Finite Group Databasebeta

← #395676

Subgroups — #395676

Order 512.

2062 conjugacy classes of subgroups.

The subgroup lattice has 653 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#419C211yesmaximal, characteristic
2C2×#2068C212yesmaximal, complemented
3C2×#1854C211yesmaximal, characteristic
4C2×#1851C211yesmaximal, characteristic
5C2×#1787C211yesmaximal, characteristic
6#13951C211yesmaximal, characteristic
7#13922C211yesmaximal, characteristic
8#11122C212yesmaximal, complemented
9#11077C212yesmaximal, complemented
10#11067C212yesmaximal, complemented
11#10722C211yesmaximal, characteristic
12C24×Q8C2211yescharacteristic
13C23×M16C2211yescharacteristic
14C22×#117C2211yescharacteristic
15C22×#107C2211yescharacteristic
16C22×#106C2211yescharacteristic
17C22×C4⋊C8C2212yes—
18C2×#419—28no—
19C2×#419—24no—
20C22×#103C2212yes—
21C22×#103C2212yes—
22C22×#103C2212yes—
23C22×#103C2212yes—
24C22×C22⋊C8C2211yescharacteristic
25C2×#391C2212yes—
26C2×#391C2212yes—
27C2×#389C2212yes—
28C2×#380C2211yescharacteristic
29C2×#380C2211yescharacteristic
30C2×C4×C22⋊C4C2211yescharacteristic
31#2068—28no—
32#1854—24no—
33#1851—24no—
34#1787—24no—
35C2×#340C2212yes—
36C2×#340C2211yescharacteristic
37C2×#340C2211yescharacteristic
38C2×#340C2212yes—
39C2×#336C2212yes—
40#1446—21no—
41#1446—21no—
42#1439—22no—
43C4⋊#95—21no—
44C4⋊#95C2212yes—
45#1434—21no—
46C22⋊#95—22no—
47C23×Q8—21no—
48C23×Q8—22no—
49C23×Q8—24no—
50C23×Q8—24no—
51C23×Q8—22no—
52C23×Q8—22no—
53C23×Q8—22no—
54C23×Q8—22no—
55C23×Q8—42no—
56C23×Q8—42no—
57C23×Q8Q814yes—
58C23×Q8C2312yes—
59C24×C4—21no—
60C24×C4C2311yescharacteristic
61C22×M16—22no—
62C22×M16—22no—
63C22×M16—22no—
64C22×M16—24no—
65C22×M16—42no—
66C23×C8C2311yescharacteristic
67C23×C8C2311yescharacteristic
68C23×C8C2312yes—
69C2×#117—22no—
70C2×#117—24no—
71C2×#117—42no—
72C2×#117D414yes—
73C22×C4⋊C4—22no—
74C22×C4⋊C4C2312yes—
75C22×C4⋊C4C2311yescharacteristic
76C22×C4⋊C4C2311yescharacteristic
77C22×C4⋊C4C2311yescharacteristic
78C22×C4⋊C4C2311yescharacteristic
79C22×C22⋊C4—21no—
80C22×C22⋊C4—21no—
81C22×C22⋊C4—21no—
82C22×C22⋊C4D412yes—
83C22×C42—21no—
84C22×C42C2312yes—
85C2×#107—24no—
86C2×#107—22no—
87C2×#106—22no—
88C2×#106—42no—
89C2×C4⋊C8—24no—
90C2×C4⋊C8—24no—
91C2×C4⋊C8—42no—
92#419—232no—
93C2×#103—24no—
94C2×#103—24no—
95C2×#103—24no—
96C2×#103—24no—
97C2×#103—24no—
98C2×#103—24no—
99C2×#103—24no—
100C2×#103—24no—
101C2×#103—24no—
102C2×#103—42no—
103C2×#103—42no—
104C2×#103D418yes—
105C2×C22⋊C8—22no—
106C2×C22⋊C8—22no—
107C2×C22⋊C8D414yes—
108#391—24no—
109#391—24no—
110#391—42no—
111#391—42no—
112#389—24no—
113#389—42no—
114#380—22no—
115#380—22no—
116#380—22no—
117#380—22no—
118C4×C22⋊C4—22no—
119C4×C22⋊C4—22no—
120C2×#95—22no—
121C2×#95—22no—
122C2×#95—22no—
123C2×#95—22no—
124C2×#95—21no—
125C2×#95—22no—
126C2×#95—22no—
127C2×#95—21no—
128C2×#95—22no—
129C2×#95D411yescharacteristic
130C2×#95D411yescharacteristic
131#340—24no—
