Finite Group Databasebeta

← #5956153

Subgroups — #5956153

Order 512.

2478 conjugacy classes of subgroups.

The subgroup lattice has 643 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2314C211yesmaximal, characteristic, complemented
2C2×#2293C211yesmaximal, characteristic
3#15235C211yesmaximal, characteristic, complemented
4#15223C211yesmaximal, characteristic, complemented
5#15196C211yesmaximal, characteristic, complemented
6#15181C212yesmaximal
7#15150C211yesmaximal, characteristic, complemented
8#15132C212yesmaximal, complemented
9#14949C214yesmaximal
10#14948C214yesmaximal, complemented
11#14784C214yesmaximal, complemented
12#14783C214yesmaximal, complemented
13#13777C212yesmaximal, complemented
14#13777C211yesmaximal, characteristic, complemented
15C4×#389C212yesmaximal, complemented
16C22×C4×D4C2211yescharacteristic
17C22×#117C2211yescharacteristic
18C22×#117C2211yescharacteristic
19C2×#397C2214yes—
20C2×#397C2214yes—
21#2314—28no—
22#2293—28no—
23C2×#390C2212yes—
24C2×#389C2212yes—
25C2×#389C2212yes—
26C2×#389C2212yes—
27C2×#388C2214yes—
28C2×#388C2212yes—
29C2×#388C2212yes—
30C2×#387C2214yes—
31C2×#384C2211yescharacteristic
32C2×#382C2211yescharacteristic
33C2×#380C2212yes—
34C2×#380C2212yes—
35C2×#380C2212yes—
36C2×#380C2212yes—
37C2×C4×C4⋊C4C2211yescharacteristic
38C2×C4×C4⋊C4C2212yes—
39C2×C4×C4⋊C4C2212yes—
40C2×C4×C4⋊C4C2214yes—
41C2×C4×C22⋊C4C2211yescharacteristic
42C2×C4×C22⋊C4C2211yescharacteristic
43C2×C43C2211yescharacteristic
44#1445C2214yes—
45#1445C2214yes—
46#1444C2214yes—
47#1443C2214yes—
48#1443C2214yes—
49#1442C2214yes—
50#1442C2214yes—
51#1442C2214yes—
52#1441C2214yes—
53#1440C2214yes—
54#1439C2214yes—
55#1439C2214yes—
56#1439C2214yes—
57#1438C2214yes—
58#1437C2214yes—
59#1437C2214yes—
60#1437C2214yes—
61#1436C2214yes—
62#1436C2214yes—
63#1436C2214yes—
64#1436C2214yes—
65#1436C2214yes—
66C4⋊#95C2212yes—
67C4⋊#95C2212yes—
68C4⋊#95C2212yes—
69C4⋊#95C2211yescharacteristic
70C4⋊#95C2212yes—
71C4⋊#95C2211yescharacteristic
72C4×#95C2214yes—
73C23×D4C2311yescharacteristic
74C24×C4C2311yescharacteristic
75C24×C4C2311yescharacteristic
76C2×C4×D4—24no—
77C2×C4×D4—24no—
78C2×C4×D4—24no—
79C2×#117—24no—
80C2×#117—28no—
81C2×#117—28no—
82C2×#117D418yes—
83C22×C4⋊C4C2312yes—
84C22×C4⋊C4C2311yescharacteristic
85C22×C4⋊C4C2312yes—
86C22×C4⋊C4C2312yes—
87C22×C4⋊C4C2312yes—
88C22×C4⋊C4C2314yes—
89C22×C4⋊C4C2312yes—
90C22×C4⋊C4C2312yes—
91C22×C4⋊C4C2312yes—
92C22×C4⋊C4C2314yes—
93C22×C4⋊C4C2314yes—
94C22×C22⋊C4C2312yes—
95C22×C22⋊C4C2311yescharacteristic
96C22×C22⋊C4C2311yescharacteristic
97C22×C22⋊C4C2312yes—
98C22×C22⋊C4C2312yes—
99C22×C22⋊C4C2312yes—
100C22×C22⋊C4C2314yes—
101C22×C22⋊C4C2314yes—
102C22×C22⋊C4C2314yes—
103C22×C22⋊C4C2314yes—
104C22×C42C2311yescharacteristic
105C22×C42C2311yescharacteristic
106C22×C42C2312yes—
107C22×C42C2312yes—
108C22×C42C2311yescharacteristic
109C22×C42C2311yescharacteristic
110C22×C42C2314yes—
111C22×C42C2314yes—
112C22×C42C2311yescharacteristic
113#397—48no—
114#397—48no—
115#390—28no—
116#389—28no—
117#389—28no—
118#389—28no—
