Finite Group Databasebeta

← #189377

Subgroups — #189377

Order 512.

819 conjugacy classes of subgroups.

The subgroup lattice has 667 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60268C211yesmaximal, characteristic
2#21993C211yesmaximal, characteristic, complemented
3C2×#1572C211yesmaximal, characteristic, complemented
4#17753C211yesmaximal, characteristic
5#17716C211yesmaximal, characteristic
6#8689C211yesmaximal, characteristic, complemented
7#8688C211yesmaximal, characteristic, complemented
8#8687C211yesmaximal, characteristic, complemented
9#8686C211yesmaximal, characteristic, complemented
10#8685C211yesmaximal, characteristic, complemented
11#8684C211yesmaximal, characteristic, complemented
12#8683C211yesmaximal, characteristic, complemented
13#8682C211yesmaximal, characteristic, complemented
14#8003C211yesmaximal, characteristic, complemented
15#8001C211yesmaximal, characteristic, complemented
16#3444C2211yescharacteristic
17#2994C412yes—
18#2994C412yes—
19#2993—21no—
20#2993—21no—
21#2991C412yes—
22#2990C412yes—
23#2989—21no—
24#2988—21no—
25#2960C2211yescharacteristic
26#2960C2211yescharacteristic
27#2933—21no—
28#2933—21no—
29#2931C2211yescharacteristic
30#2931C2211yescharacteristic
31#2925—21no—
32#2920C2211yescharacteristic
33C2×#432C2211yescharacteristic
34C2×#431C2211yescharacteristic
35C2×#431C2211yescharacteristic
36#2906—21no—
37C2×#423C2211yescharacteristic
38C2×#423C2211yescharacteristic
39#1950—21no—
40#1950—21no—
41#1855—21no—
42#1855—21no—
43#1848C2211yescharacteristic
44#1848C2211yescharacteristic
45#1837C2211yescharacteristic
46#1837C2211yescharacteristic
47#1572—21no—
48#1572—21no—
49#1572—21no—
50#1572—21no—
51C2×#335C2211yescharacteristic
52C2×#335C2211yescharacteristic
53C2×#334C2211yescharacteristic
54C2×#334C2211yescharacteristic
55#1559—21no—
56#1557—21no—
57#1554—21no—
58#1553—21no—
59#1552—21no—
60#1551—21no—
61#1550C2211yescharacteristic
62#1550C2211yescharacteristic
63#1550C2211yescharacteristic
64#1550C2211yescharacteristic
65#1547C2211yescharacteristic
66#1547C2211yescharacteristic
67#1547C2211yescharacteristic
68#1547C2211yescharacteristic
69#1544—21no—
70#1543—21no—
71#1292C2211yescharacteristic
72#1292C2211yescharacteristic
73#1292C2211yescharacteristic
74#1292C2211yescharacteristic
75#1292C2211yescharacteristic
76#1292C2211yescharacteristic
77#1292C2211yescharacteristic
78#1292C2211yescharacteristic
79C2×#131—21no—
80C2×#131—21no—
81C22×M16—21no—
82C22×M16—21no—
83#533D411yescharacteristic
84#533D411yescharacteristic
85#533C2311yescharacteristic
86#532D411yescharacteristic
87#532D411yescharacteristic
88#532C2311yescharacteristic
89#529—21no—
90#529—21no—
91#529—21no—
92#529—21no—
93#529C2×C412yes—
94#529Q812yes—
95#525—21no—
96#525—21no—
97#525Q812yes—
98C4×#41—21no—
99C4×#41—21no—
100C4×#41C2×C412yes—
