Finite Group Databasebeta

← #4390285

Subgroups — #4390285

Order 512.

2738 conjugacy classes of subgroups.

The subgroup lattice has 635 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2844C211yesmaximal, characteristic, complemented
2C2×#2328C211yesmaximal, characteristic
3#16599C212yesmaximal, complemented
4#16556C212yesmaximal, complemented
5#15508C214yesmaximal, complemented
6#15476C214yesmaximal, complemented
7#15329C214yesmaximal, complemented
8#15325C214yesmaximal, complemented
9#14825C212yesmaximal, complemented
10#14723C212yesmaximal, complemented
11#14272C212yesmaximal
12#14168C211yesmaximal, characteristic, complemented
13#13963C212yesmaximal, complemented
14C22×#127C2211yescharacteristic
15C22×C22⋊Q8C2212yes—
16C22×C4×D4C2211yescharacteristic
17C22×C4×D4C2211yescharacteristic
18#2844—28no—
19C2×#400C2214yes—
20C2×#400C2212yes—
21C2×#399C2212yes—
22C2×#396C2214yes—
23C2×#395C2214yes—
24C2×#395C2212yes—
25C2×#393C2212yes—
26C2×#393C2214yes—
27#2328—28no—
28C2×#390C2211yescharacteristic
29C2×#390C2211yescharacteristic
30C2×#390C2214yes—
31C2×#390C2214yes—
32C2×#386C2214yes—
33C2×#386C2214yes—
34C2×#385C2214yes—
35C2×#382C2212yes—
36C2×#382C2211yescharacteristic
37C2×#380C2212yes—
38C2×#376C2211yescharacteristic
39C2×#376C2212yes—
40#1444C2212yes—
41#1444C2212yes—
42#1442C2212yes—
43#1442C2212yes—
44#1441C2212yes—
45#1441C2212yes—
46#1439C2214yes—
47#1439C2214yes—
48#1439C2214yes—
49#1439C2214yes—
50#1439C2212yes—
51#1439C2214yes—
52#1439C2214yes—
53#1439C2214yes—
54#1439C2214yes—
55#1439C2212yes—
56#1438C2214yes—
57#1438C2214yes—
58#1438C2214yes—
59#1438C2214yes—
60#1436C2214yes—
61#1436C2214yes—
62#1436C2214yes—
63#1436C2214yes—
64C4⋊#95C2214yes—
65#1434C2214yes—
66C22⋊#95C2214yes—
67C22⋊#95C2212yes—
68C22⋊#95C2212yes—
69C23×Q8C2312yes—
70C23×D4C2311yescharacteristic
71C23×D4C2311yescharacteristic
72C23×D4C2314yes—
73C24×C4C2312yes—
74C24×C4C2312yes—
75C2×#127—24no—
76C2×#127—24no—
77C2×#127Q818yescomplemented
78C2×C22⋊Q8—48no—
79C2×C22⋊Q8—44no—
80C2×C4×D4—24no—
81C2×C4×D4—24no—
82C2×C4×D4—42no—
83C2×C4×D4—42no—
84C2×C4×D4—42no—
85C2×C4×D4—42no—
86C22×C4⋊C4C2312yes—
87C22×C4⋊C4C2314yes—
88C22×C4⋊C4C2314yes—
89C22×C4⋊C4C2312yes—
90C22×C4⋊C4C2312yes—
91C22×C4⋊C4C2312yes—
92C22×C4⋊C4C2314yes—
93C22×C4⋊C4C2312yes—
94C22×C4⋊C4C2311yescharacteristic
95C22×C4⋊C4C2311yescharacteristic
96C22×C22⋊C4C2312yes—
97C22×C22⋊C4C2312yes—
98C22×C22⋊C4C2314yes—
99C22×C22⋊C4C2314yes—
100C22×C22⋊C4C2314yes—
101C22×C22⋊C4C2314yes—
102C22×C22⋊C4C2314yes—
103C22×C22⋊C4C2314yes—
104C22×C22⋊C4C2314yes—
105C22×C22⋊C4C2314yes—
106C22×C22⋊C4C2314yes—
107C22×C22⋊C4C2314yes—
108C22×C42C2311yescharacteristic
109C22×C42C2314yes—
110C22×C42C2311yescharacteristic
111C22×C42C2311yescharacteristic
112#400—28no—
113#400—216no—
114#399—44no—
115#396—48no—
116#395—28no—
117#395—216no—
