Finite Group Databasebeta

← #241296

Subgroups — #241296

Order 512.

968 conjugacy classes of subgroups.

The subgroup lattice has 614 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34938C211yesmaximal, characteristic
2#21364C211yesmaximal, characteristic
3#21170C211yesmaximal, characteristic
4#20824C211yesmaximal, characteristic
5#10582C211yesmaximal, characteristic, complemented
6#10581C211yesmaximal, characteristic, complemented
7#10580C211yesmaximal, characteristic, complemented
8#10579C211yesmaximal, characteristic, complemented
9#10575C211yesmaximal, characteristic, complemented
10#10575C211yesmaximal, characteristic, complemented
11#10574C211yesmaximal, characteristic, complemented
12#10573C211yesmaximal, characteristic, complemented
13#8042C211yesmaximal, characteristic
14#7904C211yesmaximal, characteristic
15#7903C211yesmaximal, characteristic
16C2×#533—21no—
17C2×#432C2211yescharacteristic
18C2×#428—21no—
19C2×#423C2211yescharacteristic
20C2×C4×M16C2211yescharacteristic
21#2552C2211yescharacteristic
22#2552C2211yescharacteristic
23#2551C2211yescharacteristic
24#2551C2211yescharacteristic
25#2550—21no—
26#2550—21no—
27#2549—22no—
28#2440—21no—
29#2440—21no—
30#2399—22no—
31C4×C4⋊Q8—21no—
32C4×C4⋊Q8—21no—
33#2296—21no—
34#2285C2211yescharacteristic
35#2285C2211yescharacteristic
36C4×#117C2211yescharacteristic
37C2×C4×C4⋊C4C2211yescharacteristic
38#1949—21no—
39#1949—21no—
40#1948C2211yescharacteristic
41#1948C2211yescharacteristic
42#1948C2211yescharacteristic
43#1948C2211yescharacteristic
44#1947C2211yescharacteristic
45#1947C2211yescharacteristic
46#1947C2211yescharacteristic
47#1947C2211yescharacteristic
48#1854—21no—
49#1854—21no—
50#1845C2211yescharacteristic
51#1844C2211yescharacteristic
52#1834C2211yescharacteristic
53#1834C2211yescharacteristic
54#1809—22no—
55#1776—22no—
56#1776—21no—
57#1776—21no—
58#1736—21no—
59#1296C2211yescharacteristic
60#1296C2211yescharacteristic
61#1296C2211yescharacteristic
62#1296C2211yescharacteristic
63#1279C2211yescharacteristic
64#1279C2211yescharacteristic
65#1279C2211yescharacteristic
66#1279C2211yescharacteristic
67#1279C2211yescharacteristic
68#1279C2211yescharacteristic
69#1279C2211yescharacteristic
70#1279C2211yescharacteristic
71C22×M16—21no—
72C22×M16—21no—
73#533—44no—
74#533—44no—
75C2×C4⋊Q8—21no—
76C2×C4⋊Q8—21no—
77C2×C4⋊Q8—21no—
78C2×C4⋊Q8—21no—
79C2×#125—21no—
80C2×#125—21no—
81C2×C22⋊Q8—22no—
82C2×C22⋊Q8—22no—
83C2×C4×Q8—22no—
84C2×C4×Q8—22no—
85C2×#117—21no—
86C2×#117—21no—
87C2×#117D411yescharacteristic
88C2×#117D411yescharacteristic
89C22×C4⋊C4D411yescharacteristic
90C22×C4⋊C4D411yescharacteristic
91C22×C42—21no—
92C22×C42C2311yescharacteristic
93#432—22no—
94#432Q814yescomplemented
95#428—44no—
96C2×#107—22no—
97C2×#107—21no—
98C2×#107—21no—
99C2×#107—21no—
100C2×#107—21no—
101C2×#106—22no—
102C2×#106—21no—
103C2×#106—21no—
104C2×#106—21no—
105C2×#106—21no—
106#423—22no—
107#423Q814yescomplemented
