Finite Group Databasebeta

← #723037

Subgroups — #723037

Order 512.

2358 conjugacy classes of subgroups.

The subgroup lattice has 633 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2550C211yesmaximal, characteristic
2C2×#2365C211yesmaximal, characteristic
3#17080C212yesmaximal
4#16989C212yesmaximal
5#15746C214yesmaximal
6#15627C214yesmaximal
7#15218C212yesmaximal
8#15202C212yesmaximal
9#15128C212yesmaximal
10#14881C212yesmaximal
11#14282C211yesmaximal, characteristic
12#14161C211yesmaximal, characteristic
13#14046C212yesmaximal
14#14045C212yesmaximal
15#13954C211yesmaximal, characteristic
16#13856C211yesmaximal, characteristic
17#13783C211yesmaximal, characteristic
18C22×C4⋊Q8C2211yescharacteristic
19C22×C4⋊Q8C2211yescharacteristic
20C22×#125C2211yescharacteristic
21C22×C4×Q8C2212yes—
22#2550—28no—
23C2×#398C2212yes—
24C2×#395C2212yes—
25C2×#395C2214yes—
26#2365—28no—
27C2×#391C2212yes—
28C2×#391C2212yes—
29C2×#389C2212yes—
30C2×#389C2212yes—
31C2×#389C2211yescharacteristic
32C2×#389C2211yescharacteristic
33C2×#389C2214yes—
34C2×#389C2212yes—
35C2×#389C2212yes—
36C2×#389C2212yes—
37C2×#387C2214yes—
38C2×#382C2211yescharacteristic
39C2×C4×C4⋊C4C2211yescharacteristic
40C2×C4×C4⋊C4C2211yescharacteristic
41C2×C4×C4⋊C4C2211yescharacteristic
42C2×C4×C4⋊C4C2212yes—
43C2×C4×C4⋊C4C2212yes—
44C2×C4×C4⋊C4C2212yes—
45C2×C4×C4⋊C4C2211yescharacteristic
46C2×#376C2211yescharacteristic
47#1446C2214yes—
48#1446C2212yes—
49#1446C2212yes—
50#1446C2212yes—
51#1446C2212yes—
52#1445C2214yes—
53#1444C2212yes—
54#1444C2214yes—
55#1444C2214yes—
56#1444C2211yescharacteristic
57#1444C2211yescharacteristic
58#1444C2211yescharacteristic
59#1444C2212yes—
60#1444C2211yescharacteristic
61#1444C2212yes—
62#1444C2212yes—
63#1444C2214yes—
64#1444C2214yes—
65#1442C2214yes—
66#1441C2214yes—
67#1441C2214yes—
68#1441C2214yes—
69#1441C2214yes—
70#1441C2214yes—
71#1436C2214yes—
72#1436C2214yes—
73#1436C2214yes—
74C4⋊#95C2212yes—
75C4⋊#95C2214yes—
76C4⋊#95C2212yes—
77C4⋊#95C2212yes—
78C4⋊#95C2212yes—
79C4⋊#95C2212yes—
80C4⋊#95C2212yes—
81C4⋊#95C2214yes—
82C4⋊#95C2212yes—
83C4⋊#95C2212yes—
84#1434C2212yes—
85C23×Q8C2312yes—
86C23×Q8C2312yes—
87C2×C4⋊Q8—24no—
88C2×C4⋊Q8—24no—
89C2×C4⋊Q8—24no—
90C2×C4⋊Q8—44no—
91C2×C4⋊Q8Q818yescomplemented
92C2×#125—24no—
93C2×#125—44no—
94C2×C4×Q8—28no—
95C2×C4×Q8—48no—
96C22×C4⋊C4C2311yescharacteristic
97C22×C4⋊C4C2311yescharacteristic
98C22×C4⋊C4C2311yescharacteristic
99C22×C4⋊C4C2311yescharacteristic
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2314yes—
102C22×C4⋊C4C2311yescharacteristic
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2311yescharacteristic
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2312yes—
108C22×C4⋊C4C2312yes—
109C22×C4⋊C4C2314yes—
110C22×C4⋊C4C2312yes—
111C22×C4⋊C4C2312yes—
112C22×C4⋊C4C2312yes—
113C22×C4⋊C4C2312yes—
114C22×C4⋊C4C2312yes—
115C22×C4⋊C4C2311yescharacteristic
116C22×C4⋊C4C2314yes—
117C22×C4⋊C4C2312yes—
118C22×C4⋊C4C2314yes—
119C22×C4⋊C4C2311yescharacteristic
120C22×C42C2311yescharacteristic
121C22×C42C2311yescharacteristic
