Finite Group Databasebeta

← #425868

Subgroups — #425868

Order 512.

1458 conjugacy classes of subgroups.

The subgroup lattice has 613 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2076C212yesmaximal, complemented
2C2×#1852C212yesmaximal, complemented
3#13906C211yesmaximal, characteristic
4#13872C211yesmaximal, characteristic
5#11324C212yesmaximal, complemented
6#11200C212yesmaximal, complemented
7#11096C212yesmaximal, complemented
8#10964C211yesmaximal, characteristic
9#10807C212yesmaximal, complemented
10C22×C4×D4C2211yescharacteristic
11C23×C4⋊C4C2211yescharacteristic
12C22×#107C2212yes—
13C22×#107C2212yes—
14C22×C22⋊C8C2212yes—
15C22×C22⋊C8C2211yescharacteristic
16C22×C22⋊C8C2211yescharacteristic
17C2×#390—21no—
18C2×#386—21no—
19C2×#384C2211yescharacteristic
20C2×#380C2211yescharacteristic
21C2×#380C2212yes—
22C2×#380C2212yes—
23C2×#380C2212yes—
24C2×C4×C4⋊C4C2211yescharacteristic
25#2076—28no—
26#1852—28no—
27C2×#340C2212yes—
28C2×#340C2212yes—
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#336C2212yes—
34#1441—21no—
35#1439—21no—
36#1439—21no—
37#1439—21no—
38C4⋊#95—21no—
39C4⋊#95C2212yes—
40C22⋊#95—21no—
41C23×D4D411yescharacteristic
42C24×C4—21no—
43C24×C4C2311yescharacteristic
44C24×C4C2311yescharacteristic
45C23×C8C2312yes—
46C23×C8C2312yes—
47C2×C4×D4—24no—
48C2×C4×D4—24no—
49C2×C4×D4—41no—
50C2×C4×D4—41no—
51C22×C4⋊C4—24no—
52C22×C4⋊C4—21no—
53C22×C4⋊C4—21no—
54C22×C4⋊C4—21no—
55C22×C4⋊C4—41no—
56C22×C4⋊C4—41no—
57C22×C4⋊C4—42no—
58C22×C4⋊C4D411yescharacteristic
59C22×C4⋊C4C2312yes—
60C22×C4⋊C4C2312yes—
61C22×C4⋊C4C2312yes—
62C22×C4⋊C4C2312yes—
63C22×C22⋊C4—21no—
64C22×C22⋊C4—22no—
65C22×C22⋊C4—22no—
66C22×C22⋊C4—22no—
67C22×C22⋊C4D411yescharacteristic
68C22×C22⋊C4D411yescharacteristic
69C22×C42—21no—
70C22×C42C2311yescharacteristic
71C2×#107—24no—
72C2×#107—24no—
73C2×#107—24no—
74C2×#107—42no—
75C2×#107—42no—
76C2×#107—42no—
77C2×C22⋊C8—24no—
78C2×C22⋊C8—24no—
79C2×C22⋊C8—24no—
80C2×C22⋊C8—24no—
81C2×C22⋊C8—24no—
82C2×C22⋊C8—41no—
83C2×C22⋊C8—41no—
84#390—44no—
85#386—44no—
86#384—41no—
87#384—41no—
88#380—24no—
89#380—24no—
90#380—24no—
91#380—24no—
92#380—42no—
93#380—42no—
94#380—42no—
95C4×C4⋊C4—24no—
96C2×#95—22no—
97C2×#95—22no—
98C2×#95—22no—
99C2×#95—22no—
100C2×#95—21no—
101C2×#95—21no—
102C2×#95—22no—
103#340—24no—
104#340—24no—
105#340—24no—
106#340—24no—
107#340—42no—
108#340D418yescomplemented
109#336—24no—
110C25C2×D411yescharacteristic
111C25C2×D411yescharacteristic
112C22×D4—22no—
113C22×D4—22no—
114C22×D4—22no—
115C22×D4—22no—
116C22×D4—41no—
117C22×D4—41no—
118C22×D4Q1614yes—
119C23×C4—22no—
120C23×C4—22no—
121C23×C4—22no—
122C23×C4—21no—
123C23×C4—21no—