101C22×C42C2311yescharacteristic
102#446—21no—
103#446—21no—
104#446C2×C412yes—
105#446C2×C412yes—
106C8×Q8—21no—
107C8×Q8—21no—
108C8×Q8C2×C412yes—
109C8×Q8C2×C412yes—
110#436—22no—
111#436—22no—
112#436—21no—
113#436—21no—
114#436C2×C412yes—
115#436C2×C412yes—
116#435—22no—
117#435—22no—
118#435—22no—
119#435—22no—
120C8×D4—22no—
121C8×D4—22no—
122C8×D4—21no—
123C8×D4—21no—
124C8×D4C2×C412yes—
125C8×D4C2×C412yes—
126#433—22no—
127#433—22no—
128#432—22no—
129#432—21no—
130#432—21no—
131#432—21no—
132#432—21no—
133#432—21no—
134#432—21no—
135#431—22no—
136#431—21no—
137#431—21no—
138#431—21no—
139#431—21no—
140#431—21no—
141#431—21no—
142#431C2×C411yescharacteristic
143#431C2×C411yescharacteristic
144#431C2×C411yescharacteristic
145#431C2×C411yescharacteristic
146#431D411yescharacteristic
147#431D411yescharacteristic
148#431D411yescharacteristic
149#431D411yescharacteristic
150C2×#108—21no—
151C2×#108—21no—
152C2×#108—21no—
153C2×#108—21no—
154#424—21no—
155#424—21no—
156#424—22no—
157#424—22no—
158#423—21no—
159#423—21no—
160#423—22no—
161#423—21no—
162#423—21no—
163#423—21no—
164#423C2×C412yes—
165#423C2311yescharacteristic
166#423C2311yescharacteristic
167#423C2311yescharacteristic
168#423C2311yescharacteristic
169#423C2311yescharacteristic
170#423C2311yescharacteristic
171C2×C4⋊C8—21no—
172C2×C4⋊C8—22no—
173C2×C4⋊C8—21no—
174C2×C4⋊C8—21no—
175C2×C4⋊C8C2×C412yes—
176C2×C4⋊C8D411yescharacteristic
177C2×C4⋊C8D411yescharacteristic
178C2×C4⋊C8C2311yescharacteristic
179C2×C4⋊C8C2311yescharacteristic
180C2×C4⋊C8C2311yescharacteristic
181C2×C4⋊C8C2311yescharacteristic
182#408—21no—
183#408—21no—
184#407—21no—
185C2×C22⋊C8—21no—
186C2×C22⋊C8—21no—
187C2×C22⋊C8D411yescharacteristic
188C2×C22⋊C8D411yescharacteristic
189#405—21no—
190#405—21no—
191C4×M16—21no—
192C2×#97—21no—
193C2×C4×C8—21no—
194C2×C4×C8C2311yescharacteristic
195C2×C4×C8C2311yescharacteristic
196#341—21no—
197#341—21no—
198#341—21no—
199#341—21no—
200#335—21no—
201#335—21no—
202#335—21no—
203#335—21no—
204#334—21no—
205#334—21no—
206#334—21no—
207#334—21no—
208#333—21no—
209#333—21no—
210#333—21no—
211#333—21no—
212#332—21no—
213#332—21no—
214#332—21no—
215#332—21no—
216#332—21no—
217#332—21no—
218#332—21no—
219#332—21no—
220#331—21no—
221#331—21no—
222#331—21no—
223#331—21no—
224C2×#41—21no—
225C2×#41—21no—
226C2×#41C4⋊C412yes—
227C23×C4C2×D411yescharacteristic
228C23×C4C2×Q811yescharacteristic
229C23×C4#4111yescharacteristic
230#131—81no—
231#131—81no—
232#131—81no—
233#131—81no—
234C2×M16—21no—
235C2×M16—21no—
236C2×M16—22no—
237C2×M16—22no—
238C2×M16—22no—
239C2×M16—21no—
240C2×M16—21no—
241C2×M16—21no—
242C2×M16—21no—
243C2×M16—41no—
244C2×M16—81no—
245C2×M16—81no—
246C2×M16—81no—
247C2×M16—81no—
248C22×C8—22no—
249C22×C8—22no—
250C22×C8—22no—
251C22×C8—21no—
252C22×C8—21no—
253C22×C8—21no—
