Finite Group Databasebeta

← #1495522

Subgroups — #1495522

Order 512.

3256 conjugacy classes of subgroups.

The subgroup lattice has 644 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2683C211yesmaximal, characteristic, complemented
2C2×#2432C211yesmaximal, characteristic
3#15854C212yesmaximal, complemented
4#15671C212yesmaximal, complemented
5#15613C214yesmaximal, complemented
6#15385C214yesmaximal, complemented
7#15230C212yesmaximal, complemented
8#14922C214yesmaximal, complemented
9#14473C212yesmaximal, complemented
10#14229C212yesmaximal, complemented
11#14052C214yesmaximal, complemented
12#13956C212yesmaximal
13#13824C211yesmaximal, characteristic, complemented
14C23×#41C2211yescharacteristic
15C22×C4⋊Q8C2211yescharacteristic
16C22×#124C2211yescharacteristic
17C22×C22⋊Q8C2214yes—
18C22×C22⋊Q8C2211yescharacteristic
19#2683—28no—
20#2432—28no—
21C2×#397C2212yes—
22C2×#396C2214yes—
23C2×#395C2212yes—
24C2×#394C2214yes—
25C2×#393C2212yes—
26C2×#391C2211yescharacteristic
27C2×#391C2212yes—
28C2×#390C2211yescharacteristic
29C2×#389C2212yes—
30C2×#388C2214yes—
31C2×#386C2212yes—
32C2×#386C2212yes—
33C2×#386C2212yes—
34C2×#386C2212yes—
35C2×#385C2214yes—
36C2×#382C2214yes—
37C2×#381C2212yes—
38C2×C4×C22⋊C4C2211yescharacteristic
39C2×C4×C22⋊C4C2214yes—
40C22×#95C2212yes—
41#1446C2214yes—
42#1446C2214yes—
43#1446C2214yes—
44#1445C2212yes—
45#1445C2212yes—
46#1443C2214yes—
47#1442C2212yes—
48#1442C2214yes—
49#1441C2214yes—
50#1441C2214yes—
51#1441C2214yes—
52#1440C2214yes—
53#1440C2214yes—
54#1439C2214yes—
55#1439C2212yes—
56#1438C2214yes—
57#1437C2214yes—
58#1437C2212yes—
59#1436C2212yes—
60#1436C2214yes—
61#1436C2214yes—
62C4⋊#95C2214yes—
63#1434C2214yes—
64C22⋊#95C2214yes—
65C22⋊#95C2214yes—
66C22⋊#95C2214yes—
67C22⋊#95C2214yes—
68C4×#95C2212yes—
69C22×#41—24no—
70C22×#41—28no—
71C22×#41—42no—
72C22×#41—42no—
73C22×#41—44no—
74C23×Q8C2311yescharacteristic
75C23×Q8C2312yes—
76C23×Q8C2311yescharacteristic
77C23×D4C2312yes—
78C23×D4C2311yescharacteristic
79C24×C4C2311yescharacteristic
80C24×C4C2312yes—
81C24×C4C2312yes—
82C24×C4C2312yes—
83C2×C4⋊Q8—24no—
84C2×C4⋊Q8—44no—
85C2×#124—24no—
86C2×#124—28no—
87C2×C22⋊Q8—24no—
88C2×C22⋊Q8—216no—
89C2×C22⋊Q8—216no—
90C2×C22⋊Q8—42no—
91C2×C22⋊Q8—48no—
92C2×C22⋊Q8—42no—
93C22×C4⋊C4C2314yes—
94C22×C4⋊C4C2314yes—
95C22×C4⋊C4C2314yes—
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2311yescharacteristic
100C22×C22⋊C4C2312yes—
101C22×C22⋊C4C2314yes—
102C22×C22⋊C4C2312yes—
103C22×C22⋊C4C2314yes—
104C22×C22⋊C4C2314yes—
105C22×C22⋊C4C2314yes—
106C22×C22⋊C4C2314yes—
107C22×C22⋊C4C2312yes—
108C22×C22⋊C4C2314yes—
109C22×C22⋊C4C2312yes—
110C22×C42C2311yescharacteristic
111C22×C42C2314yes—
112C22×C42C2312yes—
113C22×C42C2311yescharacteristic
114#397—28no—
115#396—216no—
116#395—28no—
117#394—216no—
118#393—44no—
119#391—24no—
120#391—44no—
121#390D418yes—
122#389—28no—
123#388—48no—
124#386—28no—
125#386—28no—