118#393—44no—
119#393—48no—
120#390—24no—
121#390—24no—
122#390—48no—
123#390—48no—
124#386—216no—
125#386—216no—
126#385—48no—
127#382—28no—
128#382D418yes—
129#380—44no—
130#376—24no—
131#376—28no—
132C2×#95C2314yes—
133C2×#95C2314yes—
134C2×#95C2314yes—
135C2×#95C2314yes—
136C2×#95C2312yes—
137C2×#95C2314yes—
138C2×#95C2314yes—
139C2×#95C2312yes—
140C2×#95C2312yes—
141C2×#95C2312yes—
142C2×#95C2314yes—
143C2×#95C2312yes—
144C2×#95C2312yes—
145C2×#95C2312yes—
146C2×#95C2314yes—
147C2×#95C2312yes—
148C2×#95C2314yes—
149C2×#95C2314yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C25C2412yes—
154C25C2412yes—
155C22×Q8—24no—
156C22×Q8—44no—
157C22×Q8—42no—
158C22×Q8—44no—
159C22×Q8—42no—
160C22×D4—22no—
161C22×D4—22no—
162C22×D4—28no—
163C22×D4—22no—
164C22×D4—22no—
165C22×D4—28no—
166C22×D4—41no—
167C22×D4—41no—
168C22×D4—41no—
169C22×D4—41no—
170C22×D4—41no—
171C22×D4—41no—
172C22×D4—41no—
173C22×D4—41no—
174C22×D4—44no—
175C22×D4—44no—
176C22×D4—44no—
177C22×D4—44no—
178C22×D4C2×Q814yes—
179C22×D4C2×Q814yes—
180C22×D4C2×Q8116yes—
181C23×C4—24no—
182C23×C4—24no—
183C23×C4—24no—
184C23×C4—24no—
185C23×C4—42no—
186C23×C4—42no—
187C23×C4—44no—
188C23×C4—44no—
189C23×C4—44no—
190C23×C4—44no—
191C23×C4C2414yes—
192C23×C4C2414yes—
193C23×C4C2412yes—
194C23×C4C2414yes—
195C23×C4C2412yes—
196C23×C4C2412yes—
197C23×C4C2414yes—
198C23×C4C2411yescharacteristic
199C23×C4C2411yescharacteristic
200C23×C4C2412yes—
201C23×C4C2411yescharacteristic
202#127—232no—
203C22⋊Q8—432no—
204C4×D4—416no—
205C4×D4—416no—
206C2×C4⋊C4—28no—
207C2×C4⋊C4—28no—
208C2×C4⋊C4—28no—
209C2×C4⋊C4—24no—
210C2×C4⋊C4—24no—
211C2×C4⋊C4—28no—
212C2×C4⋊C4—24no—
213C2×C4⋊C4—41no—
214C2×C4⋊C4—41no—
215C2×C4⋊C4—41no—
216C2×C4⋊C4—41no—
217C2×C4⋊C4—42no—
218C2×C4⋊C4—44no—
219C2×C4⋊C4—44no—
220C2×C4⋊C4—44no—
221C2×C4⋊C4—44no—
222C2×C4⋊C4—42no—
223C2×C4⋊C4—42no—
224C2×C4⋊C4—44no—
225C2×C4⋊C4—44no—
226C2×C4⋊C4—44no—
227C2×C4⋊C4—44no—
228C2×C4⋊C4—44no—
229C2×C4⋊C4—42no—
230C2×C4⋊C4C2×D414yes—
231C2×C4⋊C4C2×D414yes—
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—28no—
239C2×C22⋊C4—28no—
240C2×C22⋊C4—28no—
241C2×C22⋊C4—28no—
242C2×C22⋊C4—44no—
243C2×C22⋊C4—44no—
244C2×C22⋊C4—42no—
245C2×C22⋊C4—42no—
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⋊C4—44no—
254C2×C22⋊C4—44no—
255C2×C22⋊C4—44no—
256C2×C22⋊C4—44no—
257C2×C22⋊C4—44no—
258C2×C22⋊C4—44no—
259C2×C22⋊C4—44no—
260C2×C22⋊C4—44no—
261C2×C22⋊C4—44no—
262C2×C22⋊C4—44no—
263C2×C22⋊C4—44no—
264C2×C22⋊C4—44no—
265C2×C22⋊C4—44no—
266C2×C22⋊C4—44no—
267C2×C22⋊C4—44no—
268C2×C42—22no—
269C2×C42—22no—
270C2×C42—22no—
271C2×C42—22no—
272C2×C42—28no—
273C2×C42—28no—
274C2×C42—44no—
275C2×C42—41no—
276C2×C42—41no—
277C2×C42C2×D414yes—
278C2×C42C2×Q814yes—
279C2×C42#4114yes—
280#95—24no—
281#95—24no—
282#95—28no—
283#95—28no—
284#95—24no—
285#95—28no—
