Finite Group Databasebeta

← #482164

Subgroups — #482164

Order 512.

1022 conjugacy classes of subgroups.

The subgroup lattice has 663 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35840C211yesmaximal, characteristic
2#20689C211yesmaximal, characteristic, complemented
3#17350C211yesmaximal, characteristic, complemented
4#11846C212yesmaximal, complemented
5#11843C212yesmaximal, complemented
6#11841C212yesmaximal, complemented
7#11840C212yesmaximal, complemented
8C2×#1352C212yesmaximal, complemented
9#11692C212yesmaximal, complemented
10C2×#553C411yescharacteristic, complemented
11C2×#518C2211yescharacteristic
12#2972—21no—
13#2635C411yescharacteristic, complemented
14#2633C411yescharacteristic, complemented
15#2617C2211yescharacteristic
16#2611C2211yescharacteristic
17#2598C2211yescharacteristic
18#2543C411yescharacteristic, complemented
19#2539C2211yescharacteristic
20#2487C411yescharacteristic, complemented
21#2482C2211yescharacteristic
22#2479—21no—
23#2467—21no—
24#2455—21no—
25#2443—21no—
26#2418—21no—
27#2410—21no—
28#2397C411yescharacteristic, complemented
29#2395C411yescharacteristic, complemented
30C2×#399C2211yescharacteristic
31#2361—21no—
32#2358—21no—
33#2353C411yescharacteristic, complemented
34#2308—21no—
35#1820—21no—
36#1820C2212yes—
37#1819—21no—
38#1819C2212yes—
39C2×#342—21no—
40C2×#342C2212yes—
41#1460—21no—
42#1459—21no—
43#1459—21no—
44#1458C2212yes—
45C2×#323C2212yes—
46C2×#323C2212yes—
47#1352C2214yes—
48#1352C2214yes—
49#1344C2214yes—
50#1343C2214yes—
51C23×D4D411yescharacteristic
52#553—44no—
53C2×#126—21no—
54C2×#125C2×C411yescharacteristic, complemented
55C2×#123—21no—
56C2×#123—21no—
57C2×#121—21no—
58C2×#121—21no—
59C2×#121C2×C411yescharacteristic
60C2×#121C2×C411yescharacteristic, complemented
61C2×#120—21no—
62C2×#120—21no—
63#518—44no—
64C2×C4×D4—21no—
65C2×C4×D4—21no—
66C2×C4×D4—21no—
67C2×C4×D4C2×C411yescharacteristic
68C2×C4×D4C2×C411yescharacteristic
69C2×C4×D4C2×C411yescharacteristic
70C2×C4×D4C2×C411yescharacteristic
71C2×C4×D4C2×C411yescharacteristic, complemented
72C2×C4×D4D411yescharacteristic
73C2×C4×D4D411yescharacteristic
74C2×C4×D4D411yescharacteristic
75C2×C4×D4D411yescharacteristic
76C2×#117C2×C411yescharacteristic, complemented
77C2×#117D411yescharacteristic
78C2×#117C2311yescharacteristic
79C22×C4⋊C4—21no—
80C22×C4⋊C4C2×C411yescharacteristic
81C22×C22⋊C4—21no—
82C22×C22⋊C4—22no—
83C22×C22⋊C4D411yescharacteristic
84C22×C22⋊C4D411yescharacteristic
85C22×C22⋊C4C2311yescharacteristic
86C22×C22⋊C4C2311yescharacteristic
87C8×D4—44no—
88#431—22no—
89C2×C22⋊C8—22no—
90C2×C22⋊C8—22no—
91#400C2×C411yescharacteristic, complemented
92#400C2×C411yescharacteristic, complemented
93#399—21no—
94#399—21no—
95#399—21no—