126#386—44no—
127#386—44no—
128#385—216no—
129#382—48no—
130#381—44no—
131C4×C22⋊C4—24no—
132C4×C22⋊C4—48no—
133C2×#95—44no—
134C2×#95C2314yes—
135C2×#95C2312yes—
136C2×#95C2314yes—
137C2×#95C2312yes—
138C2×#95C2314yes—
139C2×#95C2314yes—
140C2×#95C2312yes—
141C2×#95C2314yes—
142C2×#95C2314yes—
143C2×#95C2314yes—
144C2×#95C2312yes—
145C2×#95C2314yes—
146C2×#95C2314yes—
147C2×#95C2312yes—
148C2×#95C2314yes—
149C2×#95C2314yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C25C2411yescharacteristic
158C25C2412yes—
159C2×#41—416no—
160C2×#41—432no—
161C2×#41—432no—
162C2×#41—416no—
163C2×#41—416no—
164C22×Q8—22no—
165C22×Q8—22no—
166C22×Q8—24no—
167C22×Q8—24no—
168C22×Q8—28no—
169C22×Q8—24no—
170C22×Q8—24no—
171C22×Q8—41no—
172C22×Q8—41no—
173C22×Q8—41no—
174C22×Q8—41no—
175C22×Q8—44no—
176C22×Q8—42no—
177C22×Q8—42no—
178C22×Q8—42no—
179C22×D4—24no—
180C22×D4—24no—
181C22×D4—24no—
182C22×D4—24no—
183C22×D4—24no—
184C22×D4—42no—
185C22×D4—41no—
186C22×D4—41no—
187C22×D4—42no—
188C22×D4—42no—
189C22×D4C2×D414yes—
190C22×D4#4118yes—
191C23×C4—24no—
192C23×C4—22no—
193C23×C4—22no—
194C23×C4—22no—
195C23×C4—24no—
196C23×C4—24no—
197C23×C4—24no—
198C23×C4—24no—
199C23×C4—24no—
200C23×C4—24no—
201C23×C4—24no—
202C23×C4—28no—
203C23×C4—28no—
204C23×C4—42no—
205C23×C4—42no—
206C23×C4—42no—
207C23×C4—44no—
208C23×C4—42no—
209C23×C4—42no—
210C23×C4—44no—
211C23×C4—42no—
212C23×C4—42no—
213C23×C4C2412yes—
214C23×C4C2414yes—
215C23×C4C2412yes—
216C23×C4C2414yes—
217C23×C4C2412yes—
218C23×C4C2414yes—
219C23×C4C2411yescharacteristic
220C23×C4C2412yes—
221C23×C4C2414yes—
222C23×C4C2411yescharacteristic
223C23×C4C2412yes—
224C4⋊Q8—416no—
225#124—416no—
226C22⋊Q8—464no—
227C22⋊Q8—416no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—28no—
232C2×C4⋊C4—28no—
233C2×C4⋊C4—28no—
234C2×C4⋊C4—28no—
235C2×C4⋊C4—28no—
236C2×C4⋊C4—28no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—28no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—41no—
241C2×C4⋊C4—41no—
242C2×C4⋊C4—44no—
243C2×C4⋊C4—42no—
244C2×C4⋊C4—44no—
245C2×C4⋊C4—44no—
246C2×C4⋊C4C2×D414yes—
247C2×C4⋊C4#4118yescomplemented
248C2×C22⋊C4—28no—
249C2×C22⋊C4—28no—
250C2×C22⋊C4—28no—
251C2×C22⋊C4—28no—
252C2×C22⋊C4—28no—
253C2×C22⋊C4—28no—
254C2×C22⋊C4—28no—
255C2×C22⋊C4—24no—
256C2×C22⋊C4—28no—
257C2×C22⋊C4—24no—
258C2×C22⋊C4—24no—
259C2×C22⋊C4—28no—
260C2×C22⋊C4—28no—
261C2×C22⋊C4—24no—
262C2×C22⋊C4—24no—
263C2×C22⋊C4—28no—
264C2×C22⋊C4—24no—
265C2×C22⋊C4—24no—
266C2×C22⋊C4—28no—
267C2×C22⋊C4—42no—
268C2×C22⋊C4—44no—
269C2×C22⋊C4—44no—
270C2×C22⋊C4—44no—
271C2×C22⋊C4—42no—
272C2×C22⋊C4—42no—
273C2×C22⋊C4—44no—
274C2×C22⋊C4—42no—
275C2×C22⋊C4—42no—
276C2×C22⋊C4—44no—
277C2×C22⋊C4C2×D4116yes—
278C2×C42—22no—
279C2×C42—24no—
280C2×C42—28no—
281C2×C42—28no—
282C2×C42—42no—
283C2×C42—44no—
284C2×C42—44no—
285C2×C42—42no—
286C2×C42C2×D414yes—
287#95—24no—
288#95—28no—
289#95—28no—
290#95—24no—
291#95—28no—
