Finite Group Databasebeta

← #206055

Subgroups — #206055

Order 512.

1126 conjugacy classes of subgroups.

The subgroup lattice has 651 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3190C211yesmaximal, characteristic
2#21948C211yesmaximal, characteristic, complemented
3#20523C211yesmaximal, characteristic, complemented
4#17542C211yesmaximal, characteristic
5C2×#1508C211yesmaximal, characteristic, complemented
6#17381C211yesmaximal, characteristic
7C2×#1472C211yesmaximal, characteristic, complemented
8#9814C214yesmaximal, complemented
9#9813C214yesmaximal, complemented
10C2×#525C2211yescharacteristic
11C22×C22⋊Q8C2211yescharacteristic
12#3190—22no—
13#3190—22no—
14#3190—22no—
15#3190—22no—
16C2×#485C412yescomplemented
17C2×#483C411yescharacteristic, complemented
18C2×#480C411yescharacteristic, complemented
19C2×#467C2212yes—
20C2×#465C2212yes—
21C2×#452C412yescomplemented
22C2×#451C411yescharacteristic, complemented
23C2×#448C411yescharacteristic, complemented
24C2×#407C2211yescharacteristic
25C2×#407C2211yescharacteristic
26C22×C22⋊C8C2211yescharacteristic
27C22×C22⋊C8C2211yescharacteristic
28C2×C4×C22⋊C4C2211yescharacteristic
29#1855—22no—
30#1855—22no—
31#1810C2214yes—
32C8×C22⋊C4—22no—
33C8×C22⋊C4C2214yes—
34C4×C22⋊C8—22no—
35#1508—22no—
36#1508C2214yes—
37#1508C2212yes—
38#1507C2212yes—
39#1504—22no—
40#1501—22no—
41C2×#328—22no—
42#1472—22no—
43#1472C2214yes—
44#1472C2212yes—
45#1471C2212yes—
46#1468—22no—
47#1459—22no—
48C2×#324—22no—
49C22×#41C2×C411yescharacteristic
50C23×Q8C2×C411yescharacteristic
51C24×C4C2311yescharacteristic
52C22×Q16C2×C412yescomplemented
53C22×SD16C2×C412yescomplemented
54C22×M16—21no—
55C22×M16—21no—
56C23×C8—21no—
57C23×C8C2312yes—
58#525—21no—
59#525—21no—
60#525—21no—
61#525—21no—
62#525—21no—
63#525—21no—
64#525—21no—
65#525—21no—
66C2×#123—21no—
67C2×C22⋊Q8—21no—
68C2×C22⋊Q8—21no—
69C2×C22⋊Q8—21no—
70C2×C22⋊Q8—21no—
71C2×C22⋊Q8—21no—
72C2×C22⋊Q8—21no—
73C2×C22⋊Q8—21no—
74C2×C22⋊Q8—21no—
75C2×C22⋊Q8—21no—
76C2×C22⋊Q8—21no—
77C2×C22⋊Q8C2×C411yescharacteristic
78C2×C22⋊Q8C2×C412yes—
79C2×C22⋊Q8C2312yes—
80C2×C22⋊Q8C2311yescharacteristic
81C2×#121C2×C411yescharacteristic
82C2×#121C2311yescharacteristic
83C2×#120—21no—
84C2×C4×D4—21no—
85C2×C4×D4—21no—
86C2×#117C2×C411yescharacteristic
87C22×C4⋊C4—21no—
88C22×C4⋊C4C2×C411yescharacteristic
89C22×C22⋊C4—21no—
90C22×C22⋊C4—21no—
91C22×C42—21no—
92#485—24no—
93#485—24no—
94#483—22no—
95#483—22no—
96#480—22no—
97#480—22no—
98#467—24no—
99#467—24no—
100#465—24no—
101#465—24no—
102#452—24no—
103#452—24no—
104#451—22no—
105#451—22no—
106#448—22no—
107#448—22no—
108C2×#108—24no—