96#399C2×C411yescharacteristic
97#399C2×C411yescharacteristic
98#399C2×C411yescharacteristic, complemented
99#399C2×C411yescharacteristic, complemented
100#399C2311yescharacteristic
101#399C2311yescharacteristic
102#398—21no—
103#398C2×C411yescharacteristic, complemented
104#397—21no—
105#397C2×C411yescharacteristic
106#397C2×C411yescharacteristic, complemented
107#397C2311yescharacteristic
108#394—21no—
109#394—21no—
110#394—21no—
111#394—21no—
112#394—21no—
113#394C2×C411yescharacteristic, complemented
114#392—21no—
115#392—21no—
116#392C2×C411yescharacteristic
117#392C2×C411yescharacteristic, complemented
118#392C2311yescharacteristic
119#390—21no—
120#390—21no—
121#389—21no—
122#389—21no—
123#388—21no—
124#388—21no—
125#388—21no—
126#388—21no—
127#388—21no—
128#388—21no—
129#388—21no—
130#387—21no—
131#387—21no—
132#386—21no—
133#386—21no—
134#386—21no—
135#386—21no—
136#386—21no—
137#386—21no—
138#386—21no—
139#386—21no—
140#386—44no—
141#385—21no—
142#385—21no—
143#385—21no—
144#385—21no—
145#385—21no—
146#385—21no—
147#383C2×C411yescharacteristic, complemented
148#381—22no—
149#381C2×C411yescharacteristic, complemented
150#380—21no—
151#380C2×C411yescharacteristic
152#379—21no—
153#379—22no—
154C4×C4⋊C4—21no—
155C4×C22⋊C4—21no—
156C4×C22⋊C4—21no—
157C4×C22⋊C4C2×C411yescharacteristic, complemented
158#376C2×C411yescharacteristic, complemented
159C2×#95C2×C411yescharacteristic
160#342—22no—
161#342—22no—
162#342—22no—
163#342C2314yes—
164#324—22no—
165#323—22no—
166#323—22no—
167#323C2314yes—
168C25C2×D411yescharacteristic
169C25C2×D411yescharacteristic
170C22×D4—24no—
171C22×D4—24no—
172C22×D4—21no—
173C22×D4—21no—
174C22×D4—21no—
175C22×D4—21no—
176C22×D4—21no—
177C22×D4—42no—
178C22×D4C22⋊C411yescharacteristic
179C22×D4C22⋊C411yescharacteristic
180C22×D4C22⋊C411yescharacteristic
181C22×D4C22⋊C411yescharacteristic
182C22×D4C22×C411yescharacteristic
183C23×C4—21no—
184C23×C4—21no—
185C23×C4—21no—
186C23×C4—42no—
187C23×C4C22⋊C411yescharacteristic
188C23×C4C22⋊C411yescharacteristic
189C23×C4C22⋊C411yescharacteristic
190C23×C4C22⋊C411yescharacteristic
191C23×C4C22×C411yescharacteristic
192C23×C4C22×C411yescharacteristic
193C23×C4C2×D411yescharacteristic
194C23×C4C2×D411yescharacteristic
195C23×C4C2×D411yescharacteristic
196C22×C8—42no—
197C22×C8—42no—
198#126—44no—
199#125—42no—
200#123—44no—
201#123—82no—
202#121—42no—
203#121—42no—
204#121—44no—
205#121—44no—
206#120—44no—
207#120—82no—
208C4×D4—24no—
209C4×D4—24no—
210C4×D4—42no—
211C4×D4—42no—
212C4×D4—42no—
213C4×D4—42no—
214C4×D4—42no—
215C4×D4—42no—
216C4×D4—42no—
217C4×D4—44no—
218C4×D4—82no—
219C4×D4—82no—
220#117—42no—
221#117—42no—
