Finite Group Databasebeta

← #418697

Subgroups — #418697

Order 512.

1736 conjugacy classes of subgroups.

The subgroup lattice has 627 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#500C211yesmaximal, characteristic, complemented
2C2×#2452C211yesmaximal, characteristic
3C2×#2023C211yesmaximal, characteristic
4C2×#1862C212yesmaximal
5#15128C211yesmaximal, characteristic, complemented
6#11384C211yesmaximal, characteristic, complemented
7#11328C212yesmaximal, complemented
8#11219C212yesmaximal, complemented
9#11123C212yesmaximal
10#10925C211yesmaximal, characteristic
11#10874C211yesmaximal, characteristic, complemented
12C22×C4⋊Q8C2211yescharacteristic
13C22×C4⋊Q8C2211yescharacteristic
14C22×C22⋊Q8—21no—
15C22×C4×D4C2211yescharacteristic
16C2×#500—28no—
17C2×#500—24no—
18C22×#107C2212yes—
19C22×#107C2212yes—
20C22×#102C2212yes—
21C22×#102C2212yes—
22C22×C4×C8C2211yescharacteristic
23#2452—24no—
24#2452—24no—
25C2×#396—21no—
26C2×#395—21no—
27C2×#393—21no—
28C2×#390C2211yescharacteristic
29C2×#390C2211yescharacteristic
30C2×#389C2211yescharacteristic
31C2×#389C2211yescharacteristic
32C2×#389C2212yes—
33C2×#385—21no—
34C2×C4×C4⋊C4C2211yescharacteristic
35#2023—24no—
36#1862—28no—
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#340C2212yes—
40C2×#340C2212yes—
41C2×#336C2212yes—
42C2×#336C2212yes—
43C2×#336C2211yescharacteristic
44#1446—21no—
45#1444—21no—
46#1441—21no—
47C4⋊#95C2212yes—
48C4⋊#95C2211yescharacteristic
49C23×Q8—21no—
50C23×Q8—21no—
51C23×D4C2311yescharacteristic
52C24×C4—21no—
53C23×C8C2312yes—
54C23×C8C2312yes—
55C2×C4⋊Q8—24no—
56C2×C4⋊Q8—24no—
57C2×C4⋊Q8—22no—
58C2×C4⋊Q8—22no—
59C2×C4⋊Q8—22no—
60C2×C4⋊Q8—42no—
61C2×C4⋊Q8—42no—
62C2×C4⋊Q8D414yes—
63C2×C22⋊Q8—28no—
64C2×C22⋊Q8—44no—
65C2×C22⋊Q8—44no—
66C2×C4×D4—22no—
67C2×C4×D4—22no—
68C2×C4×D4—24no—
69C2×C4×D4—42no—
70C22×C4⋊C4—21no—
71C22×C4⋊C4—21no—
72C22×C4⋊C4—21no—
73C22×C4⋊C4—21no—
74C22×C4⋊C4—22no—
75C22×C4⋊C4D411yescharacteristic
76C22×C4⋊C4D411yescharacteristic
77C22×C4⋊C4C2312yes—
78C22×C4⋊C4C2311yescharacteristic
79C22×C4⋊C4C2311yescharacteristic
80C22×C4⋊C4C2312yes—
81C22×C4⋊C4C2311yescharacteristic
82C22×C22⋊C4—21no—
83C22×C22⋊C4—21no—
84C22×C22⋊C4—21no—
85C22×C22⋊C4—21no—
86C22×C22⋊C4—21no—
87C22×C42C2311yescharacteristic
88C22×C42C2311yescharacteristic
89C22×C42C2311yescharacteristic
90#500—232no—
91C2×#107—24no—
92C2×#107—24no—
93C2×#107—24no—
94C2×#107—24no—
95C2×#107—24no—
96C2×#107D418yes—
97C2×#102—24no—
98C2×#102—24no—
99C2×#102—24no—
100C2×#102—24no—
101C2×#102—42no—
102C2×#102Q818yescomplemented
103C2×C4×C8—24no—
104C2×C4×C8—22no—
105#396—28no—
106#395—44no—
107#393—28no—
108#390—22no—
109#390—22no—
110#390—22no—
111#390D414yes—
112#389—22no—
113#389—22no—
114#389—22no—
115#389—22no—
116#389—24no—
117#389—42no—
118#385—28no—
119C4×C4⋊C4—22no—
120C4×C4⋊C4—22no—
121C2×#95—21no—
122C2×#95—21no—
123C2×#95—22no—
124C2×#95—21no—
125C2×#95—21no—
126C2×#95—22no—
127C2×#95—22no—