122C22×C42C2312yes—
123C22×C42C2314yes—
124C22×C42C2312yes—
125C22×C42C2312yes—
126C22×C42C2312yes—
127C22×C42C2311yescharacteristic
128#398—28no—
129#395—28no—
130#395—48no—
131#391—28no—
132#391—28no—
133#389—28no—
134#389—24no—
135#389—24no—
136#389—28no—
137#389—48no—
138#389—44no—
139#389—44no—
140#389Q8116yescomplemented
141#387—216no—
142#382—24no—
143C4×C4⋊C4—28no—
144C4×C4⋊C4—24no—
145C4×C4⋊C4—24no—
146C4×C4⋊C4—24no—
147C4×C4⋊C4—44no—
148C4×C4⋊C4—44no—
149C4×C4⋊C4D418yes—
150#376—24no—
151C2×#95C2312yes—
152C2×#95C2312yes—
153C2×#95C2312yes—
154C2×#95C2311yescharacteristic
155C2×#95C2312yes—
156C2×#95C2311yescharacteristic
157C2×#95C2312yes—
158C2×#95C2312yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2311yescharacteristic
162C2×#95C2311yescharacteristic
163C2×#95C2312yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2314yes—
171C2×#95C2314yes—
172C2×#95C2314yes—
173C2×#95C2312yes—
174C2×#95C2314yes—
175C2×#95C2314yes—
176C2×#95C2314yes—
177C2×#95C2312yes—
178C2×#95C2312yes—
179C2×#95C2312yes—
180C2×#95C2312yes—
181C2×#95C2314yes—
182C22×Q8—24no—
183C22×Q8—24no—
184C22×Q8—24no—
185C22×Q8—24no—
186C22×Q8—44no—
187C22×Q8—44no—
188C22×Q8—44no—
189C22×Q8—42no—
190C22×Q8—44no—
191C22×Q8C2×Q818yes—
192C23×C4C2411yescharacteristic
193C23×C4C2412yes—
194C23×C4C2412yes—
195C23×C4C2412yes—
196C23×C4C2411yescharacteristic
197C23×C4C2412yes—
198C23×C4C2412yes—
199C23×C4C2414yes—
200C23×C4C2414yes—
201C23×C4C2412yes—
202C23×C4C2412yes—
203C23×C4C2411yescharacteristic
204C23×C4C2411yescharacteristic
205C23×C4C2412yes—
206C23×C4C2411yescharacteristic
207C23×C4C2411yescharacteristic
208C23×C4C2411yescharacteristic
209C4⋊Q8—232no—
210C4⋊Q8—416no—
211#125—416no—
212C4×Q8—432no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—24no—
215C2×C4⋊C4—24no—
216C2×C4⋊C4—24no—
217C2×C4⋊C4—24no—
218C2×C4⋊C4—24no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—24no—
222C2×C4⋊C4—28no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—28no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—28no—
244C2×C4⋊C4—44no—
245C2×C4⋊C4—41no—
246C2×C4⋊C4—41no—
247C2×C4⋊C4—42no—
248C2×C4⋊C4—42no—
249C2×C4⋊C4—42no—
250C2×C4⋊C4—42no—
251C2×C4⋊C4—42no—
252C2×C4⋊C4—42no—
253C2×C4⋊C4—44no—
254C2×C4⋊C4—44no—
255C2×C4⋊C4—44no—
256C2×C4⋊C4—44no—
257C2×C4⋊C4—42no—
258C2×C4⋊C4—42no—
259C2×C4⋊C4—42no—
260C2×C4⋊C4—42no—
261C2×C4⋊C4—44no—
262C2×C4⋊C4—44no—
263C2×C4⋊C4—44no—
264C2×C4⋊C4—44no—
265C2×C4⋊C4—44no—
266C2×C4⋊C4—44no—
267C2×C4⋊C4—44no—
268C2×C4⋊C4C2×D414yes—
269C2×C4⋊C4C2×D414yes—
270C2×C4⋊C4C2×Q818yes—
271C2×C4⋊C4C2×Q818yes—
272C2×C4⋊C4C2×Q818yes—
273C2×C4⋊C4C2×Q8116yes—
274C2×C4⋊C4C2×Q818yes—
275C2×C42—24no—
276C2×C42—24no—
277C2×C42—28no—
278C2×C42—22no—
279C2×C42—22no—
280C2×C42—24no—
281C2×C42—28no—
282C2×C42—24no—
283C2×C42—24no—
284C2×C42—24no—
285C2×C42—24no—
286C2×C42—24no—
287C2×C42—44no—
288C2×C42—44no—
289C2×C42—42no—
290C2×C42—42no—
291C2×C42C2×D418yes—
292C2×C42C2×D414yes—
293C2×C42C2×Q818yes—
294C2×C42C2×Q814yes—
295C2×C42#4114yes—
296#95—24no—
297#95—28no—
