Finite Group Databasebeta

← #208173

Subgroups — #208173

Order 512.

843 conjugacy classes of subgroups.

The subgroup lattice has 647 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35288C211yesmaximal, characteristic
2#21922C211yesmaximal, characteristic, complemented
3#20470C211yesmaximal, characteristic, complemented
4#9906C211yesmaximal, characteristic, complemented
5#9901C211yesmaximal, characteristic, complemented
6#9710C211yesmaximal, characteristic, complemented
7#9698C211yesmaximal, characteristic, complemented
8#9142C211yesmaximal, characteristic
9#9138C211yesmaximal, characteristic, complemented
10#9134C211yesmaximal, characteristic
11#9130C211yesmaximal, characteristic, complemented
12#8237C211yesmaximal, characteristic, complemented
13#8234C211yesmaximal, characteristic, complemented
14C4×#332C211yesmaximal, characteristic, complemented
15C4×#331C211yesmaximal, characteristic, complemented
16C2×#432C2211yescharacteristic
17C2×#431C2211yescharacteristic
18#2859C2211yescharacteristic
19#2858C2211yescharacteristic
20#2857C2211yescharacteristic
21#2856C2211yescharacteristic
22#2644—21no—
23#2624—21no—
24#2513C412yes—
25#2490C412yes—
26#2454C412yes—
27#2448—21no—
28#2447—21no—
29#2427—21no—
30#2426—21no—
31#2420C412yes—
32#2324—21no—
33#2319—21no—
34C4×C4⋊Q8C2211yescharacteristic
35C4×#127C2211yescharacteristic
36C4×#117C2211yescharacteristic
37#1923—21no—
38#1923—21no—
39#1916—21no—
40#1916—21no—
41#1842C2211yescharacteristic
42#1831C2211yescharacteristic
43#1797C2211yescharacteristic
44#1797C2211yescharacteristic
45#1767C2211yescharacteristic
46#1767C2211yescharacteristic
47C4×C4⋊C8C2211yescharacteristic
48C4×C4⋊C8C2211yescharacteristic
49C4×C4⋊C8C2211yescharacteristic
50C4×C4⋊C8C2211yescharacteristic
51#1750—21no—
52#1750—21no—
53#1746—21no—
54#1726—21no—
55C2×#332C2211yescharacteristic
56C2×#332C2211yescharacteristic
57C2×#332C2211yescharacteristic
58C2×#332C2211yescharacteristic
59C2×#331C2211yescharacteristic
60C2×#331C2211yescharacteristic
61C2×#331C2211yescharacteristic
62C2×#331C2211yescharacteristic
63#1296C2211yescharacteristic
64#1296C2211yescharacteristic
65#1296C2211yescharacteristic
66#1296C2211yescharacteristic
67#1296C2211yescharacteristic
68#1296C2211yescharacteristic
69#1296C2211yescharacteristic
70#1296C2211yescharacteristic
71C2×C4⋊Q8C2311yescharacteristic
72C2×#127C2311yescharacteristic
73C2×#125C2×C412yes—
74C2×#124—21no—
75C2×#124—21no—
76C2×C22⋊Q8—21no—
77C2×#121—21no—
78C2×#121—21no—
79C2×#121C2×C412yes—
80C2×C4×Q8—21no—
81C2×C4×D4—21no—
82C2×C4×D4—21no—
83C2×C4×D4C2×C412yes—
84C2×#117D411yescharacteristic
85C2×#117D411yescharacteristic
