Finite Group Databasebeta

← #219792

Subgroups — #219792

Order 512.

1384 conjugacy classes of subgroups.

The subgroup lattice has 628 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2452C211yesmaximal, characteristic
2#22399C212yesmaximal
3C2×#1472C212yesmaximal, complemented
4#9654C212yesmaximal, complemented
5#9085C212yesmaximal, complemented
6#9081C212yesmaximal, complemented
7#8560C211yesmaximal, characteristic, complemented
8#8308C212yesmaximal, complemented
9#8153C211yesmaximal, characteristic, complemented
10C22×C4⋊Q8C411yescharacteristic
11C22×C22⋊Q8C2212yes—
12C22×C4×D4C2211yescharacteristic
13C2×#435—22no—
14C2×#407C2212yes—
15C22×C22⋊C8C2212yes—
16#2452—22no—
17#2452—22no—
18#2452—22no—
19#2452—22no—
20C2×#396C412yes—
21C2×#393C412yes—
22C2×#390C411yescharacteristic
23C2×#390C411yescharacteristic
24C2×#389C411yescharacteristic
25C2×#389C2211yescharacteristic
26C2×#385C2212yes—
27C2×C4×C4⋊C4C2211yescharacteristic
28#1955—22no—
29#1870—22no—
30#1869—22no—
31#1797C2212yes—
32#1779C2212yes—
33#1779C2212yes—
34#1749C2212yes—
35#1472—28no—
36C2×#324C2212yes—
37C2×#324C2211yescharacteristic
38C2×#324C2212yes—
39C2×#324C2212yes—
40C2×#324C2212yes—
41C2×#324C2212yes—
42C2×#324C2211yescharacteristic
43#1280C2211yescharacteristic
44#1280C2211yescharacteristic
45#1280C2212yes—
46C23×Q8C2×C412yes—
47C23×D4C2×C411yescharacteristic
48C24×C4C2312yes—
49C22×M16—22no—
50C23×C8—22no—
51C2×C4⋊Q8—22no—
52C2×C4⋊Q8—21no—
53C2×C4⋊Q8—21no—
54C2×C4⋊Q8—44no—
55C2×C22⋊Q8—22no—
56C2×C22⋊Q8—22no—
57C2×C22⋊Q8—22no—
58C2×C22⋊Q8—22no—
59C2×C22⋊Q8—24no—
60C2×C22⋊Q8—24no—
61C2×C22⋊Q8—42no—
62C2×C22⋊Q8—42no—
63C2×C4×D4—44no—
64C2×C4×D4D412yes—
65C2×C4×D4D412yes—
66C2×C4×D4D412yes—
67C2×C4×D4D412yes—
68C22×C4⋊C4C2×C411yescharacteristic
69C22×C4⋊C4C2×C412yes—
70C22×C4⋊C4C2×C411yescharacteristic
71C22×C4⋊C4C2×C412yes—
72C22×C4⋊C4C2×C411yescharacteristic
73C22×C4⋊C4C2×C411yescharacteristic
74C22×C4⋊C4C2312yes—
75C22×C4⋊C4C2311yescharacteristic
76C22×C22⋊C4C2×C412yes—
77C22×C22⋊C4C2×C412yes—
78C22×C22⋊C4C2×C412yes—
79C22×C22⋊C4C2×C412yes—
80C22×C22⋊C4C2×C412yes—
81C22×C42C2×C411yescharacteristic
82C22×C42C2×C411yescharacteristic
83C22×C42C2311yescharacteristic
84#435—48no—
85C2×C4⋊C8—22no—
86#407—24no—
87#407—42no—
88C2×C22⋊C8—24no—
89C2×C22⋊C8—22no—
90C2×C22⋊C8—22no—
91C2×C22⋊C8—22no—
92C2×C22⋊C8—22no—
93C2×C22⋊C8—22no—
94C2×C22⋊C8—22no—
95C2×C22⋊C8—42no—
96C2×C22⋊C8C2312yes—
97C2×C22⋊C8C2312yes—
98C2×C22⋊C8C2312yes—
99C2×C22⋊C8C2312yes—
100C2×#97—22no—
101#396—22no—
102#396—22no—
103#396—22no—
104#396—22no—
105#393—22no—
106#393—22no—
107#393—22no—
108#393—22no—
109#390—22no—
110#390—21no—
111#390—21no—
112#390—22no—
113#390—21no—
114#390—21no—
115#389—21no—
116#389—22no—
117#389—21no—
118#389—22no—
119#389—22no—
120#385—22no—
121#385—22no—
122#385—22no—
123#385—22no—
124C4×C4⋊C4—22no—
