Finite Group Databasebeta

← #4462771

Subgroups — #4462771

Order 512.

2362 conjugacy classes of subgroups.

The subgroup lattice has 634 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2642C211yesmaximal, characteristic, complemented
2C2×#2398C211yesmaximal, characteristic
3#17222C214yesmaximal, complemented
4#17158C214yesmaximal, complemented
5#17153C214yesmaximal, complemented
6#17148C214yesmaximal
7#15236C212yesmaximal, complemented
8#15216C212yesmaximal, complemented
9#15201C212yesmaximal, complemented
10#15182C212yesmaximal, complemented
11#15178C212yesmaximal, complemented
12#15177C212yesmaximal
13#14348C211yesmaximal, characteristic, complemented
14C22×C4⋊Q8C2211yescharacteristic
15C22×#125C2211yescharacteristic
16C22×C22⋊Q8C2212yes—
17C22×C22⋊Q8C2212yes—
18C22×C4×Q8C2211yescharacteristic
19C22×C4×D4C2211yescharacteristic
20#2642—28no—
21#2398—28no—
22C2×#400C2214yes—
23C2×#400C2214yes—
24C2×#400C2212yes—
25C2×#397C2212yes—
26C2×#394C2212yes—
27C2×#393C2212yes—
28C2×#391C2212yes—
29C2×#391C2212yes—
30C2×#391C2212yes—
31C2×#391C2212yes—
32C2×#384C2211yescharacteristic
33C2×#382C2211yescharacteristic
34C2×#380C2212yes—
35C2×#380C2212yes—
36C2×C4×C22⋊C4C2212yes—
37C2×C4×C22⋊C4C2212yes—
38C2×#376C2211yescharacteristic
39#1446C2214yes—
40#1444C2214yes—
41#1444C2214yes—
42#1444C2214yes—
43#1442C2214yes—
44#1442C2214yes—
45#1442C2214yes—
46#1442C2214yes—
47#1442C2214yes—
48#1442C2214yes—
49#1442C2214yes—
50#1442C2214yes—
51#1442C2214yes—
52#1441C2214yes—
53#1441C2214yes—
54#1440C2214yes—
55#1440C2214yes—
56#1439C2214yes—
57#1439C2214yes—
58#1439C2214yes—
59#1439C2214yes—
60#1439C2214yes—
61#1439C2214yes—
62#1439C2214yes—
63#1439C2214yes—
64#1438C2214yes—
65C4⋊#95C2212yes—
66C4⋊#95C2212yes—
67C4⋊#95C2212yes—
68C4⋊#95C2212yes—
69C23×Q8C2312yes—
70C23×Q8C2311yescharacteristic
71C23×D4C2311yescharacteristic
72C24×C4C2312yes—
73C2×C4⋊Q8—24no—
74C2×C4⋊Q8—42no—
75C2×C4⋊Q8—42no—
76C2×#125—24no—
77C2×#125—42no—
78C2×#125—42no—
79C2×C22⋊Q8—44no—
80C2×C22⋊Q8—44no—
81C2×C22⋊Q8—44no—
82C2×C22⋊Q8—44no—
83C2×C22⋊Q8—44no—
84C2×C22⋊Q8—44no—
85C2×C4×Q8—24no—
86C2×C4×Q8—42no—
87C2×C4×Q8—42no—
88C2×C4×D4—24no—
89C2×C4×D4—42no—
90C2×C4×D4—42no—
91C22×C4⋊C4C2312yes—
92C22×C4⋊C4C2312yes—
93C22×C4⋊C4C2312yes—
94C22×C4⋊C4C2312yes—
95C22×C4⋊C4C2311yescharacteristic
96C22×C4⋊C4C2311yescharacteristic
97C22×C4⋊C4C2314yes—
98C22×C4⋊C4C2314yes—
99C22×C4⋊C4C2314yes—
100C22×C4⋊C4C2314yes—
101C22×C4⋊C4C2314yes—
102C22×C22⋊C4C2312yes—
103C22×C22⋊C4C2312yes—
104C22×C22⋊C4C2312yes—
105C22×C22⋊C4C2312yes—
106C22×C22⋊C4C2312yes—
107C22×C22⋊C4C2314yes—
108C22×C22⋊C4C2314yes—
109C22×C22⋊C4C2314yes—
110C22×C22⋊C4C2314yes—
111C22×C42C2312yes—
112C22×C42C2311yescharacteristic
113C22×C42C2311yescharacteristic
114C22×C42C2311yescharacteristic
115#400—216no—
116#400—216no—
117#400—28no—
118#397—28no—