86C22×C42C2311yescharacteristic
87#432—22no—
88#432—22no—
89#431—22no—
90#431—22no—
91C2×C4⋊C8—21no—
92C2×C4⋊C8—21no—
93C2×C4⋊C8C2311yescharacteristic
94C2×C4⋊C8C2311yescharacteristic
95C2×C4⋊C8C2311yescharacteristic
96C2×C4⋊C8C2311yescharacteristic
97C2×C4⋊C8C2311yescharacteristic
98C2×C4⋊C8C2311yescharacteristic
99C2×C4⋊C8C2311yescharacteristic
100C2×C4⋊C8C2311yescharacteristic
101C2×C22⋊C8—21no—
102C2×C22⋊C8—21no—
103C2×C22⋊C8—21no—
104C2×C22⋊C8—21no—
105C2×#97—21no—
106C2×#97—21no—
107C2×#97—21no—
108C2×C4×C8—21no—
109C2×C4×C8—21no—
110C2×C4×C8—21no—
111C2×C4×C8—21no—
112C2×C4×C8—21no—
113#399—22no—
114#397—22no—
115#397—22no—
116#397—22no—
117#397—22no—
118#396—22no—
119#394—22no—
120#394—22no—
121#391—21no—
122#391—21no—
123#390—22no—
124#390—21no—
125#390—21no—
126#390—21no—
127#390—21no—
128#390—21no—
129#390—21no—
130#390C2×C412yes—
131#390C2×C412yes—
132#390C2×C412yes—
133#390C2×C412yes—
134#389—21no—
135#389—21no—
136#389—21no—
137#389—21no—
138#389—21no—
139#389—21no—
140#389C2×C412yes—
141#389C2×C412yes—
142#388—22no—
143#388—22no—
144#388—22no—
145#386—22no—
146#385—22no—
147#384C2311yescharacteristic
148#384C2311yescharacteristic
149#382C2×C412yes—
150#381—22no—
151#380—21no—
152#380—22no—
153#380C2×C412yes—
154C4×C4⋊C4—21no—
155C4×C4⋊C4C2×C412yes—
156C4×C4⋊C4C2×C411yescharacteristic
157C4×C4⋊C4C2×C411yescharacteristic
158C4×C4⋊C4D411yescharacteristic
159C4×C4⋊C4D411yescharacteristic
160C4×C22⋊C4C2×C411yescharacteristic
161C4×C22⋊C4C2×C411yescharacteristic
162C4×C22⋊C4D411yescharacteristic
163C4×C22⋊C4D411yescharacteristic
164#376D411yescharacteristic
165#376D411yescharacteristic
166C43C2311yescharacteristic
167C43C2311yescharacteristic
168#336—21no—
169#336—21no—
170#336—21no—
171#336—21no—
172#336—21no—
173#336—21no—
174#336—21no—
175#336—21no—
176#336—21no—
177#336—21no—
178#332—22no—
179#332—22no—
180#332—22no—
181#332—22no—
182#331—22no—
183#331—22no—
184#331—22no—
185#331—22no—
186C22×Q8—21no—
187C22×D4—22no—
188C22×D4—21no—
189C22×D4—21no—
190C22×D4C22×C412yes—
191C23×C4—22no—
192C23×C4C22×C411yescharacteristic
193C23×C4C2×D411yescharacteristic
194C23×C4C2×D411yescharacteristic
195C22×C8—21no—
196C22×C8—21no—
197C22×C8—21no—
198C22×C8—21no—
199C22×C8—21no—
200C22×C8—21no—
201C22×C8—21no—
202C22×C8—21no—
203C4⋊Q8—21no—
204C4⋊Q8—21no—
205C4⋊Q8—21no—
206C4⋊Q8—21no—
207#127—21no—
208#127—21no—
209#127—21no—
210#127—21no—
211#125—82no—
212#124—81no—
213#124—81no—
214#124—81no—
215#124—81no—
216C22⋊Q8—81no—
217C22⋊Q8—81no—
218#121—42no—
219#121—42no—
220#121—81no—
221#121—81no—
222#121—82no—
223C4×Q8—81no—
224C4×Q8—81no—
225C4×D4—42no—
226C4×D4—42no—