222#117—42no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4—21no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—42no—
235C2×C4⋊C4—82no—
236C2×C4⋊C4C22⋊C411yescharacteristic
237C2×C4⋊C4C22⋊C411yescharacteristic
238C2×C4⋊C4C22×C411yescharacteristic
239C2×C4⋊C4C22×C411yescharacteristic
240C2×C4⋊C4C22×C411yescharacteristic
241C2×C4⋊C4C22×C411yescharacteristic
242C2×C4⋊C4C2×D411yescharacteristic
243C2×C22⋊C4—21no—
244C2×C22⋊C4—21no—
245C2×C22⋊C4—21no—
246C2×C22⋊C4—21no—
247C2×C22⋊C4—21no—
248C2×C22⋊C4—21no—
249C2×C22⋊C4—21no—
250C2×C22⋊C4—21no—
251C2×C22⋊C4—21no—
252C2×C22⋊C4—21no—
253C2×C22⋊C4—21no—
254C2×C22⋊C4—21no—
255C2×C22⋊C4—21no—
256C2×C22⋊C4—24no—
257C2×C22⋊C4—21no—
258C2×C22⋊C4—21no—
259C2×C22⋊C4—21no—
260C2×C22⋊C4—21no—
261C2×C22⋊C4—21no—
262C2×C22⋊C4—21no—
263C2×C22⋊C4—21no—
264C2×C22⋊C4—21no—
265C2×C22⋊C4—42no—
266C2×C22⋊C4—42no—
267C2×C22⋊C4—42no—
268C2×C22⋊C4—42no—
269C2×C22⋊C4—44no—
270C2×C22⋊C4—44no—
271C2×C22⋊C4—42no—
272C2×C22⋊C4—42no—
273C2×C22⋊C4—42no—
274C2×C22⋊C4—82no—
275C2×C22⋊C4C22⋊C411yescharacteristic
276C2×C22⋊C4C22⋊C411yescharacteristic
277C2×C22⋊C4C22⋊C411yescharacteristic
278C2×C22⋊C4C22⋊C411yescharacteristic
279C2×C22⋊C4C22×C411yescharacteristic
280C2×C22⋊C4C22×C411yescharacteristic
281C2×C22⋊C4C2×D411yescharacteristic
282C2×C22⋊C4C2×D411yescharacteristic
283C2×C22⋊C4C2×D411yescharacteristic
284C2×C22⋊C4C2×D411yescharacteristic
285C2×C22⋊C4C2×D411yescharacteristic
286C2×C22⋊C4C2411yescharacteristic
287C2×C42—21no—
288C2×C42—21no—
289C2×C42—21no—
290C2×C42C22⋊C411yescharacteristic
291C2×C42C22⋊C411yescharacteristic
292C2×C42C22×C411yescharacteristic
293C2×C42C2×D411yescharacteristic
294C4⋊C8—42no—
295C22⋊C8—24no—
296C22⋊C8—42no—
297C22⋊C8—42no—
298C4×C8—42no—
299#95—21no—
300#95—21no—
301#95—21no—
302#95—21no—
303#95—21no—
304#95—21no—
305#95—21no—
306#95—21no—
307#95—21no—
308#95—21no—
309#95—21no—
310#95—21no—
311#95—21no—
312#95—44no—
313#95C22×C411yescharacteristic
314#95C22×C411yescharacteristic
315#95C22×C411yescharacteristic
316#95C22×C411yescharacteristic
317C24—22no—
318C24—22no—
319C24—22no—
320C24—22no—
321C24—21no—
322C24—41no—
323C24—81no—
324C24—81no—
325C24—81no—
326C24—81no—
327C24C2×C22⋊C411yescharacteristic
328C24C2×C22⋊C411yescharacteristic
329C24C2×C22⋊C411yescharacteristic
330C24C2×C22⋊C411yescharacteristic
331C24#13514yes—
332C24C22×D411yescharacteristic
333C2×D4—22no—
334C2×D4—22no—
335C2×D4—22no—
336C2×D4—22no—
337C2×D4—22no—
338C2×D4—22no—
339C2×D4—22no—
340C2×D4—22no—
341C2×D4—22no—
342C2×D4—41no—
343C2×D4—41no—
344C2×D4—41no—
345C2×D4—42no—
346C2×D4—48no—
347C2×D4—48no—
348C2×D4—42no—
349C2×D4—42no—
350C2×D4—42no—
351C2×D4—42no—
352C2×D4—42no—
353C2×D4—42no—
354C2×D4—42no—
355C2×D4—81no—
356C2×D4—81no—
357C2×D4—81no—
358C2×D4—81no—
359C2×D4—81no—
