Finite Group Databasebeta

← #4308104

Subgroups — #4308104

Order 512.

2514 conjugacy classes of subgroups.

The subgroup lattice has 640 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2629C211yesmaximal, characteristic
2C2×C4×#117C211yesmaximal, characteristic
3#14845C214yesmaximal, complemented
4#14831C214yesmaximal
5#14779C214yesmaximal, complemented
6#14752C214yesmaximal
7#14712C211yesmaximal, characteristic, complemented
8#14709C211yesmaximal, characteristic, complemented
9#14699C212yesmaximal, complemented
10#14697C212yesmaximal
11#14343C212yesmaximal
12#14340C211yesmaximal, characteristic
13#14283C211yesmaximal, characteristic, complemented
14#14277C211yesmaximal, characteristic, complemented
15#14266C212yesmaximal, complemented
16C22×#125C2211yescharacteristic
17C22×#117C2211yescharacteristic
18C22×#117C2211yescharacteristic
19C23×C42C2211yescharacteristic
20#2629—28no—
21C2×#398C2214yes—
22C2×#396C2214yes—
23C2×#389C2212yes—
24C2×#389C2212yes—
25C2×#389C2212yes—
26C2×#389C2214yes—
27C2×#387C2214yes—
28C2×#385C2214yes—
29C2×#384C2211yescharacteristic
30C2×#384C2211yescharacteristic
31C2×#382C2211yescharacteristic
32C2×#380C2212yes—
33C2×#380C2212yes—
34C4×#117—28no—
35C2×C4×C4⋊C4C2212yes—
36C2×C4×C4⋊C4C2212yes—
37C2×C4×C22⋊C4C2212yes—
38C2×C4×C22⋊C4C2212yes—
39C2×#376C2211yescharacteristic
40C2×#376C2211yescharacteristic
41C2×C43C2211yescharacteristic
42C2×C43C2211yescharacteristic
43#1445C2214yes—
44#1445C2214yes—
45#1445C2214yes—
46#1444C2214yes—
47#1443C2214yes—
48#1442C2214yes—
49#1442C2214yes—
50#1441C2214yes—
51#1441C2214yes—
52#1441C2214yes—
53#1441C2214yes—
54#1440C2214yes—
55#1440C2214yes—
56#1439C2214yes—
57#1436C2214yes—
58#1436C2214yes—
59#1436C2214yes—
60#1436C2214yes—
61#1436C2214yes—
62#1436C2214yes—
63C4⋊#95C2212yes—
64C4⋊#95C2212yes—
65C4⋊#95C2212yes—
66C4⋊#95C2212yes—
67C4⋊#95C2212yes—
68C4⋊#95C2212yes—
69C4⋊#95C2212yes—
70C4⋊#95C2214yes—
71#1434C2214yes—
72C22⋊#95C2214yes—
73C24×C4C2311yescharacteristic
74C24×C4C2311yescharacteristic
75C24×C4C2311yescharacteristic
76C2×#125—24no—
77C2×#125—24no—
78C2×#125—24no—
79C2×#117—24no—
80C2×#117—24no—
81C2×#117—24no—
82C2×#117—24no—
83C2×#117D418yes—
84C2×#117Q818yes—
85C22×C4⋊C4C2314yes—
86C22×C4⋊C4C2312yes—
87C22×C4⋊C4C2312yes—
88C22×C4⋊C4C2312yes—
89C22×C4⋊C4C2312yes—
90C22×C4⋊C4C2314yes—
91C22×C4⋊C4C2312yes—
92C22×C4⋊C4C2312yes—
93C22×C4⋊C4C2314yes—
94C22×C4⋊C4C2314yes—
95C22×C4⋊C4C2314yes—
96C22×C4⋊C4C2314yes—
97C22×C22⋊C4C2312yes—
98C22×C22⋊C4C2312yes—
99C22×C22⋊C4C2314yes—
100C22×C22⋊C4C2314yes—
101C22×C42—24no—
102C22×C42—24no—
103C22×C42—24no—
104C22×C42C2311yescharacteristic
105C22×C42C2311yescharacteristic
106C22×C42C2312yes—
107C22×C42C2312yes—
108C22×C42C2312yes—
109C22×C42C2312yes—
110C22×C42C2311yescharacteristic
111C22×C42C2311yescharacteristic
112C22×C42C2311yescharacteristic
113C22×C42C2311yescharacteristic
114C22×C42C2311yescharacteristic
