Finite Group Databasebeta

← #4414328

Subgroups — #4414328

Order 512.

4597 conjugacy classes of subgroups.

The subgroup lattice has 629 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×C4⋊Q8C211yesmaximal, characteristic, complemented
2C2×#2821C212yesmaximal, complemented
3C2×#2852C211yesmaximal, characteristic
4#16603C212yesmaximal, complemented
5#16602C212yesmaximal, complemented
6#16560C212yesmaximal, complemented
7#16511C212yesmaximal, complemented
8#16507C212yesmaximal, complemented
9#16506C212yesmaximal, complemented
10#14874C212yesmaximal, complemented
11#14873C212yesmaximal
12#14843C212yesmaximal, complemented
13#14786C212yesmaximal, complemented
14#14777C212yesmaximal
15#14776C212yesmaximal
16#14683C211yesmaximal, characteristic, complemented
17#14663C212yesmaximal, complemented
18C24×Q8C2212yes—
19C22×C4⋊Q8—28no—
20C22×C4⋊Q8—216no—
21C22×C4⋊Q8C2212yes—
22C22×C4⋊Q8C2212yes—
23C22×C4⋊Q8C2212yes—
24C22×C4⋊Q8C2212yes—
25C22×#126C2211yescharacteristic
26C23×C4⋊C4C2212yes—
27C23×C4⋊C4C2212yes—
28C23×C42C2211yescharacteristic
29#2852—28no—
30#2821—216no—
31C2×#400C2214yes—
32C2×#399C2214yes—
33C2×#398C2214yes—
34C2×#397C2214yes—
35C2×#395C2214yes—
36C2×#395C2214yes—
37C2×#393C2214yes—
38C2×#393C2214yes—
39C2×#388C2211yescharacteristic
40C2×#388C2211yescharacteristic
41C2×#388C2211yescharacteristic
42C2×#387C2211yescharacteristic
43C2×#387C2211yescharacteristic
44C2×#387C2211yescharacteristic
45C2×#386C2211yescharacteristic
46C2×#386C2211yescharacteristic
47C2×#386C2211yescharacteristic
48C2×#386C2214yes—
49C2×#385C2211yescharacteristic
50C2×#385C2211yescharacteristic
51C2×#385C2211yescharacteristic
52C2×#385C2214yes—
53C2×#383C2211yescharacteristic
54C2×#383C2212yes—
55#1446C2214yes—
56#1446C2214yes—
57#1446C2214yes—
58#1446C2214yes—
59#1445C2214yes—
60#1445C2214yes—
61#1445C2214yes—
62#1445C2214yes—
63#1444C2214yes—
64#1444C2214yes—
65#1442C2214yes—
66#1442C2214yes—
67#1441C2214yes—
68#1441C2214yes—
69#1439C2214yes—
70#1439C2214yes—
71#1434C2212yes—
72#1434C2212yes—
73#1434C2212yes—
74C22⋊#95C2212yes—
75C22⋊#95C2214yes—
76C22⋊#95C2214yes—
77C22⋊#95C2212yes—
78C22⋊#95C2212yes—
79C23×Q8—216no—
80C23×Q8—28no—
81C23×Q8—48no—
82C23×Q8—48no—
83C23×Q8C2314yes—
84C23×Q8C2314yes—
85C23×D4C2311yescharacteristic
86C24×C4C2311yescharacteristic
87C24×C4C2311yescharacteristic
88C24×C4C2311yescharacteristic
89C24×C4C2314yes—
90C2×C4⋊Q8—2128no—
91C2×C4⋊Q8—216no—
92C2×C4⋊Q8—216no—
93C2×C4⋊Q8—216no—
94C2×C4⋊Q8—28no—
95C2×C4⋊Q8—28no—
96C2×C4⋊Q8—28no—
97C2×C4⋊Q8—28no—
98C2×C4⋊Q8—28no—
99C2×C4⋊Q8D4116yescomplemented
100C2×#126—28no—
101C2×#126—24no—
102C22×C4⋊C4—28no—
103C22×C4⋊C4—28no—
104C22×C4⋊C4—216no—
105C22×C4⋊C4—216no—
106C22×C4⋊C4C2311yescharacteristic
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2311yescharacteristic
109C22×C4⋊C4C2311yescharacteristic
110C22×C4⋊C4C2314yes—
111C22×C4⋊C4C2314yes—
112C22×C4⋊C4C2311yescharacteristic
113C22×C4⋊C4C2311yescharacteristic
114C22×C4⋊C4C2314yes—
115C22×C4⋊C4C2314yes—