119#394—28no—
120#393—28no—
121#391—28no—
122#391—28no—
123#391—44no—
124#391—44no—
125#384—42no—
126#382—42no—
127#380—44no—
128#380—44no—
129C4×C22⋊C4—28no—
130C4×C22⋊C4—28no—
131#376D418yes—
132C2×#95C2312yes—
133C2×#95C2312yes—
134C2×#95C2312yes—
135C2×#95C2312yes—
136C2×#95C2314yes—
137C2×#95C2314yes—
138C2×#95C2314yes—
139C2×#95C2314yes—
140C2×#95C2314yes—
141C2×#95C2314yes—
142C2×#95C2314yes—
143C2×#95C2314yes—
144C2×#95C2314yes—
145C2×#95C2314yes—
146C2×#95C2314yes—
147C2×#95C2314yes—
148C2×#95C2314yes—
149C2×#95C2314yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2312yes—
156C2×#95C2312yes—
157C25C2412yes—
158C22×Q8—22no—
159C22×Q8—24no—
160C22×Q8—24no—
161C22×Q8—24no—
162C22×Q8—24no—
163C22×Q8—24no—
164C22×Q8—24no—
165C22×Q8—24no—
166C22×Q8—24no—
167C22×Q8—41no—
168C22×Q8—41no—
169C22×Q8—42no—
170C22×D4—24no—
171C22×D4—22no—
172C22×D4—41no—
173C22×D4—41no—
174C22×D4—42no—
175C23×C4—24no—
176C23×C4—24no—
177C23×C4—24no—
178C23×C4—24no—
179C23×C4—24no—
180C23×C4—24no—
181C23×C4—24no—
182C23×C4C2414yes—
183C23×C4C2414yes—
184C23×C4C2414yes—
185C23×C4C2414yes—
186C23×C4C2412yes—
187C23×C4C2412yes—
188C23×C4C2412yes—
189C23×C4C2412yes—
190C23×C4C2411yescharacteristic
191C23×C4C2411yescharacteristic
192C23×C4C2412yes—
193C23×C4C2411yescharacteristic
194C4⋊Q8—416no—
195#125—416no—
196C22⋊Q8—432no—
197C22⋊Q8—432no—
198C4×Q8—416no—
199C4×D4—416no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—24no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—24no—
204C2×C4⋊C4—28no—
205C2×C4⋊C4—28no—
206C2×C4⋊C4—24no—
207C2×C4⋊C4—24no—
208C2×C4⋊C4—24no—
209C2×C4⋊C4—24no—
210C2×C4⋊C4—24no—
211C2×C4⋊C4—24no—
212C2×C4⋊C4—24no—
213C2×C4⋊C4—24no—
214C2×C4⋊C4—41no—
215C2×C4⋊C4—41no—
216C2×C4⋊C4—42no—
217C2×C4⋊C4—42no—
218C2×C4⋊C4—41no—
219C2×C4⋊C4—41no—
220C2×C4⋊C4—44no—
221C2×C4⋊C4—44no—
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⋊C4—44no—
231C2×C4⋊C4—44no—
232C2×C4⋊C4—44no—
233C2×C22⋊C4—24no—
234C2×C22⋊C4—24no—
235C2×C22⋊C4—24no—
236C2×C22⋊C4—24no—
237C2×C22⋊C4—24no—
238C2×C22⋊C4—24no—
239C2×C22⋊C4—24no—
240C2×C22⋊C4—24no—
241C2×C22⋊C4—24no—
242C2×C22⋊C4—28no—
243C2×C22⋊C4—42no—
244C2×C22⋊C4—42no—
245C2×C22⋊C4—42no—
246C2×C22⋊C4—42no—
247C2×C22⋊C4—42no—
248C2×C22⋊C4—42no—
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×C42—22no—
261C2×C42—22no—
262C2×C42—22no—
263C2×C42—22no—
264C2×C42—22no—
265C2×C42—22no—
266C2×C42—24no—
267C2×C42—24no—
268C2×C42—24no—
269C2×C42C2×D414yes—
270C2×C42C2×D414yes—
271C2×C42C2×D414yes—
272#95—24no—
273#95—24no—
274#95—24no—
275#95—24no—
276#95—28no—
277#95—28no—
278#95—28no—
279#95—28no—
280#95—28no—
281#95—44no—
282#95—44no—
283#95—44no—
284#95—44no—
285#95—44no—
286#95—44no—
287#95—44no—
288#95—44no—
289#95—44no—
290#95—44no—
291#95—44no—
292#95—44no—
293#95—44no—
294#95—44no—
295#95C2×D418yes—
296#95C2×D418yes—
