Finite Group Databasebeta

← #4417971

Subgroups — #4417971

Order 512.

3074 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×#2667C211yesmaximal, characteristic, complemented
2C2×#2339C211yesmaximal, characteristic
3#17079C212yesmaximal, complemented
4#16898C212yesmaximal, complemented
5#16302C214yesmaximal, complemented
6#15439C214yesmaximal, complemented
7#15324C214yesmaximal, complemented
8#15181C212yesmaximal, complemented
9#14812C212yesmaximal, complemented
10#14766C214yesmaximal, complemented
11#14675C211yesmaximal, characteristic, complemented
12#14347C211yesmaximal, characteristic, complemented
13#14337C211yesmaximal, characteristic
14#13965C211yesmaximal, characteristic, complemented
15#13956C211yesmaximal, characteristic
16C23×#41C2211yescharacteristic
17C22×C4×D4C2212yes—
18#2667—28no—
19C2×#399C2212yes—
20C2×#398C2212yes—
21C2×#398C2212yes—
22C2×#396C2212yes—
23C2×#396C2214yes—
24#2339—28no—
25C2×#391C2212yes—
26C2×#389C2212yes—
27C2×#387C2212yes—
28C2×#386C2214yes—
29C2×#386C2214yes—
30C2×#385C2214yes—
31C2×#385C2214yes—
32C2×#381C2211yescharacteristic
33C2×#381C2212yes—
34C2×#380C2211yescharacteristic
35C2×#380C2211yescharacteristic
36C2×#380C2211yescharacteristic
37C2×#380C2214yes—
38C2×#380C2214yes—
39C2×#380C2214yes—
40C2×C4×C22⋊C4C2211yescharacteristic
41C22×#95C2211yescharacteristic
42C22×#95C2212yes—
43#1446C2212yes—
44#1446C2211yescharacteristic
45#1446C2211yescharacteristic
46#1445C2214yes—
47#1445C2214yes—
48#1444C2212yes—
49#1444C2212yes—
50#1444C2212yes—
51#1442C2212yes—
52#1442C2214yes—
53#1442C2214yes—
54#1442C2212yes—
55#1442C2212yes—
56#1442C2214yes—
57#1441C2214yes—
58#1441C2214yes—
59#1441C2214yes—
60#1441C2214yes—
61#1441C2214yes—
62#1439C2214yes—
63#1439C2214yes—
64#1439C2214yes—
65#1439C2214yes—
66#1438C2212yes—
67#1438C2212yes—
68#1438C2212yes—
69#1438C2211yescharacteristic
70#1438C2211yescharacteristic
71C4⋊#95C2212yes—
72C4⋊#95C2212yes—
73C22⋊#95C2212yes—
74C22⋊#95C2212yes—
75C22⋊#95C2214yes—
76C22⋊#95C2214yes—
77C22×#41—24no—
78C22×#41—24no—
79C22×#41—48no—
80C22×#41Q818yes—
81C23×Q8C2311yescharacteristic
82C23×D4C2311yescharacteristic
83C23×D4C2312yes—
84C24×C4C2314yes—
85C24×C4C2312yes—
86C24×C4C2312yes—
87C24×C4C2311yescharacteristic
88C2×C4×D4—28no—
89C2×C4×D4—48no—
90C22×C4⋊C4C2312yes—
91C22×C4⋊C4C2312yes—
92C22×C4⋊C4C2312yes—
93C22×C4⋊C4C2312yes—
94C22×C4⋊C4C2312yes—
95C22×C4⋊C4C2312yes—
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2314yes—
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2314yes—
101C22×C4⋊C4C2314yes—
102C22×C22⋊C4C2314yes—
103C22×C22⋊C4C2312yes—
104C22×C22⋊C4C2311yescharacteristic
105C22×C22⋊C4C2312yes—
106C22×C22⋊C4C2314yes—
107C22×C22⋊C4C2314yes—
108C22×C22⋊C4C2311yescharacteristic
109C22×C22⋊C4C2312yes—
110C22×C22⋊C4C2314yes—
111C22×C22⋊C4C2314yes—
