Finite Group Databasebeta

← #4308989

Subgroups — #4308989

Order 512.

2949 conjugacy classes of subgroups.

The subgroup lattice has 627 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2614C211yesmaximal, characteristic
2C22×#391C211yesmaximal, characteristic, complemented
3#16635C214yesmaximal, complemented
4#16611C214yesmaximal, complemented
5#16610C214yesmaximal, complemented
6#16609C214yesmaximal, complemented
7#14919C212yesmaximal, complemented
8#14919C212yesmaximal, complemented
9#14912C212yesmaximal
10#14912C212yesmaximal
11#14706C211yesmaximal, characteristic
12#14705C211yesmaximal, characteristic, complemented
13#14336C211yesmaximal, characteristic, complemented
14#14263C211yesmaximal, characteristic, complemented
15#14256C211yesmaximal, characteristic
16C24×Q8C2211yescharacteristic
17C22×#123C2211yescharacteristic
18C22×#123C2211yescharacteristic
19C22×#123C2211yescharacteristic
20C23×C4⋊C4C2211yescharacteristic
21C23×C42C2211yescharacteristic
22#2614—28no—
23C2×#395C2212yes—
24C2×#395C2212yes—
25C2×#393C2212yes—
26C2×#393C2212yes—
27C2×#391—28no—
28C2×#391C2214yes—
29C2×#391C2214yes—
30C2×#386C2212yes—
31C2×#386C2212yes—
32C2×#385C2212yes—
33C2×#385C2212yes—
34C2×#381C2211yescharacteristic
35C2×#381C2211yescharacteristic
36C2×#381C2211yescharacteristic
37C2×#381C2211yescharacteristic
38C2×#381C2211yescharacteristic
39C2×#381C2212yes—
40C2×#381C2212yes—
41C22×#95C2212yes—
42C22×#95C2212yes—
43#1446C2214yes—
44#1446C2214yes—
45#1445C2214yes—
46#1445C2214yes—
47#1444C2214yes—
48#1443C2214yes—
49#1443C2214yes—
50#1443C2214yes—
51#1443C2214yes—
52#1442C2214yes—
53#1442C2214yes—
54#1442C2214yes—
55#1442C2214yes—
56#1442C2214yes—
57#1441C2214yes—
58#1441C2214yes—
59#1441C2214yes—
60#1440C2214yes—
61#1440C2214yes—
62#1439C2214yes—
63#1439C2214yes—
64#1439C2214yes—
65#1437C2214yes—
66#1436C2214yes—
67#1436C2214yes—
68#1436C2214yes—
69C22⋊#95C2212yes—
70C22⋊#95C2212yes—
71C22⋊#95C2212yes—
72C22⋊#95C2212yes—
73C23×Q8—24no—
74C23×Q8—24no—
75C23×Q8—44no—
76C23×Q8—44no—
77C23×Q8D418yes—
78C23×Q8C2312yes—
79C23×Q8C2312yes—
80C23×D4C2311yescharacteristic
81C24×C4C2311yescharacteristic
82C24×C4C2311yescharacteristic
83C24×C4C2312yes—
84C24×C4C2311yescharacteristic
85C24×C4C2312yes—
86C2×#123—24no—
87C2×#123—24no—
88C2×#123—24no—
89C2×#123—24no—
90C2×#123—24no—
91C2×#123—44no—
92C2×#123—44no—
93C22×C4⋊C4—24no—
94C22×C4⋊C4—44no—
95C22×C4⋊C4C2312yes—
96C22×C4⋊C4C2311yescharacteristic
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2311yescharacteristic
99C22×C4⋊C4C2314yes—
100C22×C4⋊C4C2314yes—
101C22×C4⋊C4C2314yes—
102C22×C22⋊C4C2312yes—
103C22×C22⋊C4C2311yescharacteristic
104C22×C22⋊C4C2311yescharacteristic
105C22×C22⋊C4C2311yescharacteristic
106C22×C22⋊C4C2312yes—