132#340—22no—
133#340—22no—
134#340—42no—
135#336—24no—
136C25C2×D411yescharacteristic
137C22×Q8—24no—
138C22×Q8—22no—
139C22×Q8—22no—
140C22×Q8—22no—
141C22×Q8—28no—
142C22×Q8—216no—
143C22×Q8—28no—
144C22×Q8—216no—
145C22×Q8—28no—
146C22×Q8—28no—
147C22×Q8—22no—
148C22×Q8—22no—
149C22×Q8—22no—
150C22×Q8—22no—
151C22×Q8—22no—
152C22×Q8—22no—
153C22×Q8—22no—
154C22×Q8—22no—
155C22×Q8—44no—
156C22×Q8—44no—
157C22×Q8—48no—
158C22×Q8—48no—
159C22×Q8—48no—
160C22×Q8—48no—
161C22×Q8—48no—
162C22×Q8—48no—
163C22×Q8—48no—
164C22×Q8—44no—
165C22×Q8—44no—
166C22×Q8—48no—
167C22×Q8—42no—
168C22×Q8—42no—
169C22×Q8—44no—
170C22×Q8—44no—
171C22×Q8C2×D414yes—
172C22×Q8C2×Q814yes—
173C22×Q8#4114yes—
174C23×C4—22no—
175C23×C4—22no—
176C23×C4—22no—
177C23×C4—22no—
178C23×C4—21no—
179C23×C4—22no—
180C23×C4—24no—
181C23×C4—21no—
182C23×C4—22no—
183C23×C4—21no—
184C23×C4—21no—
185C23×C4—44no—
186C23×C4—44no—
187C23×C4—42no—
188C23×C4—42no—
189C23×C4C2×D412yes—
190C23×C4C2×D412yes—
191C23×C4C2×D411yescharacteristic
192C23×C4C2×D412yes—
193C23×C4C2×D412yes—
194C23×C4C2×Q812yes—
195C23×C4#4112yes—
196C23×C4#4112yes—
197C23×C4#4112yes—
198C23×C4C2411yescharacteristic
199C2×M16—28no—
200C2×M16—48no—
201C2×M16—48no—
202C2×M16—44no—
203C2×M16—44no—
204C22×C8—22no—
205C22×C8—22no—
206C22×C8—22no—
207C22×C8—21no—
208C22×C8—21no—
209C22×C8—22no—
210C22×C8—22no—
211C22×C8—21no—
212C22×C8—21no—
213C22×C8—22no—
214C22×C8—22no—
215C22×C8—22no—
216C22×C8—22no—
217C22×C8—22no—
218C22×C8C2×D414yes—
219C22×C8C2×D412yes—
220C22×C8C2×D412yes—
221#117—28no—
222#117—44no—
223#117—48no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—44no—
244C2×C4⋊C4—44no—
245C2×C4⋊C4—82no—
246C2×C4⋊C4C2×D412yes—
247C2×C4⋊C4C2×D414yes—
248C2×C4⋊C4C2×D412yes—
249C2×C22⋊C4—24no—
250C2×C22⋊C4—24no—
251C2×C22⋊C4—24no—
252C2×C22⋊C4—42no—
253C2×C22⋊C4—44no—
254C2×C22⋊C4—44no—
255C2×C22⋊C4—42no—
256C2×C22⋊C4—42no—
257C2×C22⋊C4—81no—
258C2×C22⋊C4—81no—
259C2×C42—24no—
260C2×C42—22no—
261C2×C42—22no—
262C2×C42—22no—
263C2×C42—22no—
264C2×C42—22no—
265C2×C42—22no—
266C2×C42—44no—
267#107—44no—
268#106—44no—
269C4⋊C8—48no—
270#103—216no—
271#103—216no—
272#103—48no—
273#103—48no—
274C22⋊C8—28no—
275#95—22no—
276#95—22no—
277#95—44no—
278#95—44no—
279#95—44no—
280#95—44no—
281#95—44no—
282#95—42no—
283#95—42no—
284#95—44no—
285#95—82no—
286C24—22no—
287C24—22no—
288C24—22no—
289C24—22no—
290C24—21no—
291C24—21no—
292C24—21no—
293C24—21no—
294C24#12112yes—
295C24#12112yes—
296C24C22⋊Q812yes—
297C24C22×D411yescharacteristic
298C2×Q8—24no—
299C2×Q8—24no—
300C2×Q8—24no—
301C2×Q8—232no—
302C2×Q8—24no—
303C2×Q8—24no—
304C2×Q8—24no—
305C2×Q8—24no—
306C2×Q8—24no—
307C2×Q8—24no—
308C2×Q8—24no—
309C2×Q8—24no—
310C2×Q8—24no—
311C2×Q8—24no—
312C2×Q8—24no—
313C2×Q8—24no—
314C2×Q8—416no—
315C2×Q8—432no—
316C2×Q8—48no—
317C2×Q8—48no—
318C2×Q8—416no—
319C2×Q8—416no—
320C2×Q8—416no—
321C2×Q8—432no—
322C2×Q8—416no—
323C2×Q8—42no—
324C2×Q8—42no—