109C2×#107C2×C412yescomplemented
110C2×#106C2×C412yescomplemented
111C2×#103C2×C412yescomplemented
112C2×#103C2×C412yescomplemented
113C2×#103C2×C412yescomplemented
114C2×#102C2×C412yescomplemented
115#407—21no—
116#407—21no—
117#407—21no—
118#407—21no—
119#407—21no—
120#407—21no—
121#407C2312yes—
122#407C2312yes—
123C2×C22⋊C8—21no—
124C2×C22⋊C8—21no—
125C2×C22⋊C8—21no—
126C2×C22⋊C8—21no—
127C2×C22⋊C8—21no—
128C2×C22⋊C8—21no—
129C2×C22⋊C8—21no—
130C2×C22⋊C8C2×C411yescharacteristic
131C2×C22⋊C8C2×C411yescharacteristic
132C2×C22⋊C8C2×C412yes—
133C2×C22⋊C8D412yes—
134C2×C22⋊C8D412yes—
135C2×C22⋊C8D412yes—
136C2×C22⋊C8D412yes—
137C2×C22⋊C8D412yes—
138C2×C22⋊C8D412yes—
139C2×C22⋊C8C2312yes—
140C2×C4×C8—24no—
141C4×C22⋊C4—21no—
142C4×C22⋊C4—21no—
143C4×C22⋊C4D412yes—
144C4×C22⋊C4C2312yes—
145C2×#95D412yes—
146#341—24no—
147#336—24no—
148#328—28no—
149#328—24no—
150#327—24no—
151#324—28no—
152#324—24no—
153#323—24no—
154C25#4111yescharacteristic
155C2×#41—21no—
156C2×#41—21no—
157C2×#41—21no—
158C2×#41—21no—
159C2×#41—82no—
160C22×Q8—21no—
161C22×Q8—21no—
162C22×Q8—21no—
163C22×Q8—21no—
164C22×Q8—21no—
165C22×Q8—82no—
166C22×Q8C22×C412yes—
167C22×Q8C22×C411yescharacteristic
168C22×D4—21no—
169C22×D4—21no—
170C22×D4—21no—
171C22×D4C22×C411yescharacteristic
172C23×C4—22no—
173C23×C4—21no—
174C23×C4—21no—
175C23×C4—21no—
176C23×C4—82no—
177C23×C4C22×C411yescharacteristic
178C23×C4C22×C411yescharacteristic
179C23×C4C2×D411yescharacteristic
180C23×C4C2×D411yescharacteristic
181C23×C4C2×D411yescharacteristic
182C23×C4C2×D411yescharacteristic
183C23×C4C2×D411yescharacteristic
184C23×C4C2×D411yescharacteristic
185C23×C4C2×D411yescharacteristic
186C23×C4C2×D411yescharacteristic
187C23×C4#4111yescharacteristic
188C23×C4#4111yescharacteristic
189C23×C4#4111yescharacteristic
190C23×C4C2411yescharacteristic
191C2×Q16—24no—
192C2×Q16—82no—
193C2×SD16—24no—
194C2×SD16—82no—
195C2×M16—22no—
196C2×M16—22no—
197C2×M16—41no—
198C2×M16—41no—
199C2×M16—82no—
200C2×M16—82no—
201C22×C8—22no—
202C22×C8—22no—
203C22×C8—22no—
204C22×C8—41no—
205C22×C8—41no—
206C22×C8—82no—
207C22×C8—82no—
208C22×C8C22×C412yes—
209C22×C8C22×C412yes—
210C22×C8C2×D414yes—
211C22×C8C2×D414yes—
212#123—41no—
213#123—41no—
214#123—41no—
215#123—41no—
216C22⋊Q8—22no—
217C22⋊Q8—22no—
218C22⋊Q8—21no—
219C22⋊Q8—21no—
220C22⋊Q8—22no—
221C22⋊Q8—22no—
222C22⋊Q8—21no—
223C22⋊Q8—21no—
224C22⋊Q8—22no—
225C22⋊Q8—22no—
226C22⋊Q8—22no—
227C22⋊Q8—21no—
228C22⋊Q8—21no—
229C22⋊Q8—21no—
230C22⋊Q8—22no—
231C22⋊Q8—21no—
232C22⋊Q8—41no—
233C22⋊Q8—41no—
234C22⋊Q8—41no—
235C22⋊Q8—41no—
236C22⋊Q8—41no—
237C22⋊Q8—41no—
238C22⋊Q8—41no—
