Finite Group Databasebeta

← #242611

Subgroups — #242611

Order 512.

818 conjugacy classes of subgroups.

The subgroup lattice has 598 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21953C211yesmaximal, characteristic, complemented
2#21805C211yesmaximal, characteristic
3#21386C211yesmaximal, characteristic
4#21170C211yesmaximal, characteristic, complemented
5C2×#1574C211yesmaximal, characteristic, complemented
6C2×#1557C211yesmaximal, characteristic
7#10600C211yesmaximal, characteristic, complemented
8#10599C211yesmaximal, characteristic, complemented
9#10596C211yesmaximal, characteristic, complemented
10#10595C211yesmaximal, characteristic, complemented
11#10592C211yesmaximal, characteristic, complemented
12#10591C211yesmaximal, characteristic, complemented
13#10588C211yesmaximal, characteristic, complemented
14#10587C211yesmaximal, characteristic, complemented
15#8046C211yesmaximal, characteristic, complemented
16C2×#532C2211yescharacteristic
17C2×#432C2211yescharacteristic
18C2×#431C2211yescharacteristic
19C2×#431C2211yescharacteristic
20C2×#428—21no—
21C2×#423C2211yescharacteristic
22#2285C2211yescharacteristic
23C2×C4×C4⋊C4C2211yescharacteristic
24#1988C2211yescharacteristic
25#1988C2211yescharacteristic
26#1983—21no—
27#1983—21no—
28#1982C2211yescharacteristic
29#1982C2211yescharacteristic
30#1957C2211yescharacteristic
31#1957C2211yescharacteristic
32#1956C2211yescharacteristic
33#1956C2211yescharacteristic
34#1949—21no—
35#1949—21no—
36#1948C2211yescharacteristic
37#1948C2211yescharacteristic
38#1947C2211yescharacteristic
39#1947C2211yescharacteristic
40#1940—21no—
41#1940—21no—
42#1939C2211yescharacteristic
43#1939C2211yescharacteristic
44#1939C2211yescharacteristic
45#1939C2211yescharacteristic
46#1874—21no—
47#1874—21no—
48#1797—21no—
49#1797—21no—
50#1788—21no—
51#1788—21no—
52C4×#102—21no—
53C4×#102—21no—
54#1736—21no—
55#1574—24no—
56C2×#335C2211yescharacteristic
57C2×#335C2211yescharacteristic
58C2×#334C2211yescharacteristic
59C2×#334C2211yescharacteristic
60#1557—24no—
61C2×#332C2211yescharacteristic
62C2×#332C2211yescharacteristic
63C2×#331C2211yescharacteristic
64C2×#331C2211yescharacteristic
65#1296C2211yescharacteristic
66#1296C2211yescharacteristic
67#1296C2211yescharacteristic
68#1296C2211yescharacteristic
69C22×#41—21no—
70C22×M16—21no—
71#532—22no—
72#532—22no—
73#532—22no—
74#532Q814yes—
75C2×C4⋊Q8C2311yescharacteristic
76C2×#127C2311yescharacteristic
77C2×#124—21no—
78C2×#121—21no—
79C2×#121—21no—
80C2×C4×D4—21no—
81C2×C4×D4—21no—
82C2×#117—21no—
83C22×C4⋊C4D411yescharacteristic
84C22×C4⋊C4D411yescharacteristic
85C22×C42—21no—
86C22×C42C2311yescharacteristic
87#432—22no—
88#432—22no—
89#431—22no—
90#431—22no—
91#431—22no—
92#431—22no—
93#428—44no—
94C2×#107—21no—
95C2×#107—21no—
96C2×#106—21no—
97C2×#106—21no—
98#423—22no—
99#423D414yes—
100C2×C4⋊C8—21no—
101C2×C4⋊C8—21no—
102C2×C4⋊C8—21no—
103C2×C4⋊C8—21no—
104C2×C4⋊C8—21no—
105C2×C4⋊C8D411yescharacteristic
106C2×C4⋊C8D411yescharacteristic
107C2×C4⋊C8C2311yescharacteristic
108C2×C4⋊C8C2311yescharacteristic
