Finite Group Databasebeta

← #681132

Subgroups — #681132

Order 512.

2650 conjugacy classes of subgroups.

The subgroup lattice has 644 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2864C211yesmaximal, characteristic, complemented
2C2×#2349C211yesmaximal, characteristic
3#16044C214yesmaximal, complemented
4#15441C214yesmaximal, complemented
5#15187C212yesmaximal, complemented
6#15149C212yesmaximal, complemented
7#14775C214yesmaximal, complemented
8#14150C211yesmaximal, characteristic, complemented
9#14028C214yesmaximal, complemented
10#14018C212yesmaximal
11#13955C211yesmaximal, characteristic, complemented
12#13941C211yesmaximal, characteristic
13#13920C212yesmaximal
14C4×#389C211yesmaximal, characteristic, complemented
15C4×#390C211yesmaximal, characteristic
16C22×C4×D4C2212yes—
17C22×C4×D4C2211yescharacteristic
18#2864—28no—
19C2×#398C2214yes—
20#2349—28no—
21C2×#390C2211yescharacteristic
22C2×#390C2212yes—
23C2×#390C2211yescharacteristic
24C2×#389C2212yes—
25C2×#389C2212yes—
26C2×#389C2212yes—
27C2×#389C2212yes—
28C2×#389C2212yes—
29C2×#387C2214yes—
30C2×#387C2214yes—
31C2×#385C2214yes—
32C2×#385C2214yes—
33C2×#382C2211yescharacteristic
34C2×#382C2211yescharacteristic
35C2×#380C2212yes—
36C2×#380C2214yes—
37C2×C4×C4⋊C4C2211yescharacteristic
38C2×C4×C4⋊C4C2211yescharacteristic
39C2×C4×C4⋊C4C2211yescharacteristic
40C2×C4×C4⋊C4C2212yes—
41C2×C4×C22⋊C4C2214yes—
42C22×#95C2212yes—
43C2×C43C2211yescharacteristic
44#1445C2214yes—
45#1444C2212yes—
46#1443C2214yes—
47#1442C2214yes—
48#1442C2214yes—
49#1442C2212yes—
50#1442C2214yes—
51#1442C2214yes—
52#1441C2214yes—
53#1441C2214yes—
54#1439C2212yes—
55#1439C2214yes—
56#1439C2214yes—
57#1439C2214yes—
58#1439C2212yes—
59#1439C2214yes—
60#1437C2214yes—
61#1436C2214yes—
62#1436C2212yes—
63#1436C2212yes—
64C4⋊#95C2211yescharacteristic
65C4⋊#95C2212yes—
66C4⋊#95C2212yes—
67C4⋊#95C2212yes—
68C4⋊#95C2211yescharacteristic
69C4⋊#95C2212yes—
70C4⋊#95C2212yes—
71#1434C2212yes—
72#1434C2212yes—
73C22⋊#95C2214yes—
74C22⋊#95C2214yes—
75C22⋊#95C2214yes—
76C4×#95C2212yes—
77C23×D4C2311yescharacteristic
78C24×C4C2314yes—
79C24×C4C2312yes—
80C2×C4×D4—28no—
81C2×C4×D4—28no—
82C2×C4×D4—24no—
83C2×C4×D4—24no—
84C2×C4×D4—44no—
85C2×C4×D4Q818yes—
86C22×C4⋊C4C2311yescharacteristic
87C22×C4⋊C4C2311yescharacteristic
88C22×C4⋊C4C2312yes—
89C22×C4⋊C4C2311yescharacteristic
90C22×C4⋊C4C2312yes—
91C22×C4⋊C4C2314yes—
92C22×C4⋊C4C2312yes—
93C22×C4⋊C4C2312yes—
94C22×C4⋊C4C2314yes—
95C22×C4⋊C4C2312yes—
96C22×C4⋊C4C2311yescharacteristic
97C22×C4⋊C4C2314yes—
98C22×C4⋊C4C2314yes—
99C22×C4⋊C4C2311yescharacteristic
100C22×C22⋊C4C2314yes—
101C22×C22⋊C4C2314yes—
102C22×C22⋊C4C2314yes—
103C22×C22⋊C4C2314yes—
104C22×C22⋊C4C2312yes—
105C22×C22⋊C4C2314yes—
106C22×C42C2312yes—
107C22×C42C2311yescharacteristic
108C22×C42C2312yes—
109C22×C42C2311yescharacteristic
110C22×C42C2312yes—
111C22×C42C2312yes—
112C22×C42C2312yes—
113C22×C42C2311yescharacteristic
114#398—48no—
115#390—24no—
116#390—24no—
117#390—28no—
118#389—28no—
119#389—28no—