124C23×C4—22no—
125C23×C4—22no—
126C23×C4—22no—
127C23×C4—22no—
128C23×C4—22no—
129C23×C4—22no—
130C23×C4—22no—
131C23×C4—22no—
132C23×C4—21no—
133C23×C4—21no—
134C23×C4—22no—
135C23×C4—22no—
136C23×C4—22no—
137C23×C4—22no—
138C23×C4—44no—
139C23×C4—42no—
140C23×C4—42no—
141C23×C4—42no—
142C23×C4—82no—
143C23×C4C2×D411yescharacteristic
144C23×C4C2×D411yescharacteristic
145C23×C4C2×D411yescharacteristic
146C23×C4C2×D411yescharacteristic
147C23×C4#4114yes—
148C23×C4C2411yescharacteristic
149C22×C8—22no—
150C22×C8—22no—
151C22×C8—22no—
152C22×C8—22no—
153C22×C8—22no—
154C22×C8—22no—
155C22×C8—22no—
156C22×C8—22no—
157C22×C8—22no—
158C22×C8—22no—
159C22×C8—22no—
160C22×C8—22no—
161C22×C8C2×D414yes—
162C22×C8#4114yes—
163C4×D4—48no—
164C4×D4—48no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—44no—
190C2×C4⋊C4—48no—
191C2×C4⋊C4—41no—
192C2×C4⋊C4—44no—
193C2×C4⋊C4—42no—
194C2×C4⋊C4—44no—
195C2×C4⋊C4—42no—
196C2×C4⋊C4—42no—
197C2×C4⋊C4—42no—
198C2×C4⋊C4—42no—
199C2×C4⋊C4—82no—
200C2×C4⋊C4C2×D414yes—
201C2×C4⋊C4C2×D414yes—
202C2×C22⋊C4—22no—
203C2×C22⋊C4—22no—
204C2×C22⋊C4—22no—
205C2×C22⋊C4—22no—
206C2×C22⋊C4—41no—
207C2×C22⋊C4—41no—
208C2×C22⋊C4—44no—
209C2×C22⋊C4—44no—
210C2×C22⋊C4—44no—
211C2×C22⋊C4—42no—
212C2×C22⋊C4—44no—
213C2×C22⋊C4—42no—
214C2×C22⋊C4—44no—
215C2×C22⋊C4—42no—
216C2×C22⋊C4—44no—
217C2×C22⋊C4—82no—
218C2×C22⋊C4—82no—
219C2×C22⋊C4—82no—
220C2×C42—22no—
221C2×C42—21no—
222C2×C42—21no—
223C2×C42—42no—
224C2×C42—42no—
225C2×C42—42no—
226C2×C42#4114yes—
227#107—48no—
228#107—48no—
229C22⋊C8—216no—
230C22⋊C8—44no—
231C22⋊C8—44no—
232#95—44no—
233#95—44no—
234#95—44no—
235#95—42no—
236#95—42no—
237#95—44no—
238#95—82no—
239C24—21no—
240C24—21no—
241C24—22no—
242C24—22no—
243C24—21no—
244C24—21no—
245C24—22no—
246C24—22no—
247C24—21no—
248C24—21no—
249C24—21no—
250C24—22no—
251C24—22no—
252C24—21no—
253C24—21no—
254C24—21no—
255C24—22no—
256C24—22no—
257C24#12312yes—
258C24C2×Q1612yes—
259C24C2×Q1612yes—
260C24AGL(1,ℤ/8)12yes—
261C24C22×D411yescharacteristic
262C2×D4—28no—
263C2×D4—28no—
264C2×D4—44no—
265C2×D4—44no—
266C2×D4—44no—
267C2×D4—44no—
268C2×D4—44no—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—21no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—24no—
282C22×C4—24no—
283C22×C4—24no—
284C22×C4—24no—
285C22×C4—24no—
286C22×C4—24no—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—22no—
291C22×C4—24no—
292C22×C4—24no—
293C22×C4—22no—
294C22×C4—42no—
295C22×C4—42no—
296C22×C4—42no—
297C22×C4—42no—
298C22×C4—41no—
299C22×C4—42no—
300C22×C4—41no—
301C22×C4—42no—
302C22×C4—41no—
303C22×C4—41no—
304C22×C4—48no—
305C22×C4—42no—
306C22×C4—42no—
307C22×C4—42no—
308C22×C4—42no—
309C22×C4—42no—