119#388—28no—
120#388—216no—
121#388—28no—
122#387—216no—
123#384—24no—
124#382—24no—
125#380—28no—
126#380—28no—
127#380—28no—
128#380—28no—
129C4×C4⋊C4—28no—
130C4×C4⋊C4—24no—
131C4×C4⋊C4—28no—
132C4×C4⋊C4—48no—
133C4×C22⋊C4—24no—
134C4×C22⋊C4—24no—
135C2×#95C2314yes—
136C2×#95C2314yes—
137C2×#95C2314yes—
138C2×#95C2314yes—
139C2×#95C2314yes—
140C2×#95C2314yes—
141C2×#95C2314yes—
142C2×#95C2314yes—
143C2×#95C2314yes—
144C2×#95C2314yes—
145C2×#95C2314yes—
146C2×#95C2314yes—
147C2×#95C2312yes—
148C2×#95C2312yes—
149C2×#95C2312yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C43—24no—
158C25C2411yescharacteristic
159C25C2411yescharacteristic
160C22×D4—22no—
161C22×D4—22no—
162C22×D4—22no—
163C22×D4—24no—
164C22×D4—42no—
165C23×C4—22no—
166C23×C4—24no—
167C23×C4—24no—
168C23×C4—22no—
169C23×C4—22no—
170C23×C4—22no—
171C23×C4—24no—
172C23×C4—24no—
173C23×C4C2×D414yes—
174C23×C4#4114yes—
175C23×C4C2414yes—
176C23×C4C2414yes—
177C23×C4C2412yes—
178C23×C4C2414yes—
179C23×C4C2412yes—
180C23×C4C2411yescharacteristic
181C23×C4C2411yescharacteristic
182C23×C4C2414yes—
183C23×C4C2411yescharacteristic
184C23×C4C2412yes—
185C23×C4C2412yes—
186C23×C4C2411yescharacteristic
187C23×C4C2411yescharacteristic
188C4×D4—232no—
189#117—232no—
190#117—416no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—24no—
193C2×C4⋊C4—24no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—24no—
197C2×C4⋊C4—24no—
198C2×C4⋊C4—24no—
199C2×C4⋊C4—24no—
200C2×C4⋊C4—24no—
201C2×C4⋊C4—24no—
202C2×C4⋊C4—24no—
203C2×C4⋊C4—24no—
204C2×C4⋊C4—24no—
205C2×C4⋊C4—24no—
206C2×C4⋊C4—24no—
207C2×C4⋊C4—24no—
208C2×C4⋊C4—24no—
209C2×C4⋊C4—28no—
210C2×C4⋊C4—28no—
211C2×C4⋊C4—24no—
212C2×C4⋊C4—24no—
213C2×C4⋊C4—28no—
214C2×C4⋊C4—24no—
215C2×C4⋊C4—28no—
216C2×C4⋊C4—42no—
217C2×C4⋊C4—44no—
218C2×C4⋊C4—44no—
219C2×C4⋊C4—44no—
220C2×C4⋊C4—44no—
221C2×C4⋊C4—44no—
222C2×C4⋊C4C2×D418yes—
223C2×C4⋊C4#4118yes—
224C2×C22⋊C4—22no—
225C2×C22⋊C4—22no—
226C2×C22⋊C4—22no—
227C2×C22⋊C4—24no—
228C2×C22⋊C4—24no—
229C2×C22⋊C4—22no—
230C2×C22⋊C4—22no—
231C2×C22⋊C4—22no—
232C2×C22⋊C4—24no—
233C2×C22⋊C4—24no—
234C2×C22⋊C4—24no—
235C2×C22⋊C4—24no—
236C2×C22⋊C4—24no—
237C2×C22⋊C4—24no—
238C2×C22⋊C4—24no—
239C2×C22⋊C4—28no—
240C2×C22⋊C4—28no—
241C2×C22⋊C4—28no—
242C2×C22⋊C4—28no—
243C2×C22⋊C4—42no—
244C2×C22⋊C4—42no—
245C2×C22⋊C4—44no—
246C2×C22⋊C4—44no—
247C2×C22⋊C4—44no—
248C2×C22⋊C4—44no—
249C2×C22⋊C4—44no—
250C2×C22⋊C4—44no—
251C2×C22⋊C4—44no—
252C2×C22⋊C4—44no—
253C2×C22⋊C4C2×D418yes—
254C2×C42—22no—
255C2×C42—24no—
256C2×C42—22no—
257C2×C42—22no—
258C2×C42—22no—
259C2×C42—24no—
260C2×C42—24no—
261C2×C42—24no—
262C2×C42—28no—
263C2×C42—24no—
264C2×C42—24no—
265C2×C42—24no—
266C2×C42—28no—
267C2×C42—44no—
268C2×C42—42no—
269C2×C42—42no—
270C2×C42—44no—
271C2×C42—44no—
272C2×C42—44no—
273C2×C42C2×D414yes—
274C2×C42C2×D414yes—
275C2×C42#4114yes—
276C2×C42#4118yes—
277#95—24no—
278#95—24no—
279#95—28no—
280#95—24no—
281#95—28no—
282#95—28no—
283#95—28no—
284#95—28no—
285#95—28no—
286#95—28no—
287#95—28no—
288#95—28no—
289#95—28no—
290#95—28no—
