Finite Group Databasebeta

← #900732

Subgroups — #900732

Order 512.

2546 conjugacy classes of subgroups.

The subgroup lattice has 619 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2811C211yesmaximal, characteristic, complemented
2C2×C4×#124C211yesmaximal, characteristic
3#17077C212yesmaximal
4#16686C212yesmaximal, complemented
5#15851C214yesmaximal, complemented
6#15772C214yesmaximal, complemented
7#14763C214yesmaximal, complemented
8#14723C212yesmaximal
9#14703C212yesmaximal, complemented
10#14492C212yesmaximal
11#14467C211yesmaximal, characteristic, complemented
12#14220C214yesmaximal, complemented
13#13978C212yesmaximal, complemented
14C22×#124C2211yescharacteristic
15C22×C22⋊Q8C2212yes—
16C22×C4×Q8C2211yescharacteristic
17C22×C4×D4C2211yescharacteristic
18#2811—28no—
19C2×#395C2212yes—
20C2×#395C2214yes—
21C2×#393C2214yes—
22C2×#393C2212yes—
23C4×#124—28no—
24C2×#391C2212yes—
25C2×#391C2214yes—
26C2×#391C2212yes—
27C2×#391C2212yes—
28C2×#391C2212yes—
29C2×#391C2212yes—
30C2×#391C2211yescharacteristic
31C2×#390C2211yescharacteristic
32C2×#389C2212yes—
33C2×#389C2212yes—
34C2×#388C2214yes—
35C2×#385C2214yes—
36C2×#382C2211yescharacteristic
37C2×#382C2212yes—
38C2×C4×C22⋊C4C2214yes—
39C2×C43C2211yescharacteristic
40#1446C2214yes—
41#1444C2214yes—
42#1443C2214yes—
43#1442C2214yes—
44#1442C2214yes—
45#1442C2214yes—
46#1441C2212yes—
47#1441C2214yes—
48#1439C2214yes—
49#1439C2212yes—
50#1439C2214yes—
51#1439C2214yes—
52#1439C2214yes—
53#1439C2214yes—
54#1439C2214yes—
55#1438C2214yes—
56#1437C2214yes—
57#1437C2214yes—
58#1436C2214yes—
59#1436C2214yes—
60#1436C2214yes—
61#1436C2214yes—
62C4⋊#95C2212yes—
63C4⋊#95C2212yes—
64C4⋊#95C2212yes—
65#1434C2214yes—
66#1434C2214yes—
67C22⋊#95C2212yes—
68C4×#95C2212yes—
69C23×Q8C2312yes—
70C23×Q8C2312yes—
71C23×Q8C2311yescharacteristic
72C23×D4C2311yescharacteristic
73C24×C4C2312yes—
74C2×#124—24no—
75C2×#124—24no—
76C2×#124Q818yescomplemented
77C2×C22⋊Q8—48no—
78C2×C22⋊Q8—44no—
79C2×C4×Q8—24no—
80C2×C4×Q8—42no—
81C2×C4×Q8—42no—
82C2×C4×D4—24no—
83C2×C4×D4—42no—
84C2×C4×D4—42no—
85C22×C4⋊C4C2314yes—
86C22×C4⋊C4C2312yes—
87C22×C4⋊C4C2312yes—
88C22×C4⋊C4C2312yes—
89C22×C4⋊C4C2314yes—
90C22×C4⋊C4C2311yescharacteristic
91C22×C4⋊C4C2312yes—
92C22×C4⋊C4C2312yes—
93C22×C4⋊C4C2312yes—
94C22×C4⋊C4C2312yes—
95C22×C4⋊C4C2311yescharacteristic
96C22×C22⋊C4C2314yes—
97C22×C22⋊C4C2312yes—
98C22×C22⋊C4C2314yes—
99C22×C22⋊C4C2314yes—
100C22×C22⋊C4C2314yes—
101C22×C22⋊C4C2314yes—
102C22×C22⋊C4C2314yes—
103C22×C42C2311yescharacteristic
104C22×C42C2312yes—
105C22×C42C2312yes—
106C22×C42C2312yes—
107C22×C42C2311yescharacteristic
108C22×C42C2312yes—
109C22×C42C2312yes—
110C22×C42C2311yescharacteristic
111#395—28no—
112#395—216no—
113#393—216no—
114#393—44no—
115#391—28no—
116#391—28no—
117#391—28no—
118#391—28no—