227C4×D4—81no—
228C4×D4—81no—
229C4×D4—82no—
230#117—22no—
231#117—22no—
232#117—41no—
233#117—41no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4—21no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4C22⋊C411yescharacteristic
244C2×C4⋊C4C22⋊C411yescharacteristic
245C2×C4⋊C4C22⋊C411yescharacteristic
246C2×C4⋊C4C22⋊C411yescharacteristic
247C2×C4⋊C4C22×C412yes—
248C2×C4⋊C4C22×C412yes—
249C2×C4⋊C4C22×C412yes—
250C2×C22⋊C4—22no—
251C2×C22⋊C4—22no—
252C2×C22⋊C4—22no—
253C2×C22⋊C4—22no—
254C2×C22⋊C4—22no—
255C2×C22⋊C4—22no—
256C2×C22⋊C4—22no—
257C2×C22⋊C4—22no—
258C2×C22⋊C4—22no—
259C2×C22⋊C4—22no—
260C2×C22⋊C4—22no—
261C2×C22⋊C4C22⋊C411yescharacteristic
262C2×C22⋊C4C22⋊C411yescharacteristic
263C2×C22⋊C4C22⋊C411yescharacteristic
264C2×C22⋊C4C22⋊C411yescharacteristic
265C2×C42—21no—
266C2×C42—21no—
267C2×C42—21no—
268C2×C42—21no—
269C2×C42—22no—
270C2×C42C22×C411yescharacteristic
271C2×C42C22×C411yescharacteristic
272C2×C42C22×C411yescharacteristic
273C2×C42C22×C411yescharacteristic
274C2×C42C22×C411yescharacteristic
275C2×C42C2×D411yescharacteristic
276C2×C42C2×D411yescharacteristic
277C2×C42C2×D411yescharacteristic
278C2×C42C2×D411yescharacteristic
279C2×C42C2×D411yescharacteristic
280C2×C42C2×D411yescharacteristic
281C2×C42C2×D411yescharacteristic
282C2×C42C2×D411yescharacteristic
283C2×C42C2×D411yescharacteristic
284C2×C42C2×D411yescharacteristic
285C2×C42C2411yescharacteristic
286C4⋊C8—21no—
287C4⋊C8—21no—
288C4⋊C8—21no—
289C4⋊C8—21no—
290C4⋊C8—21no—
291C4⋊C8—21no—
292C4⋊C8—21no—
293C4⋊C8—21no—
294C4⋊C8—21no—
295C4⋊C8—21no—
296C4⋊C8—21no—
297C4⋊C8—21no—
298C4⋊C8—81no—
299C4⋊C8—81no—
300C4⋊C8#4112yes—
301C4⋊C8#4112yes—
302C4⋊C8#4112yes—
303C4⋊C8#4112yes—
304C22⋊C8—24no—
305C22⋊C8—24no—
306C22⋊C8—81no—
307C22⋊C8—81no—
308#97—24no—
309#97—42no—
310#97—42no—
311C4×C8—24no—
312C4×C8—42no—
313C4×C8—42no—
314C4×C8—81no—
315C4×C8—81no—
316#95—22no—
317#95—22no—
318#95—22no—
319#95—22no—
320#95—22no—
321#95—22no—
322#95C22⋊C411yescharacteristic
323#95C22⋊C411yescharacteristic
324#95C22⋊C411yescharacteristic
325#95C22⋊C411yescharacteristic
326#95C22⋊C411yescharacteristic
327#95C22⋊C411yescharacteristic
328#95C22⋊C411yescharacteristic
329#95C22⋊C411yescharacteristic
330C24—22no—
331C24C2×C22⋊C411yescharacteristic
332C2×Q8—41no—
333C2×Q8—41no—
334C2×Q8—41no—
335C2×Q8—41no—
336C2×Q8—41no—
337C2×Q8—41no—
338C2×D4—22no—
339C2×D4—22no—
340C2×D4—42no—
341C2×D4—42no—
342C2×D4—44no—
343C2×D4—41no—
344C2×D4—41no—
345C2×D4—41no—
346C2×D4—41no—
347C2×D4—41no—
348C2×D4—41no—
349C2×D4—41no—
350C2×D4—41no—
351C2×D4—41no—
352C2×D4—41no—
353C2×D4—41no—
354C2×D4—41no—
355C2×D4—82no—
356C2×D4—82no—