292#95—28no—
293#95—24no—
294#95—28no—
295#95—24no—
296#95—28no—
297#95—28no—
298#95—28no—
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—42no—
310C24—24no—
311C24—24no—
312C24—22no—
313C24—22no—
314C24—22no—
315C24—22no—
316C24—24no—
317C24—24no—
318C24—21no—
319C24—22no—
320C24—22no—
321C24—22no—
322C24—21no—
323C24—22no—
324C24—24no—
325C24C22×D414yes—
326C24C2×#4114yes—
327C24C2×#4114yes—
328C24H(5,𝔽2)12yes—
329C24C2511yescharacteristic
330#41—4128no—
331C2×Q8—28no—
332C2×Q8—48no—
333C2×Q8—48no—
334C2×Q8—416no—
335C2×Q8—48no—
336C2×Q8—44no—
337C2×Q8—44no—
338C2×Q8—44no—
339C2×Q8—44no—
340C2×Q8—44no—
341C2×Q8—84no—
342C2×Q8—84no—
343C2×Q8—84no—
344C2×Q8—88no—
345C2×Q8—84no—
346C2×D4—216no—
347C2×D4—44no—
348C2×D4—44no—
349C2×D4—48no—
350C2×D4—48no—
351C2×D4—48no—
352C2×D4—48no—
353C2×D4—48no—
354C2×D4—48no—
355C2×D4—48no—
356C2×D4—48no—
357C2×D4#124116yes—
358C22×C4—24no—
359C22×C4—22no—
360C22×C4—21no—
361C22×C4—21no—
362C22×C4—24no—
363C22×C4—28no—
364C22×C4—216no—
365C22×C4—21no—
366C22×C4—22no—
367C22×C4—24no—
368C22×C4—24no—
369C22×C4—24no—
370C22×C4—24no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—24no—
375C22×C4—24no—
376C22×C4—24no—
377C22×C4—24no—
378C22×C4—24no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—24no—
383C22×C4—24no—
384C22×C4—22no—
385C22×C4—24no—
386C22×C4—22no—
387C22×C4—24no—
388C22×C4—21no—
389C22×C4—22no—
390C22×C4—24no—
391C22×C4—22no—
392C22×C4—24no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—24no—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—24no—
403C22×C4—22no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—48no—
416C22×C4—48no—
417C22×C4—48no—
418C22×C4—48no—
419C22×C4—48no—
420C22×C4—48no—
421C22×C4—48no—
422C22×C4—48no—
423C22×C4—44no—
424C22×C4—416no—
425C22×C4—416no—
426C22×C4—48no—
427C22×C4—48no—
428C22×C4—44no—
429C22×C4—88no—
430C22×C4—88no—
431C22×C4—84no—
432C22×C4—84no—
433C22×C4—84no—
434C22×C4—84no—
435C22×C4C22×D414yes—
436C22×C4C22×D412yes—
437C22×C4C22×D414yes—
438C22×C4C2×#4118yes—
439C22×C4C2×#4114yes—
440C22×C4C2×#4114yes—
441C22×C4H(5,𝔽2)12yes—
442C22×C4H(5,𝔽2)14yes—
443C22×C4H(5,𝔽2)14yes—
444C22×C42−1+418yes—
445C4⋊C4—216no—
446C4⋊C4—44no—
447C4⋊C4—416no—
448C4⋊C4—48no—
449C4⋊C4—416no—
450C4⋊C4—416no—
451C4⋊C4—84no—
452C22⋊C4—232no—
453C22⋊C4—416no—
454C22⋊C4—416no—
455C22⋊C4—416no—
456C22⋊C4—48no—
457C22⋊C4—416no—
458C22⋊C4—48no—
459C22⋊C4—88no—
460C22⋊C4—84no—
461C22⋊C4—84no—
462C42—44no—
463C42—416no—
464C42—44no—
465C42—84no—
466C23—28no—
467C23—28no—
468C23—28no—
469C23—24no—
470C23—24no—
471C23—24no—
472C23—24no—
473C23—24no—
474C23—24no—
475C23—28no—
476C23—28no—
477C23—28no—
478C23—22no—
479C23—24no—
480C23—24no—
481C23—24no—
482C23—42no—
483C23—44no—
484C23—44no—
485C23—44no—
486C23—44no—
487C23—44no—
488C23—42no—
489C23—42no—
490C23—44no—
491C23—42no—
492C23—41no—
493C23—42no—
494C23—41no—
495C23—44no—
496C23—41no—
497C23—42no—