109C2×C4⋊C8C2311yescharacteristic
110C2×C4⋊C8C2311yescharacteristic
111C2×C4⋊C8C2311yescharacteristic
112C2×C4⋊C8C2311yescharacteristic
113C2×#103—21no—
114C2×#103—21no—
115C2×#103—21no—
116C2×#103—21no—
117C2×#102—21no—
118C2×#102—21no—
119C2×#102—21no—
120C2×#102—21no—
121C2×#102—21no—
122C2×#102—21no—
123C2×#102—21no—
124C2×#102—21no—
125C2×C22⋊C8—21no—
126C2×C22⋊C8—21no—
127C2×C22⋊C8D411yescharacteristic
128C2×C22⋊C8D411yescharacteristic
129C2×#97—21no—
130C2×C4×C8—21no—
131C2×C4×C8C2311yescharacteristic
132C2×C4×C8C2311yescharacteristic
133#384C2311yescharacteristic
134#384C2311yescharacteristic
135#382—21no—
136#380—21no—
137#380—21no—
138C4×C4⋊C4—21no—
139C4×C4⋊C4—21no—
140C4×C4⋊C4—21no—
141C4×C4⋊C4—21no—
142C4×C4⋊C4D411yescharacteristic
143C4×C4⋊C4D411yescharacteristic
144C4×C4⋊C4C2311yescharacteristic
145C4×C4⋊C4C2311yescharacteristic
146C2×#95—21no—
147#340—21no—
148#340—21no—
149#340—21no—
150#340—21no—
151#340—21no—
152#340—21no—
153#340—21no—
154#340—21no—
155#336—21no—
156#336—21no—
157#336—21no—
158#336—21no—
159#335—22no—
160#335—22no—
161#334—22no—
162#334—22no—
163#332—22no—
164#332—22no—
165#331—22no—
166#331—22no—
167C2×#41—24no—
168C2×#41—42no—
169C2×#41—81no—
170C2×#41—81no—
171C2×#41—81no—
172C2×#41—81no—
173C22×Q8—21no—
174C22×D4—21no—
175C22×D4—21no—
176C22×D4—21no—
177C22×D4—41no—
178C23×C4—21no—
179C23×C4—21no—
180C23×C4—41no—
181C23×C4C2×D411yescharacteristic
182C23×C4C2×D411yescharacteristic
183C23×C4#4111yescharacteristic
184C2×M16—24no—
185C2×M16—42no—
186C2×M16—42no—
187C22×C8—21no—
188C22×C8—21no—
189C22×C8—21no—
190C22×C8—21no—
191C22×C8—21no—
192C22×C8—21no—
193C22×C8C2×D411yescharacteristic
194C22×C8C2×D411yescharacteristic
195C22×C8C2×D411yescharacteristic
196C22×C8C2×D411yescharacteristic
197C4⋊Q8—21no—
198C4⋊Q8—21no—
199C4⋊Q8—21no—
200C4⋊Q8C2×Q812yes—
201#127—21no—
202#127—21no—
203#127—21no—
204#127C2×Q812yes—
205#124—24no—
206#124—42no—
207#121—24no—
208#121—24no—
209#121—42no—
210#121—42no—
211C4×D4—24no—
212C4×D4—24no—
213C4×D4—42no—
214C4×D4—42no—
215#117—42no—
216#117—42no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—41no—
230C2×C4⋊C4—41no—
231C2×C4⋊C4C2×D411yescharacteristic
232C2×C4⋊C4C2×D411yescharacteristic
233C2×C4⋊C4C2×D411yescharacteristic
234C2×C4⋊C4C2×D411yescharacteristic
235C2×C22⋊C4—41no—
236C2×C22⋊C4—41no—
237C2×C22⋊C4—41no—
238C2×C42—21no—
239C2×C42—21no—
240C2×C42—21no—
241C2×C42—21no—
242C2×C42—41no—
243C2×C42—81no—
244C2×C42—81no—
245C2×C42C2×D411yescharacteristic
246C2×C42C2×D411yescharacteristic
247C2×C42C2×D412yes—
248C2×C42C2×Q812yes—
249C2×C42#4112yes—
250C2×C42#4112yes—
251C2×C42#4111yescharacteristic
252C2×C42C2411yescharacteristic
253#107—42no—
254#107—42no—
255#106—42no—
256#106—42no—
257C4⋊C8—24no—
258C4⋊C8—22no—
259C4⋊C8—22no—
260C4⋊C8—21no—
261C4⋊C8—21no—
262C4⋊C8—21no—
263C4⋊C8—21no—
264C4⋊C8—21no—
265C4⋊C8—21no—
266C4⋊C8—21no—
267C4⋊C8—21no—
268C4⋊C8—21no—
269C4⋊C8—21no—
270C4⋊C8—81no—
271C4⋊C8—81no—
272C4⋊C8—81no—
273C4⋊C8—81no—
274C4⋊C8C2×D412yes—
275C4⋊C8C2×D412yes—
276#103—81no—
277#103—81no—