357C22×C4—21no—
358C22×C4—21no—
359C22×C4—21no—
360C22×C4—21no—
361C22×C4—21no—
362C22×C4—21no—
363C22×C4—21no—
364C22×C4—21no—
365C22×C4—21no—
366C22×C4—21no—
367C22×C4—21no—
368C22×C4—21no—
369C22×C4—21no—
370C22×C4—21no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—41no—
375C22×C4—41no—
376C22×C4—44no—
377C22×C4—82no—
378C22×C4—82no—
379C22×C4C2×C22⋊C411yescharacteristic
380C22×C4C2×C22⋊C411yescharacteristic
381C22×C4C2×C22⋊C411yescharacteristic
382C22×C4C2×C22⋊C411yescharacteristic
383C22×C4C2×C22⋊C411yescharacteristic
384C22×C4C2×C22⋊C411yescharacteristic
385C22×C4C2×C22⋊C411yescharacteristic
386C22×C4C2×C22⋊C411yescharacteristic
387C22×C4C2×C22⋊C411yescharacteristic
388C22×C4C2×C22⋊C411yescharacteristic
389C22×C4C2×C22⋊C411yescharacteristic
390C22×C4C23×C411yescharacteristic
391C22×C4C22×D411yescharacteristic
392C22×C4C22×D411yescharacteristic
393C2×C8—22no—
394C2×C8—22no—
395C2×C8—22no—
396C2×C8—22no—
397C2×C8—41no—
398C2×C8—41no—
399C2×C8—41no—
400C2×C8—41no—
401C2×C8—41no—
402C2×C8—41no—
403C2×C8—41no—
404C2×C8—41no—
405C2×C8—41no—
406C2×C8—41no—
407C2×C8—41no—
408C2×C8—41no—
409C2×C8—41no—
410C2×C8—41no—
411C2×C8—41no—
412C2×C8—41no—
413C2×C8—41no—
414C2×C8—41no—
415C2×C8—41no—
416C2×C8—41no—
417C4⋊C4—22no—
418C4⋊C4—22no—
419C4⋊C4—22no—
420C4⋊C4—22no—
421C4⋊C4—41no—
422C4⋊C4—41no—
423C4⋊C4—42no—
424C4⋊C4—42no—
425C4⋊C4—41no—
426C4⋊C4—41no—
427C4⋊C4—41no—
428C4⋊C4—41no—
429C4⋊C4—41no—
430C4⋊C4—41no—
431C4⋊C4—41no—
432C4⋊C4—41no—
433C4⋊C4—41no—
434C4⋊C4—41no—
435C4⋊C4—82no—
436C4⋊C4—82no—
437C4⋊C4—82no—
438C4⋊C4—82no—
439C22⋊C4—22no—
440C22⋊C4—22no—
441C22⋊C4—41no—
442C22⋊C4—41no—
443C22⋊C4—44no—
444C22⋊C4—44no—
445C22⋊C4—82no—
446C22⋊C4—82no—
447C22⋊C4—82no—
448C22⋊C4—82no—
449C22⋊C4—82no—
450C22⋊C4—82no—
451C22⋊C4—82no—
452C22⋊C4—82no—
453C22⋊C4—82no—
454C42—21no—
455C42—21no—
456C42—21no—
457C42—21no—
458C42—21no—
459C42—21no—
460C42—21no—
461C42—21no—
462C42—21no—
463C42—21no—
464C42—21no—
465C42—21no—
466C42—21no—
467C42—21no—
468C42—21no—
469C42—21no—
470C42—21no—
471C42—21no—
472C42—21no—
473C42—21no—
474C42—41no—
475C42—41no—
476C42—41no—
477C42—41no—
478C42—82no—
479C42#13512yes—
480C42#13512yes—
481C42C2×#4111yescharacteristic
482C42C2×#4111yescharacteristic
483C42C2×#4111yescharacteristic
484C42C2×#4111yescharacteristic
485C23—21no—
486C23—21no—
487C23—21no—
488C23—21no—
489C23—21no—
490C23—21no—
491C23—21no—
492C23—42no—
493C23—42no—
494C23—42no—
495C23—42no—
496C23—42no—
497C23—42no—
498C23—42no—
499C23C22×C22⋊C411yescharacteristic
500Q8—81no—
501Q8—81no—
502Q8—81no—
503Q8—81no—
504D4—82no—
505D4—82no—
506D4—81no—
507D4—81no—
508D4—81no—