115C22×C42C2311yescharacteristic
116#398—216no—
117#396—216no—
118#389—28no—
119#389—28no—
120#389—28no—
121#389—48no—
122#387—48no—
123#385—48no—
124#384—24no—
125#384—24no—
126#382—24no—
127#380—28no—
128#380—28no—
129C4×C4⋊C4—28no—
130C4×C4⋊C4—28no—
131C4×C22⋊C4—28no—
132C4×C22⋊C4—28no—
133#376—24no—
134#376—24no—
135C2×#95C2314yes—
136C2×#95C2314yes—
137C2×#95C2314yes—
138C2×#95C2314yes—
139C2×#95C2314yes—
140C2×#95C2314yes—
141C2×#95C2314yes—
142C2×#95C2314yes—
143C2×#95C2312yes—
144C2×#95C2312yes—
145C2×#95C2312yes—
146C2×#95C2312yes—
147C2×#95C2314yes—
148C2×#95C2314yes—
149C2×#95C2314yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C43—24no—
160C43—24no—
161C25C2411yescharacteristic
162C23×C4—22no—
163C23×C4—22no—
164C23×C4—22no—
165C23×C4—22no—
166C23×C4—22no—
167C23×C4—22no—
168C23×C4—22no—
169C23×C4—22no—
170C23×C4—22no—
171C23×C4—22no—
172C23×C4—24no—
173C23×C4—24no—
174C23×C4—42no—
175C23×C4—42no—
176C23×C4C2×D414yes—
177C23×C4C2×Q814yes—
178C23×C4#4114yes—
179C23×C4C2414yes—
180C23×C4C2414yes—
181C23×C4C2414yes—
182C23×C4C2414yes—
183C23×C4C2412yes—
184C23×C4C2412yes—
185C23×C4C2412yes—
186C23×C4C2411yescharacteristic
187C23×C4C2411yescharacteristic
188C23×C4C2411yescharacteristic
189C23×C4C2412yes—
190C23×C4C2411yescharacteristic
191C23×C4C2411yescharacteristic
192C23×C4C2411yescharacteristic
193#125—416no—
194#117—232no—
195#117—232no—
196C2×C4⋊C4—24no—
197C2×C4⋊C4—24no—
198C2×C4⋊C4—24no—
199C2×C4⋊C4—24no—
200C2×C4⋊C4—24no—
201C2×C4⋊C4—28no—
202C2×C4⋊C4—24no—
203C2×C4⋊C4—28no—
204C2×C4⋊C4—28no—
205C2×C4⋊C4—24no—
206C2×C4⋊C4—24no—
207C2×C4⋊C4—24no—
208C2×C4⋊C4—24no—
209C2×C4⋊C4—24no—
210C2×C4⋊C4—28no—
211C2×C4⋊C4—24no—
212C2×C4⋊C4—24no—
213C2×C4⋊C4—24no—
214C2×C4⋊C4—24no—
215C2×C4⋊C4—24no—
216C2×C4⋊C4—28no—
217C2×C4⋊C4—28no—
218C2×C4⋊C4—28no—
219C2×C4⋊C4—28no—
220C2×C4⋊C4—28no—
221C2×C4⋊C4—28no—
222C2×C4⋊C4—44no—
223C2×C4⋊C4—44no—
224C2×C4⋊C4—44no—
225C2×C4⋊C4—44no—
226C2×C4⋊C4—44no—
227C2×C4⋊C4—44no—
228C2×C4⋊C4—44no—
229C2×C4⋊C4—44no—
230C2×C4⋊C4C2×D418yes—
231C2×C4⋊C4C2×Q818yes—
232C2×C22⋊C4—24no—
233C2×C22⋊C4—24no—
234C2×C22⋊C4—24no—
235C2×C22⋊C4—24no—
236C2×C22⋊C4—28no—
237C2×C22⋊C4—28no—
238C2×C22⋊C4—44no—
239C2×C22⋊C4—44no—
240C2×C22⋊C4—44no—
241C2×C22⋊C4—44no—
242C2×C22⋊C4C2×D418yes—
243C2×C22⋊C4C2×Q818yes—
244C2×C42—232no—
245C2×C42—22no—
246C2×C42—22no—
247C2×C42—22no—
248C2×C42—22no—
249C2×C42—22no—
250C2×C42—22no—
251C2×C42—22no—
252C2×C42—22no—
253C2×C42—22no—
254C2×C42—22no—
255C2×C42—22no—
256C2×C42—22no—
257C2×C42—22no—
258C2×C42—22no—
259C2×C42—22no—
260C2×C42—22no—
261C2×C42—24no—
262C2×C42—24no—
263C2×C42—24no—
264C2×C42—24no—
265C2×C42—24no—
266C2×C42—24no—
267C2×C42—24no—
268C2×C42—24no—
269C2×C42—42no—
270C2×C42—42no—
271C2×C42—42no—
272C2×C42—42no—
273C2×C42C2×D414yes—
274C2×C42C2×D414yes—
275C2×C42C2×Q814yes—
276C2×C42C2×Q814yes—
277C2×C42#4114yes—
278C2×C42#4114yes—
279C2×C42#4114yes—
280C2×C42#4114yes—
