Finite Group Databasebeta

← #189350

Subgroups — #189350

Order 512.

847 conjugacy classes of subgroups.

The subgroup lattice has 674 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60208C211yesmaximal, characteristic
2#20831C211yesmaximal, characteristic, complemented
3C2×#1565C211yesmaximal, characteristic, complemented
4#17719C211yesmaximal, 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#8002C211yesmaximal, characteristic, complemented
15#8000C211yesmaximal, characteristic, complemented
16#3444C2211yescharacteristic
17#2971—21no—
18#2971C412yes—
19#2970—21no—
20#2970C412yes—
21#2969—21no—
22#2969—21no—
23#2969C412yes—
24#2969C412yes—
25#2960C2211yescharacteristic
26#2960C2211yescharacteristic
27#2931C2211yescharacteristic
28#2931C2211yescharacteristic
29#2930—21no—
30#2930—21no—
31#2923—21no—
32#2920C2211yescharacteristic
33#2920C2211yescharacteristic
34#2920C2211yescharacteristic
35C2×#431C2211yescharacteristic
36C22×C4⋊C8C2211yescharacteristic
37#2875—21no—
38C2×C4×M16C2211yescharacteristic
39#1848C2211yescharacteristic
40#1848C2211yescharacteristic
41#1837C2211yescharacteristic
42#1837C2211yescharacteristic
43#1811—21no—
44#1811—21no—
45#1778—21no—
46#1778—21no—
47#1565—21no—
48#1565—21no—
49#1565—21no—
50#1565—21no—
51C2×#335C2211yescharacteristic
52C2×#335C2211yescharacteristic
53C2×#334C2211yescharacteristic
54C2×#334C2211yescharacteristic
55#1554—21no—
56#1553—21no—
57#1552—21no—
58#1551—21no—
59#1550C2211yescharacteristic
60#1550C2211yescharacteristic
61#1550C2211yescharacteristic
62#1550C2211yescharacteristic
63#1547C2211yescharacteristic
64#1547C2211yescharacteristic
65#1547C2211yescharacteristic
66#1547C2211yescharacteristic
67#1544—21no—
68#1543—21no—
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×M16D411yescharacteristic
82C22×M16D411yescharacteristic
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—22no—
114#436—22no—
115#436—21no—
116#436—21no—
117#436C2×C412yes—
118#436C2×C412yes—
119#435—22no—
120#435—22no—
121#435—22no—
122#435—22no—
123C8×D4—21no—
124C8×D4—21no—
125C8×D4C2×C412yes—
126C8×D4C2×C412yes—
127#433—22no—
128#433—22no—
129#432—21no—
130#432—22no—
131#432—22no—
132#432—21no—
133#432—21no—
134#432—21no—
135#432—21no—
136#432C2×C412yes—
137#431—21no—
138#431—21no—
139#431—21no—
140#431—21no—
141#431—21no—
142#431—21no—
143#431—21no—
144#431—21no—
145#431—21no—
146#431C2×C412yes—
147C2×#108—21no—
148C2×#108—21no—
149C2×#108—21no—
150C2×#108—21no—
151#424—21no—
152#424—21no—
153#423C2311yescharacteristic
154#423C2311yescharacteristic
155#423C2311yescharacteristic
156#423C2311yescharacteristic
157#423C2311yescharacteristic
158#423C2311yescharacteristic
159C2×C4⋊C8—21no—
160C2×C4⋊C8—21no—
161C2×C4⋊C8—21no—
162C2×C4⋊C8—21no—
163C2×C4⋊C8—21no—
164C2×C4⋊C8C2311yescharacteristic