120#389—28no—
121#389—28no—
122#389—44no—
123#387—216no—
124#387—216no—
125#385—216no—
126#385—48no—
127#382—24no—
128#382D418yes—
129#380—28no—
130#380—216no—
131C4×C4⋊C4—24no—
132C4×C4⋊C4—24no—
133C4×C4⋊C4—24no—
134C4×C4⋊C4—44no—
135C4×C22⋊C4—48no—
136C2×#95—28no—
137C2×#95C2314yes—
138C2×#95C2312yes—
139C2×#95C2312yes—
140C2×#95C2312yes—
141C2×#95C2314yes—
142C2×#95C2314yes—
143C2×#95C2314yes—
144C2×#95C2311yescharacteristic
145C2×#95C2311yescharacteristic
146C2×#95C2312yes—
147C2×#95C2312yes—
148C2×#95C2314yes—
149C2×#95C2314yes—
150C2×#95C2314yes—
151C2×#95C2312yes—
152C2×#95C2312yes—
153C2×#95C2312yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2312yes—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2312yes—
163C2×#95C2312yes—
164C2×#95C2312yes—
165C43—24no—
166C25C2412yes—
167C22×D4—22no—
168C22×D4—22no—
169C22×D4—22no—
170C22×D4—22no—
171C22×D4C2×Q814yes—
172C22×D4#4114yes—
173C22×D4#4114yes—
174C23×C4—24no—
175C23×C4—24no—
176C23×C4—28no—
177C23×C4—28no—
178C23×C4—28no—
179C23×C4—44no—
180C23×C4—44no—
181C23×C4—44no—
182C23×C4—44no—
183C23×C4—44no—
184C23×C4—44no—
185C23×C4C2×Q818yes—
186C23×C4C2412yes—
187C23×C4C2414yes—
188C23×C4C2412yes—
189C23×C4C2412yes—
190C23×C4C2412yes—
191C23×C4C2414yes—
192C23×C4C2412yes—
193C23×C4C2412yes—
194C23×C4C2412yes—
195C23×C4C2412yes—
196C23×C4C2412yes—
197C23×C4C2411yescharacteristic
198C23×C4C2411yescharacteristic
199C23×C4C2411yescharacteristic
200C4×D4—232no—
201C4×D4—432no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—24no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—24no—
207C2×C4⋊C4—24no—
208C2×C4⋊C4—24no—
209C2×C4⋊C4—24no—
210C2×C4⋊C4—28no—
211C2×C4⋊C4—28no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—28no—
216C2×C4⋊C4—24no—
217C2×C4⋊C4—24no—
218C2×C4⋊C4—24no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—28no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—28no—
229C2×C4⋊C4—28no—
230C2×C4⋊C4—42no—
231C2×C4⋊C4—44no—
232C2×C4⋊C4—44no—
233C2×C4⋊C4—42no—
234C2×C4⋊C4—42no—
235C2×C4⋊C4—42no—
236C2×C4⋊C4—44no—
237C2×C4⋊C4—44no—
238C2×C4⋊C4—44no—
239C2×C4⋊C4—44no—
240C2×C4⋊C4C2×D418yes—
241C2×C4⋊C4C2×Q814yes—
242C2×C22⋊C4—24no—
243C2×C22⋊C4—24no—
244C2×C22⋊C4—28no—
245C2×C22⋊C4—28no—
246C2×C22⋊C4—28no—
247C2×C22⋊C4—28no—
248C2×C22⋊C4—28no—
249C2×C22⋊C4—28no—
250C2×C22⋊C4—28no—
251C2×C22⋊C4—28no—
252C2×C22⋊C4—28no—
253C2×C22⋊C4—44no—
254C2×C22⋊C4—44no—
255C2×C22⋊C4—44no—
256C2×C22⋊C4—44no—
257C2×C22⋊C4—44no—
258C2×C22⋊C4—44no—
259C2×C22⋊C4C2×Q818yes—
260C2×C42—24no—
261C2×C42—24no—
262C2×C42—24no—
263C2×C42—22no—
264C2×C42—22no—
265C2×C42—24no—
266C2×C42—24no—
267C2×C42—24no—
268C2×C42—22no—
269C2×C42—22no—
270C2×C42—24no—
271C2×C42—24no—
272C2×C42—24no—
273C2×C42—22no—
274C2×C42—24no—
275C2×C42—22no—
276C2×C42—24no—
277C2×C42—42no—
278C2×C42—42no—
279C2×C42—44no—
280C2×C42C2×D414yes—
281C2×C42C2×Q814yes—
282C2×C42#4114yes—
283#95—22no—
284#95—28no—
285#95—22no—
286#95—24no—
287#95—24no—
288#95—28no—
289#95—28no—
290#95—24no—
291#95—24no—
