Finite Group Databasebeta

← #360644

Subgroups — #360644

Order 512.

1482 conjugacy classes of subgroups.

The subgroup lattice has 620 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2357C211yesmaximal, characteristic
2C2×#1835C211yesmaximal, characteristic, complemented
3C2×#1804C211yesmaximal, characteristic
4C2×C4×#102C211yesmaximal, characteristic
5#13954C211yesmaximal, characteristic, complemented
6#11066C212yesmaximal
7#11066C212yesmaximal
8#10885C212yesmaximal, complemented
9#10884C212yesmaximal, complemented
10#10850C211yesmaximal, characteristic, complemented
11#10834C211yesmaximal, characteristic, complemented
12C22×C4×D4C2211yescharacteristic
13C22×C4⋊C8C2211yescharacteristic
14C22×#102C2211yescharacteristic
15C22×#102C2211yescharacteristic
16C22×#102C2212yes—
17C22×C4×C8C2211yescharacteristic
18#2357—24no—
19#2357—24no—
20C2×#390C2212yes—
21C2×#389C2212yes—
22C2×#389C2212yes—
23C2×#388—22no—
24C2×#382C2211yescharacteristic
25C2×#382C2211yescharacteristic
26C2×#381—21no—
27C2×C4×C4⋊C4C2212yes—
28C2×C4×C4⋊C4C2211yescharacteristic
29C2×C4×C4⋊C4C2211yescharacteristic
30C2×C4×C22⋊C4—21no—
31#1835—24no—
32#1804—24no—
33C4×#102C418yescomplemented
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#340C2211yescharacteristic
40C2×#340C2211yescharacteristic
41C2×#336C2212yes—
42C2×#336C2212yes—
43#1444—22no—
44C4⋊#95—21no—
45C4⋊#95—21no—
46C23×D4C2311yescharacteristic
47C24×C4—21no—
48C23×C8C2312yes—
49C23×C8C2311yescharacteristic
50C23×C8C2311yescharacteristic
51C2×C4×D4—22no—
52C2×C4×D4—22no—
53C2×C4×D4—22no—
54C2×C4×D4C2×C414yes—
55C2×C4×D4D414yes—
56C2×C4×D4Q814yes—
57C22×C4⋊C4—22no—
58C22×C4⋊C4—22no—
59C22×C4⋊C4D412yes—
60C22×C4⋊C4C2312yes—
61C22×C4⋊C4C2312yes—
62C22×C4⋊C4C2311yescharacteristic
63C22×C4⋊C4C2311yescharacteristic
64C22×C4⋊C4C2311yescharacteristic
65C22×C22⋊C4—21no—
66C22×C22⋊C4—22no—
67C22×C22⋊C4—22no—
68C22×C42—21no—
69C22×C42—22no—
70C22×C42C2312yes—
71C22×C42C2311yescharacteristic
72C2×C4⋊C8—22no—
73C2×C4⋊C8—22no—
74C2×C4⋊C8—22no—
75C2×#102—22no—
76C2×#102—22no—
77C2×#102—22no—
78C2×#102—22no—
79C2×#102—24no—
80C2×#102—42no—
81C2×#102—42no—
82C2×#102C2×C414yes—
83C2×#102C2×C414yes—
84C2×C4×C8—22no—
85C2×C4×C8—22no—
86C2×C4×C8C2×C414yes—
87#390—24no—
88#390—24no—
89#389—24no—
90#389—24no—
91#389—42no—
92#389—42no—
93#388—216no—
94#382—22no—
95#382—22no—
96#382—22no—
97#382D414yes—
98#381—28no—
99C4×C4⋊C4—24no—
100C4×C4⋊C4—24no—
101C4×C4⋊C4—22no—
102C4×C4⋊C4—22no—
103C4×C4⋊C4—22no—
104C4×C4⋊C4C2×C414yes—
105C4×C22⋊C4—28no—
106C2×#95—21no—
107C2×#95—21no—
108C2×#95—22no—
109C2×#95—21no—
110C2×#95—21no—
111C2×#95—22no—
112C2×#95—22no—
113C2×#95—21no—
114C2×#95D411yescharacteristic
115C2×#95D411yescharacteristic
116#340—24no—
117#340—24no—
118#340—24no—
119#340—24no—
120#340—24no—
121#340C2×C414yes—
122#340C2×C414yes—
123#336—42no—
124#336—42no—
125C25—21no—
126C22×D4—21no—
127C22×D4—21no—
128C22×D4—21no—
129C22×D4—21no—
130C22×D4—21no—
