Finite Group Databasebeta

← #763993

Subgroups — #763993

Order 512.

2418 conjugacy classes of subgroups.

The subgroup lattice has 639 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2547C211yesmaximal, characteristic
2C2×#2327C211yesmaximal, characteristic, complemented
3#16076C214yesmaximal, complemented
4#16071C214yesmaximal, complemented
5#15708C214yesmaximal, complemented
6#15704C214yesmaximal, complemented
7#15217C212yesmaximal
8#15192C212yesmaximal, complemented
9#14155C211yesmaximal, characteristic, complemented
10#14026C212yesmaximal
11#13955C211yesmaximal, characteristic, complemented
12#13941C211yesmaximal, characteristic
13#13931C212yesmaximal, complemented
14#13848C211yesmaximal, characteristic, complemented
15#13785C211yesmaximal, characteristic
16C22×C4⋊Q8C2211yescharacteristic
17C22×#124C2212yes—
18C22×C4×Q8C2212yes—
19C22×C4×D4C2211yescharacteristic
20#2547—28no—
21C2×#397C2212yes—
22C2×#397C2212yes—
23#2327—28no—
24C2×#391C2212yes—
25C2×#391C2212yes—
26C2×#391C2212yes—
27C2×#391C2212yes—
28C2×#390C2211yescharacteristic
29C2×#390C2211yescharacteristic
30C2×#390C2212yes—
31C2×#387C2214yes—
32C2×#387C2214yes—
33C2×#382C2211yescharacteristic
34C2×#380C2214yes—
35C2×C4×C4⋊C4C2211yescharacteristic
36C2×C4×C4⋊C4C2211yescharacteristic
37C2×C4×C4⋊C4C2211yescharacteristic
38C2×C4×C4⋊C4C2212yes—
39C2×#376C2211yescharacteristic
40#1445C2212yes—
41#1445C2212yes—
42#1445C2214yes—
43#1445C2214yes—
44#1445C2214yes—
45#1444C2212yes—
46#1443C2214yes—
47#1443C2214yes—
48#1442C2212yes—
49#1442C2214yes—
50#1442C2214yes—
51#1441C2212yes—
52#1441C2212yes—
53#1440C2214yes—
54#1440C2214yes—
55#1439C2214yes—
56#1439C2214yes—
57#1439C2214yes—
58#1439C2214yes—
59#1439C2214yes—
60#1437C2214yes—
61#1437C2214yes—
62#1437C2214yes—
63C4⋊#95C2214yes—
64C4⋊#95C2211yescharacteristic
65C4⋊#95C2212yes—
66C4⋊#95C2212yes—
67C4⋊#95C2211yescharacteristic
68C4⋊#95C2212yes—
69#1434C2212yes—
70#1434C2214yes—
71#1434C2212yes—
72#1434C2214yes—
73C22⋊#95C2212yes—
74C22⋊#95C2212yes—
75C22⋊#95C2214yes—
76C4×#95C2212yes—
77C23×Q8C2312yes—
78C23×D4C2311yescharacteristic
79C24×C4C2312yes—
80C2×C4⋊Q8—24no—
81C2×C4⋊Q8—24no—
82C2×C4⋊Q8D418yes—
83C2×#124—28no—
84C2×#124—28no—
85C2×#124—44no—
86C2×C4×Q8—28no—
87C2×C4×Q8—28no—
88C2×C4×Q8—44no—
89C2×C4×D4—24no—
90C2×C4×D4—24no—
91C2×C4×D4—24no—
92C22×C4⋊C4C2311yescharacteristic
93C22×C4⋊C4C2312yes—
94C22×C4⋊C4C2311yescharacteristic
95C22×C4⋊C4C2311yescharacteristic
96C22×C4⋊C4C2314yes—
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2312yes—
103C22×C22⋊C4C2312yes—
104C22×C22⋊C4C2314yes—
105C22×C22⋊C4C2314yes—
106C22×C22⋊C4C2314yes—
107C22×C22⋊C4C2314yes—
108C22×C22⋊C4C2314yes—
109C22×C22⋊C4C2314yes—
110C22×C42C2312yes—
111C22×C42C2312yes—
112C22×C42C2311yescharacteristic
113C22×C42C2314yes—
114C22×C42C2311yescharacteristic