297C24—22no—
298C24—22no—
299C24—22no—
300C24—22no—
301C24—22no—
302C24—22no—
303C24—22no—
304C24—22no—
305C24—22no—
306C24—22no—
307C24—22no—
308C24—22no—
309C24—22no—
310C24—22no—
311C242−1+414yes—
312C24C2511yescharacteristic
313C2×Q8—48no—
314C2×Q8—44no—
315C2×Q8—44no—
316C2×Q8—48no—
317C2×Q8—44no—
318C2×Q8—48no—
319C2×Q8—48no—
320C2×Q8—48no—
321C2×Q8—48no—
322C2×Q8—48no—
323C2×Q8—48no—
324C2×Q8—48no—
325C2×D4—48no—
326C2×D4—44no—
327C2×D4—44no—
328C2×D4—48no—
329C2×D4—44no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—21no—
335C22×C4—21no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—21no—
339C22×C4—21no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—22no—
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—24no—
358C22×C4—24no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—24no—
363C22×C4—24no—
364C22×C4—24no—
365C22×C4—24no—
366C22×C4—24no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—24no—
370C22×C4—24no—
371C22×C4—24no—
372C22×C4—24no—
373C22×C4—24no—
374C22×C4—24no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—24no—
378C22×C4—24no—
379C22×C4—22no—
380C22×C4—24no—
381C22×C4—24no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—24no—
385C22×C4—24no—
386C22×C4—22no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—24no—
392C22×C4—24no—
393C22×C4—24no—
394C22×C4—24no—
395C22×C4—24no—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—48no—
399C22×C4—48no—
400C22×C4—48no—
401C22×C4—48no—
402C22×C4—48no—
403C22×C4—48no—
404C22×C4—48no—
405C22×C4C22×D414yes—
406C22×C4C22×D412yes—
407C22×C4C22×D412yes—
408C22×C4C22×D414yes—
409C22×C4C22×D412yes—
410C22×C4H(5,𝔽2)12yes—
411C22×C4H(5,𝔽2)12yes—
412C22×C4H(5,𝔽2)12yes—
413C22×C4H(5,𝔽2)12yes—
414C22×C4H(5,𝔽2)12yes—
415C22×C42−1+414yes—
416C22×C42−1+412yes—
417C4⋊C4—44no—
418C4⋊C4—44no—
419C4⋊C4—48no—
420C4⋊C4—416no—
421C4⋊C4—416no—
422C4⋊C4—416no—
423C4⋊C4—416no—
424C4⋊C4—48no—
425C4⋊C4—48no—
426C4⋊C4—48no—
427C4⋊C4—88no—
428C22⋊C4—48no—
429C22⋊C4—48no—
430C22⋊C4—48no—
431C22⋊C4—48no—
432C22⋊C4—48no—
433C22⋊C4—416no—
434C22⋊C4—416no—
435C22⋊C4—88no—
436C22⋊C4—88no—
437C42—28no—
438C42—28no—
439C42—28no—
440C42—48no—
441C23—24no—
442C23—24no—
443C23—24no—
444C23—24no—
445C23—24no—
446C23—24no—
447C23—24no—
448C23—42no—
449C23—42no—
450C23—42no—
451C23—42no—
452C23—42no—
453C23—42no—
454C23—42no—
455C23—42no—
456C23—42no—
457C23—42no—
458C23—42no—
459C23—42no—
460C23—42no—
461C23—42no—
462C23—42no—
463C23—42no—
464C23—42no—
465C23—42no—
466C23—42no—
467C23—42no—
468C23—42no—
469C23—42no—
470C23—42no—
471C23—42no—
472C23—42no—
473C23—42no—
474C23—42no—
475C23—42no—
476C23C23×D411yescharacteristic
477C23C2×H(5,𝔽2)11yescharacteristic
478C23C2×H(5,𝔽2)11yescharacteristic
479C23C2×H(5,𝔽2)11yescharacteristic
480C23C2×H(5,𝔽2)11yescharacteristic
481C23C2×H(5,𝔽2)11yescharacteristic