278#103—81no—
279#103—81no—
280#102—42no—
281#102—42no—
282#102—42no—
283#102—42no—
284#102—81no—
285#102—81no—
286#102—81no—
287#102—81no—
288C22⋊C8—22no—
289C22⋊C8—22no—
290C22⋊C8—42no—
291C22⋊C8—42no—
292#97—42no—
293C4×C8—21no—
294C4×C8—21no—
295C4×C8—21no—
296C4×C8—21no—
297C4×C8—42no—
298#95—21no—
299#95—21no—
300#95—21no—
301#95—21no—
302#95—41no—
303#95—41no—
304C24—41no—
305C24#12111yescharacteristic
306#41—84no—
307#41—84no—
308#41—84no—
309#41—84no—
310C2×Q8—22no—
311C2×Q8—41no—
312C2×Q8—41no—
313C2×Q8—41no—
314C2×Q8—41no—
315C2×Q8—41no—
316C2×D4—22no—
317C2×D4—22no—
318C2×D4—22no—
319C2×D4—44no—
320C2×D4—41no—
321C2×D4—41no—
322C2×D4—41no—
323C2×D4—41no—
324C2×D4—41no—
325C2×D4—41no—
326C2×D4—41no—
327C2×D4—41no—
328C2×D4—41no—
329C2×D4—41no—
330C2×D4—41no—
331C2×D4—81no—
332C2×D4—81no—
333C2×D4—82no—
334C2×D4—82no—
335C22×C4—21no—
336C22×C4—21no—
337C22×C4—21no—
338C22×C4—21no—
339C22×C4—21no—
340C22×C4—21no—
341C22×C4—21no—
342C22×C4—21no—
343C22×C4—21no—
344C22×C4—21no—
345C22×C4—21no—
346C22×C4—21no—
347C22×C4—41no—
348C22×C4—41no—
349C22×C4—42no—
350C22×C4—42no—
351C22×C4—42no—
352C22×C4—42no—
353C22×C4—41no—
354C22×C4—44no—
355C22×C4—41no—
356C22×C4—41no—
357C22×C4—82no—
358C22×C4—82no—
359C22×C4—81no—
360C22×C4—81no—
361C22×C4#12112yes—
362C22×C4#12112yes—
363C22×C4#12111yescharacteristic
364C22×C4#12111yescharacteristic
365C22×C4#12111yescharacteristic
366C22×C4C22⋊Q812yes—
367C22×C4C22⋊Q812yes—
368C22×C4C22⋊Q812yes—
369C22×C4#12612yes—
370C22×C4C22×D411yescharacteristic
371C22×C4C22×D411yescharacteristic
372C22×C4C2×#4111yescharacteristic
373M16—48no—
374C2×C8—21no—
375C2×C8—21no—
376C2×C8—21no—
377C2×C8—21no—
378C2×C8—22no—
379C2×C8—22no—
380C2×C8—21no—
381C2×C8—21no—
382C2×C8—21no—
383C2×C8—21no—
384C2×C8—21no—
385C2×C8—21no—
386C2×C8—21no—
387C2×C8—21no—
388C2×C8—41no—
389C2×C8—41no—
390C2×C8—41no—
391C2×C8—41no—
392C2×C8—41no—
393C2×C8—41no—
394C2×C8—41no—
395C2×C8—41no—
396C2×C8—41no—
397C2×C8—41no—
398C2×C8—41no—
399C2×C8—41no—
400C2×C8—41no—
401C2×C8—41no—
402C2×C8—41no—
403C2×C8—41no—
404C4⋊C4—21no—
405C4⋊C4—21no—
406C4⋊C4—22no—
407C4⋊C4—22no—
408C4⋊C4—21no—
409C4⋊C4—21no—
410C4⋊C4—21no—
411C4⋊C4—21no—
412C4⋊C4—42no—
413C4⋊C4—42no—
414C4⋊C4—41no—
415C4⋊C4—41no—
416C4⋊C4—41no—
417C4⋊C4—41no—
418C4⋊C4—41no—
419C4⋊C4—41no—
420C4⋊C4—41no—
421C4⋊C4—41no—
422C4⋊C4—41no—
423C4⋊C4—41no—
424C4⋊C4—81no—
425C4⋊C4—81no—
426C4⋊C4AGL(1,ℤ/8)12yes—
427C4⋊C4#13712yes—
428C22⋊C4—44no—
429C22⋊C4—44no—
430C22⋊C4—82no—
431C42—41no—
432C42—41no—
433C42—41no—
434C42—41no—
435C42—81no—
436C42—81no—
437C42—82no—
438C42#12112yes—
439C42#12112yes—
440C42C22⋊Q812yes—
441C42C22⋊Q812yes—
442C42C22⋊Q812yes—
443C42#12612yes—
444C42C22×D411yescharacteristic
445C42C22×Q811yescharacteristic
446C42C2×#4111yescharacteristic
447C42C2×#4111yescharacteristic
448C23—21no—
449C23—21no—
450C23—21no—
451C23—21no—
452C23—21no—
453C23—21no—
454C23—42no—
455C23—81no—
456C23—81no—
457C23—81no—
458C23—81no—
459C23—81no—
460C23—81no—
461C23#39612yes—
462C23C2×#12111yescharacteristic
463Q8—42no—