298#95—22no—
299#95—22no—
300#95—24no—
301#95—24no—
302#95—28no—
303#95—24no—
304#95—24no—
305#95—22no—
306#95—22no—
307#95—24no—
308#95—24no—
309#95—44no—
310#95—44no—
311#95—44no—
312#95—44no—
313#95—44no—
314#95—42no—
315#95—42no—
316#95—42no—
317#95—42no—
318#95—44no—
319#95—44no—
320#95—44no—
321#95—44no—
322#95—44no—
323#95—44no—
324#95—44no—
325#95C2×D418yes—
326#95C2×Q8116yes—
327C24C2511yescharacteristic
328C2×Q8—216no—
329C2×Q8—416no—
330C2×Q8—48no—
331C2×Q8—48no—
332C2×Q8—416no—
333C2×Q8—48no—
334C2×Q8—48no—
335C2×Q8—84no—
336C2×Q8—88no—
337C2×Q8—88no—
338C22×C4—24no—
339C22×C4—22no—
340C22×C4—21no—
341C22×C4—21no—
342C22×C4—22no—
343C22×C4—21no—
344C22×C4—21no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—21no—
348C22×C4—24no—
349C22×C4—22no—
350C22×C4—21no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—24no—
354C22×C4—24no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—24no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—24no—
361C22×C4—24no—
362C22×C4—24no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—24no—
369C22×C4—22no—
370C22×C4—24no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—24no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—24no—
382C22×C4—24no—
383C22×C4—24no—
384C22×C4—24no—
385C22×C4—21no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—21no—
389C22×C4—22no—
390C22×C4—21no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—21no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—24no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—21no—
403C22×C4—22no—
404C22×C4—21no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—24no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—21no—
412C22×C4—22no—
413C22×C4—21no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4C22×D414yes—
417C22×C4C22×D412yes—
418C22×C4C22×D412yes—
419C22×C4C22×D414yes—
420C22×C4C22×D412yes—
421C22×C4C22×Q814yes—
422C22×C4C22×Q818yes—
423C22×C4C22×Q814yes—
424C22×C4C22×Q818yes—
425C22×C4C22×Q812yes—
426C22×C4C22×Q812yes—
427C22×C4C22×Q818yes—
428C22×C4C22×Q812yes—
429C22×C4C22×Q814yes—
430C22×C4C2×#4112yes—
431C22×C4C2×#4112yes—
432C22×C4C2×#4112yes—
433C22×C4H(5,𝔽2)14yes—
434C22×C4H(5,𝔽2)14yes—
435C22×C42−1+412yes—
436C22×C42−1+412yes—
437C4⋊C4—28no—
438C4⋊C4—216no—
439C4⋊C4—216no—
440C4⋊C4—28no—
441C4⋊C4—216no—
442C4⋊C4—416no—
443C4⋊C4—44no—
444C4⋊C4—48no—
445C4⋊C4—44no—
446C4⋊C4—48no—
447C4⋊C4—416no—
448C4⋊C4—44no—
449C4⋊C4—48no—
450C4⋊C4—48no—
451C4⋊C4—48no—
452C4⋊C4—48no—
453C4⋊C4—48no—
454C4⋊C4—48no—
455C4⋊C4—44no—
456C4⋊C4—44no—
457C4⋊C4—84no—
458C4⋊C4—88no—
459C4⋊C4—88no—
460C4⋊C4C4⋊Q8116yes—
461C42—216no—
462C42—48no—
463C42—416no—
464C42—48no—
465C42—44no—
466C42—44no—
467C42—84no—
468C42C22⋊Q8116yes—
469C23C23×D411yescharacteristic
470C23C23×Q811yescharacteristic
471C23C23×Q812yes—
472C23C22×#4111yescharacteristic
473C23C2×H(5,𝔽2)12yes—
474C23C2×H(5,𝔽2)12yes—
475C23C2×2−1+411yescharacteristic
476C23C2×2−1+411yescharacteristic
477C23#58512yes—
478C23#58511yescharacteristic
479C23#58511yescharacteristic
480Q8—432no—