310C22×C4—42no—
311C22×C4—42no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4—42no—
315C22×C4—42no—
316C22×C4—42no—
317C22×C4—42no—
318C22×C4—42no—
319C22×C4—42no—
320C22×C4—42no—
321C22×C4—42no—
322C22×C4—42no—
323C22×C4—41no—
324C22×C4—42no—
325C22×C4—42no—
326C22×C4—41no—
327C22×C4—42no—
328C22×C4—42no—
329C22×C4—42no—
330C22×C4—42no—
331C22×C4—41no—
332C22×C4—41no—
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—88no—
349C22×C4—84no—
350C22×C4—84no—
351C22×C4#12312yes—
352C22×C4#12312yes—
353C22×C4#12312yes—
354C22×C4C2×Q1612yes—
355C22×C4AGL(1,ℤ/8)12yes—
356C22×C4C22×D412yes—
357C22×C4C2×#4112yes—
358C22×C4C2×#4112yes—
359C22×C4C2×#4112yes—
360C22×C4H(5,𝔽2)11yescharacteristic
361C22×C4H(5,𝔽2)12yes—
362C22×C4H(5,𝔽2)11yescharacteristic
363C22×C42−1+412yes—
364C2×C8—24no—
365C2×C8—24no—
366C2×C8—24no—
367C2×C8—24no—
368C2×C8—24no—
369C2×C8—24no—
370C2×C8—42no—
371C2×C8—42no—
372C2×C8—42no—
373C2×C8—42no—
374C2×C8—42no—
375C2×C8—42no—
376C2×C8—42no—
377C2×C8—42no—
378C4⋊C4—24no—
379C4⋊C4—24no—
380C4⋊C4—24no—
381C4⋊C4—24no—
382C4⋊C4—44no—
383C4⋊C4—48no—
384C4⋊C4—42no—
385C4⋊C4—42no—
386C4⋊C4—42no—
387C4⋊C4—42no—
388C4⋊C4—42no—
389C4⋊C4—42no—
390C4⋊C4—42no—
391C4⋊C4—42no—
392C4⋊C4—42no—
393C4⋊C4—42no—
394C4⋊C4—42no—
395C4⋊C4—42no—
396C4⋊C4—84no—
397C4⋊C4—84no—
398C22⋊C4—44no—
399C22⋊C4—44no—
400C22⋊C4—88no—
401C22⋊C4—88no—
402C22⋊C4—88no—
403C22⋊C4—84no—
404C42—24no—
405C42—24no—
406C42—84no—
407C23—24no—
408C23—24no—
409C23—22no—
410C23—22no—
411C23—22no—
412C23—24no—
413C23—22no—
414C23—22no—
415C23—24no—
416C23—22no—
417C23—24no—
418C23—24no—
419C23—24no—
420C23—22no—
421C23—22no—
422C23—22no—
423C23—24no—
424C23—42no—
425C23—42no—
426C23—42no—
427C23—41no—
428C23—42no—
429C23—41no—
430C23—41no—
431C23—41no—
432C23—42no—
433C23—42no—
434C23—42no—
435C23—42no—
436C23—41no—
437C23—41no—
438C23—42no—
439C23—42no—
440C23—42no—
441C23—41no—
442C23—42no—
443C23—41no—
444C23—41no—
445C23—41no—
446C23—42no—
447C23—41no—
448C23—41no—
449C23—42no—
450C23—42no—
451C23—42no—
452C23#48414yes—
453C23C2×#12311yescharacteristic
454C23C2×#12311yescharacteristic
455C23#53411yescharacteristic
456C23D4212yes—
457C23#54712yes—
458C23C22×Q1611yescharacteristic
459C23C2×AGL(1,ℤ/8)11yescharacteristic
460C23C2×AGL(1,ℤ/8)11yescharacteristic
461C23#57511yescharacteristic
462C23#57612yes—
463C23#57812yes—
464D4—416no—
465C2×C4—22no—
466C2×C4—22no—
467C2×C4—22no—
468C2×C4—22no—
469C2×C4—22no—
470C2×C4—22no—
471C2×C4—24no—
472C2×C4—24no—
473C2×C4—22no—
474C2×C4—22no—
475C2×C4—42no—
476C2×C4—41no—
477C2×C4—42no—
478C2×C4—41no—
479C2×C4—41no—
480C2×C4—41no—
481C2×C4—44no—
482C2×C4—44no—
483C2×C4—44no—
484C2×C4—44no—