482C23C2×2−1+411yescharacteristic
483C23C2×2−1+412yes—
484C23C2×2−1+412yes—
485C23#58512yes—
486C23#58512yes—
487Q8—416no—
488Q8—816no—
489D4—416no—
490C2×C4—22no—
491C2×C4—24no—
492C2×C4—22no—
493C2×C4—24no—
494C2×C4—24no—
495C2×C4—24no—
496C2×C4—24no—
497C2×C4—24no—
498C2×C4—24no—
499C2×C4—24no—
500C2×C4—24no—
501C2×C4—24no—
502C2×C4—24no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—42no—
509C2×C4—41no—
510C2×C4—42no—
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—42no—
523C2×C4—44no—
524C2×C4—44no—
525C2×C4—42no—
526C2×C4—44no—
527C2×C4—42no—
528C2×C4—44no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—44no—
532C2×C4—44no—
533C2×C4—44no—
534C2×C4—44no—
535C2×C4—44no—
536C2×C4—44no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—44no—
542C2×C4—44no—
543C2×C4—42no—
544C2×C4—44no—
545C2×C4—44no—
546C2×C4—42no—
547C2×C4—44no—
548C2×C4—44no—
549C2×C4—42no—
550C2×C4—42no—
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—816no—
564C2×C4#53514yes—
565C2×C4#53514yes—
566C2×C4#53714yes—
567C2×C4H(3,𝔽4)14yes—
568C2×C4H(3,𝔽4)14yes—
569C2×C4#56214yes—
570C2×C4#56214yes—
571C22—44no—
572C22—44no—
573C22—44no—
574C22—44no—
575C22—44no—
576C22—44no—
577C22—44no—
578C22—82no—
579C22—82no—
580C22—82no—
581C22—82no—
582C22—82no—
583C22—82no—
584C22—82no—
585C22—82no—
586C22C2×#53511yescharacteristic
587C22C2×#53511yescharacteristic
588C22C2×#53711yescharacteristic
589C22#348912yes—
590C22#348912yes—
591C22C2×H(3,𝔽4)11yescharacteristic
592C22C2×H(3,𝔽4)11yescharacteristic
593C22C2×#56211yescharacteristic
594C22C2×#56211yescharacteristic
595C22#353811yescharacteristic
596C22#353811yescharacteristic
597C22#353911yescharacteristic
598C22#353911yescharacteristic
599C22#354112yes—
600C22#354112yes—
601C22#355212yes—
602C22#355212yes—
603C22#355412yes—
604C22#355512yes—
605C22#356112yes—
606C22#356212yes—
607C22#356212yes—
608C22#356312yes—
609C4—44no—
610C4—44no—
611C4—42no—
612C4—42no—
613C4—44no—
614C4—84no—
615C4—82no—
616C4—84no—
617C4—84no—
618C4—84no—
619C4—82no—
620C4#281918yes—
621C2—84no—
622C2C2×#281911yescharacteristic
623C2#3737511yescharacteristic
624C2#3737511yescharacteristic
625C2#3737812yes—
626C2#3737912yes—
627C2#5303011yescharacteristic
628C2#5303211yescharacteristic
629C2#5307411yescharacteristic
630C2#5307711yescharacteristic
631C2#5314412yes—
632C2#5314512yes—
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×#2642C2semidirect
3. #17222C2semidirect
4. #17158C2semidirect
5. #17153C2semidirect
7. #15236C2semidirect
8. #15216C2semidirect
9. #15201C2semidirect
10. #15182C2semidirect
11. #15178C2semidirect
13. #14348C2semidirect

Maximal subgroups

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

subgroupindex
C2×#26422
C2×#23982
#172222
#171582
#171532
#171482
#152362
#152162
#152012
#151822
#151782
#151772
#143482