Finite Group Databasebeta

← #539191

Subgroups — #539191

Order 512.

3714 conjugacy classes of subgroups.

The subgroup lattice has 638 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×Q82C211yesmaximal, characteristic
2C2×#2365C211yesmaximal, characteristic
3C2×#2331C211yesmaximal, characteristic
4#14450C214yesmaximal
5#14420C214yesmaximal
6#14053C212yesmaximal
7#14053C212yesmaximal
8#14024C214yesmaximal
9#13942C211yesmaximal, characteristic
10#13935C212yesmaximal
11#13927C214yesmaximal
12#13925C211yesmaximal, characteristic
13#13850C211yesmaximal, characteristic
14#13817C212yesmaximal
15#13786C211yesmaximal, characteristic
16C2×Q82—28no—
17C2×Q82—28no—
18C2×Q82C4116yescomplemented
19C22×C4⋊Q8C2212yes—
20C22×C4⋊Q8C2214yes—
21C22×C4⋊Q8C2212yes—
22C22×C4⋊Q8C2211yescharacteristic
23C22×C4×Q8C2211yescharacteristic
24C22×C4×Q8C2212yes—
25C22×C4×Q8C2212yes—
26C22×C4×Q8C2211yescharacteristic
27C22×C4×Q8C2212yes—
28#2365—28no—
29#2331—28no—
30C2×#391C2214yes—
31C2×#391C2212yes—
32C2×#391C2211yescharacteristic
33C2×#391C2212yes—
34C2×#391C2214yes—
35C2×#391C2214yes—
36C2×#391C2211yescharacteristic
37C2×#389C2212yes—
38C2×#389C2212yes—
39C2×#387C2214yes—
40C2×#387C2214yes—
41C2×#382C2211yescharacteristic
42C2×C4×C4⋊C4C2212yes—
43C2×C4×C4⋊C4C2211yescharacteristic
44C2×C4×C4⋊C4C2211yescharacteristic
45C2×C4×C4⋊C4C2212yes—
46C2×#376C2211yescharacteristic
47#1446C2212yes—
48#1446C2214yes—
49#1446C2214yes—
50#1446C2212yes—
51#1445C2212yes—
52#1445C2212yes—
53#1441C2212yes—
54#1441C2212yes—
55#1441C2214yes—
56#1441C2214yes—
57#1441C2214yes—
58#1441C2214yes—
59#1441C2212yes—
60#1441C2212yes—
61#1436C2214yes—
62#1436C2214yes—
63#1436C2214yes—
64#1436C2214yes—
65#1436C2214yes—
66#1436C2214yes—
67C4⋊#95C2212yes—
68C4⋊#95C2212yes—
69C4⋊#95C2214yes—
70C4⋊#95C2212yes—
71C4⋊#95C2211yescharacteristic
72C4⋊#95C2211yescharacteristic
73#1434C2214yes—
74#1434C2214yes—
75#1434C2214yes—
76#1434C2214yes—
77C4×#95C2212yes—
78C4×#95C2212yes—
79C4×#95C2214yes—
80C23×Q8C2311yescharacteristic
81C23×Q8C2311yescharacteristic
82C23×Q8C2314yes—
83C23×Q8C2312yes—
84Q82—2128no—
85C2×C4⋊Q8—216no—
86C2×C4⋊Q8—216no—
87C2×C4⋊Q8—28no—
88C2×C4⋊Q8—28no—
89C2×C4⋊Q8—28no—
90C2×C4⋊Q8—28no—
91C2×C4⋊Q8C2×C4132yes—
92C2×C4⋊Q8C2×C4116yes—
93C2×C4⋊Q8C2×C4116yes—
94C2×C4⋊Q8C2×C418yes—
95C2×C4⋊Q8D418yes—
96C2×C4⋊Q8D418yes—
97C2×C4×Q8—28no—
98C2×C4×Q8—28no—
99C2×C4×Q8—28no—
100C2×C4×Q8—28no—
101C2×C4×Q8—28no—
102C2×C4×Q8—24no—
103C2×C4×Q8—24no—
104C2×C4×Q8—24no—
105C2×C4×Q8—24no—
106C2×C4×Q8—48no—
107C2×C4×Q8C2×C4116yes—
108C2×C4×Q8C2×C4116yes—
109C2×C4×Q8C2×C418yes—
110C2×C4×Q8C2×C418yes—
111C22×C4⋊C4C2311yescharacteristic
112C22×C4⋊C4C2312yes—
113C22×C4⋊C4C2311yescharacteristic
114C22×C4⋊C4C2314yes—
115C22×C4⋊C4C2311yescharacteristic
116C22×C4⋊C4C2314yes—
117C22×C4⋊C4C2312yes—
118C22×C4⋊C4C2312yes—
119C22×C4⋊C4C2314yes—
120C22×C4⋊C4C2311yescharacteristic
121C22×C4⋊C4C2311yescharacteristic
122C22×C4⋊C4C2314yes—
123C22×C4⋊C4C2311yescharacteristic
124C22×C4⋊C4C2312yes—
125C22×C4⋊C4C2312yes—