128C2×#95D411yescharacteristic
129C2×#95D411yescharacteristic
130#340—24no—
131#340—24no—
132#340—24no—
133#340—42no—
134#336—24no—
135#336—24no—
136#336—22no—
137C25—21no—
138C22×Q8—24no—
139C22×Q8—44no—
140C22×Q8—44no—
141C22×Q8—42no—
142C22×Q8—42no—
143C22×Q8—42no—
144C22×Q8—42no—
145C22×Q8—42no—
146C22×Q8—82no—
147C22×Q8—82no—
148C22×D4—22no—
149C22×D4—21no—
150C22×D4—22no—
151C22×D4—22no—
152C22×D4—22no—
153C22×D4—22no—
154C22×D4C2×D412yes—
155C22×D4C2×Q814yes—
156C23×C4—21no—
157C23×C4—21no—
158C23×C4—21no—
159C23×C4—22no—
160C23×C4—22no—
161C23×C4—24no—
162C23×C4—44no—
163C23×C4—42no—
164C23×C4—42no—
165C23×C4—82no—
166C23×C4C2×D411yescharacteristic
167C23×C4C2×D411yescharacteristic
168C23×C4C2×D411yescharacteristic
169C23×C4C2×D411yescharacteristic
170C23×C4C2×D411yescharacteristic
171C23×C4C2×D411yescharacteristic
172C23×C4C2411yescharacteristic
173C22×C8—22no—
174C22×C8—22no—
175C22×C8—22no—
176C22×C8—22no—
177C22×C8—22no—
178C22×C8—22no—
179C22×C8—22no—
180C22×C8—22no—
181C22×C8—22no—
182C22×C8—22no—
183C22×C8—22no—
184C22×C8C2×D414yes—
185C22×C8C2×Q814yes—
186C22×C8#4114yes—
187C4⋊Q8—28no—
188C4⋊Q8—28no—
189C4⋊Q8—44no—
190C4⋊Q8—48no—
191C4⋊Q8—48no—
192C4⋊Q8—44no—
193C22⋊Q8—432no—
194C4×D4—48no—
195C4×D4—44no—
196C4×D4—44no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—24no—
214C2×C4⋊C4—24no—
215C2×C4⋊C4—24no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—44no—
224C2×C4⋊C4—44no—
225C2×C4⋊C4—42no—
226C2×C4⋊C4—42no—
227C2×C4⋊C4—42no—
228C2×C4⋊C4—42no—
229C2×C4⋊C4—42no—
230C2×C4⋊C4—44no—
231C2×C4⋊C4—44no—
232C2×C4⋊C4—82no—
233C2×C4⋊C4D814yes—
234C2×C4⋊C4C2×D414yes—
235C2×C4⋊C4C2×D412yes—
236C2×C4⋊C4C2×D412yes—
237C2×C4⋊C4C2×D414yes—
238C2×C4⋊C4C2×D412yes—
239C2×C4⋊C4C2×Q814yes—
240C2×C22⋊C4—24no—
241C2×C22⋊C4—24no—
242C2×C22⋊C4—24no—
243C2×C22⋊C4—24no—
244C2×C22⋊C4—24no—
245C2×C22⋊C4—44no—
246C2×C22⋊C4—42no—
247C2×C22⋊C4—42no—
248C2×C22⋊C4—42no—
249C2×C22⋊C4—42no—
250C2×C22⋊C4—42no—
251C2×C22⋊C4—42no—
252C2×C22⋊C4—82no—
253C2×C42—22no—
254C2×C42—22no—
255C2×C42—22no—
256C2×C42—21no—
257C2×C42—21no—
258C2×C42—21no—
259C2×C42—21no—
260C2×C42—22no—
261C2×C42—22no—
262C2×C42—22no—
263C2×C42C2×D412yes—
264C2×C42C2×D412yes—
265#107—216no—
266#107—216no—
267#102—216no—
268#102—48no—
269C4×C8—28no—
270#95—24no—
271#95—24no—
272#95—22no—
273#95—22no—
274#95—44no—
275#95—42no—
276#95—44no—
277#95—42no—
278#95—82no—
279C24—22no—
280C24—24no—
281C24—41no—
282C24—41no—
283C24—42no—
284C24—41no—
285C24—41no—
286C24—41no—
287C24—41no—
288C24—42no—
289C24—42no—
290C24C22×D411yescharacteristic
291C2×Q8—48no—
292C2×Q8—48no—
293C2×Q8—48no—
294C2×Q8—84no—
295C2×Q8—88no—
296C2×Q8—88no—
297C2×Q8—88no—
298C2×Q8—88no—
299C2×Q8—84no—
300C2×Q8—84no—
301C2×D4—24no—
302C2×D4—22no—
303C2×D4—24no—
304C2×D4—24no—
305C2×D4—24no—
306C2×D4—24no—
307C2×D4—24no—
308C2×D4—24no—
309C2×D4—41no—
310C2×D4—41no—
311C2×D4—42no—
312C2×D4—42no—
313C2×D4—42no—
314C2×D4—42no—
