #8314C2semidirect
13. #8207C2semidirect
14. #8203C2semidirect
15. #8146C2semidirect
100. #446D4semidirect
104. C8×Q8D4semidirect
205. C2×M16SD16semidirect

Maximal subgroups

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

subgroupindex
C2×#29622
#235682
#214862
C2×#15822
C2×C8×SD162
#89522
#89482
#89392
#88502
#88462
#88302
#83142
#82072
#82032
#81462