108C2×C4⋊C8—21no—
109C2×C4⋊C8C2311yescharacteristic
110C2×C4⋊C8C2311yescharacteristic
111C2×C4⋊C8C2311yescharacteristic
112C2×C4⋊C8C2311yescharacteristic
113C2×C22⋊C8—21no—
114C4×M16—22no—
115C4×M16Q814yescomplemented
116C2×#97—21no—
117C2×#97—21no—
118C2×C4×C8C2311yescharacteristic
119C2×C4×C8C2311yescharacteristic
120#398—22no—
121#398—22no—
122#398—22no—
123#398—22no—
124#395—22no—
125#395—22no—
126#395—22no—
127#395—22no—
128#391—22no—
129#391—22no—
130#389—22no—
131#389—22no—
132#389—21no—
133#389—21no—
134#389—21no—
135#389—21no—
136#389—21no—
137#389—21no—
138#389—21no—
139#389—21no—
140#385—22no—
141#385—22no—
142#384—21no—
143#384—21no—
144#384C2311yescharacteristic
145#384C2311yescharacteristic
146#384C2311yescharacteristic
147#384C2311yescharacteristic
148#382—21no—
149#382—21no—
150#380—21no—
151#380—21no—
152#380—21no—
153#380D412yes—
154C4×C4⋊C4—22no—
155C4×C4⋊C4—22no—
156C4×C4⋊C4—21no—
157C4×C4⋊C4—21no—
158C4×C4⋊C4—21no—
159C4×C4⋊C4—21no—
160C4×C4⋊C4—21no—
161C4×C4⋊C4D411yescharacteristic
162C4×C4⋊C4D411yescharacteristic
163C4×C4⋊C4D412yes—
164C4×C4⋊C4D411yescharacteristic
165C4×C4⋊C4D411yescharacteristic
166C4×C4⋊C4C2311yescharacteristic
167C4×C4⋊C4C2311yescharacteristic
168C4×C4⋊C4C2311yescharacteristic
169C4×C4⋊C4C2311yescharacteristic
170C4×C22⋊C4—21no—
171C4×C22⋊C4—21no—
172#376—21no—
173C2×#95—21no—
174C43—21no—
175#340—22no—
176#340—22no—
177#340—21no—
178#340—21no—
179#340—21no—
180#340—21no—
181#340—21no—
182#340—21no—
183#340—21no—
184#340—21no—
185C22×Q8—22no—
186C22×Q8—22no—
187C23×C4—21no—
188C23×C4—21no—
189C23×C4C2×D411yescharacteristic
190C23×C4C2×D411yescharacteristic
191C23×C4C2×D411yescharacteristic
192C2×M16—24no—
193C2×M16—24no—
194C2×M16—42no—
195C2×M16—42no—
196C2×M16—42no—
197C2×M16—42no—
198C22×C8—21no—
199C22×C8—21no—
200C22×C8—21no—
201C22×C8—21no—
202C22×C8—21no—
203C22×C8—21no—
204C4⋊Q8—42no—
205C4⋊Q8—42no—
206C4⋊Q8—42no—
207C4⋊Q8—42no—
208C4⋊Q8—42no—
209C4⋊Q8—42no—
210C4⋊Q8—42no—
211C4⋊Q8—42no—
212#125—42no—
213#125—42no—
214#125—42no—
215#125—42no—
216C22⋊Q8—48no—
217C22⋊Q8—48no—
218C4×Q8—48no—
219C4×Q8—48no—
220#117—22no—
221#117—22no—
222#117—22no—
223#117—22no—
224#117—42no—
225#117—42no—
226#117—42no—
227#117—42no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—21no—
239C2×C4⋊C4—21no—
240C2×C4⋊C4—21no—
241C2×C4⋊C4—21no—
242C2×C4⋊C4—21no—
243C2×C4⋊C4—21no—
244C2×C4⋊C4—21no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—41no—
253C2×C4⋊C4—41no—
254C2×C4⋊C4C2×D411yescharacteristic
255C2×C4⋊C4C2×D411yescharacteristic
256C2×C4⋊C4C2×D411yescharacteristic
257C2×C4⋊C4C2×D411yescharacteristic
258C2×C4⋊C4C2×D411yescharacteristic
259C2×C4⋊C4C2×D411yescharacteristic
260C2×C4⋊C4C2×D411yescharacteristic
261C2×C4⋊C4C2×D411yescharacteristic
262C2×C4⋊C4C2×D412yes—
263C2×C4⋊C4C2×D412yes—
264C2×C22⋊C4—21no—
265C2×C22⋊C4—21no—
266C2×C22⋊C4—22no—
267C2×C22⋊C4—41no—
268C2×C42—21no—
269C2×C42—21no—
270C2×C42—22no—
271C2×C42—21no—
272C2×C42—21no—