254C22×C8—21no—
255C22×C8—41no—
256C22×C8C2×D411yescharacteristic
257C22×C8C2×D411yescharacteristic
258C22×C8C2×D411yescharacteristic
259C22×C8C2×D411yescharacteristic
260C4⋊Q8C2×D411yescharacteristic
261C4⋊Q8C2×D411yescharacteristic
262C4⋊Q8C2×D411yescharacteristic
263#127C2×D411yescharacteristic
264#126—22no—
265#126—22no—
266#125—21no—
267#125—21no—
268#125C4⋊C412yes—
269#124—21no—
270#124—21no—
271#124C4⋊C412yes—
272#123—22no—
273#123—22no—
274#123—22no—
275C22⋊Q8—21no—
276C22⋊Q8—21no—
277C22⋊Q8—21no—
278C22⋊Q8—21no—
279C22⋊Q8C4⋊C412yes—
280C22⋊Q8C2×Q812yes—
281#121—21no—
282#121—21no—
283#121—21no—
284#121—21no—
285#121C4⋊C412yes—
286#121C2×Q812yes—
287#120—22no—
288C4×Q8—21no—
289C4×Q8—21no—
290C4×Q8—21no—
291C4×Q8—21no—
292C4×Q8C4⋊C412yes—
293C4×Q8C22×C412yes—
294C4×D4—22no—
295C4×D4—22no—
296C4×D4—22no—
297C4×D4—22no—
298C4×D4—22no—
299C4×D4—22no—
300C4×D4—21no—
301C4×D4—21no—
302C4×D4—21no—
303C4×D4—21no—
304C4×D4C4⋊C412yes—
305C4×D4C22×C412yes—
306#117—21no—
307#117—21no—
308#117—22no—
309#117—22no—
310#117—22no—
311#117C4⋊C412yes—
312C2×C42—22no—
313C2×C42C22×C411yescharacteristic
314C2×C42C22×C411yescharacteristic
315C2×C42C2×D411yescharacteristic
316C2×C42C2×D411yescharacteristic
317C2×C42C2×D411yescharacteristic
318C2×C42C2×D411yescharacteristic
319C2×C42C2×D411yescharacteristic
320C2×C42C2×Q811yescharacteristic
321C2×C42#4111yescharacteristic
322C2×C42#4111yescharacteristic
323C2×C42#4111yescharacteristic
324C2×C42C2411yescharacteristic
325#108—81no—
326#108—81no—
327#108—81no—
328#108—81no—
329C4⋊C8—22no—
330C4⋊C8—22no—
331C4⋊C8—22no—
332C4⋊C8—22no—
333C4⋊C8—22no—
334C4⋊C8—22no—
335C4⋊C8—21no—
336C4⋊C8—21no—
337C4⋊C8—21no—
338C4⋊C8—21no—
339C4⋊C8—21no—
340C4⋊C8—21no—
341C4⋊C8—21no—
342C4⋊C8—21no—
343C4⋊C8—21no—
344C4⋊C8—21no—
345C4⋊C8—21no—
346C4⋊C8—21no—
347C4⋊C8—21no—
348C4⋊C8—21no—
349C4⋊C8—21no—
350C4⋊C8—21no—
351C4⋊C8—21no—
352C4⋊C8—21no—
353C4⋊C8—41no—
354C4⋊C8—41no—
355C4⋊C8C22⋊C411yescharacteristic
356C4⋊C8C22⋊C411yescharacteristic
357C4⋊C8C22⋊C411yescharacteristic
358C4⋊C8C22⋊C411yescharacteristic
359C4⋊C8C22⋊C411yescharacteristic
360C4⋊C8C22⋊C411yescharacteristic
361C4⋊C8C22⋊C411yescharacteristic
362C4⋊C8C22⋊C411yescharacteristic
363C4⋊C8C22×C412yes—
364C4⋊C8C22×C412yes—
365C22⋊C8—22no—
366C22⋊C8—22no—
367C22⋊C8—22no—
368C22⋊C8—22no—
369C22⋊C8—41no—
370C22⋊C8—41no—
371C22⋊C8—41no—
372C22⋊C8—41no—
373C22⋊C8—41no—
374C22⋊C8—41no—
375C22⋊C8C22⋊C411yescharacteristic
376C22⋊C8C22⋊C411yescharacteristic
377C22⋊C8C22⋊C411yescharacteristic
378C22⋊C8C22⋊C411yescharacteristic
379C22⋊C8C22⋊C411yescharacteristic
380C22⋊C8C22⋊C411yescharacteristic
381C22⋊C8C22⋊C411yescharacteristic
382C22⋊C8C22⋊C411yescharacteristic
383#97—22no—
384#97—22no—
385#97—41no—
386#97—41no—
387#97—41no—
388C4×C8—22no—
389C4×C8—22no—
390C4×C8—41no—
391C4×C8—41no—
392C4×C8—41no—