119#391—216no—
120#391—44no—
121#391D418yes—
122#390—24no—
123#389—44no—
124#389—44no—
125#388—216no—
126#385—48no—
127#382—24no—
128#382—44no—
129C4×C22⋊C4—216no—
130C2×#95C2314yes—
131C2×#95C2314yes—
132C2×#95C2314yes—
133C2×#95C2314yes—
134C2×#95C2312yes—
135C2×#95C2314yes—
136C2×#95C2314yes—
137C2×#95C2314yes—
138C2×#95C2314yes—
139C2×#95C2314yes—
140C2×#95C2314yes—
141C2×#95C2314yes—
142C2×#95C2314yes—
143C2×#95C2314yes—
144C2×#95C2312yes—
145C2×#95C2314yes—
146C2×#95C2314yes—
147C2×#95C2314yes—
148C2×#95C2314yes—
149C2×#95C2314yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C43—24no—
153C25C2412yes—
154C22×Q8—24no—
155C22×Q8—24no—
156C22×Q8—24no—
157C22×Q8—41no—
158C22×Q8—41no—
159C22×Q8—41no—
160C22×Q8—41no—
161C22×Q8—44no—
162C22×Q8—42no—
163C22×Q8—42no—
164C22×Q8—42no—
165C22×Q8—42no—
166C22×Q8—42no—
167C22×Q8—44no—
168C22×Q8—42no—
169C22×Q8C2×D414yes—
170C22×Q8C2×Q814yes—
171C22×Q8#4118yes—
172C22×Q8#4114yes—
173C22×D4—22no—
174C22×D4—22no—
175C22×D4—41no—
176C22×D4—41no—
177C22×D4—41no—
178C22×D4—41no—
179C22×D4C2×Q814yes—
180C23×C4—24no—
181C23×C4—24no—
182C23×C4—28no—
183C23×C4—42no—
184C23×C4—44no—
185C23×C4C2414yes—
186C23×C4C2414yes—
187C23×C4C2412yes—
188C23×C4C2414yes—
189C23×C4C2412yes—
190C23×C4C2412yes—
191C23×C4C2414yes—
192C23×C4C2411yescharacteristic
193C23×C4C2412yes—
194C23×C4C2411yescharacteristic
195C23×C4C2412yes—
196C23×C4C2411yescharacteristic
197#124—232no—
198C22⋊Q8—432no—
199C4×Q8—416no—
200C4×D4—416no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—24no—
203C2×C4⋊C4—24no—
204C2×C4⋊C4—24no—
205C2×C4⋊C4—24no—
206C2×C4⋊C4—28no—
207C2×C4⋊C4—28no—
208C2×C4⋊C4—28no—
209C2×C4⋊C4—28no—
210C2×C4⋊C4—24no—
211C2×C4⋊C4—24no—
212C2×C4⋊C4—24no—
213C2×C4⋊C4—42no—
214C2×C4⋊C4—42no—
215C2×C4⋊C4—41no—
216C2×C4⋊C4—41no—
217C2×C4⋊C4—42no—
218C2×C4⋊C4—41no—
219C2×C4⋊C4—41no—
220C2×C4⋊C4—44no—
221C2×C4⋊C4—42no—
222C2×C4⋊C4—42no—
223C2×C4⋊C4—42no—
224C2×C4⋊C4—42no—
225C2×C4⋊C4—42no—
226C2×C4⋊C4—44no—
227C2×C4⋊C4—42no—
228C2×C4⋊C4—44no—
229C2×C4⋊C4—42no—
230C2×C4⋊C4—42no—
231C2×C4⋊C4—42no—
232C2×C4⋊C4C2×D414yes—
233C2×C22⋊C4—28no—
234C2×C22⋊C4—28no—
235C2×C22⋊C4—24no—
236C2×C22⋊C4—28no—
237C2×C22⋊C4—28no—
238C2×C22⋊C4—28no—
239C2×C22⋊C4—24no—
240C2×C22⋊C4—28no—
241C2×C22⋊C4—28no—
242C2×C22⋊C4—44no—
243C2×C22⋊C4—42no—
244C2×C22⋊C4—44no—
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×Q8116yes—
254C2×C42—22no—
255C2×C42—22no—
256C2×C42—22no—
257C2×C42—22no—
258C2×C42—22no—
259C2×C42—24no—
260C2×C42—24no—
261C2×C42—24no—
262C2×C42—24no—
263C2×C42—24no—
264C2×C42—24no—
265C2×C42—24no—
266C2×C42—24no—
267C2×C42—24no—
268C2×C42—42no—
269C2×C42—42no—
270C2×C42—41no—
271C2×C42—41no—
272C2×C42—42no—
273C2×C42—42no—
274C2×C42—42no—
275C2×C42—42no—
276C2×C42C2×D414yes—
277C2×C42C2×Q814yes—
278#95—28no—
279#95—24no—
280#95—28no—