126C22×C42C2311yescharacteristic
127C22×C42C2312yes—
128C22×C42C2314yes—
129C22×C42C2312yes—
130C22×C42C2314yes—
131C22×C42C2312yes—
132C22×C42C2311yescharacteristic
133C22×C42C2312yes—
134C22×C42C2314yes—
135C22×C42C2311yescharacteristic
136#391—216no—
137#391—28no—
138#391—24no—
139#391—28no—
140#391—48no—
141#391—48no—
142#391Q818yescomplemented
143#389—28no—
144#389—28no—
145#387—216no—
146#387—216no—
147#382—24no—
148C4×C4⋊C4—28no—
149C4×C4⋊C4—28no—
150C4×C4⋊C4—24no—
151C4×C4⋊C4—24no—
152#376—24no—
153C2×#95C2312yes—
154C2×#95C2312yes—
155C2×#95C2312yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2312yes—
159C2×#95C2312yes—
160C2×#95C2312yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2312yes—
164C2×#95C2312yes—
165C2×#95C2312yes—
166C2×#95C2312yes—
167C2×#95C2314yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2314yes—
171C2×#95C2314yes—
172C2×#95C2314yes—
173C2×#95C2312yes—
174C2×#95C2312yes—
175C2×#95C2312yes—
176C2×#95C2312yes—
177C2×#95C2314yes—
178C2×#95C2314yes—
179C2×#95C2314yes—
180C2×#95C2314yes—
181C2×#95C2314yes—
182C2×#95C2314yes—
183C22×Q8—22no—
184C22×Q8—22no—
185C22×Q8—22no—
186C22×Q8—22no—
187C22×Q8—28no—
188C22×Q8—28no—
189C22×Q8—24no—
190C22×Q8—24no—
191C22×Q8—22no—
192C22×Q8—22no—
193C22×Q8—44no—
194C22×Q8—44no—
195C22×Q8—44no—
196C22×Q8—44no—
197C22×Q8—44no—
198C22×Q8—44no—
199C22×Q8C22×C414yes—
200C22×Q8C22×C414yes—
201C22×Q8C22×C4116yes—
202C22×Q8C22×C418yes—
203C22×Q8C2×D418yes—
204C22×Q8C2×D418yes—
205C22×Q8C2×Q814yes—
206C22×Q8#4114yes—
207C23×C4C2412yes—
208C23×C4C2412yes—
209C23×C4C2414yes—
210C23×C4C2412yes—
211C23×C4C2412yes—
212C23×C4C2412yes—
213C23×C4C2412yes—
214C23×C4C2412yes—
215C23×C4C2412yes—
216C23×C4C2414yes—
217C23×C4C2411yescharacteristic
218C23×C4C2411yescharacteristic
219C23×C4C2412yes—
220C23×C4C2412yes—
221C23×C4C2411yescharacteristic
222C4⋊Q8—2128no—
223C4⋊Q8—264no—
224C4⋊Q8—264no—
225C4⋊Q8C22⋊C4164yes—
226C4×Q8—264no—
227C4×Q8—264no—
228C4×Q8—232no—
229C4×Q8—232no—
230C4×Q8—432no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—28no—
235C2×C4⋊C4—28no—
236C2×C4⋊C4—28no—
237C2×C4⋊C4—28no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—28no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—28no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—28no—
257C2×C4⋊C4—42no—
258C2×C4⋊C4—44no—
259C2×C4⋊C4—44no—
260C2×C4⋊C4—44no—
261C2×C4⋊C4C22×C414yes—
262C2×C4⋊C4C22×C414yes—
263C2×C4⋊C4C22×C4116yes—
264C2×C4⋊C4C22×C418yes—
265C2×C4⋊C4C22×C4116yes—
266C2×C4⋊C4C22×C418yes—
267C2×C4⋊C4C22×C418yes—
268C2×C4⋊C4C22×C418yes—
269C2×C4⋊C4C2×D418yes—
270C2×C4⋊C4C2×D418yes—
271C2×C4⋊C4C2×D418yes—
272C2×C4⋊C4C2×D418yes—
273C2×C4⋊C4C2×Q814yes—
274C2×C42—28no—
275C2×C42—28no—
276C2×C42—24no—
277C2×C42—24no—
278C2×C42—24no—
279C2×C42—24no—
280C2×C42—22no—
281C2×C42—22no—
282C2×C42—28no—
283C2×C42—24no—
284C2×C42—22no—
285C2×C42—24no—
286C2×C42—24no—
287C2×C42—22no—
288C2×C42—28no—
289C2×C42—24no—
290C2×C42—22no—
291C2×C42—28no—
292C2×C42—44no—
293C2×C42—44no—