281#95—24no—
282#95—24no—
283#95—24no—
284#95—24no—
285#95—28no—
286#95—28no—
287#95—28no—
288#95—28no—
289#95—28no—
290#95—28no—
291#95—28no—
292#95—28no—
293#95—44no—
294#95—44no—
295#95—44no—
296#95—44no—
297#95—44no—
298#95—44no—
299#95—44no—
300#95—44no—
301#95—44no—
302#95—44no—
303#95—44no—
304#95—44no—
305C24—21no—
306C24—21no—
307C24—21no—
308C24—21no—
309C24—22no—
310C24—22no—
311C24—22no—
312C24—22no—
313C24C22×D412yes—
314C24C22×Q812yes—
315C24C2×#4112yes—
316C24C2511yescharacteristic
317C22×C4—21no—
318C22×C4—21no—
319C22×C4—21no—
320C22×C4—21no—
321C22×C4—21no—
322C22×C4—21no—
323C22×C4—21no—
324C22×C4—21no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—28no—
332C22×C4—28no—
333C22×C4—28no—
334C22×C4—28no—
335C22×C4—28no—
336C22×C4—28no—
337C22×C4—28no—
338C22×C4—28no—
339C22×C4—21no—
340C22×C4—21no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—21no—
344C22×C4—21no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—24no—
366C22×C4—24no—
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—24no—
376C22×C4—24no—
377C22×C4—24no—
378C22×C4—24no—
379C22×C4—24no—
380C22×C4—24no—
381C22×C4—24no—
382C22×C4—24no—
383C22×C4—24no—
384C22×C4—24no—
385C22×C4—24no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—24no—
392C22×C4—24no—
393C22×C4—48no—
394C22×C4—48no—
395C22×C4—48no—
396C22×C4—48no—
397C22×C4—44no—
398C22×C4—44no—
399C22×C4—44no—
400C22×C4—44no—
401C22×C4C22⋊Q8116yes—
402C22×C4C22×D414yes—
403C22×C4C22×D412yes—
404C22×C4C22×D414yes—
405C22×C4C22×D412yes—
406C22×C4C22×Q814yes—
407C22×C4C22×Q814yes—
408C22×C4C22×Q812yes—
409C22×C4C22×Q812yes—
410C22×C4C2×#4112yes—
411C22×C4C2×#4112yes—
412C22×C4C2×#4112yes—
413C22×C4C2×#4114yes—
414C22×C4C2×#4114yes—
415C22×C4C2×#4112yes—
416C22×C4C2×#4112yes—
417C22×C4C2×#4112yes—
418C22×C4C2×#4112yes—
419C22×C4C2×#4112yes—
420C22×C4C2×#4112yes—
421C22×C4C2×#4112yes—
422C22×C4H(5,𝔽2)14yes—
423C22×C42−1+414yes—
424C4⋊C4—216no—
425C4⋊C4—216no—
426C4⋊C4—416no—
427C4⋊C4—416no—
428C4⋊C4—48no—
429C4⋊C4—48no—
430C4⋊C4—48no—
431C4⋊C4—48no—
432C4⋊C4—416no—
433C4⋊C4—88no—
434C4⋊C4—88no—
435C4⋊C4—88no—
436C22⋊C4—216no—
437C22⋊C4—216no—
438C22⋊C4—416no—
439C22⋊C4—88no—
440C42—28no—
441C42—28no—
442C42—28no—
443C42—28no—
444C42—28no—
445C42—28no—
446C42—28no—
447C42—48no—
448C42—48no—
449C42—48no—
450C42—48no—
451C42#123116yes—
452C23—22no—
453C23—22no—
454C23—22no—
455C23—22no—
456C23—22no—
457C23—22no—
458C23—24no—
459C23—24no—
460C23—24no—
461C23—24no—
462C23—24no—
463C23—24no—
464C23—41no—
465C23—41no—
466C23—41no—
467C23—41no—
468C23—42no—
469C23—42no—
470C23—42no—
471C23—42no—
472C23—42no—
473C23—42no—
474C23C2×C22⋊Q814yes—
475C23C23×D411yescharacteristic
476C23C23×Q811yescharacteristic
477C23C22×#4111yescharacteristic
478C23C22×#4111yescharacteristic
479C23C22×#4111yescharacteristic
480C23C22×#4111yescharacteristic
481C23C22×#4111yescharacteristic
482C23C2×H(5,𝔽2)12yes—