125C4×C4⋊C4—22no—
126C2×#95C2×C411yescharacteristic
127C2×#95C2×C412yes—
128C2×#95C2×C412yes—
129C2×#95C2311yescharacteristic
130#336—22no—
131#336—22no—
132#324—24no—
133#324—22no—
134#324—24no—
135#324—24no—
136#324—42no—
137#324—42no—
138#324D414yescomplemented
139C25C22×C412yes—
140C22×Q8—22no—
141C22×Q8—22no—
142C22×Q8—24no—
143C22×Q8—42no—
144C22×Q8—44no—
145C22×Q8—82no—
146C22×D4—24no—
147C22×D4—42no—
148C22×D4—41no—
149C22×D4—41no—
150C22×D4C22⋊C412yes—
151C22×D4C22⋊C411yescharacteristic
152C22×D4C22⋊C411yescharacteristic
153C23×C4—22no—
154C23×C4—22no—
155C23×C4—22no—
156C23×C4—22no—
157C23×C4—44no—
158C23×C4—82no—
159C23×C4C22×C412yes—
160C23×C4C22×C411yescharacteristic
161C23×C4C22×C411yescharacteristic
162C23×C4C22×C412yes—
163C23×C4C22×C412yes—
164C23×C4C22×C411yescharacteristic
165C23×C4C22×C411yescharacteristic
166C23×C4C22×C411yescharacteristic
167C23×C4C22×C411yescharacteristic
168C23×C4C2×D412yes—
169C23×C4C2×D412yes—
170C23×C4C2×D412yes—
171C23×C4C2×D412yes—
172C23×C4C2411yescharacteristic
173C2×M16—44no—
174C2×M16—44no—
175C2×M16—82no—
176C22×C8—22no—
177C22×C8—22no—
178C22×C8—22no—
179C22×C8—22no—
180C22×C8—44no—
181C22×C8—44no—
182C22×C8—82no—
183C4⋊Q8—48no—
184C4⋊Q8—44no—
185C4⋊Q8—44no—
186C22⋊Q8—44no—
187C22⋊Q8—44no—
188C22⋊Q8—44no—
189C22⋊Q8—44no—
190C22⋊Q8—44no—
191C22⋊Q8—44no—
192C22⋊Q8—44no—
193C22⋊Q8—44no—
194C4×D4—28no—
195C4×D4—44no—
196C4×D4—44no—
197C4×D4—44no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—24no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—24no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—41no—
219C2×C4⋊C4—41no—
220C2×C4⋊C4—41no—
221C2×C4⋊C4—42no—
222C2×C4⋊C4—42no—
223C2×C4⋊C4—44no—
224C2×C4⋊C4C22⋊C411yescharacteristic
225C2×C4⋊C4C22⋊C412yes—
226C2×C4⋊C4C22⋊C411yescharacteristic
227C2×C4⋊C4C2×D412yes—
228C2×C4⋊C4#4112yes—
229C2×C22⋊C4—22no—
230C2×C22⋊C4—22no—
231C2×C22⋊C4—22no—
232C2×C22⋊C4—22no—
233C2×C22⋊C4—22no—
234C2×C22⋊C4—22no—
235C2×C22⋊C4—24no—
236C2×C22⋊C4—24no—
237C2×C22⋊C4—24no—
238C2×C22⋊C4—22no—
239C2×C22⋊C4—22no—
240C2×C22⋊C4—44no—
241C2×C22⋊C4—42no—
242C2×C22⋊C4—42no—
243C2×C22⋊C4—42no—
244C2×C22⋊C4—82no—
245C2×C22⋊C4C22⋊C412yes—
246C2×C22⋊C4C22⋊C412yes—
247C2×C22⋊C4C22⋊C412yes—
248C2×C22⋊C4C22⋊C412yes—
249C2×C42—21no—
250C2×C42—21no—
251C2×C42—21no—
252C2×C42—21no—
253C2×C42—42no—
254C2×C42—41no—
255C2×C42—41no—
256C2×C42—42no—
257C2×C42C2×D412yes—
258C2×C42C2×D412yes—
259C4⋊C8—44no—
260C22⋊C8—22no—
261C22⋊C8—22no—
262C22⋊C8—22no—
263C22⋊C8—22no—
264C22⋊C8—22no—
265C22⋊C8—22no—
266C22⋊C8—22no—
267C22⋊C8—44no—
268C22⋊C8—44no—
269C22⋊C8—44no—
270C22⋊C8—44no—
271C22⋊C8—82no—
272C22⋊C8—82no—
273C22⋊C8C2×D414yes—
274#97—44no—
275#95—22no—
276#95—22no—
277#95—22no—
278#95—22no—
279#95—22no—
280#95—22no—
281C24—24no—
282C24—22no—
283C24—22no—
284C24—22no—
285C24—22no—
286C24—22no—
