Finite Group Databasebeta

← #405911

Subgroups — #405911

Order 512.

1550 conjugacy classes of subgroups.

The subgroup lattice has 630 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1875C211yesmaximal, characteristic
2C2×#1874C211yesmaximal, characteristic
3C2×#1852C211yesmaximal, characteristic, complemented
4C2×#1798C211yesmaximal, characteristic, complemented
5#14257C211yesmaximal, characteristic
6#13795C211yesmaximal, characteristic, complemented
7#11316C212yesmaximal, complemented
8#11232C212yesmaximal, complemented
9#11123C212yesmaximal, complemented
10#11076C212yesmaximal, complemented
11#10731C211yesmaximal, characteristic, complemented
12C22×#121C2211yescharacteristic
13C22×#117C2211yescharacteristic
14C22×#117C2211yescharacteristic
15C23×C4⋊C4C2211yescharacteristic
16C22×#107C2212yes—
17C22×C4⋊C8C2212yes—
18C22×#102C2212yes—
19C22×#102C2212yes—
20C22×C22⋊C8C2211yescharacteristic
21C22×C22⋊C8C2211yescharacteristic
22C2×#390C2211yescharacteristic
23C2×#390C2211yescharacteristic
24C2×#380C2211yescharacteristic
25#1875—24no—
26#1874—24no—
27#1852—24no—
28#1798—24no—
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#336C2212yes—
35#1439—22no—
36#1438—22no—
37C4⋊#95—22no—
38C4⋊#95C2211yescharacteristic
39C4⋊#95C2212yes—
40C4⋊#95C2212yes—
41C4⋊#95C2211yescharacteristic
42C22⋊#95—22no—
43C23×D4—21no—
44C23×D4C2311yescharacteristic
45C24×C4—21no—
46C24×C4C2311yescharacteristic
47C23×C8C2312yes—
48C23×C8C2312yes—
49C2×#121—22no—
50C2×#121—22no—
51C2×#121—22no—
52C2×#121—22no—
53C2×#121—22no—
54C2×#121Q814yes—
55C2×#117—22no—
56C2×#117—22no—
57C2×#117—24no—
58C2×#117—24no—
59C2×#117—22no—
60C2×#117—22no—
61C2×#117—42no—
62C2×#117—42no—
63C22×C4⋊C4—21no—
64C22×C4⋊C4—22no—
65C22×C4⋊C4—42no—
66C22×C4⋊C4—42no—
67C22×C4⋊C4D412yes—
68C22×C4⋊C4D414yes—
69C22×C4⋊C4C2311yescharacteristic
70C22×C4⋊C4C2311yescharacteristic
71C22×C4⋊C4C2312yes—
72C22×C4⋊C4C2311yescharacteristic
73C22×C4⋊C4C2312yes—
74C22×C22⋊C4—21no—
75C22×C22⋊C4—21no—
76C22×C22⋊C4—21no—
77C22×C22⋊C4—22no—
78C22×C22⋊C4—22no—
79C22×C22⋊C4D412yes—
80C22×C42—21no—
81C22×C42C2311yescharacteristic
82C22×C42C2311yescharacteristic
83C2×#107—24no—
84C2×#107—42no—
85C2×#107—42no—
86C2×C4⋊C8—24no—
87C2×C4⋊C8—24no—
88C2×C4⋊C8—42no—
89C2×#102—24no—
90C2×#102—24no—
91C2×#102—24no—
92C2×#102—42no—
93C2×#102—42no—
94C2×#102—42no—
95C2×C22⋊C8—22no—
96C2×C22⋊C8—22no—
97C2×C22⋊C8—22no—
98C2×C22⋊C8—22no—
99C2×C22⋊C8—22no—
100C2×C22⋊C8D414yes—
101#390—24no—
102#390—42no—
103#380—22no—
104#380—22no—
105C2×#95—21no—
106C2×#95—22no—
107C2×#95—22no—
108C2×#95—22no—
109C2×#95—22no—
110C2×#95—22no—
111C2×#95—22no—
112#340—24no—
113#340—24no—
114#340—24no—
115#340—24no—
116#340—42no—
117#336—24no—
118C25—21no—
119C25C2×D411yescharacteristic
120C22×D4—24no—
121C22×D4—21no—
122C22×D4—21no—
123C22×D4—21no—
124C22×D4—21no—
125C22×D4—22no—