112C22×C22⋊C4C2312yes—
113C22×C42C2312yes—
114#399—28no—
115#398—28no—
116#398—28no—
117#396—28no—
118#396—48no—
119#391—28no—
120#389—28no—
121#387—28no—
122#386—216no—
123#386—48no—
124#385—48no—
125#385—48no—
126#381—24no—
127#381—28no—
128#380—24no—
129#380—24no—
130#380—24no—
131#380—48no—
132#380—48no—
133#380—48no—
134C4×C22⋊C4D418yes—
135C2×#95—24no—
136C2×#95—28no—
137C2×#95C2312yes—
138C2×#95C2312yes—
139C2×#95C2312yes—
140C2×#95C2312yes—
141C2×#95C2314yes—
142C2×#95C2311yescharacteristic
143C2×#95C2311yescharacteristic
144C2×#95C2312yes—
145C2×#95C2312yes—
146C2×#95C2311yescharacteristic
147C2×#95C2311yescharacteristic
148C2×#95C2312yes—
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—
163C2×#95C2311yescharacteristic
164C2×#95C2311yescharacteristic
165C25C2411yescharacteristic
166C25C2412yes—
167C2×#41—232no—
168C2×#41—432no—
169C2×#41—432no—
170C2×#41—432no—
171C22×Q8—22no—
172C22×Q8—22no—
173C22×Q8—44no—
174C22×Q8C2×Q814yes—
175C22×D4—28no—
176C22×D4—24no—
177C22×D4—24no—
178C22×D4—28no—
179C22×D4—22no—
180C22×D4—22no—
181C22×D4—44no—
182C22×D4C2×Q814yes—
183C22×D4C2×Q818yes—
184C23×C4—28no—
185C23×C4—24no—
186C23×C4—24no—
187C23×C4—28no—
188C23×C4—24no—
189C23×C4—28no—
190C23×C4—28no—
191C23×C4—28no—
192C23×C4—44no—
193C23×C4—44no—
194C23×C4—44no—
195C23×C4—44no—
196C23×C4—44no—
197C23×C4—42no—
198C23×C4—42no—
199C23×C4—44no—
200C23×C4—44no—
201C23×C4C2×D414yes—
202C23×C4C2×Q818yes—
203C23×C4C2×Q814yes—
204C23×C4#4114yes—
205C23×C4C2412yes—
206C23×C4C2412yes—
207C23×C4C2412yes—
208C23×C4C2412yes—
209C23×C4C2414yes—
210C23×C4C2414yes—
211C23×C4C2412yes—
212C23×C4C2412yes—
213C23×C4C2412yes—
214C23×C4C2412yes—
215C23×C4C2411yescharacteristic
216C23×C4C2411yescharacteristic
217C23×C4C2412yes—
218C4×D4—432no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—24no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—28no—
233C2×C4⋊C4—44no—
234C2×C4⋊C4—44no—
235C2×C4⋊C4—44no—
236C2×C4⋊C4—44no—
237C2×C4⋊C4—44no—
238C2×C4⋊C4—44no—
239C2×C4⋊C4—44no—
240C2×C4⋊C4—44no—
241C2×C4⋊C4—44no—
242C2×C4⋊C4—44no—
243C2×C4⋊C4—44no—
244C2×C4⋊C4—44no—
245C2×C4⋊C4—44no—
246C2×C4⋊C4—44no—
247C2×C4⋊C4—44no—
248C2×C22⋊C4—24no—
249C2×C22⋊C4—24no—
250C2×C22⋊C4—28no—
251C2×C22⋊C4—28no—
252C2×C22⋊C4—28no—
253C2×C22⋊C4—24no—
254C2×C22⋊C4—22no—
255C2×C22⋊C4—22no—
256C2×C22⋊C4—24no—
257C2×C22⋊C4—24no—
258C2×C22⋊C4—28no—
259C2×C22⋊C4—28no—
260C2×C22⋊C4—28no—
261C2×C22⋊C4—44no—
262C2×C22⋊C4—44no—
263C2×C22⋊C4—44no—
264C2×C22⋊C4—44no—
265C2×C22⋊C4—42no—
266C2×C22⋊C4—44no—
267C2×C22⋊C4—42no—
268C2×C22⋊C4—42no—
269C2×C22⋊C4—44no—
270C2×C22⋊C4—42no—
271C2×C22⋊C4—44no—
272C2×C22⋊C4—44no—
273C2×C22⋊C4—44no—
274C2×C22⋊C4—44no—
275C2×C22⋊C4—44no—
276C2×C22⋊C4C2×D418yes—
277C2×C42—44no—
278C2×C42C2×D418yes—
279#95—24no—
280#95—28no—
281#95—24no—
282#95—28no—
283#95—22no—