291#95—28no—
292#95—44no—
293#95—44no—
294#95—44no—
295#95—44no—
296#95—44no—
297#95—44no—
298#95—44no—
299C24—21no—
300C24—22no—
301C24—22no—
302C24—22no—
303C24—22no—
304C24—21no—
305C24—21no—
306C24—21no—
307C24—22no—
308C24—22no—
309C24—22no—
310C24—22no—
311C24—22no—
312C24—22no—
313C24—22no—
314C24—22no—
315C24C22×D412yes—
316C24C2×#4112yes—
317C24C2511yescharacteristic
318C2×D4—28no—
319C2×D4—48no—
320C2×D4—48no—
321C2×D4—44no—
322C2×D4—44no—
323C22×C4—21no—
324C22×C4—21no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—21no—
331C22×C4—22no—
332C22×C4—21no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—24no—
337C22×C4—24no—
338C22×C4—21no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—21no—
343C22×C4—21no—
344C22×C4—21no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—24no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—24no—
354C22×C4—24no—
355C22×C4—24no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—28no—
361C22×C4—28no—
362C22×C4—28no—
363C22×C4—216no—
364C22×C4—28no—
365C22×C4—24no—
366C22×C4—22no—
367C22×C4—24no—
368C22×C4—24no—
369C22×C4—24no—
370C22×C4—24no—
371C22×C4—24no—
372C22×C4—24no—
373C22×C4—24no—
374C22×C4—24no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—24no—
378C22×C4—24no—
379C22×C4—24no—
380C22×C4—24no—
381C22×C4—22no—
382C22×C4—24no—
383C22×C4—24no—
384C22×C4—24no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—24no—
392C22×C4—24no—
393C22×C4—44no—
394C22×C4—48no—
395C22×C4—48no—
396C22×C4—48no—
397C22×C4—44no—
398C22×C4C22×D412yes—
399C22×C4C22×D414yes—
400C22×C4C22×D414yes—
401C22×C4C22×D412yes—
402C22×C4C2×#4112yes—
403C22×C4C2×#4118yes—
404C22×C4C2×#4112yes—
405C22×C4C2×#4114yes—
406C22×C4C2×#4114yes—
407C22×C4C2×#4112yes—
408C22×C4C2×#4114yes—
409C22×C4C2×#4112yes—
410C22×C4C2×#4112yes—
411C22×C4C2×#4114yes—
412C22×C4H(5,𝔽2)14yes—
413C22×C42−1+414yes—
414C4⋊C4—28no—
415C4⋊C4—216no—
416C4⋊C4—216no—
417C4⋊C4—48no—
418C4⋊C4—416no—
419C4⋊C4—416no—
420C4⋊C4—48no—
421C4⋊C4—48no—
422C4⋊C4—48no—
423C4⋊C4—48no—
424C4⋊C4—88no—
425C22⋊C4—28no—
426C22⋊C4—28no—
427C22⋊C4—216no—
428C22⋊C4—48no—
429C22⋊C4—48no—
430C22⋊C4—48no—
431C22⋊C4—88no—
432C22⋊C4—88no—
433C22⋊C4—88no—
434C22⋊C4—88no—
435C42—28no—
436C42—28no—
437C42—48no—
438C42—48no—
439C42—416no—
440C42—44no—
441C42—44no—
442C42—88no—
443C42#123116yes—
444C23—22no—
445C23—22no—
446C23—22no—
447C23—24no—
448C23—22no—
449C23—24no—
450C23—24no—
451C23—24no—
452C23—22no—
453C23—22no—
454C23—41no—
455C23—42no—
456C23—42no—
457C23—42no—
458C23—41no—
459C23—41no—
460C23—41no—
461C23—41no—
462C23—41no—
463C23—42no—
464C23—42no—
465C23—42no—
466C23—42no—
467C23—42no—
468C23—42no—
469C23—42no—
470C23—42no—
471C23—42no—
472C23—41no—
473C23—42no—
474C23—42no—
475C23—42no—
476C23—41no—
477C23—42no—
478C23—42no—
479C23—42no—
480C23—42no—
481C23—42no—
482C23—42no—
483C23—42no—
484C23—42no—
485C23—42no—
486C23C23×D411yescharacteristic
487C23C22×#4112yes—
488C23C22×#4111yescharacteristic
489C23C22×#4111yescharacteristic
490C23C22×#4112yes—
491C23C2×H(5,𝔽2)12yes—
492C23C2×2−1+412yes—