116C22×C4⋊C4C2318yes—
117C22×C22⋊C4C2311yescharacteristic
118C22×C22⋊C4C2311yescharacteristic
119C22×C22⋊C4C2311yescharacteristic
120C22×C22⋊C4C2311yescharacteristic
121C22×C22⋊C4C2314yes—
122C22×C22⋊C4C2311yescharacteristic
123C22×C22⋊C4C2311yescharacteristic
124C22×C22⋊C4C2311yescharacteristic
125C22×C22⋊C4C2311yescharacteristic
126C22×C22⋊C4C2311yescharacteristic
127C22×C22⋊C4C2318yes—
128C22×C42—28no—
129C22×C42D418yes—
130C22×C42C2311yescharacteristic
131C22×C42C2311yescharacteristic
132C22×C42C2311yescharacteristic
133C22×C42C2311yescharacteristic
134#400—216no—
135#399—216no—
136#398—216no—
137#397—216no—
138#395—216no—
139#395—48no—
140#393—216no—
141#393—48no—
142#388—24no—
143#388—24no—
144#388—24no—
145#387—24no—
146#387—24no—
147#387—24no—
148#386—24no—
149#386—24no—
150#386—24no—
151#386—48no—
152#385—24no—
153#385—24no—
154#385—24no—
155#385—48no—
156#383—28no—
157#383—24no—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2318yes—
161C2×#95C2318yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2311yescharacteristic
167C2×#95C2311yescharacteristic
168C2×#95C2311yescharacteristic
169C2×#95C2311yescharacteristic
170C2×#95C2311yescharacteristic
171C2×#95C2311yescharacteristic
172C2×#95C2311yescharacteristic
173C2×#95C2311yescharacteristic
174C2×#95C2311yescharacteristic
175C2×#95C2311yescharacteristic
176C2×#95C2311yescharacteristic
177C2×#95C2311yescharacteristic
178C2×#95C2318yes—
179C2×#95C2318yes—
180C2×#95C2318yes—
181C2×#95C2318yes—
182C25C2411yescharacteristic
183C25C2411yescharacteristic
184C22×Q8—264no—
185C22×Q8—216no—
186C22×Q8—28no—
187C22×Q8—28no—
188C22×Q8—28no—
189C22×Q8—28no—
190C22×Q8—28no—
191C22×Q8—48no—
192C22×Q8—48no—
193C22×Q8—48no—
194C22×Q8—4128no—
195C22×Q8—432no—
196C22×Q8—432no—
197C22×Q8C2×D4116yes—
198C22×D4—22no—
199C22×D4—24no—
200C22×D4—24no—
201C22×D4—24no—
202C23×C4—24no—
203C23×C4—24no—
204C23×C4—24no—
205C23×C4—24no—
206C23×C4—24no—
207C23×C4—24no—
208C23×C4—24no—
209C23×C4—24no—
210C23×C4—24no—
211C23×C4—28no—
212C23×C4—216no—
213C23×C4—28no—
214C23×C4—28no—
215C23×C4—48no—
216C23×C4C2×D414yes—
217C23×C4C2×D414yes—
218C23×C4C2×D414yes—
219C23×C4C2411yescharacteristic
220C23×C4C2411yescharacteristic
221C23×C4C2411yescharacteristic
222C23×C4C2418yes—
223C23×C4C2411yescharacteristic
224C23×C4C2411yescharacteristic
225C23×C4C2411yescharacteristic
226C23×C4C2411yescharacteristic
227C23×C4C2418yes—
228C23×C4C2411yescharacteristic
229C23×C4C2411yescharacteristic
230C23×C4C2411yescharacteristic
231C23×C4C2411yescharacteristic
232C23×C4C2411yescharacteristic
233C23×C4C2411yescharacteristic
234C4⋊Q8—264no—
235C4⋊Q8—264no—
236C4⋊Q8—264no—
237C4⋊Q8—264no—
238#126—416no—
239C2×C4⋊C4—264no—
240C2×C4⋊C4—264no—
241C2×C4⋊C4—216no—
242C2×C4⋊C4—216no—
243C2×C4⋊C4—28no—
244C2×C4⋊C4—28no—
245C2×C4⋊C4—28no—
246C2×C4⋊C4—28no—
247C2×C4⋊C4—28no—
248C2×C4⋊C4—28no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—216no—
262C2×C4⋊C4—48no—
263C2×C4⋊C4—48no—
264C2×C4⋊C4C2×D4116yes—
265C2×C4⋊C4C2×D4116yes—