239C22⋊Q8—41no—
240#121—21no—
241#121—21no—
242#121—21no—
243#121—21no—
244#121—21no—
245#121—21no—
246#121—21no—
247#121—21no—
248#120—41no—
249#120—41no—
250#120—41no—
251#120—41no—
252C4×D4—41no—
253C4×D4—41no—
254C4×D4—41no—
255C4×D4—41no—
256C4×D4—41no—
257C4×D4—41no—
258C4×D4—41no—
259C4×D4—41no—
260#117—21no—
261#117—21no—
262#117—21no—
263#117—21no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—21no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—21no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—21no—
270C2×C4⋊C4—21no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—21no—
273C2×C4⋊C4—41no—
274C2×C4⋊C4—41no—
275C2×C4⋊C4—41no—
276C2×C4⋊C4—41no—
277C2×C4⋊C4C22×C412yes—
278C2×C4⋊C4C22×C411yescharacteristic
279C2×C4⋊C4C22×C411yescharacteristic
280C2×C22⋊C4—28no—
281C2×C22⋊C4—22no—
282C2×C22⋊C4—22no—
283C2×C22⋊C4—22no—
284C2×C22⋊C4—21no—
285C2×C22⋊C4—21no—
286C2×C22⋊C4—22no—
287C2×C22⋊C4—21no—
288C2×C22⋊C4—21no—
289C2×C22⋊C4—21no—
290C2×C22⋊C4—41no—
291C2×C22⋊C4—41no—
292C2×C22⋊C4—41no—
293C2×C22⋊C4—41no—
294C2×C42—21no—
295C2×C42—22no—
296C2×C42—22no—
297C2×C42—82no—
298#108—84no—
299#107—24no—
300#106—24no—
301#103—24no—
302#103—24no—
303#103—24no—
304#102—24no—
305C22⋊C8—22no—
306C22⋊C8—22no—
307C22⋊C8—22no—
308C22⋊C8—22no—
309C22⋊C8—22no—
310C22⋊C8—22no—
311C22⋊C8—22no—
312C22⋊C8—41no—
313C22⋊C8—41no—
314C22⋊C8—41no—
315C22⋊C8—41no—
316C22⋊C8C22⋊C412yes—
317C22⋊C8C22⋊C412yes—
318C22⋊C8C22⋊C412yes—
319C22⋊C8C22⋊C412yes—
320C22⋊C8C22⋊C414yes—
321C22⋊C8C22⋊C414yes—
322C22⋊C8C2×D414yes—
323C4×C8—84no—
324#95—22no—
325#95C2×D414yes—
326C24—24no—
327C24—21no—
328C24—42no—
329C24—81no—
330C24C4×D411yescharacteristic
331C24C4×D411yescharacteristic
332C24#12111yescharacteristic
333C24#12111yescharacteristic
334C24#12411yescharacteristic
335C24#12411yescharacteristic
336C24C2×#4111yescharacteristic
337#41—82no—
338#41—82no—
339#41—82no—
340#41—82no—
341C2×Q8—22no—
342C2×Q8—21no—
343C2×Q8—22no—
344C2×Q8—21no—
345C2×Q8—42no—
346C2×Q8—42no—
347C2×Q8—42no—
348C2×Q8—41no—
349C2×Q8—41no—
350C2×Q8—82no—
351C2×Q8—82no—
352C2×Q8—82no—
353C2×Q8—82no—
354C2×Q8—82no—
355C2×D4—21no—
356C2×D4—21no—
357C2×D4—42no—
358C2×D4—41no—
359C2×D4—41no—
360C2×D4—41no—
361C2×D4—41no—
362C2×D4—41no—
363C2×D4—41no—
364C2×D4—44no—
365C2×D4—82no—
366C2×D4—82no—
367C22×C4—24no—
368C22×C4—28no—
369C22×C4—21no—
370C22×C4—22no—
371C22×C4—21no—
372C22×C4—21no—
373C22×C4—22no—
374C22×C4—42no—
375C22×C4—42no—
376C22×C4—42no—
377C22×C4—42no—
378C22×C4—42no—
379C22×C4—42no—
380C22×C4—42no—
381C22×C4—41no—
382C22×C4—41no—
383C22×C4—41no—
384C22×C4—41no—
385C22×C4—44no—