131C22×D4—21no—
132C22×D4—21no—
133C22×D4—21no—
134C22×D4—21no—
135C22×D4—21no—
136C22×D4—21no—
137C22×D4C22×C412yes—
138C22×D4C2×D412yes—
139C22×D4C2×Q812yes—
140C23×C4—21no—
141C23×C4—21no—
142C23×C4—21no—
143C23×C4—24no—
144C23×C4—24no—
145C23×C4—24no—
146C23×C4—22no—
147C23×C4—22no—
148C23×C4—82no—
149C23×C4—82no—
150C23×C4C2×D412yes—
151C23×C4C2×D412yes—
152C23×C4C2×D411yescharacteristic
153C23×C4C2×D411yescharacteristic
154C23×C4C2411yescharacteristic
155C22×C8—21no—
156C22×C8—21no—
157C22×C8—21no—
158C22×C8—21no—
159C22×C8—21no—
160C22×C8—21no—
161C22×C8—21no—
162C22×C8—21no—
163C22×C8—21no—
164C22×C8—21no—
165C22×C8—21no—
166C22×C8—21no—
167C22×C8—22no—
168C22×C8—22no—
169C22×C8—22no—
170C22×C8—22no—
171C22×C8—22no—
172C22×C8—22no—
173C22×C8—22no—
174C22×C8C22×C412yes—
175C22×C8C22×C412yes—
176C4×D4—28no—
177C4×D4—28no—
178C4×D4—28no—
179C4×D4C4⋊C4116yescomplemented
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—24no—
194C2×C4⋊C4—24no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—42no—
203C2×C4⋊C4—44no—
204C2×C4⋊C4—44no—
205C2×C4⋊C4—44no—
206C2×C4⋊C4—44no—
207C2×C4⋊C4—82no—
208C2×C4⋊C4—82no—
209C2×C4⋊C4C22×C412yes—
210C2×C4⋊C4C22×C412yes—
211C2×C4⋊C4C22×C412yes—
212C2×C4⋊C4C2×D414yes—
213C2×C4⋊C4C2×D412yes—
214C2×C4⋊C4C2×Q812yes—
215C2×C4⋊C4#4112yes—
216C2×C4⋊C4#4112yes—
217C2×C4⋊C4#4112yes—
218C2×C4⋊C4#4112yes—
219C2×C22⋊C4—24no—
220C2×C22⋊C4—24no—
221C2×C22⋊C4—24no—
222C2×C22⋊C4—28no—
223C2×C22⋊C4—28no—
224C2×C22⋊C4—44no—
225C2×C22⋊C4—44no—
226C2×C22⋊C4—82no—
227C2×C22⋊C4—82no—
228C2×C42—22no—
229C2×C42—22no—
230C2×C42—22no—
231C2×C42—22no—
232C2×C42—22no—
233C2×C42—22no—
234C2×C42—28no—
235C2×C42—24no—
236C2×C42—82no—
237C2×C42—82no—
238C2×C42—82no—
239C2×C42C22×C412yes—
240C2×C42C2×D412yes—
241C2×C42C2×D412yes—
242C2×C42C2×Q812yes—
243C2×C42#4112yes—
244C2×C42#4112yes—
245C4⋊C8—44no—
246#102—28no—
247#102—28no—
248#102—48no—
249C4×C8—28no—
250#95—28no—
251#95—24no—
252#95—24no—
253#95—24no—
254#95—22no—
255#95—22no—
256#95—44no—
257#95—44no—
258#95—42no—
259#95—42no—
260C24—22no—
261C24—22no—
262C24—22no—
263C24—41no—
264C24—41no—
265C24—41no—
266C24—41no—
267C24—42no—
268C24—42no—
269C24—42no—
270C24—42no—
271C24C22×D411yescharacteristic
272C2×D4—22no—
273C2×D4—22no—
274C2×D4—22no—
275C2×D4—22no—
276C2×D4—22no—
277C2×D4—22no—
278C2×D4—22no—
279C2×D4—22no—
280C2×D4—22no—
281C2×D4—22no—
282C2×D4—22no—
283C2×D4—22no—
284C2×D4—22no—
285C2×D4—22no—
286C2×D4—22no—
287C2×D4—41no—
288C2×D4—41no—
289C2×D4—41no—
290C2×D4—41no—
291C2×D4—41no—
292C2×D4—41no—
293C2×D4—41no—
294C2×D4—41no—
295C2×D4—41no—
296C2×D4—41no—
297C2×D4—41no—
298C2×D4—41no—
299C2×D4C2×C4⋊C414yes—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—216no—
320C22×C4—22no—
321C22×C4—24no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—42no—
329C22×C4—42no—
330C22×C4—42no—
331C22×C4—42no—
332C22×C4—42no—
333C22×C4—42no—
334C22×C4—42no—
335C22×C4—42no—
336C22×C4—42no—
337C22×C4—42no—