165C2×C4⋊C8C2311yescharacteristic
166C2×C4⋊C8C2311yescharacteristic
167C2×C4⋊C8C2311yescharacteristic
168#408—21no—
169#408—21no—
170#407—21no—
171#407—21no—
172#407—21no—
173C2×C22⋊C8—21no—
174#405—21no—
175#405—21no—
176#405—22no—
177#405—22no—
178C4×M16—21no—
179C4×M16—22no—
180C4×M16—21no—
181C4×M16—21no—
182C4×M16C2×C411yescharacteristic
183C4×M16C2×C411yescharacteristic
184C4×M16C2×C411yescharacteristic
185C4×M16C2×C411yescharacteristic
186C4×M16D411yescharacteristic
187C4×M16D411yescharacteristic
188C4×M16D411yescharacteristic
189C4×M16D411yescharacteristic
190C2×#97—22no—
191C2×#97D411yescharacteristic
192C2×#97D411yescharacteristic
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—42no—
231#131—42no—
232#131—42no—
233#131—42no—
234C2×M16—21no—
235C2×M16—21no—
236C2×M16—22no—
237C2×M16—22no—
238C2×M16—22no—
239C2×M16—22no—
240C2×M16—21no—
241C2×M16—21no—
242C2×M16—21no—
243C2×M16—21no—
244C2×M16—41no—
245C2×M16—41no—
246C2×M16C22⋊C411yescharacteristic
247C2×M16C22⋊C411yescharacteristic
248C2×M16C22⋊C411yescharacteristic
249C2×M16C22⋊C411yescharacteristic
250C2×M16C22⋊C411yescharacteristic
251C2×M16C22⋊C411yescharacteristic
252C2×M16C22⋊C411yescharacteristic
253C2×M16C22⋊C411yescharacteristic
254C22×C8—22no—
255C22×C8—22no—
256C22×C8—21no—
257C22×C8—21no—
258C22×C8—21no—
259C22×C8—21no—
260C22×C8—41no—
261C22×C8—41no—
262C22×C8—82no—
263C22×C8C2×D411yescharacteristic
264C22×C8C2×D411yescharacteristic
265C22×C8C2×D411yescharacteristic
266C22×C8C2×D411yescharacteristic
267C4⋊Q8C2×D411yescharacteristic
268C4⋊Q8C2×D411yescharacteristic
269C4⋊Q8C2×D411yescharacteristic
270#127C2×D411yescharacteristic
271#126—22no—
272#126—22no—
273#125—21no—
274#125—21no—
275#125C4⋊C412yes—
276#124—21no—
277#124—21no—
278#124C4⋊C412yes—
279#123—22no—
280#123—22no—
281#123—22no—
282C22⋊Q8—21no—
283C22⋊Q8—21no—
284C22⋊Q8—21no—
285C22⋊Q8—21no—
286C22⋊Q8C4⋊C412yes—
287C22⋊Q8C2×Q812yes—
288#121—21no—
289#121—21no—
290#121—21no—
291#121—21no—
292#121C4⋊C412yes—
293#121C2×Q812yes—
294#120—22no—
295C4×Q8—21no—
296C4×Q8—21no—
297C4×Q8—21no—
298C4×Q8—21no—
299C4×Q8C4⋊C412yes—
300C4×Q8C22×C412yes—
301C4×D4—22no—
302C4×D4—22no—
303C4×D4—22no—
304C4×D4—22no—
305C4×D4—22no—
306C4×D4—22no—
307C4×D4—21no—
308C4×D4—21no—
309C4×D4—21no—
310C4×D4—21no—
311C4×D4C4⋊C412yes—
312C4×D4C22×C412yes—
313#117—21no—
314#117—21no—
315#117—22no—
316#117—22no—
317#117—22no—
318#117C4⋊C412yes—
319C2×C42—22no—
320C2×C42C22×C411yescharacteristic
321C2×C42C22×C411yescharacteristic
322C2×C42C2×D411yescharacteristic
323C2×C42C2×D411yescharacteristic
324C2×C42C2×D411yescharacteristic
325C2×C42C2×D411yescharacteristic
326C2×C42C2×D411yescharacteristic
327C2×C42C2×Q811yescharacteristic
328C2×C42#4111yescharacteristic
329C2×C42#4111yescharacteristic