315C2×D4—42no—
316C2×D4C4⋊Q814yes—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—21no—
322C22×C4—22no—
323C22×C4—21no—
324C22×C4—21no—
325C22×C4—21no—
326C22×C4—22no—
327C22×C4—24no—
328C22×C4—21no—
329C22×C4—21no—
330C22×C4—21no—
331C22×C4—21no—
332C22×C4—22no—
333C22×C4—21no—
334C22×C4—21no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—21no—
338C22×C4—21no—
339C22×C4—22no—
340C22×C4—24no—
341C22×C4—24no—
342C22×C4—21no—
343C22×C4—21no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—48no—
348C22×C4—48no—
349C22×C4—41no—
350C22×C4—41no—
351C22×C4—42no—
352C22×C4—42no—
353C22×C4—41no—
354C22×C4—42no—
355C22×C4—41no—
356C22×C4—42no—
357C22×C4—42no—
358C22×C4—42no—
359C22×C4—42no—
360C22×C4—42no—
361C22×C4—41no—
362C22×C4—41no—
363C22×C4—42no—
364C22×C4—41no—
365C22×C4—41no—
366C22×C4—41no—
367C22×C4—42no—
368C22×C4—41no—
369C22×C4—42no—
370C22×C4—42no—
371C22×C4—42no—
372C22×C4—42no—
373C22×C4—42no—
374C22×C4—88no—
375C22×C4—84no—
376C22×C4—88no—
377C22×C4#12012yes—
378C22×C4#12012yes—
379C22×C4C2×D812yes—
380C22×C4C2×D812yes—
381C22×C4C2×D812yes—
382C22×C4#13512yes—
383C22×C4AGL(1,ℤ/8)12yes—
384C22×C4#13712yes—
385C22×C4C22×D412yes—
386C22×C4C22×D411yescharacteristic
387C22×C4C22×D411yescharacteristic
388C22×C4C22×Q812yes—
389C22×C4C2×#4112yes—
390C22×C4H(5,𝔽2)12yes—
391C22×C42−1+412yes—
392C22×C42−1+412yes—
393C2×C8—24no—
394C2×C8—24no—
395C2×C8—24no—
396C2×C8—24no—
397C2×C8—24no—
398C2×C8—24no—
399C2×C8—42no—
400C2×C8—42no—
401C2×C8—42no—
402C2×C8—42no—
403C2×C8—42no—
404C2×C8—42no—
405C2×C8—42no—
406C2×C8C22⋊Q818yescomplemented
407C4⋊C4—22no—
408C4⋊C4—24no—
409C4⋊C4—28no—
410C4⋊C4—22no—
411C4⋊C4—24no—
412C4⋊C4—24no—
413C4⋊C4—22no—
414C4⋊C4—24no—
415C4⋊C4—24no—
416C4⋊C4—24no—
417C4⋊C4—48no—
418C4⋊C4—48no—
419C4⋊C4—48no—
420C4⋊C4—48no—
421C4⋊C4—42no—
422C4⋊C4—42no—
423C4⋊C4—42no—
424C4⋊C4—42no—
425C4⋊C4—42no—
426C4⋊C4—42no—
427C4⋊C4—44no—
428C4⋊C4—88no—
429C4⋊C4#12014yes—
430C22⋊C4—48no—
431C22⋊C4—48no—
432C22⋊C4—48no—
433C22⋊C4—48no—
434C22⋊C4—84no—
435C42—22no—
436C42—22no—
437C42—24no—
438C42—41no—
439C42—41no—
440C42—42no—
441C42—42no—
442C42—42no—
443C23—24no—
444C23—44no—
445C23—44no—
446C23—44no—
447C23—44no—
448C23—42no—
449C23—44no—
450C23—42no—
451C23—44no—
452C23—42no—
453C23—42no—
454C23—42no—
455C23—42no—
456C23—81no—
457C23—82no—
458C23—81no—
459C23—81no—
460C23—81no—
461C23—82no—
462C23—82no—
463C23—82no—
464C23—82no—
465C23—82no—
466C23C2×#12011yescharacteristic
467C23#53211yescharacteristic
468C23#53611yescharacteristic
469C23D4212yes—
470C23D4×Q812yes—
471C23C22×D811yescharacteristic
472C23C2×#13511yescharacteristic
473C23C2×AGL(1,ℤ/8)11yescharacteristic
474C23C2×#13711yescharacteristic
475C23#57612yes—
476C23#57812yes—
477Q8—816no—
478Q8—816no—
479D4—28no—
480D4—44no—
481D4—44no—
482D4—44no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—22no—
486C2×C4—24no—
487C2×C4—22no—
488C2×C4—24no—
489C2×C4—24no—
490C2×C4—24no—
491C2×C4—22no—