107C22×C22⋊C4C2312yes—
108C22×C22⋊C4C2311yescharacteristic
109C22×C22⋊C4C2312yes—
110C22×C22⋊C4C2311yescharacteristic
111C22×C22⋊C4C2314yes—
112C22×C22⋊C4C2314yes—
113C22×C42—24no—
114C22×C42—44no—
115C22×C42C2314yes—
116#395—28no—
117#395—28no—
118#393—28no—
119#393—28no—
120#391—216no—
121#391—216no—
122#386—44no—
123#386—44no—
124#385—44no—
125#385—44no—
126#381—24no—
127#381—24no—
128#381—24no—
129#381—24no—
130#381—24no—
131#381—44no—
132#381—44no—
133C2×#95—28no—
134C2×#95—28no—
135C2×#95C2312yes—
136C2×#95C2312yes—
137C2×#95C2312yes—
138C2×#95C2312yes—
139C2×#95C2312yes—
140C2×#95C2312yes—
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×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C25C2411yescharacteristic
164C25C2411yescharacteristic
165C22×Q8—232no—
166C22×Q8—24no—
167C22×Q8—24no—
168C22×Q8—24no—
169C22×Q8—24no—
170C22×Q8—416no—
171C22×Q8—416no—
172C22×Q8—42no—
173C22×Q8—42no—
174C22×Q8—42no—
175C22×Q8—42no—
176C22×Q8—42no—
177C22×Q8—42no—
178C22×Q8—42no—
179C22×Q8—42no—
180C22×Q8—432no—
181C22×Q8—432no—
182C22×Q8C2×D418yes—
183C22×Q8C2×D418yes—
184C22×D4—22no—
185C22×D4—22no—
186C22×D4—22no—
187C22×D4—24no—
188C22×D4—24no—
189C23×C4—24no—
190C23×C4—24no—
191C23×C4—24no—
192C23×C4—22no—
193C23×C4—22no—
194C23×C4—22no—
195C23×C4—22no—
196C23×C4—24no—
197C23×C4—24no—
198C23×C4—24no—
199C23×C4—24no—
200C23×C4—28no—
201C23×C4—24no—
202C23×C4—24no—
203C23×C4—24no—
204C23×C4—24no—
205C23×C4—24no—
206C23×C4—44no—
207C23×C4—44no—
208C23×C4—44no—
209C23×C4C2×D414yes—
210C23×C4C2×D414yes—
211C23×C4C2×D414yes—
212C23×C4#4114yes—
213C23×C4#4114yes—
214C23×C4C2411yescharacteristic
215C23×C4C2412yes—
216C23×C4C2412yes—
217C23×C4C2411yescharacteristic
218C23×C4C2414yes—
219C23×C4C2414yes—
220C23×C4C2414yes—
221C23×C4C2411yescharacteristic
222C23×C4C2411yescharacteristic
223C23×C4C2412yes—
224C23×C4C2411yescharacteristic
225C23×C4C2414yes—
226C23×C4C2412yes—
227#123—416no—
228#123—416no—
229#123—416no—
230C2×C4⋊C4—28no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—28no—
238C2×C4⋊C4—44no—
239C2×C4⋊C4—44no—
240C2×C4⋊C4—42no—
241C2×C4⋊C4—42no—
242C2×C4⋊C4—42no—
243C2×C4⋊C4—42no—
244C2×C4⋊C4—416no—
245C2×C4⋊C4—44no—
246C2×C4⋊C4—44no—
247C2×C4⋊C4—44no—
248C2×C4⋊C4—44no—
249C2×C4⋊C4—44no—
250C2×C22⋊C4—22no—
251C2×C22⋊C4—22no—
252C2×C22⋊C4—22no—
253C2×C22⋊C4—22no—
254C2×C22⋊C4—22no—
255C2×C22⋊C4—24no—
256C2×C22⋊C4—24no—
257C2×C22⋊C4—24no—
258C2×C22⋊C4—24no—
259C2×C22⋊C4—24no—
260C2×C22⋊C4—24no—
261C2×C22⋊C4—24no—
262C2×C22⋊C4—24no—
263C2×C22⋊C4—22no—
264C2×C22⋊C4—24no—
265C2×C22⋊C4—22no—
266C2×C22⋊C4—24no—
267C2×C22⋊C4—24no—
268C2×C22⋊C4—24no—
269C2×C22⋊C4—28no—
270C2×C22⋊C4—42no—
271C2×C22⋊C4—42no—
272C2×C22⋊C4—42no—
273C2×C22⋊C4—42no—
274C2×C22⋊C4—44no—