464Q8—42no—
465Q8—81no—
466Q8—81no—
467D4—42no—
468D4—42no—
469D4—42no—
470D4—42no—
471D4—82no—
472D4—82no—
473D4—88no—
474D4—81no—
475D4—81no—
476D4—81no—
477D4—81no—
478C2×C4—21no—
479C2×C4—21no—
480C2×C4—21no—
481C2×C4—21no—
482C2×C4—21no—
483C2×C4—21no—
484C2×C4—21no—
485C2×C4—21no—
486C2×C4—22no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—22no—
490C2×C4—22no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—41no—
499C2×C4—41no—
500C2×C4—41no—
501C2×C4—41no—
502C2×C4—42no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—88no—
510C2×C4—84no—
511C2×C4—84no—
512C2×C4—81no—
513C2×C4—81no—
514C2×C4—81no—
515C2×C4—81no—
516C2×C4—81no—
517C2×C4—81no—
518C2×C4—81no—
519C2×C4—81no—
520C2×C4#39612yes—
521C2×C4#39612yes—
522C2×C4#39612yes—
523C2×C4C2×#12111yescharacteristic
524C2×C4C2×#12111yescharacteristic
525C2×C4C2×C22⋊Q811yescharacteristic
526C2×C4C2×C22⋊Q811yescharacteristic
527C2×C4C2×C22⋊Q811yescharacteristic
528C2×C4C2×#12611yescharacteristic
529C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
530C2×C4C2×#13711yescharacteristic
531C2×C4#57511yescharacteristic
532C2×C4#57511yescharacteristic
533C2×C4#57611yescharacteristic
534C2×C4#57611yescharacteristic
535C2×C4#57711yescharacteristic
536C2×C4#57711yescharacteristic
537C8—42no—
538C8—42no—
539C8—41no—
540C8—41no—
541C8—41no—
542C8—41no—
543C8—81no—
544C8—81no—
545C8—81no—
546C8—81no—
547C22—22no—
548C22—22no—
549C22—22no—
550C22—41no—
551C22—41no—
552C22—41no—
553C22—41no—
554C22—82no—
555C22—82no—
556C22—82no—
557C22—161no—
558C22—161no—
559C22—161no—
560C22—161no—
561C22C2×#39611yescharacteristic
562C22#304111yescharacteristic
563C22#304311yescharacteristic
564C22#305511yescharacteristic
565C22#305611yescharacteristic
566C22#306711yescharacteristic
567C22#306711yescharacteristic
568C4—21no—
569C4—21no—
570C4—22no—
571C4—22no—
572C4—22no—
573C4—42no—
574C4—42no—
575C4—81no—
576C4—81no—
577C4—82no—
578C4—161no—
579C4—161no—
580C4#303611yescharacteristic
581C4#303711yescharacteristic
582C4#306211yescharacteristic
583C4#306211yescharacteristic
584C4#307911yescharacteristic
585C4#308011yescharacteristic
586C4#308111yescharacteristic
587C4#308111yescharacteristic
588C2—42no—
589C2—162no—
590C2#1946011yescharacteristic
591C2#1946111yescharacteristic
592C2#1947211yescharacteristic
593C2#1947311yescharacteristic
594C2#2360111yescharacteristic
595C2#2361411yescharacteristic
596C2#2361711yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutator, Frattini, cosocleC2×C42C24
prime radicalC2×#532C22
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #21953C2semidirect
4. #21170C2semidirect
5. C2×#1574C2semidirect
7. #10600C2semidirect
8. #10599C2semidirect
9. #10596C2semidirect
10. #10595C2semidirect
11. #10592C2semidirect
12. #10591C2semidirect
13. #10588C2semidirect
14. #10587C2semidirect
15. #8046C2semidirect

Maximal subgroups

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

subgroupindex
#219532
#218052
#213862
#211702
C2×#15742
C2×#15572
#106002
#105992
#105962
#105952
#105922
#105912
#105882
#105872
#80462