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—42no—
299#95C2×D4116yes—
300C24—22no—
301C24—24no—
302C24—22no—
303C24—22no—
304C24—24no—
305C24—22no—
306C24—24no—
307C24—24no—
308C24—22no—
309C24—22no—
310C24C22×Q814yes—
311C24C2511yescharacteristic
312C2×Q8—216no—
313C2×Q8—44no—
314C2×Q8—44no—
315C2×Q8—44no—
316C2×Q8—44no—
317C2×Q8—44no—
318C2×Q8—44no—
319C2×Q8—48no—
320C2×Q8—48no—
321C2×Q8—48no—
322C2×Q8—84no—
323C2×Q8—84no—
324C2×Q8—84no—
325C2×Q8—84no—
326C2×Q8—84no—
327C2×Q8—88no—
328C2×Q8—84no—
329C2×Q8—88no—
330C2×Q8C22⋊Q8116yescomplemented
331C2×D4—28no—
332C2×D4—44no—
333C2×D4—44no—
334C2×D4—44no—
335C2×D4—44no—
336C2×D4—44no—
337C2×D4—44no—
338C22×C4—21no—
339C22×C4—21no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—21no—
343C22×C4—21no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—21no—
352C22×C4—21no—
353C22×C4—24no—
354C22×C4—22no—
355C22×C4—21no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—24no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—24no—
363C22×C4—21no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—24no—
368C22×C4—22no—
369C22×C4—24no—
370C22×C4—24no—
371C22×C4—24no—
372C22×C4—24no—
373C22×C4—24no—
374C22×C4—22no—
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—22no—
385C22×C4—22no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—24no—
392C22×C4—24no—
393C22×C4—24no—
394C22×C4—24no—
395C22×C4—24no—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—24no—
399C22×C4—48no—
400C22×C4—416no—
401C22×C4—48no—
402C22×C4—84no—
403C22×C4—88no—
404C22×C4C22×D414yes—
405C22×C4C22×D412yes—
406C22×C4C22×D414yes—
407C22×C4C22×D414yes—
408C22×C4C22×Q814yes—
409C22×C4C22×Q812yes—
410C22×C4C22×Q814yes—
411C22×C4C2×#4118yes—
412C22×C4C2×#4112yes—
413C22×C4C2×#4114yes—
414C22×C4C2×#4114yes—
415C22×C4H(5,𝔽2)12yes—
416C22×C4H(5,𝔽2)14yes—
417C22×C4H(5,𝔽2)12yes—
418C22×C42−1+414yes—
419C4⋊C4—44no—
420C4⋊C4—48no—
421C4⋊C4—48no—
422C4⋊C4—48no—
423C4⋊C4—44no—
424C4⋊C4—48no—
425C4⋊C4—416no—
426C4⋊C4—84no—
427C4⋊C4—84no—
428C4⋊C4—84no—
429C4⋊C4—88no—
430C22⋊C4—232no—
431C22⋊C4—48no—
432C22⋊C4—416no—
433C22⋊C4—416no—
434C22⋊C4—416no—
435C22⋊C4—88no—
436C22⋊C4—88no—
437C42—28no—
438C42—28no—
439C42—44no—
440C42—48no—
441C42—48no—
442C42—48no—
443C42—48no—
444C42—84no—
445C23—24no—
446C23—28no—
447C23—24no—
448C23—24no—
449C23—28no—
450C23—44no—
451C23—42no—
452C23—42no—
453C23—44no—
454C23—42no—
455C23—44no—
456C23—42no—
457C23—42no—
458C23—44no—
459C23—44no—
460C23—44no—
461C23—44no—
462C23—42no—
463C23—42no—
464C23—42no—
465C23—44no—
466C23—44no—
467C23—44no—
468C23C23×D411yescharacteristic
469C23C23×Q811yescharacteristic
470C23C22×#4111yescharacteristic
471C23C22×#4112yes—
472C23C2×H(5,𝔽2)11yescharacteristic
473C23C2×H(5,𝔽2)12yes—
474C23C2×H(5,𝔽2)11yescharacteristic
475C23C2×2−1+412yes—
