Finite Group Databasebeta

← #189382

Subgroups — #189382

Order 512.

833 conjugacy classes of subgroups.

The subgroup lattice has 670 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60268C211yesmaximal, characteristic
2#21993C211yesmaximal, characteristic, complemented
3C2×#1573C211yesmaximal, characteristic, complemented
4#17753C211yesmaximal, characteristic
5#17719C211yesmaximal, 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
15C2×#1292C211yesmaximal, characteristic, complemented
16#3444C2211yescharacteristic
17#2994—21no—
18#2994—21no—
19#2993C412yes—
20#2993C412yes—
21#2991—21no—
22#2990—21no—
23#2989C412yes—
24#2988C412yes—
25#2960C2211yescharacteristic
26#2960C2211yescharacteristic
27#2933—21no—
28#2933—21no—
29#2931C2211yescharacteristic
30#2931C2211yescharacteristic
31#2925—21no—
32#2920C2211yescharacteristic
33C2×#432C2211yescharacteristic
34C2×#431C2211yescharacteristic
35#2906—21no—
36C2×#423C2211yescharacteristic
37C2×#423C2211yescharacteristic
38C22×C4⋊C8C2211yescharacteristic
39#1950—21no—
40#1950—21no—
41#1855—21no—
42#1855—21no—
43#1848C2211yescharacteristic
44#1848C2211yescharacteristic
45#1837C2211yescharacteristic
46#1837C2211yescharacteristic
47#1573—21no—
48#1573—21no—
49#1573—21no—
50#1573—21no—
51C2×#335C2211yescharacteristic
52C2×#335C2211yescharacteristic
53C2×#334C2211yescharacteristic
54C2×#334C2211yescharacteristic
55#1559—21no—
56#1557—21no—
57#1552—21no—
58#1552—21no—
59#1551—21no—
60#1551—21no—
61#1550C2211yescharacteristic
62#1550C2211yescharacteristic
63#1550C2211yescharacteristic
64#1550C2211yescharacteristic
65#1547C2211yescharacteristic
66#1547C2211yescharacteristic
67#1547C2211yescharacteristic
68#1547C2211yescharacteristic
69#1542—21no—
70#1540—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—
83C23×C8—21no—
84#533D411yescharacteristic
85#533D411yescharacteristic
86#533C2311yescharacteristic
87#532D411yescharacteristic
88#532D411yescharacteristic
89#532C2311yescharacteristic
90#529—21no—
91#529—21no—
92#529—21no—
93#529—21no—
94#529C2×C412yes—
95#529Q812yes—
96#525—21no—
97#525—21no—
98#525Q812yes—
99C4×#41—21no—
100C4×#41—21no—
101C4×#41C2×C412yes—
102C22×C42C2311yescharacteristic
103#446—21no—
104#446—21no—
105#446C2×C412yes—
106#446C2×C412yes—
107C8×Q8—21no—
108C8×Q8—21no—
109C8×Q8C2×C412yes—
110C8×Q8C2×C412yes—
111#436—22no—
112#436—22no—
113#436—21no—
114#436—21no—
115#436C2×C412yes—
116#436C2×C412yes—
117#435—22no—
118#435—22no—
119#435—22no—
120#435—22no—
121C8×D4—22no—
122C8×D4—22no—
123C8×D4—21no—
124C8×D4—21no—
125C8×D4C2×C412yes—
126C8×D4C2×C412yes—
127#433—22no—
128#433—22no—
129#432—22no—
130#432—21no—
131#432—21no—
132#432—21no—
133#432—21no—
134#432—21no—
135#432—21no—
136#431—22no—
137#431—21no—
138#431—21no—
139#431C2×C411yescharacteristic
140#431C2×C411yescharacteristic
141#431C2×C411yescharacteristic
142#431C2×C411yescharacteristic
143#431D411yescharacteristic
144#431D411yescharacteristic
145#431D411yescharacteristic
146#431D411yescharacteristic
147C2×#108—21no—
148C2×#108—21no—
149C2×#108—21no—
150C2×#108—21no—
151#424—21no—
152#424—21no—