126C22×D4—22no—
127C22×D4—21no—
128C22×D4—22no—
129C22×D4—22no—
130C22×D4—42no—
131C22×D4—42no—
132C22×D4—82no—
133C22×D4—82no—
134C22×D4C2×Q812yes—
135C23×C4—22no—
136C23×C4—22no—
137C23×C4—22no—
138C23×C4—22no—
139C23×C4—21no—
140C23×C4—22no—
141C23×C4—22no—
142C23×C4—24no—
143C23×C4—21no—
144C23×C4—21no—
145C23×C4—42no—
146C23×C4—42no—
147C23×C4—82no—
148C23×C4—82no—
149C23×C4C2×D412yes—
150C23×C4C2×D412yes—
151C23×C4C2×D411yescharacteristic
152C23×C4C2×D412yes—
153C23×C4C2×D412yes—
154C23×C4C2×Q812yes—
155C23×C4#4112yes—
156C23×C4#4112yes—
157C23×C4#4112yes—
158C23×C4C2411yescharacteristic
159C22×C8—22no—
160C22×C8—22no—
161C22×C8—22no—
162C22×C8—22no—
163C22×C8—22no—
164C22×C8—22no—
165C22×C8—22no—
166C22×C8—22no—
167C22×C8—22no—
168C22×C8—22no—
169C22×C8—22no—
170C22×C8—22no—
171C22×C8—22no—
172C22×C8C2×D414yes—
173#121—28no—
174#121—28no—
175#121—44no—
176#121—44no—
177#117—48no—
178#117—48no—
179#117—48no—
180#117—48no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—24no—
194C2×C4⋊C4—24no—
195C2×C4⋊C4—24no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—48no—
208C2×C4⋊C4—42no—
209C2×C4⋊C4—42no—
210C2×C4⋊C4—42no—
211C2×C4⋊C4—48no—
212C2×C4⋊C4C2×D414yes—
213C2×C4⋊C4C2×Q812yes—
214C2×C4⋊C4#4114yes—
215C2×C22⋊C4—24no—
216C2×C22⋊C4—24no—
217C2×C22⋊C4—24no—
218C2×C22⋊C4—24no—
219C2×C22⋊C4—42no—
220C2×C22⋊C4—42no—
221C2×C22⋊C4—42no—
222C2×C22⋊C4—42no—
223C2×C22⋊C4—42no—
224C2×C22⋊C4—44no—
225C2×C22⋊C4—42no—
226C2×C22⋊C4—44no—
227C2×C22⋊C4—82no—
228C2×C22⋊C4—82no—
229C2×C22⋊C4—82no—
230C2×C22⋊C4—82no—
231C2×C22⋊C4—82no—
232C2×C42—21no—
233C2×C42—21no—
234C2×C42—22no—
235C2×C42—24no—
236C2×C42—21no—
237C2×C42—22no—
238C2×C42—21no—
239C2×C42—22no—
240C2×C42—42no—
241C2×C42—42no—
242C2×C42#4114yes—
243#107—48no—
244C4⋊C8—48no—
245#102—48no—
246#102—48no—
247C22⋊C8—28no—
248C22⋊C8—44no—
249#95—42no—
250#95—82no—
251#95—82no—
252#95—82no—
253#95—82no—
254#95—82no—
255#95—82no—
256C24—21no—
257C24—21no—
258C24—21no—
259C24—21no—
260C24—22no—
261C24—22no—
262C24—22no—
263C24—22no—
264C24—22no—
265C24—41no—
266C24—41no—
267C24—41no—
268C24—41no—
269C24—41no—
270C24—41no—
271C24—42no—
272C24—42no—
273C24—42no—
274C24—42no—
275C24#12112yes—
276C24#12112yes—
277C24C22⋊Q812yes—
278C24C22×D411yescharacteristic
279C2×D4—22no—
280C2×D4—22no—
281C2×D4—24no—
282C2×D4—24no—
283C2×D4—48no—
284C2×D4—41no—
285C2×D4—41no—
286C2×D4—42no—
287C2×D4—42no—
288C2×D4—42no—
289C2×D4—42no—
290C2×D4—42no—
291C2×D4—42no—
292C2×D4—42no—
293C2×D4—42no—
294C2×D4—42no—
295C2×D4—42no—
296C2×D4—88no—
297C2×D4—84no—
298C2×D4—84no—
299C2×D4—88no—
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—24no—
305C22×C4—24no—
306C22×C4—24no—
307C22×C4—24no—
308C22×C4—24no—
309C22×C4—24no—
310C22×C4—24no—
311C22×C4—24no—
312C22×C4—24no—
313C22×C4—24no—
314C22×C4—24no—