286#95—24no—
287#95—24no—
288#95—28no—
289#95—24no—
290#95—28no—
291#95—28no—
292#95—28no—
293#95—44no—
294#95—44no—
295#95—44no—
296#95—44no—
297#95—42no—
298#95—44no—
299#95—42no—
300#95C2×D4116yes—
301C24—22no—
302C24—22no—
303C24—22no—
304C24—22no—
305C24—24no—
306C24—24no—
307C24—22no—
308C24—22no—
309C24—24no—
310C24—24no—
311C24—22no—
312C24—22no—
313C24—22no—
314C24—22no—
315C24—22no—
316C24—22no—
317C24—24no—
318C24—24no—
319C24—24no—
320C24—24no—
321C24C22×Q814yes—
322C24C22×Q814yes—
323C24C2511yescharacteristic
324C2×Q8—48no—
325C2×Q8—84no—
326C2×Q8—88no—
327C2×Q8—84no—
328C2×Q8—88no—
329C2×D4—28no—
330C2×D4—28no—
331C2×D4—232no—
332C2×D4—44no—
333C2×D4—44no—
334C2×D4—44no—
335C2×D4—44no—
336C2×D4—44no—
337C2×D4—44no—
338C2×D4—44no—
339C2×D4—44no—
340C2×D4—44no—
341C2×D4—44no—
342C2×D4—44no—
343C2×D4—44no—
344C2×D4—416no—
345C2×D4—416no—
346C2×D4—88no—
347C2×D4—88no—
348C2×D4—88no—
349C2×D4—88no—
350C22×C4—21no—
351C22×C4—21no—
352C22×C4—24no—
353C22×C4—21no—
354C22×C4—21no—
355C22×C4—24no—
356C22×C4—21no—
357C22×C4—21no—
358C22×C4—24no—
359C22×C4—24no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—24no—
363C22×C4—24no—
364C22×C4—21no—
365C22×C4—21no—
366C22×C4—24no—
367C22×C4—24no—
368C22×C4—22no—
369C22×C4—22no—
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—22no—
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—22no—
393C22×C4—24no—
394C22×C4—24no—
395C22×C4—24no—
396C22×C4—24no—
397C22×C4—22no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—48no—
403C22×C4—48no—
404C22×C4—48no—
405C22×C4—48no—
406C22×C4—88no—
407C22×C4—88no—
408C22×C4—84no—
409C22×C4—84no—
410C22×C4—88no—
411C22×C4—88no—
412C22×C4C22×D414yes—
413C22×C4C22×D414yes—
414C22×C4C22×D414yes—
415C22×C4C22×D412yes—
416C22×C4C22×Q812yes—
417C22×C4C22×Q814yes—
418C22×C4C2×#4112yes—
419C22×C4C2×#4114yes—
420C22×C4H(5,𝔽2)12yes—
421C22×C4H(5,𝔽2)18yes—
422C22×C4H(5,𝔽2)12yes—
423C4⋊C4—44no—
424C4⋊C4—44no—
425C4⋊C4—48no—
426C4⋊C4—416no—
427C4⋊C4—84no—
428C4⋊C4—84no—
429C4⋊C4—84no—
430C4⋊C4—88no—
431C4⋊C4—88no—
432C4⋊C4—84no—
433C22⋊C4—48no—
434C22⋊C4—48no—
435C22⋊C4—416no—
436C22⋊C4—416no—
437C22⋊C4—88no—
438C22⋊C4—88no—
439C22⋊C4—88no—
440C22⋊C4—88no—
441C22⋊C4—88no—
442C22⋊C4—88no—
443C22⋊C4—88no—
444C22⋊C4—88no—
445C42—28no—
446C42—416no—
447C42—44no—
448C42C22⋊Q8116yescomplemented
449C23—24no—
450C23—24no—
451C23—24no—
452C23—24no—
453C23—24no—
454C23—24no—
455C23—28no—
456C23—28no—
457C23—28no—
458C23—28no—
459C23—42no—
460C23—42no—
461C23—42no—
462C23—42no—
463C23—42no—
464C23—42no—
465C23—42no—
466C23—42no—
467C23—44no—
468C23—44no—
469C23—44no—
470C23—44no—
471C23—44no—
472C23—44no—
473C23—44no—
474C23—44no—
475C23—42no—
476C23—44no—
477C23—42no—
478C23—44no—
479C23—44no—
480C23—42no—
481C23—42no—
482C23—44no—
483C23—44no—
484C23—44no—
485C23—42no—
486C23—44no—