153#424—22no—
154#424—22no—
155#423—21no—
156#423—21no—
157#423—22no—
158#423—21no—
159#423—21no—
160#423—21no—
161#423C2×C412yes—
162#423C2311yescharacteristic
163#423C2311yescharacteristic
164#423C2311yescharacteristic
165#423C2311yescharacteristic
166#423C2311yescharacteristic
167#423C2311yescharacteristic
168C2×C4⋊C8—21no—
169C2×C4⋊C8—22no—
170C2×C4⋊C8—21no—
171C2×C4⋊C8—21no—
172C2×C4⋊C8—21no—
173C2×C4⋊C8—21no—
174C2×C4⋊C8—21no—
175C2×C4⋊C8—21no—
176C2×C4⋊C8—21no—
177C2×C4⋊C8C2×C412yes—
178C2×C4⋊C8D411yescharacteristic
179C2×C4⋊C8D411yescharacteristic
180C2×C4⋊C8C2311yescharacteristic
181C2×C4⋊C8C2311yescharacteristic
182C2×C4⋊C8C2311yescharacteristic
183C2×C4⋊C8C2311yescharacteristic
184#408—21no—
185#408—21no—
186#407—21no—
187C2×C22⋊C8—21no—
188C2×C22⋊C8D411yescharacteristic
189C2×C22⋊C8D411yescharacteristic
190#405—21no—
191#405—21no—
192C4×M16—21no—
193C2×#97—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×C8—41no—
257C22×C8—41no—
258C22×C8—82no—
259C22×C8C2×D411yescharacteristic
260C22×C8C2×D411yescharacteristic
261C22×C8C2×D411yescharacteristic
262C22×C8C2×D411yescharacteristic
263C4⋊Q8C2×D411yescharacteristic
264C4⋊Q8C2×D411yescharacteristic
265C4⋊Q8C2×D411yescharacteristic
266#127C2×D411yescharacteristic
267#126—22no—
268#126—22no—
269#125—21no—
270#125—21no—
271#125C4⋊C412yes—
272#124—21no—
273#124—21no—
274#124C4⋊C412yes—
275#123—22no—
276#123—22no—
277#123—22no—
278C22⋊Q8—21no—
279C22⋊Q8—21no—
280C22⋊Q8—21no—
281C22⋊Q8—21no—
282C22⋊Q8C4⋊C412yes—
283C22⋊Q8C2×Q812yes—
284#121—21no—
285#121—21no—
286#121—21no—
287#121—21no—
288#121C4⋊C412yes—
289#121C2×Q812yes—
290#120—22no—
291C4×Q8—21no—
292C4×Q8—21no—
293C4×Q8—21no—
294C4×Q8—21no—
295C4×Q8C4⋊C412yes—
296C4×Q8C22×C412yes—
297C4×D4—22no—
298C4×D4—22no—
299C4×D4—22no—
300C4×D4—22no—
301C4×D4—22no—
302C4×D4—22no—
303C4×D4—21no—
304C4×D4—21no—
305C4×D4—21no—
306C4×D4—21no—
307C4×D4C4⋊C412yes—
308C4×D4C22×C412yes—
309#117—21no—
310#117—21no—
311#117—22no—
312#117—22no—
313#117—22no—
314#117C4⋊C412yes—
315C2×C42—22no—
316C2×C42C22×C411yescharacteristic
317C2×C42C22×C411yescharacteristic
318C2×C42C2×D411yescharacteristic
319C2×C42C2×D411yescharacteristic
320C2×C42C2×D411yescharacteristic
321C2×C42C2×D411yescharacteristic
322C2×C42C2×D411yescharacteristic
323C2×C42C2×Q811yescharacteristic
324C2×C42#4111yescharacteristic
325C2×C42#4111yescharacteristic
326C2×C42#4111yescharacteristic
327C2×C42C2411yescharacteristic
328#108—81no—
329#108—81no—
330#108—81no—
331#108—81no—
332C4⋊C8—22no—
333C4⋊C8—22no—
334C4⋊C8—22no—
335C4⋊C8—22no—
336C4⋊C8—22no—
337C4⋊C8—22no—
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—21no—
354C4⋊C8—21no—
355C4⋊C8—21no—
356C4⋊C8—41no—
357C4⋊C8—41no—
358C4⋊C8—41no—
359C4⋊C8—41no—
360C4⋊C8C22⋊C411yescharacteristic
361C4⋊C8C22⋊C411yescharacteristic
362C4⋊C8C22⋊C411yescharacteristic
363C4⋊C8C22⋊C411yescharacteristic
364C4⋊C8C22⋊C411yescharacteristic
365C4⋊C8C22⋊C411yescharacteristic
366C4⋊C8C22⋊C411yescharacteristic
367C4⋊C8C22⋊C411yescharacteristic
368C4⋊C8C22×C412yes—
369C4⋊C8C22×C412yes—
370C22⋊C8—22no—
371C22⋊C8—22no—