266C2×C22⋊C4—22no—
267C2×C22⋊C4—22no—
268C2×C22⋊C4—22no—
269C2×C22⋊C4—24no—
270C2×C22⋊C4—28no—
271C2×C22⋊C4—24no—
272C2×C22⋊C4—24no—
273C2×C22⋊C4—28no—
274C2×C22⋊C4—28no—
275C2×C22⋊C4—22no—
276C2×C22⋊C4—22no—
277C2×C22⋊C4—22no—
278C2×C22⋊C4—24no—
279C2×C22⋊C4—24no—
280C2×C22⋊C4—24no—
281C2×C22⋊C4—22no—
282C2×C22⋊C4—22no—
283C2×C22⋊C4—22no—
284C2×C22⋊C4—24no—
285C2×C22⋊C4—24no—
286C2×C22⋊C4—24no—
287C2×C22⋊C4—216no—
288C2×C22⋊C4—48no—
289C2×C22⋊C4—48no—
290C2×C42—24no—
291C2×C42—24no—
292C2×C42—24no—
293C2×C42—232no—
294C2×C42C2×D414yes—
295C2×C42C2×D414yes—
296C2×C42C2×D414yes—
297C2×C42C2×D414yes—
298C2×C42C2×D418yes—
299#95—22no—
300#95—22no—
301#95—22no—
302#95—22no—
303#95—22no—
304#95—22no—
305#95—28no—
306#95—28no—
307#95—28no—
308#95—22no—
309#95—22no—
310#95—22no—
311#95—22no—
312#95—22no—
313#95—22no—
314#95—216no—
315#95—216no—
316#95—28no—
317#95—28no—
318#95—28no—
319#95—48no—
320#95—48no—
321#95—48no—
322#95—48no—
323C24—22no—
324C24—22no—
325C24—22no—
326C24—22no—
327C24—22no—
328C24—21no—
329C24—22no—
330C24—22no—
331C24—22no—
332C24—22no—
333C24—22no—
334C24—22no—
335C24C22×D412yes—
336C24H(5,𝔽2)14yes—
337C242−1+414yes—
338C242−1+414yes—
339C24C2511yescharacteristic
340C2×Q8—232no—
341C2×Q8—232no—
342C2×Q8—432no—
343C2×Q8—432no—
344C2×Q8—432no—
345C2×Q8—4256no—
346C2×Q8—416no—
347C2×Q8—416no—
348C2×Q8—416no—
349C2×Q8—432no—
350C2×Q8—416no—
351C2×D4—44no—
352C2×D4—44no—
353C2×D4—44no—
354C2×D4—416no—
355C22×C4—232no—
356C22×C4—232no—
357C22×C4—232no—
358C22×C4—28no—
359C22×C4—28no—
360C22×C4—28no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—28no—
374C22×C4—28no—
375C22×C4—28no—
376C22×C4—28no—
377C22×C4—28no—
378C22×C4—28no—
379C22×C4—232no—
380C22×C4—28no—
381C22×C4—28no—
382C22×C4—28no—
383C22×C4—28no—
384C22×C4—28no—
385C22×C4—28no—
386C22×C4—21no—
387C22×C4—28no—
388C22×C4—21no—
389C22×C4—21no—
390C22×C4—21no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—21no—
394C22×C4—21no—
395C22×C4—21no—
396C22×C4—22no—
397C22×C4—28no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—28no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—28no—
406C22×C4—28no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—28no—
414C22×C4—28no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—432no—
422C22×C4—432no—
423C22×C4—416no—
424C22×C4—416no—
425C22×C4C22×D4116yes—
426C22×C4C22×D414yes—
427C22×C4C22×D414yes—
428C22×C4C22×D414yes—
429C22×C4C22×D412yes—
430C22×C4C22×D412yes—
431C22×C4C22×D412yes—
432C22×C4C22×D412yes—
433C22×C4C22×D412yes—
434C22×C4C22×D412yes—
435C22×C4H(5,𝔽2)14yes—
436C22×C42−1+414yes—
437C22×C42−1+414yes—
438C4⋊C4—232no—
439C4⋊C4—232no—
440C4⋊C4—232no—
441C4⋊C4—232no—
442C4⋊C4—44no—
443C4⋊C4—44no—
444C4⋊C4—44no—
445C4⋊C4—44no—
446C4⋊C4—44no—
447C4⋊C4—44no—
448C4⋊C4—816no—
449C22⋊C4—44no—
450C22⋊C4—44no—
451C22⋊C4—44no—
452C22⋊C4—416no—
453C22⋊C4—44no—
454C22⋊C4—44no—
455C22⋊C4—44no—
456C22⋊C4—44no—
457C22⋊C4—44no—
458C22⋊C4—44no—
459C22⋊C4—816no—
460C42—28no—
461C42—28no—