498C23—42no—
499C23—44no—
500C23—44no—
501C23—44no—
502C23—42no—
503C23—41no—
504C23C2×#12418yes—
505C23C23×D411yescharacteristic
506C23C22×#4112yes—
507C23C22×#4112yes—
508C23C2×H(5,𝔽2)12yes—
509C23C2×H(5,𝔽2)11yescharacteristic
510C23C2×H(5,𝔽2)12yes—
511C23C2×H(5,𝔽2)11yescharacteristic
512C23C2×2−1+414yes—
513Q8—416no—
514Q8—88no—
515Q8—816no—
516D4—416no—
517D4—432no—
518C2×C4—24no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—24no—
522C2×C4—24no—
523C2×C4—24no—
524C2×C4—28no—
525C2×C4—28no—
526C2×C4—28no—
527C2×C4—28no—
528C2×C4—28no—
529C2×C4—24no—
530C2×C4—22no—
531C2×C4—24no—
532C2×C4—24no—
533C2×C4—24no—
534C2×C4—24no—
535C2×C4—28no—
536C2×C4—28no—
537C2×C4—28no—
538C2×C4—28no—
539C2×C4—416no—
540C2×C4—432no—
541C2×C4—41no—
542C2×C4—42no—
543C2×C4—44no—
544C2×C4—42no—
545C2×C4—41no—
546C2×C4—42no—
547C2×C4—44no—
548C2×C4—42no—
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—42no—
558C2×C4—44no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—44no—
562C2×C4—42no—
563C2×C4—44no—
564C2×C4—44no—
565C2×C4—42no—
566C2×C4—44no—
567C2×C4—42no—
568C2×C4—44no—
569C2×C4—44no—
570C2×C4—44no—
571C2×C4—44no—
572C2×C4—42no—
573C2×C4—44no—
574C2×C4—44no—
575C2×C4—44no—
576C2×C4—44no—
577C2×C4—44no—
578C2×C4—44no—
579C2×C4—44no—
580C2×C4—44no—
581C2×C4—44no—
582C2×C4—44no—
583C2×C4—816no—
584C2×C4—816no—
585C2×C4C2×#12414yes—
586C2×C4#54618yes—
587C22—28no—
588C22—216no—
589C22—44no—
590C22—48no—
591C22—44no—
592C22—44no—
593C22—44no—
594C22—48no—
595C22—48no—
596C22—44no—
597C22—44no—
598C22—42no—
599C22—42no—
600C22—42no—
601C22—48no—
602C22—82no—
603C22—82no—
604C22—82no—
605C22—82no—
606C22C22×#12411yescharacteristic
607C22C2×#54612yes—
608C22#348111yescharacteristic
609C22#348214yes—
610C22#348411yescharacteristic
611C22#348411yescharacteristic
612C22#349012yes—
613C22#349012yes—
614C22#349912yes—
615C22#351114yes—
616C22#351912yes—
617C22#352014yes—
618C22#353614yes—
619C22#353812yes—
620C22#354412yes—
621C22#354911yescharacteristic
622C4—24no—
623C4—28no—
624C4—42no—
625C4—44no—
626C4—48no—
627C4—44no—
628C4—48no—
629C4—82no—
630C4—84no—
631C4—82no—
632C4—84no—
633C2—48no—
634C2—82no—
635C2#3536711yescharacteristic
636C2#3630312yes—
637C2#4347514yes—
638C2#4388812yes—
639C2#4966012yes—
640C2#5154211yescharacteristic
641C2#5904111yescharacteristic
642C2#6003512yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC23×#41C22
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×#2683C2semidirect
3. #15854C2semidirect
4. #15671C2semidirect
5. #15613C2semidirect
6. #15385C2semidirect
7. #15230C2semidirect
8. #14922C2semidirect
9. #14473C2semidirect
10. #14229C2semidirect
11. #14052C2semidirect
13. #13824C2semidirect
247. C2×C4⋊C4#41semidirect

Maximal subgroups

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

subgroupindex
C2×#26832
C2×#24322
#158542
#156712
#156132
#153852
#152302
#149222
#144732
#142292
#140522
#139562
#138242