284#95—22no—
285#95—24no—
286#95—24no—
287#95—24no—
288#95—22no—
289#95—22no—
290#95—24no—
291#95—24no—
292#95—28no—
293#95—44no—
294#95—44no—
295#95—44no—
296#95—44no—
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#95C2×D414yes—
306#95C2×D414yes—
307C24—22no—
308C24—22no—
309C24—22no—
310C24—24no—
311C24—24no—
312C24—24no—
313C24—22no—
314C24—24no—
315C24—24no—
316C24—22no—
317C24—22no—
318C24—24no—
319C24C22×D412yes—
320C24C22×Q812yes—
321C24C22×Q814yes—
322C24C2×#4112yes—
323C24H(5,𝔽2)18yes—
324C24C2511yescharacteristic
325#41—4128no—
326C2×Q8—28no—
327C2×Q8—48no—
328C2×Q8—48no—
329C2×Q8—48no—
330C2×D4—216no—
331C2×D4—216no—
332C2×D4—216no—
333C2×D4—216no—
334C2×D4—216no—
335C2×D4—28no—
336C2×D4—48no—
337C2×D4—416no—
338C2×D4—416no—
339C22×C4—24no—
340C22×C4—24no—
341C22×C4—24no—
342C22×C4—24no—
343C22×C4—216no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—24no—
348C22×C4—24no—
349C22×C4—24no—
350C22×C4—22no—
351C22×C4—24no—
352C22×C4—216no—
353C22×C4—22no—
354C22×C4—24no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—24no—
361C22×C4—24no—
362C22×C4—24no—
363C22×C4—24no—
364C22×C4—24no—
365C22×C4—24no—
366C22×C4—24no—
367C22×C4—24no—
368C22×C4—24no—
369C22×C4—24no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—24no—
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—22no—
383C22×C4—22no—
384C22×C4—24no—
385C22×C4—24no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—24no—
394C22×C4—24no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—48no—
398C22×C4—48no—
399C22×C4—48no—
400C22×C4—48no—
401C22×C4—416no—
402C22×C4—416no—
403C22×C4—416no—
404C22×C4—416no—
405C22×C4—416no—
406C22×C4—48no—
407C22×C4—416no—
408C22×C4—88no—
409C22×C4—88no—
410C22×C4—88no—
411C22×C4—84no—
412C22×C4—84no—
413C22×C4—88no—
414C22×C4—88no—
415C22×C4C22⋊Q8116yes—
416C22×C4C22×D414yes—
417C22×C4C22×D412yes—
418C22×C4C22×D412yes—
419C22×C4C22×D414yes—
420C22×C4C22×Q814yes—
421C22×C4C22×Q812yes—
422C22×C4C22×Q812yes—
423C22×C4C2×#4112yes—
424C22×C4C2×#4112yes—
425C22×C4H(5,𝔽2)14yes—
426C22×C42−1+414yes—
427C4⋊C4—48no—
428C4⋊C4—48no—
429C4⋊C4—48no—
430C4⋊C4—84no—
431C4⋊C4—84no—
432C4⋊C4—84no—
433C4⋊C4—88no—
434C4⋊C4—84no—
435C4⋊C4—84no—
436C4⋊C4—84no—
437C4⋊C4—88no—
438C4⋊C4—88no—
439C22⋊C4—216no—
440C22⋊C4—416no—
441C22⋊C4—416no—
442C22⋊C4—84no—
443C22⋊C4—82no—
444C22⋊C4—82no—
445C22⋊C4—84no—
446C22⋊C4—84no—
447C22⋊C4—88no—
448C22⋊C4—88no—
449C22⋊C4—88no—
450C42—48no—
451C23—24no—
452C23—24no—
453C23—24no—
454C23—24no—
455C23—28no—
456C23—24no—
457C23—24no—
458C23—28no—
459C23—24no—
460C23—22no—
461C23—22no—
462C23—28no—
463C23—24no—
464C23—28no—
465C23—24no—
466C23—22no—
467C23—22no—
468C23—28no—
469C23—28no—
470C23—28no—
471C23—28no—
472C23—44no—
473C23—44no—
474C23—42no—
475C23—42no—
476C23—42no—
477C23—42no—
478C23—44no—
479C23—44no—