338C22×C4—42no—
339C22×C4—42no—
340C22×C4—42no—
341C22×C4—42no—
342C22×C4—42no—
343C22×C4—42no—
344C22×C4—42no—
345C22×C4—42no—
346C22×C4—42no—
347C22×C4—88no—
348C22×C4—84no—
349C22×C4—84no—
350C22×C4—88no—
351C22×C4C4×D412yes—
352C22×C4C4×D412yes—
353C22×C4#12012yes—
354C22×C4#12012yes—
355C22×C4C22⋊Q812yes—
356C22×C4C22⋊Q812yes—
357C22×C4#13514yes—
358C22×C4C23×C411yescharacteristic
359C22×C4C22×D411yescharacteristic
360C22×C4C22×D411yescharacteristic
361C22×C4C22×Q811yescharacteristic
362C22×C4C2×#4111yescharacteristic
363C22×C4C2×#4111yescharacteristic
364C22×C4C2×#4111yescharacteristic
365C22×C4C2×#4111yescharacteristic
366C22×C4C2×#4111yescharacteristic
367C22×C4C2×#4111yescharacteristic
368C22×C4H(5,𝔽2)11yescharacteristic
369C22×C4H(5,𝔽2)11yescharacteristic
370C22×C42−1+411yescharacteristic
371C22×C42−1+411yescharacteristic
372C2×C8—22no—
373C2×C8—22no—
374C2×C8—22no—
375C2×C8—22no—
376C2×C8—22no—
377C2×C8—22no—
378C2×C8—41no—
379C2×C8—41no—
380C2×C8—41no—
381C2×C8—41no—
382C2×C8—41no—
383C2×C8—41no—
384C2×C8—41no—
385C2×C8—41no—
386C2×C8—42no—
387C2×C8—42no—
388C2×C8—42no—
389C2×C8—42no—
390C2×C8—42no—
391C2×C8—42no—
392C2×C8—42no—
393C4⋊C4—22no—
394C4⋊C4—22no—
395C4⋊C4—22no—
396C4⋊C4—22no—
397C4⋊C4—22no—
398C4⋊C4—22no—
399C4⋊C4—22no—
400C4⋊C4—22no—
401C4⋊C4—22no—
402C4⋊C4—24no—
403C4⋊C4—24no—
404C4⋊C4—48no—
405C4⋊C4—42no—
406C4⋊C4—42no—
407C4⋊C4—42no—
408C4⋊C4—42no—
409C4⋊C4—42no—
410C4⋊C4—42no—
411C4⋊C4—88no—
412C4⋊C4—88no—
413C4⋊C4C2×C4⋊C414yes—
414C4⋊C4#11714yes—
415C4⋊C4#11714yes—
416C22⋊C4—216no—
417C22⋊C4—416no—
418C22⋊C4—88no—
419C42—42no—
420C42—42no—
421C42—42no—
422C42—42no—
423C42—84no—
424C42—88no—
425C42C2×C4⋊C414yes—
426C42C4×D414yes—
427C42C22⋊Q814yes—
428C42#12314yes—
429C23—24no—
430C23—44no—
431C23—44no—
432C23—42no—
433C23—42no—
434C23—42no—
435C23—42no—
436C23—42no—
437C23—42no—
438C23—42no—
439C23—42no—
440C23—42no—
441C23—42no—
442C23—44no—
443C23—44no—
444C23—82no—
445C23—82no—
446C23—82no—
447C23—82no—
448C23—82no—
449C23—82no—
450C23—82no—
451C23—82no—
452C23C2×C4×D411yescharacteristic
453C23C2×#12011yescharacteristic
454C23C2×C22⋊Q811yescharacteristic
455C23#54611yescharacteristic
456C23#54611yescharacteristic
457C23#54811yescharacteristic
458C23#54811yescharacteristic
459C23C2×#13512yes—
460C23#57512yes—
461C23#57612yes—
462C23#57712yes—
463D4—24no—
464D4—24no—
465D4—42no—
466D4—42no—
467D4—42no—
468D4—42no—
469D4—42no—
470D4—42no—
471C2×C4—24no—
472C2×C4—24no—
473C2×C4—22no—
474C2×C4—22no—
475C2×C4—22no—
476C2×C4—22no—
477C2×C4—24no—
478C2×C4—24no—
479C2×C4—24no—
480C2×C4—24no—
481C2×C4—24no—
482C2×C4—24no—
483C2×C4—24no—
484C2×C4—24no—
485C2×C4—44no—
486C2×C4—44no—
487C2×C4—44no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—44no—
497C2×C4—44no—
498C2×C4—44no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—44no—
503C2×C4—44no—
504C2×C4—42no—
505C2×C4—44no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—816no—