481Q8—816no—
482C2×C4—24no—
483C2×C4—24no—
484C2×C4—22no—
485C2×C4—22no—
486C2×C4—22no—
487C2×C4—22no—
488C2×C4—24no—
489C2×C4—28no—
490C2×C4—24no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—28no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—28no—
497C2×C4—24no—
498C2×C4—24no—
499C2×C4—24no—
500C2×C4—28no—
501C2×C4—24no—
502C2×C4—24no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—24no—
506C2×C4—24no—
507C2×C4—28no—
508C2×C4—28no—
509C2×C4—28no—
510C2×C4—24no—
511C2×C4—24no—
512C2×C4—24no—
513C2×C4—28no—
514C2×C4—24no—
515C2×C4—24no—
516C2×C4—24no—
517C2×C4—24no—
518C2×C4—41no—
519C2×C4—42no—
520C2×C4—41no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—44no—
524C2×C4—44no—
525C2×C4—44no—
526C2×C4—44no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—44no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—44no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—44no—
537C2×C4—42no—
538C2×C4—44no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—44no—
543C2×C4—42no—
544C2×C4—44no—
545C2×C4—42no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—41no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—41no—
555C2×C4—42no—
556C2×C4—41no—
557C2×C4—42no—
558C2×C4—41no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—44no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—44no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—44no—
571C2×C4—42no—
572C2×C4—44no—
573C2×C4C2×C22⋊Q814yes—
574C2×C4C2×C22⋊Q814yes—
575C2×C4C2×C22⋊Q814yes—
576C2×C4C2×C4⋊Q818yes—
577C2×C4C2×C4⋊Q814yes—
578C2×C4#53514yes—
579C2×C4#55418yes—
580C2×C4Q8218yes—
581C22C22×C22⋊Q811yescharacteristic
582C22C22×C4⋊Q811yescharacteristic
583C22C2×#53511yescharacteristic
584C22#346012yes—
585C22C2×#55412yes—
586C22C2×Q8212yes—
587C22#348411yescharacteristic
588C22#348711yescharacteristic
589C22#348911yescharacteristic
590C22#348911yescharacteristic
591C22#350112yes—
592C22#350112yes—
593C22#350212yes—
594C22#350211yescharacteristic
595C22#350211yescharacteristic
596C22#350312yes—
597C22#350412yes—
598C22#350911yescharacteristic
599C22#351011yescharacteristic
600C22#353612yes—
601C22#354012yes—
602C22#354112yes—
603C22#354212yes—
604C4—24no—
605C4—28no—
606C4—24no—
607C4—44no—
608C4—44no—
609C4—48no—
610C4—48no—
611C4—42no—
612C4—44no—
613C4—44no—
614C4—42no—
615C4—81no—
616C4—82no—
617C4—82no—
618C4—81no—
619C4—82no—
620C4#256718yes—
621C2C2×#256711yescharacteristic
622C2#3517411yescharacteristic
623C2#3566211yescharacteristic
624C2#3592811yescharacteristic
625C2#3595711yescharacteristic
626C2#3969112yes—
627C2#3969412yes—
628C2#4247811yescharacteristic
629C2#4288711yescharacteristic
630C2#4495012yes—
631C2#4536412yes—
subgroupG/H
center, commutator, Frattini, socle, prime radical, cosocleC24C25
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
91. C2×C4⋊Q8Q8semidirect
140. #389Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#25502
C2×#23652
#170802
#169892
#157462
#156272
#152182
#152022
#151282
#148812
#142822
#141612
#140462
#140452
#139542
#138562
#137832