294C2×C42—44no—
295C2×C42—44no—
296C2×C42C22×C418yes—
297C2×C42C22×C418yes—
298C2×C42C22×C4116yes—
299C2×C42C22×C414yes—
300C2×C42C2×D414yes—
301C2×C42C2×D414yes—
302C2×C42C2×Q814yes—
303#95—24no—
304#95—24no—
305#95—24no—
306#95—24no—
307#95—28no—
308#95—28no—
309#95—24no—
310#95—24no—
311#95—28no—
312#95—28no—
313#95—28no—
314#95—28no—
315#95—28no—
316#95—24no—
317#95—28no—
318#95—24no—
319#95—24no—
320#95—24no—
321#95—28no—
322#95—24no—
323#95—24no—
324#95—28no—
325#95—42no—
326#95—44no—
327#95—42no—
328#95—44no—
329#95—44no—
330#95—44no—
331#95—44no—
332#95C2×Q8116yes—
333C24C2511yescharacteristic
334C2×Q8—28no—
335C2×Q8—28no—
336C2×Q8—216no—
337C2×Q8—28no—
338C2×Q8—28no—
339C2×Q8—232no—
340C2×Q8—216no—
341C2×Q8—28no—
342C2×Q8—28no—
343C2×Q8—28no—
344C2×Q8—28no—
345C2×Q8—28no—
346C2×Q8—48no—
347C2×Q8—416no—
348C2×Q8—48no—
349C2×Q8—416no—
350C2×Q8—416no—
351C2×Q8—416no—
352C2×Q8—88no—
353C2×Q8—88no—
354C2×Q8—88no—
355C2×Q8—88no—
356C2×Q8C2×C22⋊C4132yes—
357C2×Q8C4×Q8116yescomplemented
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—24no—
363C22×C4—22no—
364C22×C4—24no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—24no—
372C22×C4—22no—
373C22×C4—24no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—24no—
385C22×C4—22no—
386C22×C4—24no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—24no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—24no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—24no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—24no—
407C22×C4—22no—
408C22×C4—24no—
409C22×C4—22no—
410C22×C4—24no—
411C22×C4—22no—
412C22×C4—24no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—24no—
416C22×C4—22no—
417C22×C4—24no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—24no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—24no—
425C22×C4—22no—
426C22×C4C23×C414yes—
427C22×C4C23×C414yes—
428C22×C4C23×C412yes—
429C22×C4C23×C418yes—
430C22×C4C23×C412yes—
431C22×C4C23×C414yes—
432C22×C4C23×C414yes—
433C22×C4C23×C412yes—
434C22×C4C22×D412yes—
435C22×C4C22×D412yes—
436C22×C4C22×D412yes—
437C22×C4C22×D418yes—
438C22×C4C22×D418yes—
439C22×C4C22×D412yes—
440C22×C4C22×D412yes—
441C22×C4C22×D412yes—
442C22×C4C22×Q814yes—
443C22×C4C22×Q814yes—
444C22×C4C22×Q814yes—
445C22×C4C22×Q812yes—
446C22×C4C2×#4114yes—
447C22×C4C2×#4112yes—
448C22×C4H(5,𝔽2)14yes—
449C22×C42−1+414yes—
450C22×C42−1+414yes—
451C22×C42−1+412yes—
452C22×C42−1+412yes—
453C4⋊C4—28no—
454C4⋊C4—232no—
455C4⋊C4—232no—
456C4⋊C4—216no—
457C4⋊C4—216no—
458C4⋊C4—28no—
459C4⋊C4—28no—
460C4⋊C4—28no—
461C4⋊C4—416no—
462C4⋊C4—44no—
463C4⋊C4—44no—
464C4⋊C4—48no—
465C4⋊C4—88no—
466C4⋊C4C2×C22⋊C4132yes—
467C4⋊C4C2×C22⋊C4132yes—
468C42—232no—
469C42—216no—
470C42—216no—
471C42—28no—
472C42—416no—
473C42—48no—
474C42—48no—
475C42—44no—
476C42—88no—
477C42C2×C22⋊C4116yes—
478C23C24×C411yescharacteristic
479C23C23×D411yescharacteristic
480C23C23×D411yescharacteristic
481C23C23×Q811yescharacteristic
482C23C22×#4111yescharacteristic
483C23C2×H(5,𝔽2)12yes—
