Finite Group Databasebeta

← #4933528

Subgroups — #4933528

Order 512.

2748 conjugacy classes of subgroups.

The subgroup lattice has 616 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2809C211yesmaximal, characteristic, complemented
2C2×#2432C211yesmaximal, characteristic
3#16558C212yesmaximal, complemented
4#16371C214yesmaximal, complemented
5#15820C212yesmaximal, complemented
6#15670C212yesmaximal, complemented
7#15639C214yesmaximal, complemented
8#15431C214yesmaximal, complemented
9#15045C212yesmaximal, complemented
10#14803C214yesmaximal, complemented
11#14450C212yesmaximal, complemented
12#13937C212yesmaximal
13#13834C211yesmaximal, characteristic, complemented
14C23×#41C2211yescharacteristic
15C22×C4⋊Q8C2211yescharacteristic
16C22×#124C2211yescharacteristic
17C22×C22⋊Q8C2211yescharacteristic
18#2809—28no—
19#2432—28no—
20C2×#400C2212yes—
21C2×#398C2214yes—
22C2×#397C2214yes—
23C2×#397C2212yes—
24C2×#394C2214yes—
25C2×#391C2212yes—
26C2×#391C2212yes—
27C2×#391C2211yescharacteristic
28C2×#390C2211yescharacteristic
29C2×#390C2212yes—
30C2×#390C2212yes—
31C2×#389C2212yes—
32C2×#387C2214yes—
33C2×#385C2214yes—
34C2×#382C2212yes—
35C2×#380C2212yes—
36C2×C4×C22⋊C4C2211yescharacteristic
37C2×C4×C22⋊C4C2212yes—
38#1446C2214yes—
39#1445C2212yes—
40#1442C2214yes—
41#1442C2212yes—
42#1442C2214yes—
43#1442C2212yes—
44#1441C2214yes—
45#1441C2214yes—
46#1440C2214yes—
47#1440C2214yes—
48#1440C2214yes—
49#1439C2212yes—
50#1439C2214yes—
51#1439C2214yes—
52#1439C2214yes—
53#1439C2214yes—
54#1439C2214yes—
55#1439C2212yes—
56#1438C2214yes—
57#1438C2214yes—
58#1437C2212yes—
59#1437C2214yes—
60#1436C2214yes—
61#1436C2212yes—
62#1436C2214yes—
63#1436C2214yes—
64#1434C2212yes—
65#1434C2214yes—
66#1434C2214yes—
67C22⋊#95C2214yes—
68C22⋊#95C2214yes—
69C22×#41—24no—
70C22×#41—28no—
71C22×#41—42no—
72C22×#41—42no—
73C22×#41—44no—
74C23×Q8C2311yescharacteristic
75C23×Q8C2311yescharacteristic
76C23×D4C2312yes—
77C23×D4C2311yescharacteristic
78C24×C4C2311yescharacteristic
79C24×C4C2312yes—
80C2×C4⋊Q8—24no—
81C2×C4⋊Q8—28no—
82C2×#124—24no—
83C2×#124—44no—
84C2×C22⋊Q8—24no—
85C2×C22⋊Q8—42no—
86C2×C22⋊Q8—42no—
87C22×C4⋊C4C2314yes—
88C22×C4⋊C4C2312yes—
89C22×C4⋊C4C2312yes—
90C22×C4⋊C4C2312yes—
91C22×C4⋊C4C2312yes—
92C22×C4⋊C4C2312yes—
93C22×C4⋊C4C2314yes—
94C22×C4⋊C4C2312yes—
95C22×C4⋊C4C2311yescharacteristic
96C22×C22⋊C4C2312yes—
97C22×C22⋊C4C2312yes—
98C22×C22⋊C4C2314yes—
99C22×C22⋊C4C2314yes—
100C22×C22⋊C4C2314yes—
101C22×C22⋊C4C2314yes—
102C22×C22⋊C4C2314yes—
103C22×C22⋊C4C2314yes—
104C22×C22⋊C4C2314yes—
105C22×C22⋊C4C2314yes—
106C22×C42C2311yescharacteristic
107C22×C42C2312yes—
108C22×C42C2312yes—
109C22×C42C2312yes—
110C22×C42C2311yescharacteristic
111#400—28no—
112#398—216no—
113#397—44no—
114#397—48no—
115#394—216no—
116#391—28no—
117#391—28no—
118#391D418yes—
119#390—24no—
120#390—44no—
121#390—44no—
122#389—28no—
123#387—216no—
124#385—216no—
125#382—44no—
126#380—44no—
127C4×C22⋊C4—24no—
128C4×C22⋊C4—44no—
129C2×#95C2314yes—
130C2×#95C2314yes—
131C2×#95C2314yes—