480C23—44no—
481C23—44no—
482C23—44no—
483C23—44no—
484C23—44no—
485C23—44no—
486C23C2×C22⋊Q814yes—
487C23#54318yes—
488C23C23×D411yescharacteristic
489C23C23×Q811yescharacteristic
490C23C22×#4111yescharacteristic
491C23C2×H(5,𝔽2)12yes—
492C23C2×H(5,𝔽2)14yes—
493C23C2×2−1+412yes—
494C23#58512yes—
495C23#58512yes—
496Q8—416no—
497D4—232no—
498D4—432no—
499C2×C4—24no—
500C2×C4—24no—
501C2×C4—24no—
502C2×C4—24no—
503C2×C4—24no—
504C2×C4—24no—
505C2×C4—24no—
506C2×C4—24no—
507C2×C4—28no—
508C2×C4—28no—
509C2×C4—24no—
510C2×C4—24no—
511C2×C4—24no—
512C2×C4—24no—
513C2×C4—24no—
514C2×C4—24no—
515C2×C4—416no—
516C2×C4—432no—
517C2×C4—44no—
518C2×C4—44no—
519C2×C4—44no—
520C2×C4—44no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—44no—
526C2×C4—44no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—44no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—44no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—44no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—44no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—44no—
546C2×C4—44no—
547C2×C4—44no—
548C2×C4—44no—
549C2×C4—44no—
550C2×C4—44no—
551C2×C4—44no—
552C2×C4—44no—
553C2×C4—44no—
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—832no—
566C2×C4—816no—
567C2×C4C2×C22⋊Q814yes—
568C2×C4C2×C22⋊Q814yes—
569C2×C4#53718yes—
570C22—216no—
571C22—24no—
572C22—28no—
573C22—24no—
574C22—24no—
575C22—44no—
576C22—48no—
577C22—48no—
578C22—48no—
579C22—44no—
580C22—48no—
581C22—44no—
582C22—44no—
583C22—48no—
584C22—44no—
585C22—48no—
586C22—44no—
587C22C22×C22⋊Q811yescharacteristic
588C22C2×#53712yes—
589C22C2×#54312yes—
590C22#348412yes—
591C22#349012yes—
592C22#349312yes—
593C22#350011yescharacteristic
594C22#350011yescharacteristic
595C22#350312yes—
596C22#350412yes—
597C22#350811yescharacteristic
598C22#350911yescharacteristic
599C22#353414yes—
600C22#353514yes—
601C22#354214yes—
602C22#355914yes—
603C4—28no—
604C4—24no—
605C4—24no—
606C4—44no—
607C4—44no—
608C4—82no—
609C4—84no—
610C4—82no—
611C4—82no—
612C4—82no—
613C4—84no—
614C4—82no—
615C4—82no—
616C2—48no—
617C2—44no—
618C2#3498211yescharacteristic
619C2#3528011yescharacteristic
620C2#3562011yescharacteristic
621C2#4049212yes—
622C2#4199814yes—
623C2#4242412yes—
624C2#4788512yes—
625C2#4959512yes—
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×#2667C2semidirect
3. #17079C2semidirect
4. #16898C2semidirect
5. #16302C2semidirect
6. #15439C2semidirect
7. #15324C2semidirect
8. #15181C2semidirect
9. #14812C2semidirect
10. #14766C2semidirect
11. #14675C2semidirect
12. #14347C2semidirect
14. #13965C2semidirect

Maximal subgroups

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

subgroupindex
C2×#26672
C2×#23392
#170792
#168982
#163022
#154392
#153242
#151812
#148122
#147662
#146752
#143472
#143372
#139652
#139562