330C2×C42#4111yescharacteristic
331C2×C42C2411yescharacteristic
332#108—42no—
333#108—42no—
334#108—42no—
335#108—42no—
336C4⋊C8—22no—
337C4⋊C8—22no—
338C4⋊C8—22no—
339C4⋊C8—22no—
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—21no—
357C4⋊C8—21no—
358C4⋊C8—41no—
359C4⋊C8—41no—
360C4⋊C8C22×C412yes—
361C4⋊C8C22×C412yes—
362C22⋊C8—22no—
363C22⋊C8—22no—
364C22⋊C8—22no—
365C22⋊C8—22no—
366C22⋊C8—22no—
367C22⋊C8—22no—
368C22⋊C8—41no—
369C22⋊C8—41no—
370C22⋊C8—41no—
371C22⋊C8—41no—
372C22⋊C8—41no—
373C22⋊C8—41no—
374#97—22no—
375#97—22no—
376#97—22no—
377#97—41no—
378#97—41no—
379#97C22⋊C411yescharacteristic
380#97C22⋊C411yescharacteristic
381#97C22⋊C411yescharacteristic
382#97C22⋊C411yescharacteristic
383#97C22⋊C411yescharacteristic
384#97C22⋊C411yescharacteristic
385#97C22⋊C411yescharacteristic
386#97C22⋊C411yescharacteristic
387C4×C8—22no—
388C4×C8—22no—
389C4×C8—22no—
390C4×C8—41no—
391C4×C8—41no—
392C4×C8—41no—
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—41no—
404#41—41no—
405#41—41no—
406#41—41no—
407#41—81no—
408#41—81no—
409#41—82no—
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—81no—
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—22no—
448M16—22no—
449M16—22no—
450M16—22no—
451M16—41no—
452M16—41no—
453M16—41no—
454M16—41no—
455M16—44no—
456M16—41no—
457M16—41no—
458M16—41no—
459M16—41no—
460M16—44no—
461M16—41no—
462M16—41no—
463M16—41no—
464M16—41no—
465M16—41no—
466M16—41no—
467M16—41no—
468M16—41no—
469C2×C8—21no—
470C2×C8—21no—
471C2×C8—21no—
472C2×C8—21no—
473C2×C8—21no—
474C2×C8—21no—
475C2×C8—22no—
476C2×C8—22no—
477C2×C8—44no—
478C2×C8—44no—
479C2×C8—41no—
480C2×C8—41no—
481C2×C8—41no—
482C2×C8—41no—
483C2×C8—41no—
484C2×C8—41no—
485C2×C8—41no—
486C2×C8—41no—
487C2×C8—82no—
488C2×C8—82no—
489C2×C8—81no—
490C2×C8—81no—
491C2×C8—81no—
492C2×C8—81no—
493C2×C8C2×C22⋊C411yescharacteristic
494C2×C8C2×C22⋊C411yescharacteristic
495C2×C8C2×C22⋊C411yescharacteristic
496C2×C8C2×C22⋊C411yescharacteristic
497C2×C8C2×C22⋊C411yescharacteristic
498C2×C8C2×C22⋊C411yescharacteristic
499C2×C8C2×C22⋊C411yescharacteristic
500C2×C8C2×C22⋊C411yescharacteristic
501C2×C8#13512yes—
502C2×C8#13512yes—
503C4⋊C4—22no—
504C4⋊C4—21no—
505C4⋊C4—21no—
506C4⋊C4—21no—
507C4⋊C4—21no—
508C4⋊C4—22no—
509C4⋊C4—22no—
510C4⋊C4—22no—
511C4⋊C4—22no—
512C4⋊C4—22no—
513C4⋊C4C2×C4⋊C412yes—
514C4⋊C4C2×C4⋊C412yes—
515C22⋊C4—22no—
516C22⋊C4—22no—
517C22⋊C4—22no—
518C22⋊C4—22no—
519C22⋊C4—22no—
520C22⋊C4—22no—
521C22⋊C4—22no—
522C22⋊C4—22no—
523C22⋊C4—22no—
524C42—22no—
525C42—22no—
526C42—22no—
527C42C2×C22⋊C411yescharacteristic
528C42C2×C22⋊C411yescharacteristic
529C42C2×C4⋊C411yescharacteristic
530C42C2×C4⋊C411yescharacteristic
531C42#11711yescharacteristic
532C42#11711yescharacteristic
533C42#12111yescharacteristic