493C23#58512yes—
494C23#58512yes—
495D4—416no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—24no—
499C2×C4—24no—
500C2×C4—24no—
501C2×C4—24no—
502C2×C4—24no—
503C2×C4—24no—
504C2×C4—24no—
505C2×C4—24no—
506C2×C4—24no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—28no—
512C2×C4—24no—
513C2×C4—22no—
514C2×C4—24no—
515C2×C4—24no—
516C2×C4—24no—
517C2×C4—24no—
518C2×C4—28no—
519C2×C4—24no—
520C2×C4—28no—
521C2×C4—24no—
522C2×C4—41no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—41no—
528C2×C4—44no—
529C2×C4—44no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—44no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—41no—
536C2×C4—42no—
537C2×C4—44no—
538C2×C4—41no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—416no—
543C2×C4—416no—
544C2×C4—42no—
545C2×C4—44no—
546C2×C4—44no—
547C2×C4—44no—
548C2×C4—44no—
549C2×C4—44no—
550C2×C4—44no—
551C2×C4—44no—
552C2×C4—42no—
553C2×C4—44no—
554C2×C4—42no—
555C2×C4—44no—
556C2×C4—44no—
557C2×C4—44no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—44no—
561C2×C4—44no—
562C2×C4—44no—
563C2×C4—42no—
564C2×C4—44no—
565C2×C4—42no—
566C2×C4—44no—
567C2×C4—44no—
568C2×C4—44no—
569C2×C4—44no—
570C2×C4—44no—
571C2×C4—44no—
572C2×C4C2×#12318yes—
573C2×C4C2×#12314yes—
574C2×C4#52514yes—
575C2×C4#52918yes—
576C2×C4#52918yes—
577C2×C4#53214yes—
578C2×C4#54114yes—
579C22—44no—
580C22—44no—
581C22—42no—
582C22—42no—
583C22—44no—
584C22—42no—
585C22—42no—
586C22—42no—
587C22—42no—
588C22—44no—
589C22—82no—
590C22—82no—
591C22—82no—
592C22—82no—
593C22—82no—
594C22—82no—
595C22—82no—
596C22—82no—
597C22C22×#12311yescharacteristic
598C22C2×#52511yescharacteristic
599C22C2×#52912yes—
600C22C2×#52912yes—
601C22C2×#53211yescharacteristic
602C22C2×#54111yescharacteristic
603C22#345511yescharacteristic
604C22#346012yes—
605C22#348812yes—
606C22#348912yes—
607C22#349512yes—
608C22#350612yes—
609C22#350712yes—
610C22#350811yescharacteristic
611C22#351112yes—
612C22#351211yescharacteristic
613C22#351212yes—
614C22#351312yes—
615C22#351812yes—
616C22#352112yes—
617C22#352212yes—
618C4—28no—
619C4—24no—
620C4—48no—
621C4—42no—
622C4—44no—
623C4—42no—
624C4—42no—
625C4—44no—
626C4—84no—
627C4—84no—
628C4—82no—
629C4—84no—
630C4#239518yes—
631C2—82no—
632C2—82no—
633C2C2×#239511yescharacteristic
634C2#3518512yes—
635C2#3521812yes—
636C2#3525911yescharacteristic
637C2#3531011yescharacteristic
638C2#3551712yes—
639C2#3556912yes—
640C2#3559812yes—
641C2#3565212yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC23×D4C23
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2314C2semidirect
3. #15235C2semidirect
4. #15223C2semidirect
5. #15196C2semidirect
7. #15150C2semidirect
8. #15132C2semidirect
10. #14948C2semidirect
11. #14784C2semidirect
12. #14783C2semidirect
13. #13777C2semidirect
14. #13777C2semidirect
15. C4×#389C2semidirect

Maximal subgroups

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

subgroupindex
C2×#23142
C2×#22932
#152352
#152232
#151962
#151812
#151502
#151322
#149492
#149482
#147842
#147832
#137772
C4×#3892