509D4—81no—
510D4—81no—
511D4—81no—
512D4—81no—
513D4—81no—
514D4—164no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
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—21no—
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—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—81no—
566C2×C4—81no—
567C2×C4—164no—
568C2×C4#41612yes—
569C2×C4#41612yes—
570C2×C4C4×#4111yescharacteristic
571C2×C4C4×#4111yescharacteristic
572C2×C4#52511yescharacteristic
573C2×C4#52511yescharacteristic
574C2×C4#53211yescharacteristic
575C2×C4#53211yescharacteristic
576C2×C4C2×#13511yescharacteristic
577C2×C4C2×#13511yescharacteristic
578C2×C4#57511yescharacteristic
579C2×C4#57511yescharacteristic
580C2×C4#57611yescharacteristic
581C2×C4#57611yescharacteristic
582C2×C4#57711yescharacteristic
583C2×C4#57711yescharacteristic
584C8—42no—
585C8—42no—
586C8—42no—
587C8—42no—
588C8—81no—
589C8—81no—
590C8—81no—
591C8—81no—
592C22—41no—
593C22—41no—
594C22—41no—
595C22—41no—
596C22—41no—
597C22—41no—
598C22—41no—
599C22—82no—
600C22—82no—
601C22—82no—
602C22—82no—
603C22—82no—
604C22—82no—
605C22—82no—
606C22#229711yescharacteristic
607C22#288811yescharacteristic
608C22#288911yescharacteristic
609C22C2×#41611yescharacteristic
610C22#289611yescharacteristic
611C22#289711yescharacteristic
612C22#289711yescharacteristic
613C4—22no—
614C4—22no—
615C4—21no—
616C4—21no—
617C4—41no—
618C4—41no—
619C4—41no—
620C4—41no—
621C4—41no—
622C4—41no—
623C4—41no—
624C4—41no—
625C4—41no—
626C4—82no—
627C4—82no—
628C4—82no—
629C4#310211yescharacteristic
630C4#310211yescharacteristic
631C4#310211yescharacteristic
632C4#310211yescharacteristic
633C4#317411yescharacteristic
634C4#317611yescharacteristic
635C4#317911yescharacteristic
636C4#317911yescharacteristic
637C2—81no—
638C2—162no—
639C2#1772411yescharacteristic
640C2#1772711yescharacteristic
641C2#1776611yescharacteristic
642C2#1776911yescharacteristic
643C2#2138711yescharacteristic
644C2#2141411yescharacteristic
645C2#2142711yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radical#2856C22
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C4×#117 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21922C2semidirect
3. #20470C2semidirect
4. #9906C2semidirect
5. #9901C2semidirect
6. #9710C2semidirect
7. #9698C2semidirect
9. #9138C2semidirect
11. #9130C2semidirect
12. #8237C2semidirect
13. #8234C2semidirect
14. C4×#332C2semidirect
15. C4×#331C2semidirect

Maximal subgroups

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

subgroupindex
#352882
#219222
#204702
#99062
#99012
#97102
#96982
#91422
#91382
#91342
#91302
#82372
#82342
C4×#3322
C4×#3312