386C22×C4—41no—
387C22×C4—41no—
388C22×C4—81no—
389C22×C4—81no—
390C22×C4—81no—
391C22×C4—81no—
392C22×C4—81no—
393C22×C4—81no—
394C22×C4—81no—
395C22×C4—82no—
396C22×C4—82no—
397C22×C4—82no—
398C22×C4—82no—
399C22×C4C2×C22⋊C411yescharacteristic
400C22×C4C2×C22⋊C411yescharacteristic
401C22×C4C2×C22⋊C411yescharacteristic
402C22×C4C2×C22⋊C411yescharacteristic
403C22×C4C4×D411yescharacteristic
404C22×C4C4×D411yescharacteristic
405C22×C4C4×D411yescharacteristic
406C22×C4C4×D411yescharacteristic
407C22×C4C4×D411yescharacteristic
408C22×C4C4×D411yescharacteristic
409C22×C4#12111yescharacteristic
410C22×C4#12111yescharacteristic
411C22×C4#12111yescharacteristic
412C22×C4#12111yescharacteristic
413C22×C4#12111yescharacteristic
414C22×C4#12111yescharacteristic
415C22×C4#12411yescharacteristic
416C22×C4#12411yescharacteristic
417C22×C4#12711yescharacteristic
418C22×C4#12711yescharacteristic
419C22×C4#12711yescharacteristic
420C22×C4#12711yescharacteristic
421C22×C4C23×C411yescharacteristic
422C22×C4C22×D411yescharacteristic
423C22×C4C22×D411yescharacteristic
424C22×C4C22×D411yescharacteristic
425C22×C4C22×D411yescharacteristic
426C22×C4C2×#4111yescharacteristic
427Q16—84no—
428SD16—84no—
429M16—44no—
430M16—44no—
431M16—82no—
432M16—82no—
433C2×C8—24no—
434C2×C8—44no—
435C2×C8—44no—
436C2×C8—42no—
437C2×C8—42no—
438C2×C8—44no—
439C2×C8—42no—
440C2×C8—42no—
441C2×C8—82no—
442C2×C8—82no—
443C2×C8—82no—
444C2×C8—82no—
445C2×C8C2×C22⋊C414yes—
446C2×C8C2×C22⋊C414yes—
447C4⋊C4—22no—
448C4⋊C4—21no—
449C4⋊C4—21no—
450C4⋊C4—22no—
451C4⋊C4—21no—
452C4⋊C4—21no—
453C4⋊C4—42no—
454C4⋊C4—41no—
455C4⋊C4—42no—
456C4⋊C4—41no—
457C4⋊C4—41no—
458C4⋊C4—41no—
459C4⋊C4—42no—
460C4⋊C4—42no—
461C4⋊C4—41no—
462C4⋊C4—41no—
463C22⋊C4—28no—
464C22⋊C4—28no—
465C22⋊C4—42no—
466C22⋊C4—41no—
467C22⋊C4—42no—
468C22⋊C4—41no—
469C22⋊C4—42no—
470C22⋊C4—42no—
471C22⋊C4—41no—
472C22⋊C4—41no—
473C22⋊C4—41no—
474C22⋊C4—41no—
475C22⋊C4—41no—
476C22⋊C4—41no—
477C22⋊C4—41no—
478C22⋊C4—41no—
479C42—44no—
480C42—41no—
481C42—41no—
482C42—82no—
483C23—22no—
484C23—22no—
485C23—22no—
486C23—22no—
487C23—22no—
488C23—22no—
489C23—22no—
490C23—22no—
491C23—22no—
492C23—41no—
493C23—41no—
494C23—41no—
495C23—41no—
496C23—41no—
497C23—41no—
498C23—41no—
499C23—41no—
500C23—41no—
501C23—41no—
502C23—41no—
503C23—41no—
504C23—42no—
505C23—81no—
506C23—81no—
507C23—88no—
508C23#39011yescharacteristic
509C23#39011yescharacteristic
510C23#39011yescharacteristic
511C23#39011yescharacteristic
512C23#44314yes—
513C23C2×C4×D411yescharacteristic
514C23C2×#12111yescharacteristic
515C23C2×#12411yescharacteristic
516Q8—42no—