393C4×C8C22×C411yescharacteristic
394C4×C8C22×C411yescharacteristic
395C4×C8C22×C411yescharacteristic
396C4×C8C22×C411yescharacteristic
397C4×C8C2×D411yescharacteristic
398C4×C8C2×D411yescharacteristic
399C4×C8C2×D411yescharacteristic
400C4×C8C2×D411yescharacteristic
401C24C22⋊Q811yescharacteristic
402#41—42no—
403#41—42no—
404#41—41no—
405#41—41no—
406#41—41no—
407#41—41no—
408#41—82no—
409C2×Q8—21no—
410C2×Q8—21no—
411C2×Q8C2×C4⋊C412yes—
412C2×D4—22no—
413C2×D4—22no—
414C2×D4—22no—
415C2×D4—21no—
416C2×D4—21no—
417C2×D4C2×C4⋊C412yes—
418C22×C4—21no—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—41no—
426C22×C4—41no—
427C22×C4—81no—
428C22×C4C2×C22⋊C411yescharacteristic
429C22×C4C2×C22⋊C411yescharacteristic
430C22×C4C2×C4⋊C411yescharacteristic
431C22×C4C2×C4⋊C411yescharacteristic
432C22×C4#11711yescharacteristic
433C22×C4#11711yescharacteristic
434C22×C4#12111yescharacteristic
435C22×C4#12111yescharacteristic
436C22×C4#12111yescharacteristic
437C22×C4#12111yescharacteristic
438C22×C4C22⋊Q811yescharacteristic
439C22×C4C22⋊Q811yescharacteristic
440C22×C4C22⋊Q811yescharacteristic
441C22×C4C22⋊Q811yescharacteristic
442C22×C4C22⋊Q811yescharacteristic
443C22×C4C22×D411yescharacteristic
444C22×C4C22×Q811yescharacteristic
445C22×C4C2×#4111yescharacteristic
446M16—41no—
447M16—41no—
448M16—41no—
449M16—41no—
450M16—41no—
451M16—41no—
452M16—41no—
453M16—41no—
454M16—41no—
455M16—41no—
456M16—41no—
457M16—41no—
458M16—82no—
459M16—82no—
460M16—82no—
461M16—82no—
462C2×C8—21no—
463C2×C8—21no—
464C2×C8—21no—
465C2×C8—21no—
466C2×C8—21no—
467C2×C8—21no—
468C2×C8—21no—
469C2×C8—21no—
470C2×C8—22no—
471C2×C8—22no—
472C2×C8—41no—
473C2×C8—41no—
474C2×C8—41no—
475C2×C8—41no—
476C2×C8—41no—
477C2×C8—41no—
478C2×C8—41no—
479C2×C8—41no—
480C2×C8—82no—
481C2×C8—82no—
482C2×C8—82no—
483C2×C8—82no—
484C2×C8—81no—
485C2×C8—81no—
486C2×C8—81no—
487C2×C8—81no—
488C2×C8C2×C22⋊C411yescharacteristic
489C2×C8C2×C22⋊C411yescharacteristic
490C2×C8C2×C22⋊C411yescharacteristic
491C2×C8C2×C22⋊C411yescharacteristic
492C2×C8C2×C22⋊C411yescharacteristic
493C2×C8C2×C22⋊C411yescharacteristic
494C2×C8C2×C22⋊C411yescharacteristic
495C2×C8C2×C22⋊C411yescharacteristic
496C4⋊C4—22no—
497C4⋊C4—21no—
498C4⋊C4—21no—
499C4⋊C4—21no—
500C4⋊C4—21no—
501C4⋊C4—22no—
502C4⋊C4—22no—
503C4⋊C4—22no—
504C4⋊C4—22no—
505C4⋊C4—22no—
506C4⋊C4C2×C4⋊C412yes—
507C4⋊C4C2×C4⋊C412yes—
508C22⋊C4—22no—
509C22⋊C4—22no—
510C22⋊C4—22no—
511C22⋊C4—22no—
512C22⋊C4—22no—
513C22⋊C4—22no—
514C22⋊C4—22no—
515C22⋊C4—22no—
516C22⋊C4—22no—
517C42—22no—
518C42—22no—
519C42—22no—
520C42C2×C22⋊C411yescharacteristic
521C42C2×C22⋊C411yescharacteristic
522C42C2×C4⋊C411yescharacteristic
523C42C2×C4⋊C411yescharacteristic
524C42#11711yescharacteristic
525C42#11711yescharacteristic
526C42#12111yescharacteristic
527C42#12111yescharacteristic