287C24—22no—
288C24—42no—
289C24—42no—
290C24—42no—
291C24C2×C22⋊C412yes—
292C24C2×C22⋊C412yes—
293C24C23×C411yescharacteristic
294C2×Q8—44no—
295C2×Q8—44no—
296C2×Q8—44no—
297C2×Q8—44no—
298C2×Q8—82no—
299C2×Q8—82no—
300C2×Q8—88no—
301C2×Q8—88no—
302C2×D4—24no—
303C2×D4—24no—
304C2×D4—22no—
305C2×D4—41no—
306C2×D4—41no—
307C2×D4—41no—
308C2×D4—42no—
309C2×D4—41no—
310C2×D4—48no—
311C2×D4—48no—
312C2×D4#10114yes—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—21no—
318C22×C4—21no—
319C22×C4—21no—
320C22×C4—22no—
321C22×C4—21no—
322C22×C4—21no—
323C22×C4—22no—
324C22×C4—21no—
325C22×C4—21no—
326C22×C4—21no—
327C22×C4—21no—
328C22×C4—21no—
329C22×C4—21no—
330C22×C4—21no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—24no—
336C22×C4—24no—
337C22×C4—24no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—21no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—21no—
354C22×C4—21no—
355C22×C4—41no—
356C22×C4—42no—
357C22×C4—42no—
358C22×C4—41no—
359C22×C4—42no—
360C22×C4—42no—
361C22×C4—42no—
362C22×C4—41no—
363C22×C4—42no—
364C22×C4—42no—
365C22×C4—41no—
366C22×C4—41no—
367C22×C4—41no—
368C22×C4—41no—
369C22×C4—41no—
370C22×C4—41no—
371C22×C4—42no—
372C22×C4—42no—
373C22×C4—41no—
374C22×C4—84no—
375C22×C4—84no—
376C22×C4—84no—
377C22×C4—84no—
378C22×C4C2×C22⋊C412yes—
379C22×C4C2×C22⋊C412yes—
380C22×C4C2×C22⋊C412yes—
381C22×C4C2×C22⋊C411yescharacteristic
382C22×C4C2×C22⋊C411yescharacteristic
383C22×C4C4×D412yes—
384C22×C4C4×D412yes—
385C22×C4#13112yes—
386C22×C4C22×D412yes—
387C22×C4C22×D411yescharacteristic
388C22×C4C2×#4111yescharacteristic
389C22×C4H(5,𝔽2)12yes—
390M16—88no—
391C2×C8—24no—
392C2×C8—42no—
393C2×C8—42no—
394C2×C8—42no—
395C2×C8—42no—
396C2×C8—42no—
397C2×C8—42no—
398C2×C8—42no—
399C2×C8—42no—
400C2×C8—42no—
401C2×C8—42no—
402C2×C8—42no—
403C2×C8—88no—
404C4⋊C4—24no—
405C4⋊C4—24no—
406C4⋊C4—24no—
407C4⋊C4—22no—
408C4⋊C4—44no—
409C4⋊C4—44no—
410C4⋊C4—44no—
411C4⋊C4—42no—
412C4⋊C4—42no—
413C4⋊C4—42no—
414C4⋊C4—44no—
415C4⋊C4—42no—
416C4⋊C4—42no—
417C4⋊C4—42no—
418C4⋊C4—42no—
419C4⋊C4—44no—
420C4⋊C4—42no—
421C4⋊C4#9914yes—
422C22⋊C4—24no—
423C22⋊C4—24no—
424C22⋊C4—42no—
425C22⋊C4—42no—
426C22⋊C4—44no—
427C22⋊C4—44no—
428C22⋊C4—44no—
429C22⋊C4—44no—
430C22⋊C4—44no—
431C22⋊C4—44no—
432C22⋊C4—82no—
433C22⋊C4—82no—
434C42—24no—
435C42—42no—
436C42—42no—
437C42—42no—
438C42—42no—
439C42—42no—
440C23—21no—
441C23—21no—
442C23—21no—
443C23—21no—
444C23—22no—
445C23—22no—
446C23—22no—
447C23—22no—
448C23—22no—
449C23—22no—
450C23—22no—
451C23—22no—
452C23—22no—
453C23—22no—
454C23—22no—
455C23—22no—
456C23—22no—
457C23—44no—
458C23—44no—
459C23—44no—
460C23—44no—
461C23—44no—
462C23—44no—
463C23—42no—
464C23—42no—
465C23—42no—
466C23—42no—
467C23—82no—
468C23—82no—
469C23—82no—
470C23—82no—
471C23—82no—
472C23—82no—
473C23C2×#10114yes—
474C23C22×C22⋊C411yescharacteristic