512C2×C4—82no—
513C2×C4—82no—
514C2×C4—82no—
515C2×C4—82no—
516C2×C4—82no—
517C2×C4—82no—
518C2×C4—82no—
519C2×C4—82no—
520C2×C4—82no—
521C2×C4—82no—
522C2×C4—82no—
523C2×C4#38514yes—
524C2×C4#38514yes—
525C2×C4C22×C4⋊C411yescharacteristic
526C2×C4C2×#11711yescharacteristic
527C2×C4C2×#11711yescharacteristic
528C2×C4C2×C4×D411yescharacteristic
529C2×C4#52011yescharacteristic
530C2×C4#52011yescharacteristic
531C2×C4C2×C22⋊Q811yescharacteristic
532C2×C4C2×#12311yescharacteristic
533C2×C4#53211yescharacteristic
534C2×C4#53211yescharacteristic
535C2×C4#53311yescharacteristic
536C2×C4#53311yescharacteristic
537C2×C4#53511yescharacteristic
538C2×C4#53611yescharacteristic
539C2×C4#54411yescharacteristic
540C2×C4#54411yescharacteristic
541C2×C4#54711yescharacteristic
542C2×C4#54711yescharacteristic
543C2×C4#55011yescharacteristic
544C2×C4#55011yescharacteristic
545C2×C4#55211yescharacteristic
546C2×C4#55211yescharacteristic
547C2×C4#55311yescharacteristic
548C2×C4#55311yescharacteristic
549C2×C4#55511yescharacteristic
550C2×C4#55511yescharacteristic
551C2×C4#55611yescharacteristic
552C2×C4#55611yescharacteristic
553C8—42no—
554C8—42no—
555C8—82no—
556C22—44no—
557C22—44no—
558C22—44no—
559C22—84no—
560C22—84no—
561C22—84no—
562C22—84no—
563C22—82no—
564C22—82no—
565C22—82no—
566C22—82no—
567C22C2×#38511yescharacteristic
568C22#235211yescharacteristic
569C22#235211yescharacteristic
570C22#243111yescharacteristic
571C22#243211yescharacteristic
572C22#243911yescharacteristic
573C22#243911yescharacteristic
574C22C4×#13511yescharacteristic
575C22#294311yescharacteristic
576C22#294811yescharacteristic
577C22#294911yescharacteristic
578C22#300212yes—
579C22#301212yes—
580C22#307612yes—
581C22#308112yes—
582C22#329712yes—
583C22#330112yes—
584C22#332212yes—
585C22#332612yes—
586C22#336112yes—
587C22#336512yes—
588C22#337012yes—
589C22#337112yes—
590C4—24no—
591C4—24no—
592C4—44no—
593C4—44no—
594C4—44no—
595C4—44no—
596C4—84no—
597C4—82no—
598C4—162no—
599C4#228511yescharacteristic
600C4#231711yescharacteristic
601C4#235411yescharacteristic
602C4#235411yescharacteristic
603C4#244111yescharacteristic
604C4#244111yescharacteristic
605C4#249911yescharacteristic
606C4#250011yescharacteristic
607C2—84no—
608C2#1387911yescharacteristic
609C2#2087211yescharacteristic
610C2#2087911yescharacteristic
611C2#2276411yescharacteristic
612C2#2276511yescharacteristic
613C2#2277211yescharacteristic
614C2#2277311yescharacteristic
615C2#2557512yes—
616C2#2565312yes—
617C2#2602812yes—
618C2#2603012yes—
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#1835C2semidirect
5. #13954C2semidirect
8. #10885C2semidirect
9. #10884C2semidirect
10. #10850C2semidirect
11. #10834C2semidirect
33. C4×#102C4semidirect
179. C4×D4C4⋊C4semidirect

Maximal subgroups

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

subgroupindex
C2×#23572
C2×#18352
C2×#18042
C2×C4×#1022
#139542
#110662
#108852
#108842
#108502
#108342