315C22×C4—24no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—24no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—24no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—48no—
333C22×C4—42no—
334C22×C4—41no—
335C22×C4—41no—
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—42no—
348C22×C4—42no—
349C22×C4—41no—
350C22×C4—41no—
351C22×C4—41no—
352C22×C4—41no—
353C22×C4—41no—
354C22×C4—41no—
355C22×C4—42no—
356C22×C4—42no—
357C22×C4—42no—
358C22×C4—42no—
359C22×C4—84no—
360C22×C4—84no—
361C22×C4—88no—
362C22×C4—88no—
363C22×C4#12114yes—
364C22×C4#12112yes—
365C22×C4#12112yes—
366C22×C4C22⋊Q814yes—
367C22×C4C22⋊Q814yes—
368C22×C4C22⋊Q812yes—
369C22×C4#12314yes—
370C22×C4#12614yes—
371C22×C4C22×D411yescharacteristic
372C22×C4C22×D411yescharacteristic
373C22×C4C22×Q811yescharacteristic
374C22×C4C2×#4111yescharacteristic
375C22×C4C2×#4111yescharacteristic
376C22×C4C2×#4111yescharacteristic
377C22×C4C2×#4112yes—
378C22×C4C2×#4112yes—
379C22×C4H(5,𝔽2)12yes—
380C22×C4H(5,𝔽2)12yes—
381C2×C8—24no—
382C2×C8—24no—
383C2×C8—24no—
384C2×C8—42no—
385C2×C8—42no—
386C2×C8—42no—
387C2×C8—42no—
388C2×C8—42no—
389C2×C8—42no—
390C2×C8—42no—
391C2×C8—42no—
392C2×C8—42no—
393C2×C8—42no—
394C2×C8—42no—
395C4⋊C4—22no—
396C4⋊C4—22no—
397C4⋊C4—24no—
398C4⋊C4—24no—
399C4⋊C4—24no—
400C4⋊C4—24no—
401C4⋊C4—48no—
402C4⋊C4—48no—
403C4⋊C4—41no—
404C4⋊C4—41no—
405C4⋊C4—42no—
406C4⋊C4—42no—
407C4⋊C4—42no—
408C4⋊C4—42no—
409C4⋊C4—42no—
410C4⋊C4—42no—
411C4⋊C4—42no—
412C4⋊C4—42no—
413C22⋊C4—48no—
414C22⋊C4—48no—
415C22⋊C4—48no—
416C22⋊C4—88no—
417C22⋊C4—88no—
418C22⋊C4—84no—
419C42—24no—
420C42—24no—
421C42—48no—
422C42—42no—
423C42—42no—
424C23—24no—
425C23—24no—
426C23—22no—
427C23—22no—
428C23—22no—
429C23—22no—
430C23—22no—
431C23—22no—
432C23—22no—
433C23—22no—
434C23—22no—
435C23—22no—
436C23—24no—
437C23—24no—
438C23—42no—
439C23—42no—
440C23—42no—
441C23—42no—
442C23—42no—
443C23—42no—
444C23—42no—
445C23—42no—
446C23—44no—
447C23—44no—
448C23—42no—
449C23—42no—
450C23—42no—
451C23—82no—
452C23—82no—
453C23—81no—
454C23—82no—
455C23—82no—
456C23—81no—
457C23—81no—
458C23—81no—
459C23—82no—
460C23—82no—
461C23—82no—
462C23—82no—
463C23—81no—
464C23—81no—
465C23—81no—
466C23—81no—
467C23—82no—
468C23—82no—
469C23#39614yes—
470C23C2×#12111yescharacteristic
471C23C2×#12111yescharacteristic
472C23C2×C22⋊Q811yescharacteristic
473C23C2×#12311yescharacteristic
474C23#52511yescharacteristic
475C23#53411yescharacteristic
476C23#53511yescharacteristic
477C23#57612yes—
478C23#57612yes—
479C23#57712yes—
480C23#57812yes—
481D4—44no—
482D4—44no—
483D4—816no—
484D4—82no—
485D4—82no—
486C2×C4—22no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—22no—
490C2×C4—22no—
491C2×C4—22no—
492C2×C4—28no—
493C2×C4—28no—
494C2×C4—28no—
495C2×C4—28no—
496C2×C4—24no—
497C2×C4—24no—
498C2×C4—24no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—44no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—44no—