484C23C2×2−1+411yescharacteristic
485C23C2×2−1+411yescharacteristic
486C23C2×2−1+412yes—
487C23C2×2−1+412yes—
488C23#58512yes—
489Q8—232no—
490Q8—232no—
491Q8—432no—
492Q8—832no—
493C2×C4—24no—
494C2×C4—24no—
495C2×C4—24no—
496C2×C4—24no—
497C2×C4—24no—
498C2×C4—24no—
499C2×C4—28no—
500C2×C4—28no—
501C2×C4—24no—
502C2×C4—24no—
503C2×C4—24no—
504C2×C4—24no—
505C2×C4—22no—
506C2×C4—28no—
507C2×C4—22no—
508C2×C4—28no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—24no—
512C2×C4—24no—
513C2×C4—24no—
514C2×C4—24no—
515C2×C4—24no—
516C2×C4—28no—
517C2×C4—28no—
518C2×C4—24no—
519C2×C4—24no—
520C2×C4—24no—
521C2×C4—24no—
522C2×C4—28no—
523C2×C4—24no—
524C2×C4—24no—
525C2×C4—24no—
526C2×C4—24no—
527C2×C4—24no—
528C2×C4—24no—
529C2×C4—28no—
530C2×C4—24no—
531C2×C4—24no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—44no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—44no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—44no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—44no—
559C2×C4—42no—
560C2×C4—44no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—44no—
566C2×C4—42no—
567C2×C4—44no—
568C2×C4—44no—
569C2×C4—44no—
570C2×C4—44no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—44no—
576C2×C4C22×C22⋊C414yes—
577C2×C4C22×C22⋊C4116yes—
578C2×C4C22×C22⋊C414yes—
579C2×C4C22×C22⋊C414yes—
580C2×C4C2×C4×Q818yes—
581C2×C4C2×C4×Q814yes—
582C2×C4#51918yes—
583C2×C4#51914yes—
584C2×C4#54814yes—
585C2×C4#54814yes—
586C2×C4D4×Q814yes—
587C2×C4D4×Q814yes—
588C22C23×C22⋊C411yescharacteristic
589C22C22×C4×Q811yescharacteristic
590C22C2×#51912yes—
591C22C2×#51911yescharacteristic
592C22C4×H(5,𝔽2)12yes—
593C22C2×#54811yescharacteristic
594C22C2×#54811yescharacteristic
595C22C2×D4×Q811yescharacteristic
596C22C2×D4×Q811yescharacteristic
597C22#348411yescharacteristic
598C22#348511yescharacteristic
599C22#348611yescharacteristic
600C22#348611yescharacteristic
601C22#348612yes—
602C22#349612yes—
603C22#350212yes—
604C22#350312yes—
605C22#351812yes—
606C22#352212yes—
607C22#354412yes—
608C22#355012yes—
609C22#355512yes—
610C22#355512yes—
611C4—28no—
612C4—24no—
613C4—28no—
614C4—24no—
615C4—216no—
616C4—44no—
617C4—44no—
618C4—44no—
619C4—44no—
620C4—44no—
621C4—82no—
622C4—82no—
623C4—84no—
624C4—82no—
625C4Q8×C22⋊C418yes—
626C2C2×Q8×C22⋊C411yescharacteristic
627C2#3417311yescharacteristic
628C2#3425611yescharacteristic
629C2#3433512yes—
630C2#3449112yes—
631C2#4489412yes—
632C2#4489612yes—
633C2#4493411yescharacteristic
634C2#4493711yescharacteristic
635C2#4533811yescharacteristic
636C2#4538911yescharacteristic
subgroupG/H
center, Frattini, socle, prime radical, cosocleC24C25
commutatorC23C24×C4
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
18. C2×Q82C4semidirect
142. #391Q8semidirect
357. C2×Q8C4×Q8semidirect

Maximal subgroups

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

subgroupindex
C22×Q822
C2×#23652
C2×#23312
#144502
#144202
#140532
#140242
#139422
#139352
#139272
#139252
#138502
#138172
#137862