115C22×C42C2312yes—
116C22×C42C2311yescharacteristic
117#397—28no—
118#397—28no—
119#391—28no—
120#391—28no—
121#391—44no—
122#391—44no—
123#390—24no—
124#390—24no—
125#390—44no—
126#387—216no—
127#387—216no—
128#382—24no—
129#380—216no—
130C4×C4⋊C4—24no—
131C4×C4⋊C4—24no—
132C4×C4⋊C4—24no—
133C4×C4⋊C4—44no—
134#376—24no—
135C2×#95C2311yescharacteristic
136C2×#95C2311yescharacteristic
137C2×#95C2314yes—
138C2×#95C2314yes—
139C2×#95C2312yes—
140C2×#95C2311yescharacteristic
141C2×#95C2312yes—
142C2×#95C2312yes—
143C2×#95C2312yes—
144C2×#95C2311yescharacteristic
145C2×#95C2312yes—
146C2×#95C2312yes—
147C2×#95C2314yes—
148C2×#95C2314yes—
149C2×#95C2312yes—
150C2×#95C2312yes—
151C2×#95C2312yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2312yes—
165C2×#95C2312yes—
166C25C2412yes—
167C22×Q8—24no—
168C22×Q8—24no—
169C22×Q8—24no—
170C22×Q8—24no—
171C22×Q8—24no—
172C22×Q8—24no—
173C22×Q8C2×D418yes—
174C22×D4—22no—
175C22×D4—22no—
176C22×D4—22no—
177C22×D4—22no—
178C22×D4—22no—
179C22×D4—22no—
180C22×D4#4114yes—
181C23×C4—24no—
182C23×C4—24no—
183C23×C4—24no—
184C23×C4—44no—
185C23×C4—44no—
186C23×C4C2412yes—
187C23×C4C2414yes—
188C23×C4C2412yes—
189C23×C4C2412yes—
190C23×C4C2412yes—
191C23×C4C2412yes—
192C23×C4C2412yes—
193C23×C4C2412yes—
194C23×C4C2414yes—
195C23×C4C2412yes—
196C23×C4C2412yes—
197C23×C4C2411yescharacteristic
198C23×C4C2411yescharacteristic
199C23×C4C2411yescharacteristic
200C4⋊Q8—232no—
201#124—432no—
202C4×Q8—432no—
203C4×D4—416no—
204C2×C4⋊C4—24no—
205C2×C4⋊C4—24no—
206C2×C4⋊C4—24no—
207C2×C4⋊C4—24no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—24no—
210C2×C4⋊C4—28no—
211C2×C4⋊C4—24no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—28no—
215C2×C4⋊C4—24no—
216C2×C4⋊C4—24no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—24no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—42no—
227C2×C4⋊C4—42no—
228C2×C4⋊C4—42no—
229C2×C4⋊C4—42no—
230C2×C4⋊C4—42no—
231C2×C4⋊C4—42no—
232C2×C4⋊C4—44no—
233C2×C4⋊C4—42no—
234C2×C4⋊C4—42no—
235C2×C4⋊C4C2×D418yes—
236C2×C4⋊C4C2×D418yes—
237C2×C22⋊C4—28no—
238C2×C22⋊C4—28no—
239C2×C22⋊C4—24no—
240C2×C22⋊C4—28no—
241C2×C22⋊C4—28no—
242C2×C22⋊C4—24no—
243C2×C22⋊C4—24no—
244C2×C22⋊C4—28no—
245C2×C22⋊C4—28no—
246C2×C22⋊C4—28no—
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⋊C4—44no—
254C2×C22⋊C4—44no—
255C2×C22⋊C4—44no—
256C2×C22⋊C4—44no—
257C2×C22⋊C4—44no—
258C2×C42—22no—
259C2×C42—22no—
260C2×C42—28no—
261C2×C42—24no—
262C2×C42—22no—
263C2×C42—28no—
264C2×C42—24no—
265C2×C42—24no—
266C2×C42—22no—
267C2×C42—24no—
268C2×C42—22no—
269C2×C42—24no—
270C2×C42—24no—
271C2×C42—24no—
272C2×C42—44no—
273C2×C42—44no—
274C2×C42C2×D414yes—
275C2×C42#4114yes—
276C2×C42#4114yes—
277C2×C42#4114yes—
278#95—24no—
279#95—24no—
280#95—22no—
281#95—22no—
282#95—28no—
283#95—28no—
284#95—24no—