462C42—28no—
463C42#120116yes—
464C23—22no—
465C23—22no—
466C23—22no—
467C23—22no—
468C23—28no—
469C23—28no—
470C23—28no—
471C23—28no—
472C23—28no—
473C23—28no—
474C23—41no—
475C23—41no—
476C23—41no—
477C23—41no—
478C23—41no—
479C23—41no—
480C23—41no—
481C23—44no—
482C23—44no—
483C23—44no—
484C23—44no—
485C23—44no—
486C23—44no—
487C23—44no—
488C23#53514yes—
489C23#53614yes—
490C23#53614yes—
491C23#563116yes—
492C23C23×D411yescharacteristic
493C23C23×D412yes—
494C23C2×H(5,𝔽2)12yes—
495C23C2×H(5,𝔽2)12yes—
496C23C2×2−1+412yes—
497C23C2×2−1+412yes—
498C23C2×2−1+412yes—
499C23C2×2−1+412yes—
500Q8—464no—
501Q8—464no—
502D4—88no—
503C2×C4—232no—
504C2×C4—232no—
505C2×C4—232no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—28no—
519C2×C4—28no—
520C2×C4—28no—
521C2×C4—28no—
522C2×C4—28no—
523C2×C4—28no—
524C2×C4—216no—
525C2×C4—216no—
526C2×C4—216no—
527C2×C4—464no—
528C2×C4—48no—
529C2×C4—48no—
530C2×C4—48no—
531C2×C4—48no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—48no—
554C2×C4—44no—
555C2×C4—44no—
556C2×C4—44no—
557C2×C4—48no—
558C2×C4—48no—
559C2×C4—48no—
560C2×C4—44no—
561C2×C4—44no—
562C2×C4—48no—
563C2×C4—44no—
564C2×C4—44no—
565C2×C4—48no—
566C2×C4—48no—
567C2×C4C2×#12014yes—
568C2×C4C2×#12014yes—
569C2×C4C2×#12014yes—
570C2×C4#53514yes—
571C2×C4#53614yes—
572C2×C4#53614yes—
573C22—24no—
574C22—24no—
575C22—24no—
576C22—24no—
577C22—24no—
578C22—24no—
579C22—232no—
580C22—81no—
581C22—81no—
582C22—81no—
583C22—81no—
584C22—81no—
585C22—81no—
586C22—81no—
587C22—88no—
588C22#282118yescomplemented
589C22C22×#12011yescharacteristic
590C22C2×#53511yescharacteristic
591C22C2×#53511yescharacteristic
592C22C2×#53611yescharacteristic
593C22C2×#53611yescharacteristic
594C22C2×#53611yescharacteristic
595C22C2×#53611yescharacteristic
596C22#348414yes—
597C22#348614yes—
598C22#348614yes—
599C22C2×#56314yes—
600C22#354414yes—
601C22#355214yes—
602C22#355214yes—
603C4—24no—
604C4—24no—
605C4—24no—
606C4—24no—
607C4—24no—
608C4—24no—
609C4—416no—
610C4—81no—
611C4—81no—
612C4—81no—
613C4—81no—
614C4—81no—
615C4—81no—
616C4—81no—
617C4—88no—
618C2—28no—
619C2—161no—
620C2#3522211yescharacteristic
621C2#3524211yescharacteristic
622C2#3524211yescharacteristic
623C2C2×#282111yescharacteristic
624C2#3761511yescharacteristic
625C2#3761711yescharacteristic
626C2#3763211yescharacteristic
627C2#4632518yes—
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. C23×C4⋊Q8C2semidirect
2. C2×#2821C2semidirect
4. #16603C2semidirect
5. #16602C2semidirect
6. #16560C2semidirect
7. #16511C2semidirect
8. #16507C2semidirect
9. #16506C2semidirect
10. #14874C2semidirect
12. #14843C2semidirect
13. #14786C2semidirect
16. #14683C2semidirect
17. #14663C2semidirect
99. C2×C4⋊Q8D4semidirect
588. C22#2821semidirect

Maximal subgroups

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

subgroupindex
C23×C4⋊Q82
C2×#28212
C2×#28522
#166032
#166022
#165602
#165112
#165072
#165062
#148742
#148732
#148432
#147862
#147772
#147762
#146832
#146632