517Q8—42no—
518Q8—42no—
519Q8—82no—
520Q8—81no—
521Q8—82no—
522Q8—82no—
523D4—42no—
524D4—81no—
525D4—82no—
526D4—82no—
527D4—82no—
528D4—82no—
529D4—82no—
530D4—82no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—28no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
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—44no—
561C2×C4—44no—
562C2×C4—44no—
563C2×C4—42no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—42no—
567C2×C4—41no—
568C2×C4—42no—
569C2×C4—82no—
570C2×C4—82no—
571C2×C4—88no—
572C2×C4—82no—
573C2×C4—82no—
574C2×C4—82no—
575C2×C4—82no—
576C2×C4—81no—
577C2×C4—82no—
578C2×C4—81no—
579C2×C4#39011yescharacteristic
580C2×C4#39011yescharacteristic
581C2×C4#39011yescharacteristic
582C2×C4#39011yescharacteristic
583C2×C4C22×C22⋊C411yescharacteristic
584C2×C4C2×C4×D411yescharacteristic
585C2×C4C2×#12111yescharacteristic
586C2×C4C2×#12711yescharacteristic
587C8—24no—
588C8—24no—
589C8—44no—
590C8—82no—
591C8—82no—
592C22—21no—
593C22—21no—
594C22—21no—
595C22—21no—
596C22—21no—
597C22—21no—
598C22—21no—
599C22—21no—
600C22—21no—
601C22—21no—
602C22—21no—
603C22—21no—
604C22—21no—
605C22—21no—
606C22—21no—
607C22—21no—
608C22—21no—
609C22—44no—
610C22—44no—
611C22—44no—
612C22—44no—
613C22—82no—
614C22—82no—
615C22—82no—
616C22—81no—
617C22—81no—
618C22—81no—
619C22—162no—
620C22#197412yes—
621C22#197412yes—
622C22C2×#39011yescharacteristic
623C22C2×#44312yes—
624C22#306912yes—
625C4—21no—
626C4—21no—
627C4—21no—
628C4—21no—
629C4—21no—
630C4—21no—
631C4—21no—
632C4—48no—
633C4—81no—
634C4—81no—
635C4—82no—
636C4—82no—
637C4—81no—
638C4#288312yes—
639C2—22no—
640C2—22no—
641C2—22no—
642C2—22no—
643C2—41no—
644C2—41no—
645C2—41no—
646C2—161no—
647C2#2121311yescharacteristic
648C2C2×#197411yescharacteristic
649C2#2125011yescharacteristic
subgroupG/H
center, socleC22C2×#390
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC2×#525C22
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C24×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21948C2semidirect
3. #20523C2semidirect
5. C2×#1508C2semidirect
7. C2×#1472C2semidirect
8. #9814C2semidirect
9. #9813C2semidirect
16. C2×#485C4semidirect
17. C2×#483C4semidirect
18. C2×#480C4semidirect
21. C2×#452C4semidirect
22. C2×#451C4semidirect
23. C2×#448C4semidirect
52. C22×Q16C2×C4semidirect
53. C22×SD16C2×C4semidirect
109. C2×#107C2×C4semidirect
110. C2×#106C2×C4semidirect
111. C2×#103C2×C4semidirect
112. C2×#103C2×C4semidirect
113. C2×#103C2×C4semidirect
114. C2×#102C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#31902
#219482
#205232
#175422
C2×#15082
#173812
C2×#14722
#98142
#98132