292#95—28no—
293#95—24no—
294#95—28no—
295#95—42no—
296#95—42no—
297#95—44no—
298#95—44no—
299#95—44no—
300#95—44no—
301#95—42no—
302#95—42no—
303#95—44no—
304#95—44no—
305#95—44no—
306#95—44no—
307#95—44no—
308#95C2×D418yes—
309#95C2×D418yes—
310#95#4118yes—
311C24—22no—
312C24—24no—
313C24—22no—
314C24—24no—
315C24—22no—
316C24—22no—
317C24—24no—
318C24—24no—
319C24C22×Q814yes—
320C24C2×#4114yes—
321C24C2×#4114yes—
322C24C2511yescharacteristic
323C2×D4—28no—
324C2×D4—28no—
325C2×D4—28no—
326C2×D4—28no—
327C2×D4—28no—
328C2×D4—28no—
329C2×D4#125116yes—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—24no—
347C22×C4—24no—
348C22×C4—24no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—24no—
356C22×C4—24no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—24no—
361C22×C4—24no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—24no—
367C22×C4—22no—
368C22×C4—216no—
369C22×C4—24no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—24no—
375C22×C4—24no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—24no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—24no—
383C22×C4—22no—
384C22×C4—24no—
385C22×C4—24no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—24no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—24no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—416no—
401C22×C4—416no—
402C22×C4—416no—
403C22×C4—416no—
404C22×C4—88no—
405C22×C4—84no—
406C22×C4—84no—
407C22×C4—88no—
408C22×C4—88no—
409C22×C4—88no—
410C22×C4—88no—
411C22×C4C22×D418yes—
412C22×C4C22×D412yes—
413C22×C4C22×D412yes—
414C22×C4C22×D412yes—
415C22×C4C22×Q814yes—
416C22×C4C22×Q812yes—
417C22×C4C22×Q812yes—
418C22×C4C22×Q812yes—
419C22×C4C2×#4114yes—
420C22×C4C2×#4114yes—
421C22×C4C2×#4114yes—
422C22×C4C2×#4112yes—
423C22×C4C2×#4112yes—
424C22×C4C2×#4112yes—
425C22×C4C2×#4112yes—
426C22×C4C2×#4112yes—
427C22×C4H(5,𝔽2)14yes—
428C22×C4H(5,𝔽2)12yes—
429C22×C42−1+412yes—
430C4⋊C4—28no—
431C4⋊C4—28no—
432C4⋊C4—216no—
433C4⋊C4—48no—
434C4⋊C4—416no—
435C4⋊C4—416no—
436C4⋊C4—48no—
437C4⋊C4—48no—
438C4⋊C4—44no—
439C4⋊C4—48no—
440C4⋊C4—44no—
441C4⋊C4—44no—
442C4⋊C4—88no—
443C4⋊C4—88no—
444C22⋊C4—216no—
445C22⋊C4—416no—
446C22⋊C4—416no—
447C22⋊C4—416no—
448C22⋊C4—416no—
449C22⋊C4—88no—
450C42—48no—
451C42—48no—
452C42—44no—
453C42—48no—
454C42—44no—
455C42—48no—
456C42—84no—
457C42C22⋊Q8116yes—
458C23—24no—
459C23—24no—
460C23—24no—
461C23—24no—
462C23—24no—
463C23—24no—
464C23—28no—
465C23—28no—
466C23—28no—
467C23—28no—
468C23—28no—
469C23—28no—
470C23—42no—
471C23—42no—
472C23—44no—
473C23—44no—
474C23—44no—
475C23—42no—
476C23—42no—
477C23—44no—
478C23—44no—
479C23—44no—
480C23C2×#12518yes—
481C23C23×D411yescharacteristic
482C23C23×Q811yescharacteristic
483C23C22×#4111yescharacteristic
484C23C22×#4111yescharacteristic
485C23C22×#4111yescharacteristic
486C23C22×#4112yes—
487C23C2×H(5,𝔽2)11yescharacteristic
488C23C2×H(5,𝔽2)12yes—
489C23C2×2−1+411yescharacteristic
490C23#58512yes—
491C23#58512yes—
492D4—232no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—24no—