534C42#12111yescharacteristic
535C42#12111yescharacteristic
536C42#12111yescharacteristic
537C42C22⋊Q811yescharacteristic
538C42C22⋊Q811yescharacteristic
539C42C23×C411yescharacteristic
540C42C22×D411yescharacteristic
541C42C22×D411yescharacteristic
542C42C2×#4111yescharacteristic
543C23—21no—
544C23—21no—
545C23—22no—
546C23—41no—
547C23—41no—
548C23#38011yescharacteristic
549C23#38011yescharacteristic
550C23C2×C22⋊Q811yescharacteristic
551Q8—42no—
552Q8—41no—
553Q8—41no—
554Q8—41no—
555Q8—41no—
556D4—42no—
557D4—42no—
558D4—42no—
559D4—41no—
560D4—41no—
561D4—41no—
562D4—41no—
563D4—82no—
564D4—82no—
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—21no—
580C2×C4—21no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—82no—
596C2×C4—82no—
597C2×C4#38011yescharacteristic
598C2×C4#38011yescharacteristic
599C2×C4#38011yescharacteristic
600C2×C4#38011yescharacteristic
601C2×C4#38011yescharacteristic
602C2×C4#38011yescharacteristic
603C2×C4C22×C22⋊C411yescharacteristic
604C2×C4C22×C4⋊C411yescharacteristic
605C2×C4C2×#11711yescharacteristic
606C2×C4C2×#12111yescharacteristic
607C2×C4C2×#12111yescharacteristic
608C2×C4C2×C22⋊Q811yescharacteristic
609C2×C4C2×#13511yescharacteristic
610C2×C4C2×#13511yescharacteristic
611C2×C4#57511yescharacteristic
612C2×C4#57511yescharacteristic
613C8—21no—
614C8—21no—
615C8—21no—
616C8—21no—
617C8—21no—
618C8—21no—
619C8—42no—
620C8—42no—
621C8—81no—
622C8—81no—
623C8—81no—
624C8—81no—
625C8—81no—
626C8—81no—
627C8—81no—
628C8—81no—
629C8#41612yes—
630C8#41612yes—
631C22—21no—
632C22—21no—
633C22—21no—
634C22—21no—
635C22—21no—
636C22—21no—
637C22—21no—
638C22—21no—
639C22—42no—
640C22—42no—
641C22—42no—
642C22—42no—
643C22—42no—
644C22C2×#38011yescharacteristic
645C22#307611yescharacteristic
646C22#307611yescharacteristic
647C4—21no—
648C4—21no—
649C4—21no—
650C4—21no—
651C4—21no—
652C4—21no—
653C4—21no—
654C4—21no—
655C4—21no—
656C4—42no—
657C4—42no—
658C4—42no—
659C4C2×#41611yescharacteristic
660C4#289611yescharacteristic
661C4#303511yescharacteristic
662C4#303511yescharacteristic
663C4#306811yescharacteristic
664C4#306911yescharacteristic
665C2—21no—
666C2—21no—
667C2—41no—
668C2—41no—
669C2—82no—
670C2#2064511yescharacteristic
671C2#2067311yescharacteristic
672C2#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. #20831C2semidirect
3. C2×#1565C2semidirect
6. #8689C2semidirect
7. #8688C2semidirect
8. #8687C2semidirect
9. #8686C2semidirect
10. #8685C2semidirect
11. #8684C2semidirect
12. #8683C2semidirect
13. #8682C2semidirect
14. #8002C2semidirect
15. #8000C2semidirect

Maximal subgroups

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

subgroupindex
#602082
#208312
C2×#15652
#177192
#177162
#86892
#86882
#86872
#86862
#86852
#86842
#86832
#86822
#80022
#80002