275C2×C22⋊C4—44no—
276C2×C22⋊C4—44no—
277C2×C22⋊C4—44no—
278C2×C22⋊C4—44no—
279C2×C42—28no—
280C2×C42—44no—
281C2×C42—44no—
282C2×C42—416no—
283#95—28no—
284#95—28no—
285#95—28no—
286#95—28no—
287#95—24no—
288#95—24no—
289#95—24no—
290#95—24no—
291#95—24no—
292#95—24no—
293#95—28no—
294#95—28no—
295#95—28no—
296#95—28no—
297#95—44no—
298#95—44no—
299#95—44no—
300#95—44no—
301#95—44no—
302#95—44no—
303#95—44no—
304#95—44no—
305#95—44no—
306#95—44no—
307#95—44no—
308#95—44no—
309#95—44no—
310#95—44no—
311C24—24no—
312C24—24no—
313C24—21no—
314C24—21no—
315C24—21no—
316C24—22no—
317C24—22no—
318C24—24no—
319C24—24no—
320C24C22×D412yes—
321C24C2×#4112yes—
322C24C2×#4112yes—
323C24H(5,𝔽2)14yes—
324C24H(5,𝔽2)14yes—
325C24C2511yescharacteristic
326C2×Q8—216no—
327C2×Q8—216no—
328C2×Q8—48no—
329C2×Q8—48no—
330C2×Q8—48no—
331C2×Q8—48no—
332C2×Q8—4128no—
333C2×Q8—48no—
334C2×Q8—48no—
335C2×Q8—48no—
336C2×Q8—48no—
337C2×Q8—48no—
338C2×Q8—48no—
339C2×Q8—48no—
340C2×Q8—48no—
341C2×D4—44no—
342C2×D4—44no—
343C2×D4—44no—
344C2×D4—48no—
345C2×D4—48no—
346C22×C4—28no—
347C22×C4—28no—
348C22×C4—216no—
349C22×C4—216no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—216no—
355C22×C4—28no—
356C22×C4—28no—
357C22×C4—28no—
358C22×C4—28no—
359C22×C4—28no—
360C22×C4—28no—
361C22×C4—24no—
362C22×C4—21no—
363C22×C4—22no—
364C22×C4—21no—
365C22×C4—21no—
366C22×C4—22no—
367C22×C4—21no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—24no—
373C22×C4—24no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—24no—
379C22×C4—24no—
380C22×C4—24no—
381C22×C4—22no—
382C22×C4—21no—
383C22×C4—22no—
384C22×C4—21no—
385C22×C4—24no—
386C22×C4—21no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—21no—
392C22×C4—21no—
393C22×C4—24no—
394C22×C4—24no—
395C22×C4—24no—
396C22×C4—24no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—24no—
402C22×C4—24no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—24no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—416no—
424C22×C4—416no—
425C22×C4—48no—
426C22×C4—48no—
427C22×C4—48no—
428C22×C4—48no—
429C22×C4—48no—
430C22×C4—48no—
431C22×C4—48no—
432C22×C4—48no—
433C22×C4—48no—
434C22×C4—88no—
435C22×C4—88no—
436C22×C4#123116yes—
437C22×C4C22×D414yes—
438C22×C4C22×D412yes—
439C22×C4C22×D412yes—
440C22×C4C22×D414yes—
441C22×C4C2×#4114yes—
442C22×C4C2×#4114yes—
443C22×C42−1+418yes—
444C4⋊C4—416no—
445C4⋊C4—48no—
446C4⋊C4—44no—
447C4⋊C4—44no—
448C4⋊C4—48no—
449C4⋊C4—88no—
450C4⋊C4—88no—
451C22⋊C4—48no—
452C22⋊C4—44no—
453C22⋊C4—44no—
454C22⋊C4—44no—
455C22⋊C4—44no—
456C22⋊C4—44no—
457C22⋊C4—48no—
458C22⋊C4—48no—
459C22⋊C4—48no—
460C22⋊C4—88no—
461C22⋊C4—88no—
462C42—416no—
463C23—28no—
464C23—24no—
465C23—24no—
466C23—28no—
467C23—28no—
468C23—28no—