487C23—42no—
488C23—44no—
489C23—44no—
490C23—44no—
491C23—42no—
492C23—42no—
493C23—44no—
494C23—44no—
495C23C23×D411yescharacteristic
496C23C23×Q811yescharacteristic
497C23C22×#4111yescharacteristic
498C23C2×H(5,𝔽2)11yescharacteristic
499C23C2×H(5,𝔽2)11yescharacteristic
500C23C2×H(5,𝔽2)14yes—
501C23#58511yescharacteristic
502C23#58511yescharacteristic
503C23#58514yes—
504Q8—816no—
505D4—416no—
506D4—416no—
507D4—832no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—28no—
513C2×C4—24no—
514C2×C4—24no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—28no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—24no—
521C2×C4—24no—
522C2×C4—28no—
523C2×C4—28no—
524C2×C4—28no—
525C2×C4—28no—
526C2×C4—24no—
527C2×C4—28no—
528C2×C4—24no—
529C2×C4—44no—
530C2×C4—44no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—44no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—44no—
540C2×C4—44no—
541C2×C4—44no—
542C2×C4—42no—
543C2×C4—44no—
544C2×C4—44no—
545C2×C4—42no—
546C2×C4—44no—
547C2×C4—44no—
548C2×C4—44no—
549C2×C4—42no—
550C2×C4—44no—
551C2×C4—44no—
552C2×C4—44no—
553C2×C4—42no—
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×C4—44no—
569C2×C4—816no—
570C2×C4—816no—
571C2×C4C2×C22⋊Q814yes—
572C2×C4C2×C22⋊Q818yes—
573C22—44no—
574C22—44no—
575C22—44no—
576C22—48no—
577C22—44no—
578C22—48no—
579C22—48no—
580C22—48no—
581C22—44no—
582C22—44no—
583C22—82no—
584C22—82no—
585C22—82no—
586C22—84no—
587C22—82no—
588C22—82no—
589C22—82no—
590C22—84no—
591C22—84no—
592C22—82no—
593C22—82no—
594C22—84no—
595C22C22×C22⋊Q811yescharacteristic
596C22#346011yescharacteristic
597C22#346011yescharacteristic
598C22#348411yescharacteristic
599C22#348711yescharacteristic
600C22#349414yes—
601C22#349812yes—
602C22#350012yes—
603C22#350111yescharacteristic
604C22#350111yescharacteristic
605C22#350114yes—
606C22#353212yes—
607C22#353312yes—
608C22#353314yes—
609C22#354912yes—
610C22#355112yes—
611C22#355114yes—
612C4—24no—
613C4—28no—
614C4—44no—
615C4—42no—
616C4—48no—
617C4—42no—
618C4—44no—
619C4—84no—
620C4—82no—
621C4—84no—
622C4—84no—
623C2—84no—
624C2—84no—
625C2#3517411yescharacteristic
626C2#3552912yes—
627C2#4205012yes—
628C2#4222212yes—
629C2#4781412yes—
630C2#4781812yes—
631C2#5190211yescharacteristic
632C2#5190412yes—
633C2#5197811yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC22×#127C22
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×#2844C2semidirect
3. #16599C2semidirect
4. #16556C2semidirect
5. #15508C2semidirect
6. #15476C2semidirect
7. #15329C2semidirect
8. #15325C2semidirect
9. #14825C2semidirect
10. #14723C2semidirect
12. #14168C2semidirect
13. #13963C2semidirect
77. C2×#127Q8semidirect
448. C42C22⋊Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#28442
C2×#23282
#165992
#165562
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