485C2×C4—44no—
486C2×C4—41no—
487C2×C4—44no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—41no—
491C2×C4—41no—
492C2×C4—44no—
493C2×C4—44no—
494C2×C4—44no—
495C2×C4—44no—
496C2×C4—816no—
497C2×C4—82no—
498C2×C4—82no—
499C2×C4—82no—
500C2×C4—82no—
501C2×C4—82no—
502C2×C4—82no—
503C2×C4—82no—
504C2×C4—82no—
505C2×C4—82no—
506C2×C4—82no—
507C2×C4—82no—
508C2×C4—82no—
509C2×C4—82no—
510C2×C4—82no—
511C2×C4—82no—
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#48414yes—
524C2×C4#52512yes—
525C2×C4#53212yes—
526C2×C4#53512yes—
527C2×C4#53912yes—
528C2×C4#54212yes—
529C2×C4#54212yes—
530C2×C4D4211yescharacteristic
531C2×C4#54712yes—
532C2×C4#54812yes—
533C2×C4#55111yescharacteristic
534C2×C4#55112yes—
535C2×C4#55112yes—
536C2×C4#55312yes—
537C2×C4#55312yes—
538C2×C4#56011yescharacteristic
539C2×C4#56311yescharacteristic
540C8—44no—
541C8—44no—
542C22—28no—
543C22—24no—
544C22—42no—
545C22—42no—
546C22—42no—
547C22—44no—
548C22—42no—
549C22—44no—
550C22—44no—
551C22—42no—
552C22—42no—
553C22—42no—
554C22—44no—
555C22—42no—
556C22—44no—
557C22—42no—
558C22—81no—
559C22—82no—
560C22—81no—
561C22—82no—
562C22—81no—
563C22—81no—
564C22#242212yes—
565C22#244312yes—
566C22#248311yescharacteristic
567C22#266911yescharacteristic
568C22#271711yescharacteristic
569C22C2×#48411yescharacteristic
570C22C2×#48411yescharacteristic
571C22#309411yescharacteristic
572C22#309411yescharacteristic
573C22#309812yes—
574C22#309812yes—
575C22#318911yescharacteristic
576C22#319211yescharacteristic
577C22#319911yescharacteristic
578C22#320211yescharacteristic
579C22#326412yes—
580C22#326912yes—
581C22#327412yes—
582C22D4×Q1612yes—
583C22#333812yes—
584C22#334312yes—
585C22#334712yes—
586C22#335312yes—
587C4—24no—
588C4—42no—
589C4—42no—
590C4—84no—
591C4—84no—
592C4—81no—
593C4—162no—
594C4—162no—
595C4#246112yes—
596C4#260312yes—
597C4#272312yes—
598C4#275712yes—
599C2—44no—
600C2—82no—
601C2#1534211yescharacteristic
602C2#2501612yes—
603C2#2502712yes—
604C2#2613212yes—
605C2#2615712yes—
606C2#2760311yescharacteristic
607C2#2760811yescharacteristic
608C2#3050711yescharacteristic
609C2#3050911yescharacteristic
610C2#3226111yescharacteristic
611C2#3227111yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC23×D4D4
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×C4×D4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2076C2semidirect
2. C2×#1852C2semidirect
5. #11324C2semidirect
6. #11200C2semidirect
7. #11096C2semidirect
9. #10807C2semidirect
108. #340D4semidirect

Maximal subgroups

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

subgroupindex
C2×#20762
C2×#18522
#139062
#138722
#113242
#112002
#110962
#109642
#108072