285#95—24no—
286#95—28no—
287#95—28no—
288#95—22no—
289#95—22no—
290#95—24no—
291#95—24no—
292#95—44no—
293#95—42no—
294#95—44no—
295#95—42no—
296#95—42no—
297#95—44no—
298#95—44no—
299#95—44no—
300#95—44no—
301#95—44no—
302#95—44no—
303#95—42no—
304#95—42no—
305#95—44no—
306#95—44no—
307#95—44no—
308#95—44no—
309C24—22no—
310C24—22no—
311C24—22no—
312C24—22no—
313C24—22no—
314C24—22no—
315C24—24no—
316C24—24no—
317C24—24no—
318C24—24no—
319C24C2×#4114yes—
320C24C2511yescharacteristic
321C2×Q8—216no—
322C2×Q8—216no—
323C2×Q8—216no—
324C2×Q8—48no—
325C2×Q8—48no—
326C2×Q8—48no—
327C2×Q8—48no—
328C2×D4—28no—
329C2×D4—28no—
330C2×D4—28no—
331C2×D4—44no—
332C2×D4—44no—
333C2×D4—44no—
334C2×D4—44no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—24no—
342C22×C4—24no—
343C22×C4—24no—
344C22×C4—24no—
345C22×C4—22no—
346C22×C4—22no—
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—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—24no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—24no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—24no—
381C22×C4—22no—
382C22×C4—24no—
383C22×C4—22no—
384C22×C4—22no—
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—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—24no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—416no—
408C22×C4—48no—
409C22×C4—88no—
410C22×C4—84no—
411C22×C4—84no—
412C22×C4C22×D418yes—
413C22×C4C22×D412yes—
414C22×C4C22×D412yes—
415C22×C4C22×D412yes—
416C22×C4C2×#4114yes—
417C22×C4C2×#4112yes—
418C22×C4C2×#4112yes—
419C22×C4C2×#4112yes—
420C22×C4C2×#4112yes—
421C22×C4C2×#4112yes—
422C22×C4C2×#4112yes—
423C22×C4C2×#4112yes—
424C22×C4C2×#4112yes—
425C22×C4H(5,𝔽2)12yes—
426C22×C4H(5,𝔽2)12yes—
427C22×C42−1+414yes—
428C4⋊C4—216no—
429C4⋊C4—216no—
430C4⋊C4—48no—
431C4⋊C4—416no—
432C4⋊C4—48no—
433C4⋊C4—48no—
434C4⋊C4—48no—
435C4⋊C4—44no—
436C4⋊C4—48no—
437C4⋊C4—44no—
438C4⋊C4—44no—
439C22⋊C4—416no—
440C22⋊C4—416no—
441C22⋊C4—48no—
442C22⋊C4—88no—
443C22⋊C4—88no—
444C22⋊C4—88no—
445C22⋊C4—88no—
446C42—28no—
447C42—416no—
448C42—48no—
449C42—44no—
450C42—48no—
451C42—84no—
452C42#123116yes—
453C23—24no—
454C23—24no—
455C23—24no—
456C23—28no—
457C23—28no—
458C23—42no—
459C23—42no—
460C23—42no—
461C23—42no—
462C23—44no—
463C23—44no—
464C23—44no—
465C23—44no—
466C23—42no—
467C23—44no—
468C23—44no—
469C23—42no—
470C23—42no—
471C23—42no—
472C23—44no—
473C23—44no—
474C23—44no—
475C23—44no—
476C23C23×D411yescharacteristic
477C23C22×#4111yescharacteristic
478C23C22×#4111yescharacteristic
479C23C22×#4111yescharacteristic
480C23C22×#4111yescharacteristic
481C23C2×H(5,𝔽2)11yescharacteristic
482C23C2×H(5,𝔽2)11yescharacteristic
483C23C2×2−1+412yes—
484C23#58512yes—
485C23#58512yes—
486C23#58512yes—
487Q8—432no—
488D4—416no—
489C2×C4—24no—
490C2×C4—24no—