476C23#58511yescharacteristic
477C23#58511yescharacteristic
478C23#58512yes—
479Q8—416no—
480Q8—816no—
481Q8—816no—
482D4—416no—
483C2×C4—24no—
484C2×C4—24no—
485C2×C4—24no—
486C2×C4—22no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—22no—
490C2×C4—24no—
491C2×C4—22no—
492C2×C4—24no—
493C2×C4—24no—
494C2×C4—24no—
495C2×C4—24no—
496C2×C4—24no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—24no—
500C2×C4—24no—
501C2×C4—22no—
502C2×C4—24no—
503C2×C4—24no—
504C2×C4—24no—
505C2×C4—24no—
506C2×C4—28no—
507C2×C4—28no—
508C2×C4—24no—
509C2×C4—22no—
510C2×C4—24no—
511C2×C4—28no—
512C2×C4—28no—
513C2×C4—24no—
514C2×C4—24no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—41no—
519C2×C4—44no—
520C2×C4—42no—
521C2×C4—44no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—41no—
526C2×C4—44no—
527C2×C4—44no—
528C2×C4—44no—
529C2×C4—44no—
530C2×C4—44no—
531C2×C4—44no—
532C2×C4—44no—
533C2×C4—44no—
534C2×C4—44no—
535C2×C4—44no—
536C2×C4—44no—
537C2×C4—44no—
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—42no—
551C2×C4—44no—
552C2×C4—44no—
553C2×C4—44no—
554C2×C4—44no—
555C2×C4—816no—
556C2×C4C2×C22⋊Q814yes—
557C2×C4C2×C22⋊Q818yes—
558C2×C4#54818yes—
559C2×C4H(3,𝔽4)14yes—
560C22—48no—
561C22—44no—
562C22—48no—
563C22—44no—
564C22—44no—
565C22—82no—
566C22—82no—
567C22—84no—
568C22—82no—
569C22—84no—
570C22—82no—
571C22C22×C22⋊Q811yescharacteristic
572C22#345912yes—
573C22#345911yescharacteristic
574C22#345911yescharacteristic
575C22C2×#54812yes—
576C22Q8×#4112yes—
577C22#348311yescharacteristic
578C22#348312yes—
579C22#348411yescharacteristic
580C22#348511yescharacteristic
581C22#348812yes—
582C22#349112yes—
583C22#350111yescharacteristic
584C22#350111yescharacteristic
585C22#350312yes—
586C22#351812yes—
587C22#352212yes—
588C22C2×H(3,𝔽4)11yescharacteristic
589C22#354511yescharacteristic
590C22#355011yescharacteristic
591C22#355812yes—
592C22#356112yes—
593C22#356412yes—
594C4—28no—
595C4—24no—
596C4—24no—
597C4—28no—
598C4—44no—
599C4—48no—
600C4—44no—
601C4—42no—
602C4—84no—
603C4—84no—
604C4—84no—
605C4—82no—
606C2—84no—
607C2#3498511yescharacteristic
608C2#3645612yes—
609C2#3720811yescharacteristic
610C2#3775412yes—
611C2#3883012yes—
612C2#3970212yes—
613C2#4597711yescharacteristic
614C2#4600411yescharacteristic
615C2#5230711yescharacteristic
616C2#5908411yescharacteristic
617C2#5992811yescharacteristic
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×#2811C2semidirect
4. #16686C2semidirect
5. #15851C2semidirect
6. #15772C2semidirect
7. #14763C2semidirect
9. #14703C2semidirect
11. #14467C2semidirect
12. #14220C2semidirect
13. #13978C2semidirect
76. C2×#124Q8semidirect
330. C2×Q8C22⋊Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#28112
C2×C4×#1242
#170772
#166862
#158512
#157722
#147632
#147232
#147032
#144922
#144672
#142202
#139782