496C2×C4—24no—
497C2×C4—24no—
498C2×C4—24no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—24no—
502C2×C4—24no—
503C2×C4—28no—
504C2×C4—24no—
505C2×C4—24no—
506C2×C4—28no—
507C2×C4—24no—
508C2×C4—24no—
509C2×C4—24no—
510C2×C4—24no—
511C2×C4—28no—
512C2×C4—28no—
513C2×C4—24no—
514C2×C4—24no—
515C2×C4—24no—
516C2×C4—24no—
517C2×C4—24no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—44no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—44no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—44no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—44no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—44no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—44no—
546C2×C4—42no—
547C2×C4—44no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—44no—
553C2×C4—44no—
554C2×C4—44no—
555C2×C4—42no—
556C2×C4—44no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—44no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—44no—
564C2×C4—42no—
565C2×C4—44no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—44no—
571C2×C4—44no—
572C2×C4—816no—
573C2×C4—832no—
574C2×C4C2×C22⋊Q814yes—
575C2×C4C2×C22⋊Q814yes—
576C2×C4C2×C22⋊Q814yes—
577C2×C4C2×#12514yes—
578C2×C4#54414yes—
579C2×C4#54614yes—
580C2×C4#54814yes—
581C2×C4#55214yes—
582C2×C4#55314yes—
583C22—28no—
584C22—216no—
585C22—44no—
586C22—48no—
587C22—44no—
588C22—44no—
589C22—44no—
590C22—48no—
591C22—44no—
592C22—44no—
593C22—48no—
594C22C22×C22⋊Q811yescharacteristic
595C22C22×#12511yescharacteristic
596C22#344411yescharacteristic
597C22#345511yescharacteristic
598C22C2×#54411yescharacteristic
599C22C2×#54611yescharacteristic
600C22C2×#54811yescharacteristic
601C22D4×#4112yes—
602C22C2×#55211yescharacteristic
603C22C2×#55311yescharacteristic
604C22#347512yes—
605C22Q8×#4112yes—
606C22#348112yes—
607C22#348812yes—
608C22#350112yes—
609C22#350712yes—
610C22#350811yescharacteristic
611C22#351011yescharacteristic
612C22#351312yes—
613C22#351912yes—
614C22#352112yes—
615C22#355112yes—
616C22#355412yes—
617C4—24no—
618C4—24no—
619C4—44no—
620C4—44no—
621C4—48no—
622C4—44no—
623C4—44no—
624C4—44no—
625C4—82no—
626C4—82no—
627C4—82no—
628C4—82no—
629C4—84no—
630C4#244218yes—
631C2—48no—
632C2C2×#244211yescharacteristic
633C2#3474511yescharacteristic
634C2#3551511yescharacteristic
635C2#3623912yes—
636C2#3639411yescharacteristic
637C2#3649811yescharacteristic
638C2#3896911yescharacteristic
639C2#3918711yescharacteristic
640C2#3997212yes—
641C2#5904412yes—
642C2#5908312yes—
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×#2864C2semidirect
3. #16044C2semidirect
4. #15441C2semidirect
5. #15187C2semidirect
6. #15149C2semidirect
7. #14775C2semidirect
8. #14150C2semidirect
9. #14028C2semidirect
11. #13955C2semidirect
14. C4×#389C2semidirect

Maximal subgroups

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

subgroupindex
C2×#28642
C2×#23492
#160442
#154412
#151872
#151492
#147752
#141502
#140282
#140182
#139552
#139412
#139202
C4×#3892
C4×#3902