360C2×D4—81no—
361C2×D4—81no—
362C2×D4—81no—
363C2×D4#9914yes—
364C22×C4—21no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—21no—
372C22×C4—21no—
373C22×C4—21no—
374C22×C4—21no—
375C22×C4—21no—
376C22×C4—21no—
377C22×C4—41no—
378C22×C4—41no—
379C22×C4—41no—
380C22×C4—41no—
381C22×C4—41no—
382C22×C4—42no—
383C22×C4—42no—
384C22×C4—42no—
385C22×C4—42no—
386C22×C4—44no—
387C22×C4—81no—
388C22×C4—81no—
389C22×C4—81no—
390C22×C4—81no—
391C22×C4—81no—
392C22×C4—81no—
393C22×C4—81no—
394C22×C4—81no—
395C22×C4—81no—
396C22×C4—81no—
397C22×C4—81no—
398C22×C4—81no—
399C22×C4—81no—
400C22×C4—84no—
401C22×C4—162no—
402C22×C4—162no—
403C22×C4C2×C22⋊C411yescharacteristic
404C22×C4C2×C22⋊C411yescharacteristic
405C22×C4C2×C22⋊C411yescharacteristic
406C22×C4C2×C22⋊C411yescharacteristic
407C22×C4C2×C22⋊C411yescharacteristic
408C22×C4C2×C22⋊C411yescharacteristic
409C22×C4C2×C22⋊C411yescharacteristic
410C22×C4C2×C22⋊C411yescharacteristic
411C22×C4C23×C411yescharacteristic
412C22×C4C22×D411yescharacteristic
413C2×C8—44no—
414C2×C8—162no—
415C2×C8—162no—
416C4⋊C4—22no—
417C4⋊C4—22no—
418C4⋊C4—41no—
419C4⋊C4—41no—
420C4⋊C4—41no—
421C4⋊C4—41no—
422C4⋊C4—41no—
423C4⋊C4—42no—
424C4⋊C4—42no—
425C4⋊C4—42no—
426C4⋊C4—42no—
427C4⋊C4—42no—
428C4⋊C4—42no—
429C4⋊C4—81no—
430C4⋊C4—81no—
431C4⋊C4—81no—
432C4⋊C4—81no—
433C4⋊C4—81no—
434C22⋊C4—22no—
435C22⋊C4—22no—
436C22⋊C4—22no—
437C22⋊C4—22no—
438C22⋊C4—41no—
439C22⋊C4—41no—
440C22⋊C4—41no—
441C22⋊C4—41no—
442C22⋊C4—41no—
443C22⋊C4—41no—
444C22⋊C4—41no—
445C22⋊C4—41no—
446C22⋊C4—42no—
447C22⋊C4—42no—
448C22⋊C4—42no—
449C22⋊C4—42no—
450C22⋊C4—42no—
451C22⋊C4—42no—
452C22⋊C4—42no—
453C22⋊C4—42no—
454C22⋊C4—42no—
455C22⋊C4—42no—
456C22⋊C4—42no—
457C22⋊C4—42no—
458C22⋊C4—42no—
459C22⋊C4—44no—
460C22⋊C4—44no—
461C22⋊C4—81no—
462C22⋊C4—81no—
463C22⋊C4—81no—
464C22⋊C4—81no—
465C22⋊C4—81no—
466C22⋊C4—81no—
467C22⋊C4—81no—
468C22⋊C4—81no—
469C22⋊C4—84no—
470C22⋊C4—162no—
471C22⋊C4—162no—
472C42—22no—
473C42—22no—
474C42—41no—
475C42—41no—
476C42—42no—
477C42—42no—
478C42—81no—
479C23—21no—
480C23—21no—
481C23—21no—
482C23—21no—
483C23—21no—
484C23—21no—
485C23—24no—
486C23—24no—
487C23—21no—
488C23—21no—
489C23—21no—
490C23—41no—
491C23—41no—
492C23—41no—
493C23—41no—
494C23—44no—
495C23—41no—
496C23—41no—
497C23—41no—
498C23—41no—
499C23—41no—
500C23—41no—
501C23—41no—
502C23—41no—
503C23—41no—
504C23—41no—
505C23—41no—
506C23—41no—
507C23—41no—
508C23—82no—
509C23—82no—
510C23—82no—
511C23—82no—
512C23—84no—
513C23—162no—
514C23—162no—
515C23—162no—
516C23C2×#9912yes—
517C23C2×#9912yes—
518C23#41612yes—
519C23#41612yes—
520C23#42112yes—
521C23C22×C22⋊C411yescharacteristic
522C23C2×#13512yes—
523D4—28no—
524D4—44no—
525D4—44no—
526D4—44no—