372C22⋊C8—22no—
373C22⋊C8—22no—
374C22⋊C8—41no—
375C22⋊C8—41no—
376C22⋊C8—41no—
377C22⋊C8—41no—
378C22⋊C8C22⋊C411yescharacteristic
379C22⋊C8C22⋊C411yescharacteristic
380C22⋊C8C22⋊C411yescharacteristic
381C22⋊C8C22⋊C411yescharacteristic
382C22⋊C8C22⋊C411yescharacteristic
383C22⋊C8C22⋊C411yescharacteristic
384C22⋊C8C22⋊C411yescharacteristic
385C22⋊C8C22⋊C411yescharacteristic
386#97—22no—
387#97—22no—
388#97—41no—
389#97—41no—
390#97—41no—
391C4×C8—22no—
392C4×C8—22no—
393C4×C8—41no—
394C4×C8C22×C411yescharacteristic
395C4×C8C22×C411yescharacteristic
396C4×C8C22×C411yescharacteristic
397C4×C8C22×C411yescharacteristic
398C4×C8C2×D411yescharacteristic
399C4×C8C2×D411yescharacteristic
400C4×C8C2×D411yescharacteristic
401C4×C8C2×D411yescharacteristic
402C24C22⋊Q811yescharacteristic
403#41—44no—
404#41—41no—
405#41—41no—
406#41—41no—
407#41—41no—
408#41—81no—
409#41—81no—
410C2×Q8—21no—
411C2×Q8—21no—
412C2×Q8C2×C4⋊C412yes—
413C2×D4—22no—
414C2×D4—22no—
415C2×D4—22no—
416C2×D4—21no—
417C2×D4—21no—
418C2×D4C2×C4⋊C412yes—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—41no—
427C22×C4—41no—
428C22×C4—42no—
429C22×C4C2×C22⋊C411yescharacteristic
430C22×C4C2×C22⋊C411yescharacteristic
431C22×C4C2×C4⋊C411yescharacteristic
432C22×C4C2×C4⋊C411yescharacteristic
433C22×C4#11711yescharacteristic
434C22×C4#11711yescharacteristic
435C22×C4#12111yescharacteristic
436C22×C4#12111yescharacteristic
437C22×C4#12111yescharacteristic
438C22×C4#12111yescharacteristic
439C22×C4C22⋊Q811yescharacteristic
440C22×C4C22⋊Q811yescharacteristic
441C22×C4C22⋊Q811yescharacteristic
442C22×C4C22⋊Q811yescharacteristic
443C22×C4C22⋊Q811yescharacteristic
444C22×C4C22×D411yescharacteristic
445C22×C4C22×Q811yescharacteristic
446C22×C4C2×#4111yescharacteristic
447M16—41no—
448M16—41no—
449M16—41no—
450M16—41no—
451M16—41no—
452M16—41no—
453M16—41no—
454M16—41no—
455M16—41no—
456M16—41no—
457M16—41no—
458M16—41no—
459M16—82no—
460M16—82no—
461M16—82no—
462M16—82no—
463C2×C8—21no—
464C2×C8—21no—
465C2×C8—21no—
466C2×C8—21no—
467C2×C8—21no—
468C2×C8—21no—
469C2×C8—21no—
470C2×C8—21no—
471C2×C8—22no—
472C2×C8—22no—
473C2×C8—41no—
474C2×C8—41no—
475C2×C8—41no—
476C2×C8—41no—
477C2×C8—41no—
478C2×C8—41no—
479C2×C8—41no—
480C2×C8—41no—
481C2×C8—82no—
482C2×C8—82no—
483C2×C8—82no—
484C2×C8—82no—
485C2×C8—82no—
486C2×C8—82no—
487C2×C8—81no—
488C2×C8—81no—
489C2×C8—81no—
490C2×C8—81no—
491C2×C8C2×C22⋊C411yescharacteristic
492C2×C8C2×C22⋊C411yescharacteristic
493C2×C8C2×C22⋊C411yescharacteristic
494C2×C8C2×C22⋊C411yescharacteristic
495C2×C8C2×C22⋊C411yescharacteristic
496C2×C8C2×C22⋊C411yescharacteristic
497C2×C8C2×C22⋊C411yescharacteristic
498C2×C8C2×C22⋊C411yescharacteristic
499C4⋊C4—22no—
500C4⋊C4—21no—
501C4⋊C4—21no—
502C4⋊C4—21no—
503C4⋊C4—21no—
504C4⋊C4—22no—
505C4⋊C4—22no—
506C4⋊C4—22no—
507C4⋊C4—22no—
508C4⋊C4—22no—
509C4⋊C4C2×C4⋊C412yes—
510C4⋊C4C2×C4⋊C412yes—
511C22⋊C4—22no—
512C22⋊C4—22no—
513C22⋊C4—22no—
514C22⋊C4—22no—
515C22⋊C4—22no—
516C22⋊C4—22no—
517C22⋊C4—22no—
518C22⋊C4—22no—
519C22⋊C4—22no—
520C42—22no—
521C42—22no—
522C42—22no—