273C2×C42—21no—
274C2×C42—21no—
275C2×C42—21no—
276C2×C42—21no—
277C2×C42—21no—
278C2×C42—21no—
279C2×C42—42no—
280C2×C42—42no—
281C2×C42—41no—
282C2×C42C2×D411yescharacteristic
283C2×C42C2×D411yescharacteristic
284C2×C42C2×D411yescharacteristic
285C2×C42C2×Q812yes—
286C2×C42C2×Q812yes—
287C2×C42C2×Q812yes—
288C2×C42#4112yes—
289C2×C42C2411yescharacteristic
290#107—42no—
291#107—44no—
292#107—42no—
293#107—42no—
294#107—42no—
295#106—42no—
296#106—44no—
297#106—42no—
298#106—42no—
299#106—42no—
300C4⋊C8—24no—
301C4⋊C8—21no—
302C4⋊C8—21no—
303C4⋊C8—21no—
304C4⋊C8—21no—
305C4⋊C8C2×Q812yes—
306C4⋊C8C2×Q812yes—
307C4⋊C8C2×Q812yes—
308C4⋊C8C2×Q812yes—
309C22⋊C8—24no—
310#97—24no—
311#97—24no—
312C4×C8—21no—
313C4×C8—21no—
314C4×C8C2×Q812yes—
315C4×C8C2×Q812yes—
316#95—22no—
317#95—22no—
318#95—22no—
319#95—22no—
320#95—21no—
321#95—21no—
322#95—21no—
323#95—21no—
324#95—22no—
325#95—22no—
326#95—21no—
327#95—21no—
328#95—21no—
329#95—21no—
330#95—41no—
331#95—41no—
332C24#12011yescharacteristic
333C2×Q8—44no—
334C2×Q8—44no—
335C2×Q8—44no—
336C2×Q8—44no—
337C2×Q8—82no—
338C2×Q8—82no—
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—21no—
348C22×C4—21no—
349C22×C4—21no—
350C22×C4—21no—
351C22×C4—21no—
352C22×C4—21no—
353C22×C4—21no—
354C22×C4—21no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—41no—
358C22×C4—41no—
359C22×C4—42no—
360C22×C4—42no—
361C22×C4—42no—
362C22×C4—42no—
363C22×C4—41no—
364C22×C4—41no—
365C22×C4—81no—
366C22×C4—81no—
367C22×C4#12011yescharacteristic
368C22×C4#12011yescharacteristic
369C22×C4#12011yescharacteristic
370C22×C4C22⋊Q812yes—
371C22×C4C22⋊Q812yes—
372C22×C4C22⋊Q812yes—
373C22×C4C4⋊Q812yes—
374C22×C4C4⋊Q812yes—
375C22×C4C4⋊Q812yes—
376C22×C4C22×D411yescharacteristic
377C22×C4C22×D411yescharacteristic
378C22×C4C22×D411yescharacteristic
379M16—48no—
380M16—48no—
381C2×C8—22no—
382C2×C8—22no—
383C2×C8—22no—
384C2×C8—22no—
385C2×C8—22no—
386C2×C8—22no—
387C2×C8—22no—
388C2×C8—22no—
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—
399C4⋊C4—21no—
400C4⋊C4—21no—
401C4⋊C4—21no—
402C4⋊C4—21no—
403C4⋊C4—21no—
404C4⋊C4—21no—
405C4⋊C4—21no—
406C4⋊C4—21no—
407C4⋊C4—21no—
408C4⋊C4—21no—
409C4⋊C4—42no—
410C4⋊C4—42no—
411C4⋊C4—42no—
412C4⋊C4—42no—
413C4⋊C4—44no—
414C4⋊C4—44no—
415C4⋊C4—44no—
416C4⋊C4—44no—
417C4⋊C4—44no—
418C4⋊C4—44no—
419C4⋊C4—44no—
420C4⋊C4—44no—
421C4⋊C4—42no—
422C4⋊C4—42no—
423C4⋊C4—41no—
424C4⋊C4—41no—
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—41no—
436C4⋊C4—41no—
437C4⋊C4—41no—
438C4⋊C4—41no—
439C4⋊C4—82no—
440C4⋊C4—82no—
441C4⋊C4AGL(1,ℤ/8)12yes—
442C4⋊C4AGL(1,ℤ/8)12yes—
443C4⋊C4AGL(1,ℤ/8)12yes—
444C4⋊C4#13712yes—
445C4⋊C4#13712yes—
446C4⋊C4#13712yes—
447C22⋊C4—24no—
448C22⋊C4—24no—
449C22⋊C4—44no—
450C22⋊C4—82no—
451C42—24no—
452C42—24no—
453C42—42no—
454C42—42no—
455C42—44no—
456C42—41no—
457C42—41no—
458C42—41no—
459C42—41no—
460C42—41no—
461C42—41no—
462C42—41no—
463C42—41no—
464C42—81no—
465C42—81no—
466C42—82no—