475C23C2×C4×D411yescharacteristic
476C23#51811yescharacteristic
477C23C2×#13111yescharacteristic
478C23#56811yescharacteristic
479C23#56811yescharacteristic
480C23#56811yescharacteristic
481Q8—88no—
482Q8—88no—
483D4—44no—
484D4—48no—
485D4—44no—
486C2×C4—24no—
487C2×C4—24no—
488C2×C4—22no—
489C2×C4—21no—
490C2×C4—21no—
491C2×C4—22no—
492C2×C4—21no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—21no—
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—41no—
508C2×C4—41no—
509C2×C4—42no—
510C2×C4—44no—
511C2×C4—44no—
512C2×C4—44no—
513C2×C4—44no—
514C2×C4—44no—
515C2×C4—44no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—41no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—41no—
535C2×C4—42no—
536C2×C4—41no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—82no—
542C2×C4—82no—
543C2×C4—82no—
544C2×C4—82no—
545C2×C4—81no—
546C2×C4—81no—
547C2×C4—81no—
548C2×C4—81no—
549C2×C4—82no—
550C2×C4—82no—
551C2×C4C2×#9912yes—
552C2×C4C2×#9914yes—
553C2×C4C2×#10112yes—
554C2×C4D4212yes—
555C2×C4#54612yes—
556C8—44no—
557C8—82no—
558C8—82no—
559C8—82no—
560C22—21no—
561C22—21no—
562C22—21no—
563C22—21no—
564C22—21no—
565C22—21no—
566C22—21no—
567C22—21no—
568C22—21no—
569C22—21no—
570C22—21no—
571C22—21no—
572C22—24no—
573C22—24no—
574C22—24no—
575C22—42no—
576C22—42no—
577C22—42no—
578C22—42no—
579C22—42no—
580C22—42no—
581C22—42no—
582C22—42no—
583C22—42no—
584C22—42no—
585C22—42no—
586C22—84no—
587C22—84no—
588C22—84no—
589C22—84no—
590C22D4×C22⋊C411yescharacteristic
591C22#287911yescharacteristic
592C22#288011yescharacteristic
593C22C22×#9911yescharacteristic
594C22#288311yescharacteristic
595C22C22×#10111yescharacteristic
596C22#288911yescharacteristic
597C22#293211yescharacteristic
598C22#293511yescharacteristic
599C22#298011yescharacteristic
600C22#298111yescharacteristic
601C4—22no—
602C4—22no—
603C4—24no—
604C4—22no—
605C4—22no—
606C4—42no—
607C4—42no—
608C4—42no—
609C4—42no—
610C4—42no—
611C4—82no—
612C4—82no—
613C4—81no—
614C2—21no—
615C2—21no—
616C2—21no—
617C2—21no—
618C2—44no—
619C2—82no—
620C2#1739711yescharacteristic
621C2#1740711yescharacteristic
622C2#1741411yescharacteristic
623C2#1742611yescharacteristic
624C2#2240311yescharacteristic
625C2D4×#9911yescharacteristic
626C2D4×#10111yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC24C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×D4C2×C4
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×D4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C2×#1472C2semidirect
4. #9654C2semidirect
5. #9085C2semidirect
6. #9081C2semidirect
7. #8560C2semidirect
8. #8308C2semidirect
9. #8153C2semidirect
138. #324D4semidirect

Maximal subgroups

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

subgroupindex
C2×#24522
#223992
C2×#14722
#96542
#90852
#90812
#85602
#83082
#81532