491C2×C4—24no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—24no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—28no—
498C2×C4—24no—
499C2×C4—28no—
500C2×C4—28no—
501C2×C4—24no—
502C2×C4—24no—
503C2×C4—24no—
504C2×C4—24no—
505C2×C4—24no—
506C2×C4—44no—
507C2×C4—42no—
508C2×C4—44no—
509C2×C4—44no—
510C2×C4—44no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—44no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—44no—
529C2×C4—42no—
530C2×C4—44no—
531C2×C4—42no—
532C2×C4—44no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—44no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—44no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—44no—
544C2×C4—44no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—44no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—44no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—44no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—44no—
567C2×C4—44no—
568C2×C4—816no—
569C2×C4C2×#12314yes—
570C2×C4C2×#12314yes—
571C2×C4C2×#12314yes—
572C2×C4#52914yes—
573C2×C4#54014yes—
574C2×C4#54614yes—
575C2×C4#54614yes—
576C2×C4#55314yes—
577C2×C4#55514yes—
578C22—48no—
579C22—44no—
580C22—44no—
581C22—44no—
582C22—48no—
583C22—84no—
584C22—82no—
585C22—82no—
586C22—82no—
587C22—82no—
588C22—84no—
589C22C22×#12311yescharacteristic
590C22C2×#52911yescharacteristic
591C22C2×#54011yescharacteristic
592C22C2×#54611yescharacteristic
593C22C2×#54611yescharacteristic
594C22C2×#55311yescharacteristic
595C22C2×#55511yescharacteristic
596C22#348712yes—
597C22#349512yes—
598C22#349712yes—
599C22#350712yes—
600C22#350911yescharacteristic
601C22#351012yes—
602C22#351012yes—
603C22#351011yescharacteristic
604C22#351011yescharacteristic
605C22#351011yescharacteristic
606C22#351512yes—
607C22#352112yes—
608C22#353912yes—
609C22#354112yes—
610C22#356112yes—
611C22#356212yes—
612C4—24no—
613C4—24no—
614C4—44no—
615C4—48no—
616C4—44no—
617C4—82no—
618C4—82no—
619C4—82no—
620C4—82no—
621C4—84no—
622C4—82no—
623C4—82no—
624C4—82no—
625C4#242018yes—
626C2—84no—
627C2C2×#242011yescharacteristic
628C2#3563212yes—
629C2#3567311yescharacteristic
630C2#3567911yescharacteristic
631C2#3740912yes—
632C2#4515111yescharacteristic
633C2#4520011yescharacteristic
634C2#4522111yescharacteristic
635C2#4524012yes—
636C2#4524312yes—
637C2#4542811yescharacteristic
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
2. C2×#2327C2semidirect
3. #16076C2semidirect
4. #16071C2semidirect
5. #15708C2semidirect
6. #15704C2semidirect
8. #15192C2semidirect
9. #14155C2semidirect
11. #13955C2semidirect
13. #13931C2semidirect
14. #13848C2semidirect

Maximal subgroups

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

subgroupindex
C2×#25472
C2×#23272
#160762
#160712
#157082
#157042
#152172
#151922
#141552
#140262
#139552
#139412
#139312
#138482
#137852