467C42C22⋊Q812yes—
468C42C22⋊Q812yes—
469C42C22⋊Q812yes—
470C42C4⋊Q812yes—
471C42C4⋊Q812yes—
472C42C4⋊Q812yes—
473C42C22×Q811yescharacteristic
474C42C22×Q811yescharacteristic
475C42C22×Q811yescharacteristic
476C42C2×#4111yescharacteristic
477C23—21no—
478C23—21no—
479C23—21no—
480C23—21no—
481C23—21no—
482C23—21no—
483C23#39512yes—
484C23C2×#12011yescharacteristic
485Q8—88no—
486Q8—88no—
487C2×C4—21no—
488C2×C4—21no—
489C2×C4—21no—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—41no—
528C2×C4—42no—
529C2×C4—41no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—84no—
536C2×C4—84no—
537C2×C4—81no—
538C2×C4—81no—
539C2×C4—81no—
540C2×C4—81no—
541C2×C4—81no—
542C2×C4—81no—
543C2×C4#39512yes—
544C2×C4#39512yes—
545C2×C4#39512yes—
546C2×C4C2×C22⋊Q811yescharacteristic
547C2×C4C2×C22⋊Q811yescharacteristic
548C2×C4C2×C22⋊Q811yescharacteristic
549C2×C4C2×C4⋊Q811yescharacteristic
550C2×C4C2×C4⋊Q811yescharacteristic
551C2×C4C2×C4⋊Q811yescharacteristic
552C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
553C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
554C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
555C2×C4C2×#13711yescharacteristic
556C2×C4C2×#13711yescharacteristic
557C2×C4C2×#13711yescharacteristic
558C2×C4#57511yescharacteristic
559C2×C4#57511yescharacteristic
560C8—24no—
561C8—42no—
562C8—42no—
563C8—42no—
564C8—42no—
565C8—42no—
566C22—22no—
567C22—22no—
568C22—22no—
569C22—41no—
570C22—41no—
571C22—41no—
572C22—41no—
573C22C2×#39511yescharacteristic
574C22#300311yescharacteristic
575C22#300411yescharacteristic
576C22#300611yescharacteristic
577C22#300711yescharacteristic
578C22#301811yescharacteristic
579C22#301811yescharacteristic
580C4—22no—
581C4—22no—
582C4—21no—
583C4—21no—
584C4—22no—
585C4—42no—
586C4—42no—
587C4—42no—
588C4—42no—
589C4—42no—
590C4—42no—
591C4—81no—
592C4—82no—
593C4—81no—
594C4—82no—
595C4—161no—
596C4—161no—
597C4#307711yescharacteristic
598C4#307811yescharacteristic
599C4#307911yescharacteristic
600C4#307911yescharacteristic
601C4#316511yescharacteristic
602C4#316511yescharacteristic
603C4#316511yescharacteristic
604C4#316611yescharacteristic
605C2—42no—
606C2#1855411yescharacteristic
607C2#1855511yescharacteristic
608C2#1855611yescharacteristic
609C2#1855711yescharacteristic
610C2#2355211yescharacteristic
611C2#2355411yescharacteristic
612C2#2356011yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC2×C42C24
prime radicalC24#120
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
5. #10582C2semidirect
6. #10581C2semidirect
7. #10580C2semidirect
8. #10579C2semidirect
9. #10575C2semidirect
10. #10575C2semidirect
11. #10574C2semidirect
12. #10573C2semidirect
94. #432Q8semidirect
107. #423Q8semidirect
115. C4×M16Q8semidirect

Maximal subgroups

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

subgroupindex
#349382
#213642
#211702
#208242
#105822
#105812
#105802
#105792
#105752
#105742
#105732
#80422
#79042
#79032