483C23C2×2−1+412yes—
484C23#58512yes—
485C23#58512yes—
486C2×C4—22no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—22no—
490C2×C4—22no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—24no—
497C2×C4—24no—
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—232no—
507C2×C4—24no—
508C2×C4—24no—
509C2×C4—24no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—24no—
513C2×C4—24no—
514C2×C4—24no—
515C2×C4—24no—
516C2×C4—24no—
517C2×C4—24no—
518C2×C4—24no—
519C2×C4—24no—
520C2×C4—416no—
521C2×C4—416no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—44no—
541C2×C4—44no—
542C2×C4—44no—
543C2×C4—44no—
544C2×C4—44no—
545C2×C4—44no—
546C2×C4—44no—
547C2×C4—44no—
548C2×C4—44no—
549C2×C4—44no—
550C2×C4—44no—
551C2×C4—44no—
552C2×C4—44no—
553C2×C4—44no—
554C2×C4—44no—
555C2×C4—44no—
556C2×C4—44no—
557C2×C4—44no—
558C2×C4—44no—
559C2×C4—44no—
560C2×C4—44no—
561C2×C4—44no—
562C2×C4—44no—
563C2×C4—44no—
564C2×C4—44no—
565C2×C4—44no—
566C2×C4—44no—
567C2×C4—44no—
568C2×C4C2×C22⋊Q814yes—
569C2×C4C2×C22⋊Q814yes—
570C2×C4C2×#12318yes—
571C2×C4C2×#12314yes—
572C2×C4#52514yes—
573C2×C4#52914yes—
574C2×C4#52914yes—
575C2×C4#53314yes—
576C2×C4#53314yes—
577C2×C4#54014yes—
578C2×C4#54114yes—
579C22—24no—
580C22—28no—
581C22—42no—
582C22—42no—
583C22—42no—
584C22—44no—
585C22—42no—
586C22—44no—
587C22—44no—
588C22—42no—
589C22—42no—
590C22C22×C22⋊Q811yescharacteristic
591C22C22×#12311yescharacteristic
592C22C2×#52511yescharacteristic
593C22C2×#52911yescharacteristic
594C22C2×#52911yescharacteristic
595C22C2×#53311yescharacteristic
596C22C2×#53311yescharacteristic
597C22C2×#54011yescharacteristic
598C22C2×#54111yescharacteristic
599C22#345511yescharacteristic
600C22#345511yescharacteristic
601C22#346012yes—
602C22#346012yes—
603C22#346112yes—
604C22#346112yes—
605C22#348812yes—
606C22#348912yes—
607C22#349412yes—
608C22#349512yes—
609C22#350312yes—
610C22#350412yes—
611C22#351112yes—
612C22#351212yes—
613C4—24no—
614C4—28no—
615C4—24no—
616C4—24no—
617C4—44no—
618C4—44no—
619C4—44no—
620C4—42no—
621C4—42no—
622C4—84no—
623C4—84no—
624C4—84no—
625C4—84no—
626C4#285518yes—
627C2—44no—
628C2#3462111yescharacteristic
629C2#3462911yescharacteristic
630C2#3525512yes—
631C2#3525712yes—
632C2#3526111yescharacteristic
633C2#3526912yes—
634C2#3528212yes—
635C2#3528411yescharacteristic
636C2#3531011yescharacteristic
637C2#3533611yescharacteristic
638C2C2×#285511yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC25C24
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #14845C2semidirect
5. #14779C2semidirect
7. #14712C2semidirect
8. #14709C2semidirect
9. #14699C2semidirect
13. #14283C2semidirect
14. #14277C2semidirect
15. #14266C2semidirect

Maximal subgroups

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

subgroupindex
C2×#26292
C2×C4×#1172
#148452
#148312
#147792
#147522
#147122
#147092
#146992
#146972
#143432
#143402
#142832
#142772
#142662