469C23—28no—
470C23—28no—
471C23—41no—
472C23—41no—
473C23—41no—
474C23—42no—
475C23—42no—
476C23—44no—
477C23—42no—
478C23—42no—
479C23—44no—
480C23—44no—
481C23—44no—
482C23—44no—
483C23—44no—
484C23C2×#12314yes—
485C23#53414yes—
486C23#53414yes—
487C23#53818yes—
488C23#53818yes—
489C23C23×D411yescharacteristic
490C23C22×#4111yescharacteristic
491C23C22×#4111yescharacteristic
492C23C2×H(5,𝔽2)12yes—
493C23C2×H(5,𝔽2)12yes—
494C23C2×2−1+414yes—
495C23#58514yes—
496Q8—432no—
497Q8—432no—
498D4—88no—
499C2×C4—232no—
500C2×C4—24no—
501C2×C4—24no—
502C2×C4—24no—
503C2×C4—24no—
504C2×C4—24no—
505C2×C4—24no—
506C2×C4—24no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—24no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—24no—
515C2×C4—28no—
516C2×C4—28no—
517C2×C4—28no—
518C2×C4—432no—
519C2×C4—416no—
520C2×C4—416no—
521C2×C4—44no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—44no—
531C2×C4—44no—
532C2×C4—44no—
533C2×C4—42no—
534C2×C4—41no—
535C2×C4—42no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—44no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—44no—
542C2×C4—44no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—44no—
546C2×C4—44no—
547C2×C4—44no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
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—44no—
570C2×C4—44no—
571C2×C4—44no—
572C2×C4—44no—
573C2×C4—44no—
574C2×C4—44no—
575C2×C4—816no—
576C2×C4C2×#12318yes—
577C22—216no—
578C22—28no—
579C22—24no—
580C22—28no—
581C22—24no—
582C22—216no—
583C22—84no—
584C22—82no—
585C22—81no—
586C22—82no—
587C22—81no—
588C22—84no—
589C22—81no—
590C22#262218yes—
591C22C22×#12311yescharacteristic
592C22C2×#53411yescharacteristic
593C22C2×#53411yescharacteristic
594C22C2×#53812yes—
595C22C2×#53812yes—
596C22#346114yes—
597C22#348612yes—
598C22#348712yes—
599C22#350714yes—
600C22#353514yes—
601C22#353514yes—
602C22#354414yes—
603C22#354514yes—
604C4—28no—
605C4—48no—
606C4—44no—
607C4—42no—
608C4—42no—
609C4—84no—
610C4—82no—
611C4—81no—
612C4—84no—
613C4—82no—
614C4—81no—
615C4—84no—
616C4—81no—
617C2—28no—
618C2—161no—
619C2C2×#262211yescharacteristic
620C2#3518212yes—
621C2#3714014yes—
622C2#3718612yes—
623C2#3718711yescharacteristic
624C2#3725811yescharacteristic
625C2#3751814yes—
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
2. C22×#391C2semidirect
3. #16635C2semidirect
4. #16611C2semidirect
5. #16610C2semidirect
6. #16609C2semidirect
7. #14919C2semidirect
8. #14919C2semidirect
12. #14705C2semidirect
13. #14336C2semidirect
14. #14263C2semidirect

Maximal subgroups

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

subgroupindex
C2×#26142
C22×#3912
#166352
#166112
#166102
#166092
#149192
#149122
#147062
#147052
#143362
#142632
#142562