492C2×C4—24no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—24no—
497C2×C4—41no—
498C2×C4—41no—
499C2×C4—41no—
500C2×C4—41no—
501C2×C4—42no—
502C2×C4—41no—
503C2×C4—41no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—44no—
507C2×C4—44no—
508C2×C4—44no—
509C2×C4—44no—
510C2×C4—41no—
511C2×C4—42no—
512C2×C4—44no—
513C2×C4—42no—
514C2×C4—41no—
515C2×C4—42no—
516C2×C4—44no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—42no—
524C2×C4—44no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—42no—
528C2×C4—44no—
529C2×C4—42no—
530C2×C4—816no—
531C2×C4—81no—
532C2×C4—82no—
533C2×C4—82no—
534C2×C4—81no—
535C2×C4—81no—
536C2×C4—81no—
537C2×C4—82no—
538C2×C4—82no—
539C2×C4—82no—
540C2×C4—82no—
541C2×C4—82no—
542C2×C4—82no—
543C2×C4—82no—
544C2×C4—82no—
545C2×C4#44714yes—
546C2×C4#45214yes—
547C2×C4C2×#12011yescharacteristic
548C2×C4C2×C22⋊Q812yes—
549C2×C4C2×C4⋊Q811yescharacteristic
550C2×C4#53411yescharacteristic
551C2×C4#53612yes—
552C2×C4#53611yescharacteristic
553C2×C4#54212yes—
554C2×C4#54412yes—
555C2×C4D4212yes—
556C2×C4#54712yes—
557C2×C4#54812yes—
558C2×C4#54812yes—
559C2×C4D4×Q812yes—
560C2×C4D4×Q812yes—
561C2×C4#55412yes—
562C2×C4#56312yes—
563C8—28no—
564C8—82no—
565C22—44no—
566C22—44no—
567C22—44no—
568C22—82no—
569C22—82no—
570C22—82no—
571C22—82no—
572C22—84no—
573C22—84no—
574C22—84no—
575C22—84no—
576C22#240311yescharacteristic
577C22#243211yescharacteristic
578C22#243411yescharacteristic
579C22#245212yes—
580C22#267712yes—
581C22C2×#44711yescharacteristic
582C22C2×#45211yescharacteristic
583C22#300811yescharacteristic
584C22#301111yescharacteristic
585C22#316911yescharacteristic
586C22#317512yes—
587C22#318211yescharacteristic
588C22#322611yescharacteristic
589C22#322811yescharacteristic
590C22#324312yes—
591C22#326412yes—
592C22#327012yes—
593C22D4×D812yes—
594C22#328512yes—
595C22Q8×D812yes—
596C22#338012yes—
597C22#338512yes—
598C22#338612yes—
599C4—24no—
600C4—24no—
601C4—42no—
602C4—44no—
603C4—42no—
604C4—82no—
605C4—84no—
606C4—81no—
607C4—82no—
608C4—81no—
609C4—84no—
610C4#240912yes—
611C4#244012yes—
612C4#267412yes—
613C4#272912yes—
614C2—84no—
615C2#1526511yescharacteristic
616C2#2384811yescharacteristic
617C2#2385911yescharacteristic
618C2#2567311yescharacteristic
619C2#2568111yescharacteristic
620C2#2583811yescharacteristic
621C2#2583911yescharacteristic
622C2#2648412yes—
623C2#2649512yes—
624C2#3094612yes—
625C2#3095512yes—
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC23×D4C23
LCSG ▷ C23×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
1. C22×#500C2semidirect
5. #15128C2semidirect
6. #11384C2semidirect
7. #11328C2semidirect
8. #11219C2semidirect
11. #10874C2semidirect
102. C2×#102Q8semidirect
406. C2×C8C22⋊Q8semidirect

Maximal subgroups

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

subgroupindex
C22×#5002
C2×#24522
C2×#20232
C2×#18622
#151282
#113842
#113282
#112192
#111232
#109252
#108742