523C42C2×C22⋊C411yescharacteristic
524C42C2×C22⋊C411yescharacteristic
525C42C2×C4⋊C411yescharacteristic
526C42C2×C4⋊C411yescharacteristic
527C42#11711yescharacteristic
528C42#11711yescharacteristic
529C42#12111yescharacteristic
530C42#12111yescharacteristic
531C42#12111yescharacteristic
532C42#12111yescharacteristic
533C42C22⋊Q811yescharacteristic
534C42C22⋊Q811yescharacteristic
535C42C23×C411yescharacteristic
536C42C22×D411yescharacteristic
537C42C22×D411yescharacteristic
538C42C2×#4111yescharacteristic
539C23—21no—
540C23—21no—
541C23—22no—
542C23—22no—
543C23—41no—
544C23#38011yescharacteristic
545C23#38011yescharacteristic
546C23C2×C22⋊Q811yescharacteristic
547Q8—42no—
548Q8—41no—
549Q8—41no—
550Q8—41no—
551Q8—41no—
552D4—42no—
553D4—42no—
554D4—42no—
555D4—44no—
556D4—41no—
557D4—41no—
558D4—41no—
559D4—41no—
560D4—82no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—22no—
564C2×C4—22no—
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—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—44no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—82no—
593C2×C4#38011yescharacteristic
594C2×C4#38011yescharacteristic
595C2×C4#38011yescharacteristic
596C2×C4#38011yescharacteristic
597C2×C4#38011yescharacteristic
598C2×C4#38011yescharacteristic
599C2×C4C22×C22⋊C411yescharacteristic
600C2×C4C22×C4⋊C411yescharacteristic
601C2×C4C2×#11711yescharacteristic
602C2×C4C2×#12111yescharacteristic
603C2×C4C2×#12111yescharacteristic
604C2×C4C2×C22⋊Q811yescharacteristic
605C2×C4#57611yescharacteristic
606C2×C4#57711yescharacteristic
607C2×C4#57711yescharacteristic
608C2×C4#57811yescharacteristic
609C8—21no—
610C8—21no—
611C8—21no—
612C8—21no—
613C8—21no—
614C8—21no—
615C8—21no—
616C8—21no—
617C8—42no—
618C8—42no—
619C8—81no—
620C8—81no—
621C8—81no—
622C8—81no—
623C8—81no—
624C8—81no—
625C8—81no—
626C8—81no—
627C22—21no—
628C22—21no—
629C22—21no—
630C22—21no—
631C22—21no—
632C22—21no—
633C22—21no—
634C22—42no—
635C22—42no—
636C22—42no—
637C22—42no—
638C22—42no—
639C22#181312yes—
640C22C2×#38011yescharacteristic
641C22#308311yescharacteristic
642C22#308411yescharacteristic
643C4—21no—
644C4—21no—
645C4—21no—
646C4—21no—
647C4—21no—
648C4—21no—
649C4—21no—
650C4—21no—
651C4—21no—
652C4—42no—
653C4—42no—
654C4—42no—
655C4#289711yescharacteristic
656C4#289811yescharacteristic
657C4#304511yescharacteristic
658C4#304611yescharacteristic
659C4#306811yescharacteristic
660C4#306911yescharacteristic
661C2—21no—
662C2—21no—
663C2—22no—
664C2—41no—
665C2—82no—
666C2#2065211yescharacteristic
667C2C2×#181311yescharacteristic
668C2#2067911yescharacteristic
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×#1573C2semidirect
6. #8689C2semidirect
7. #8688C2semidirect
8. #8687C2semidirect
9. #8686C2semidirect
10. #8685C2semidirect
11. #8684C2semidirect
12. #8683C2semidirect
13. #8682C2semidirect
14. #8003C2semidirect
15. C2×#1292C2semidirect

Maximal subgroups

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

subgroupindex
#602682
#219932
C2×#15732
#177532
#177192
#86892
#86882
#86872
#86862
#86852
#86842
#86832
#86822
#80032
C2×#12922