325C2×Q8—42no—
326C2×Q8—42no—
327C2×Q8—42no—
328C2×Q8—42no—
329C2×Q8—42no—
330C2×Q8—42no—
331C2×Q8—42no—
332C2×Q8—42no—
333C2×Q8—42no—
334C2×Q8—42no—
335C2×Q8—48no—
336C2×Q8C22⋊Q818yes—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—24no—
341C22×C4—22no—
342C22×C4—24no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—24no—
346C22×C4—24no—
347C22×C4—24no—
348C22×C4—24no—
349C22×C4—24no—
350C22×C4—24no—
351C22×C4—24no—
352C22×C4—24no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—24no—
356C22×C4—22no—
357C22×C4—24no—
358C22×C4—24no—
359C22×C4—22no—
360C22×C4—24no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—416no—
373C22×C4—48no—
374C22×C4—48no—
375C22×C4—416no—
376C22×C4—42no—
377C22×C4—42no—
378C22×C4—42no—
379C22×C4—42no—
380C22×C4—42no—
381C22×C4—41no—
382C22×C4—41no—
383C22×C4—48no—
384C22×C4—42no—
385C22×C4—42no—
386C22×C4—41no—
387C22×C4—41no—
388C22×C4—41no—
389C22×C4—41no—
390C22×C4—42no—
391C22×C4—42no—
392C22×C4—42no—
393C22×C4—42no—
394C22×C4—42no—
395C22×C4—42no—
396C22×C4—42no—
397C22×C4—42no—
398C22×C4—42no—
399C22×C4—42no—
400C22×C4—42no—
401C22×C4—41no—
402C22×C4—41no—
403C22×C4—41no—
404C22×C4—41no—
405C22×C4#12114yes—
406C22×C4#12112yes—
407C22×C4#12112yes—
408C22×C4C22⋊Q814yes—
409C22×C4C22⋊Q814yes—
410C22×C4C22⋊Q812yes—
411C22×C4#12614yes—
412C22×C4#13714yes—
413C22×C4C22×D411yescharacteristic
414C22×C4C22×D412yes—
415C22×C4C22×D411yescharacteristic
416C22×C4C22×Q811yescharacteristic
417C22×C4C2×#4111yescharacteristic
418C22×C4C2×#4111yescharacteristic
419C22×C4C2×#4112yes—
420C22×C4C2×#4111yescharacteristic
421C22×C4H(5,𝔽2)12yes—
422C22×C4H(5,𝔽2)12yes—
423M16—416no—
424C2×C8—24no—
425C2×C8—22no—
426C2×C8—22no—
427C2×C8—24no—
428C2×C8—24no—
429C2×C8—24no—
430C2×C8—24no—
431C2×C8—41no—
432C2×C8—42no—
433C2×C8—41no—
434C2×C8—41no—
435C2×C8—41no—
436C2×C8—42no—
437C2×C8—42no—
438C2×C8—42no—
439C2×C8—42no—
440C2×C8—42no—
441C4⋊C4—22no—
442C4⋊C4—24no—
443C4⋊C4—22no—
444C4⋊C4—22no—
445C4⋊C4—42no—
446C4⋊C4—41no—
447C4⋊C4—41no—
448C4⋊C4—42no—
449C4⋊C4—41no—
450C4⋊C4—41no—
451C4⋊C4—42no—
452C4⋊C4—42no—
453C4⋊C4—42no—
454C4⋊C4—42no—
455C4⋊C4—42no—
456C4⋊C4—88no—
457C4⋊C4#12014yes—
458C22⋊C4—48no—
459C22⋊C4—48no—
460C22⋊C4—84no—
461C22⋊C4—84no—
462C42—48no—
463C42—42no—
464C42—42no—
465C42—42no—
466C42—42no—
467C23—24no—
468C23—22no—
469C23—22no—
470C23—22no—
471C23—22no—
472C23—24no—
473C23—24no—
474C23—22no—
475C23—22no—
476C23—22no—
477C23—22no—
478C23—24no—
479C23—22no—
480C23—22no—
481C23—42no—
482C23—42no—
483C23—42no—
484C23—42no—
485C23—42no—
486C23—42no—
487C23—42no—
488C23—42no—
489C23#39614yes—
490C23C2×#12111yescharacteristic
491C23C2×#12111yescharacteristic
492C23C2×C22⋊Q811yescharacteristic
493C23D4212yes—
494C23#54612yes—
495C23C2×#13712yes—
496C23#57512yes—
497C23#57711yescharacteristic
498C23#57711yescharacteristic
499C23#57811yescharacteristic
500C23#57811yescharacteristic
501Q8—28no—
502Q8—28no—
503Q8—432no—