528C42#12111yescharacteristic
529C42#12111yescharacteristic
530C42C22⋊Q811yescharacteristic
531C42C22⋊Q811yescharacteristic
532C42C23×C411yescharacteristic
533C42C22×D411yescharacteristic
534C42C22×D411yescharacteristic
535C42C2×#4111yescharacteristic
536C23—21no—
537C23—21no—
538C23—22no—
539C23—41no—
540C23—41no—
541C23#38011yescharacteristic
542C23#38011yescharacteristic
543C23C2×C22⋊Q811yescharacteristic
544Q8—42no—
545Q8—41no—
546Q8—41no—
547Q8—41no—
548Q8—41no—
549D4—42no—
550D4—44no—
551D4—42no—
552D4—42no—
553D4—41no—
554D4—41no—
555D4—41no—
556D4—41no—
557D4—82no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—44no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—82no—
590C2×C4#38011yescharacteristic
591C2×C4#38011yescharacteristic
592C2×C4#38011yescharacteristic
593C2×C4#38011yescharacteristic
594C2×C4#38011yescharacteristic
595C2×C4#38011yescharacteristic
596C2×C4C22×C22⋊C411yescharacteristic
597C2×C4C22×C4⋊C411yescharacteristic
598C2×C4C2×#11711yescharacteristic
599C2×C4C2×#12111yescharacteristic
600C2×C4C2×#12111yescharacteristic
601C2×C4C2×C22⋊Q811yescharacteristic
602C2×C4#57611yescharacteristic
603C2×C4#57711yescharacteristic
604C2×C4#57711yescharacteristic
605C2×C4#57811yescharacteristic
606C8—21no—
607C8—21no—
608C8—21no—
609C8—21no—
610C8—21no—
611C8—21no—
612C8—21no—
613C8—21no—
614C8—42no—
615C8—42no—
616C8—81no—
617C8—81no—
618C8—81no—
619C8—81no—
620C8—81no—
621C8—81no—
622C8—81no—
623C8—81no—
624C22—21no—
625C22—21no—
626C22—21no—
627C22—21no—
628C22—21no—
629C22—21no—
630C22—21no—
631C22—21no—
632C22—42no—
633C22—42no—
634C22—42no—
635C22—81no—
636C22—81no—
637C22C2×#38011yescharacteristic
638C22#308311yescharacteristic
639C22#308411yescharacteristic
640C4—21no—
641C4—21no—
642C4—21no—
643C4—21no—
644C4—21no—
645C4—21no—
646C4—21no—
647C4—21no—
648C4—21no—
649C4—42no—
650C4—42no—
651C4—42no—
652C4#289711yescharacteristic
653C4#289811yescharacteristic
654C4#304511yescharacteristic
655C4#304611yescharacteristic
656C4#306711yescharacteristic
657C4#306711yescharacteristic
658C2—21no—
659C2—21no—
660C2—41no—
661C2—41no—
662C2—82no—
663C2#2065211yescharacteristic
664C2#2067311yescharacteristic
665C2#2067811yescharacteristic
subgroupG/H
center, socleC22C2×#380
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radical#3444C22
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21993C2semidirect
3. C2×#1572C2semidirect
6. #8689C2semidirect
7. #8688C2semidirect
8. #8687C2semidirect
9. #8686C2semidirect
10. #8685C2semidirect
11. #8684C2semidirect
12. #8683C2semidirect
13. #8682C2semidirect
14. #8003C2semidirect
15. #8001C2semidirect

Maximal subgroups

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

subgroupindex
#602682
#219932
C2×#15722
#177532
#177162
#86892
#86882
#86872
#86862
#86852
#86842
#86832
#86822
#80032
#80012