527D4—84no—
528D4—82no—
529D4—84no—
530D4—84no—
531D4—162no—
532D4—162no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—22no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—44no—
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—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—82no—
577C2×C4—82no—
578C2×C4—82no—
579C2×C4—82no—
580C2×C4—84no—
581C2×C4—84no—
582C2×C4—84no—
583C2×C4—84no—
584C2×C4—161no—
585C2×C4—161no—
586C2×C4—161no—
587C2×C4—161no—
588C2×C4—162no—
589C2×C4—162no—
590C2×C4—168no—
591C2×C4C2×#9912yes—
592C8—84no—
593C22—21no—
594C22—22no—
595C22—21no—
596C22—22no—
597C22—22no—
598C22—22no—
599C22—22no—
600C22—22no—
601C22—22no—
602C22—42no—
603C22—42no—
604C22—42no—
605C22—42no—
606C22—42no—
607C22—41no—
608C22—41no—
609C22—41no—
610C22—41no—
611C22—41no—
612C22—41no—
613C22—41no—
614C22—84no—
615C22—81no—
616C22—81no—
617C22—81no—
618C22—81no—
619C22—84no—
620C22—81no—
621C22—81no—
622C22—81no—
623C22—81no—
624C22—81no—
625C22—81no—
626C22—81no—
627C22—81no—
628C22—164no—
629C22#151614yes—
630C22C22×#9911yescharacteristic
631C22C2×#41611yescharacteristic
632C22C2×#42111yescharacteristic
633C4—22no—
634C4—42no—
635C4—41no—
636C4—41no—
637C4—41no—
638C4—41no—
639C4—41no—
640C4—41no—
641C4—41no—
642C4—41no—
643C4—41no—
644C4—81no—
645C4—81no—
646C4—81no—
647C4—81no—
648C4—81no—
649C4—81no—
650C4—164no—
651C2—21no—
652C2—21no—
653C2—24no—
654C2—42no—
655C2—42no—
656C2—42no—
657C2—81no—
658C2—161no—
659C2C2×#151611yescharacteristic
660C2#3335011yescharacteristic
661C2#3335911yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC22×C4C23×C4
Frattini, cosocleC2×C22⋊C4C24
prime radical#35840C2
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×D4 ◁ C2×#518 ◁ G
LECSG ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #20689C2semidirect
3. #17350C2semidirect
4. #11846C2semidirect
5. #11843C2semidirect
6. #11841C2semidirect
7. #11840C2semidirect
8. C2×#1352C2semidirect
9. #11692C2semidirect
10. C2×#553C4semidirect
13. #2635C4semidirect
14. #2633C4semidirect
18. #2543C4semidirect
20. #2487C4semidirect
28. #2397C4semidirect
29. #2395C4semidirect
33. #2353C4semidirect
54. C2×#125C2×C4semidirect
60. C2×#121C2×C4semidirect
71. C2×C4×D4C2×C4semidirect
76. C2×#117C2×C4semidirect
91. #400C2×C4semidirect
92. #400C2×C4semidirect
98. #399C2×C4semidirect
99. #399C2×C4semidirect
103. #398C2×C4semidirect
106. #397C2×C4semidirect
113. #394C2×C4semidirect
117. #392C2×C4semidirect
147. #383C2×C4semidirect
149. #381C2×C4semidirect
157. C4×C22⋊C4C2×C4semidirect
158. #376C2×C4semidirect

Maximal subgroups

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

subgroupindex
#358402
#206892
#173502
#118462
#118432
#118412
#118402
C2×#13522
#116922