504Q8—44no—
505Q8—44no—
506Q8—44no—
507Q8—44no—
508Q8—44no—
509Q8—44no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—24no—
513C2×C4—24no—
514C2×C4—24no—
515C2×C4—24no—
516C2×C4—24no—
517C2×C4—28no—
518C2×C4—28no—
519C2×C4—24no—
520C2×C4—24no—
521C2×C4—28no—
522C2×C4—24no—
523C2×C4—24no—
524C2×C4—24no—
525C2×C4—432no—
526C2×C4—44no—
527C2×C4—44no—
528C2×C4—44no—
529C2×C4—42no—
530C2×C4—44no—
531C2×C4—44no—
532C2×C4—44no—
533C2×C4—42no—
534C2×C4—44no—
535C2×C4—44no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—82no—
551C2×C4—82no—
552C2×C4—82no—
553C2×C4—81no—
554C2×C4—81no—
555C2×C4—82no—
556C2×C4—82no—
557C2×C4—82no—
558C2×C4—82no—
559C2×C4—82no—
560C2×C4—81no—
561C2×C4—81no—
562C2×C4—81no—
563C2×C4—81no—
564C2×C4—82no—
565C2×C4—82no—
566C2×C4—82no—
567C2×C4—82no—
568C2×C4—82no—
569C2×C4#39614yes—
570C2×C4C2×#12011yescharacteristic
571C2×C4C2×#12111yescharacteristic
572C2×C4C2×C22⋊Q811yescharacteristic
573C2×C4C2×C22⋊Q812yes—
574C2×C4C2×C22⋊Q811yescharacteristic
575C2×C4#52512yes—
576C2×C4#52511yescharacteristic
577C2×C4C2×#12611yescharacteristic
578C2×C4#53411yescharacteristic
579C2×C4#53511yescharacteristic
580C2×C4#53812yes—
581C2×C4#54312yes—
582C2×C4D4212yes—
583C2×C4#54612yes—
584C2×C4#54612yes—
585C2×C4#54612yes—
586C2×C4#55112yes—
587C2×C4#55112yes—
588C8—42no—
589C8—44no—
590C8—42no—
591C22—24no—
592C22—24no—
593C22—24no—
594C22—44no—
595C22—44no—
596C22—42no—
597C22—42no—
598C22—44no—
599C22—44no—
600C22—42no—
601C22—42no—
602C22C2×#39611yescharacteristic
603C22#241011yescharacteristic
604C22#241111yescharacteristic
605C22#242412yes—
606C22#243712yes—
607C22#303911yescharacteristic
608C22#304011yescharacteristic
609C22#304611yescharacteristic
610C22#304611yescharacteristic
611C22#305612yes—
612C22#305711yescharacteristic
613C22#305911yescharacteristic
614C22#307812yes—
615C22#308211yescharacteristic
616C22#308211yescharacteristic
617C22#326912yes—
618C22#327012yes—
619C22#327512yes—
620C22#327812yes—
621C22#331212yes—
622C22#331312yes—
623C22#332612yes—
624C22#332912yes—
625C4—24no—
626C4—44no—
627C4—44no—
628C4—48no—
629C4—82no—
630C4—82no—
631C4—162no—
632C4—161no—
633C4—161no—
634C4#240811yescharacteristic
635C4#243712yes—
636C4#248012yes—
637C4#253411yescharacteristic
638C4#262611yescharacteristic
639C4#284811yescharacteristic
640C2—44no—
641C2#1472611yescharacteristic
642C2#2361311yescharacteristic
643C2#2361911yescharacteristic
644C2#2437411yescharacteristic
645C2#2437711yescharacteristic
646C2#2449711yescharacteristic
647C2#2449811yescharacteristic
648C2#2511812yes—
649C2#2513812yes—
650C2#2597312yes—
651C2#2597912yes—
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2068C2semidirect
8. #11122C2semidirect
9. #11077C2semidirect
10. #11067C2semidirect

Maximal subgroups

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

subgroupindex
C22×#4192
C2×#20682
C2×#18542
C2×#18512
C2×#17872
#139512
#139222
#111222
#110772
#110672
#107222