507C2×C4—44no—
508C2×C4—44no—
509C2×C4—42no—
510C2×C4—44no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—44no—
515C2×C4—44no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—816no—
519C2×C4—82no—
520C2×C4—82no—
521C2×C4—82no—
522C2×C4—82no—
523C2×C4—82no—
524C2×C4—82no—
525C2×C4—81no—
526C2×C4—81no—
527C2×C4—82no—
528C2×C4—82no—
529C2×C4—82no—
530C2×C4—82no—
531C2×C4—81no—
532C2×C4—82no—
533C2×C4—81no—
534C2×C4—82no—
535C2×C4—82no—
536C2×C4—81no—
537C2×C4—81no—
538C2×C4—81no—
539C2×C4—81no—
540C2×C4#39614yes—
541C2×C4C2×#12111yescharacteristic
542C2×C4C2×C22⋊Q811yescharacteristic
543C2×C4C2×C22⋊Q811yescharacteristic
544C2×C4C2×#12611yescharacteristic
545C2×C4#52912yes—
546C2×C4#53212yes—
547C2×C4#53412yes—
548C2×C4#53812yes—
549C2×C4#54612yes—
550C2×C4#54612yes—
551C2×C4#55112yes—
552C2×C4#55112yes—
553C2×C4#55312yes—
554C2×C4#55312yes—
555C2×C4#55412yes—
556C2×C4#55412yes—
557C8—44no—
558C8—82no—
559C22—24no—
560C22—24no—
561C22—24no—
562C22—42no—
563C22—42no—
564C22—44no—
565C22—44no—
566C22—42no—
567C22—42no—
568C22—44no—
569C22—44no—
570C22—84no—
571C22—82no—
572C22—82no—
573C22—84no—
574C22—82no—
575C22—161no—
576C22—161no—
577C22—161no—
578C22—161no—
579C22—162no—
580C22—162no—
581C22C2×#39611yescharacteristic
582C22#241111yescharacteristic
583C22#241211yescharacteristic
584C22#248711yescharacteristic
585C22#257911yescharacteristic
586C22#261711yescharacteristic
587C22#285411yescharacteristic
588C22#304111yescharacteristic
589C22#304111yescharacteristic
590C22#304311yescharacteristic
591C22#304411yescharacteristic
592C22#306112yes—
593C22#306212yes—
594C22#308112yes—
595C22#308312yes—
596C22#309612yes—
597C22#309812yes—
598C22#311912yes—
599C22#312412yes—
600C22#319912yes—
601C22#320112yes—
602C22#322012yes—
603C22#322512yes—
604C4—24no—
605C4—44no—
606C4—84no—
607C4—82no—
608C4—84no—
609C4—82no—
610C4—161no—
611C4—161no—
612C4#245012yes—
613C4#251012yes—
614C4#254312yes—
615C4#256412yes—
616C2—44no—
617C2—162no—
618C2#1492111yescharacteristic
619C2#2361111yescharacteristic
620C2#2361411yescharacteristic
621C2#2451312yes—
622C2#2456512yes—
623C2#2768712yes—
624C2#2924712yes—
625C2#2972111yescharacteristic
626C2#2972811yescharacteristic
627C2#3311811yescharacteristic
628C2#3312011yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC22×#121C22
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×#117 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C2×#1852C2semidirect
4. C2×#1798C2semidirect
6. #13795C2semidirect
7. #11316C2semidirect
8. #11232C2semidirect
9. #11123C2semidirect
10. #11076C2semidirect
11. #10731C2semidirect

Maximal subgroups

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

subgroupindex
C2×#18752
C2×#18742
C2×#18522
C2×#17982
#142572
#137952
#113162
#112322
#111232
#110762
#107312