132C2×#95C2312yes—
133C2×#95C2314yes—
134C2×#95C2314yes—
135C2×#95C2314yes—
136C2×#95C2314yes—
137C2×#95C2312yes—
138C2×#95C2314yes—
139C2×#95C2312yes—
140C2×#95C2314yes—
141C2×#95C2314yes—
142C2×#95C2314yes—
143C2×#95C2314yes—
144C2×#95C2312yes—
145C2×#95C2312yes—
146C2×#95C2314yes—
147C2×#95C2314yes—
148C2×#95C2314yes—
149C2×#95C2314yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C25C2411yescharacteristic
153C25C2412yes—
154C2×#41—416no—
155C2×#41—432no—
156C2×#41—416no—
157C2×#41—432no—
158C2×#41—416no—
159C22×Q8—22no—
160C22×Q8—24no—
161C22×Q8—42no—
162C22×Q8—41no—
163C22×Q8—41no—
164C22×Q8—41no—
165C22×Q8—41no—
166C22×Q8—42no—
167C22×Q8C2×D414yes—
168C22×Q8#4118yes—
169C22×D4—24no—
170C22×D4—24no—
171C22×D4—24no—
172C22×D4—22no—
173C22×D4—24no—
174C22×D4—24no—
175C22×D4—24no—
176C22×D4—24no—
177C22×D4—42no—
178C22×D4—42no—
179C22×D4—41no—
180C22×D4—41no—
181C23×C4—22no—
182C23×C4—24no—
183C23×C4—22no—
184C23×C4—24no—
185C23×C4—22no—
186C23×C4—24no—
187C23×C4—24no—
188C23×C4—24no—
189C23×C4—24no—
190C23×C4—42no—
191C23×C4—42no—
192C23×C4—42no—
193C23×C4C2414yes—
194C23×C4C2412yes—
195C23×C4C2414yes—
196C23×C4C2414yes—
197C23×C4C2414yes—
198C23×C4C2412yes—
199C23×C4C2412yes—
200C23×C4C2411yescharacteristic
201C23×C4C2412yes—
202C23×C4C2412yes—
203C23×C4C2411yescharacteristic
204C4⋊Q8—416no—
205#124—416no—
206C22⋊Q8—416no—
207C2×C4⋊C4—24no—
208C2×C4⋊C4—28no—
209C2×C4⋊C4—28no—
210C2×C4⋊C4—24no—
211C2×C4⋊C4—28no—
212C2×C4⋊C4—28no—
213C2×C4⋊C4—24no—
214C2×C4⋊C4—24no—
215C2×C4⋊C4—24no—
216C2×C4⋊C4—24no—
217C2×C4⋊C4—24no—
218C2×C4⋊C4—44no—
219C2×C4⋊C4—44no—
220C2×C4⋊C4—44no—
221C2×C4⋊C4—41no—
222C2×C4⋊C4—41no—
223C2×C4⋊C4—42no—
224C2×C4⋊C4—42no—
225C2×C4⋊C4—42no—
226C2×C4⋊C4—42no—
227C2×C4⋊C4—42no—
228C2×C4⋊C4—42no—
229C2×C4⋊C4—44no—
230C2×C4⋊C4—42no—
231C2×C4⋊C4C2×D414yes—
232C2×C22⋊C4—24no—
233C2×C22⋊C4—28no—
234C2×C22⋊C4—28no—
235C2×C22⋊C4—24no—
236C2×C22⋊C4—28no—
237C2×C22⋊C4—24no—
238C2×C22⋊C4—28no—
239C2×C22⋊C4—24no—
240C2×C22⋊C4—28no—
241C2×C22⋊C4—28no—
242C2×C22⋊C4—28no—
243C2×C22⋊C4—42no—
244C2×C22⋊C4—44no—
245C2×C22⋊C4—44no—
246C2×C22⋊C4—44no—
247C2×C22⋊C4—42no—
248C2×C22⋊C4—44no—
249C2×C22⋊C4—44no—
250C2×C22⋊C4—44no—
251C2×C22⋊C4—44no—
252C2×C22⋊C4—44no—
253C2×C22⋊C4—44no—
254C2×C22⋊C4—44no—
255C2×C22⋊C4—44no—
256C2×C22⋊C4—44no—
257C2×C22⋊C4—44no—
258C2×C22⋊C4—44no—
259C2×C22⋊C4—44no—
260C2×C22⋊C4—44no—
261C2×C22⋊C4—44no—
262C2×C42—22no—
263C2×C42—24no—
264C2×C42—24no—
265C2×C42—24no—
266C2×C42—42no—
267C2×C42—42no—
268C2×C42—44no—
269C2×C42—42no—
270C2×C42—42no—
271C2×C42—42no—
272C2×C42—42no—
273C2×C42C2×D414yes—
274#95—24no—
275#95—24no—
276#95—28no—
277#95—28no—
278#95—28no—
279#95—28no—
280#95—28no—
281#95—28no—
282#95—28no—
283#95—44no—
284#95—44no—
285#95—44no—
286#95—42no—
287#95—42no—
288#95—44no—
289#95—44no—
290#95—44no—
291#95—44no—
292#95—42no—
293#95—44no—
294#95—44no—
295#95—44no—
296#95C2×D4116yes—
297C24—21no—
298C24—22no—
299C24—22no—
300C24—22no—
301C24—22no—
302C24—21no—
303C24—22no—
304C24—22no—
305C24—22no—
306C24—22no—
307C24—24no—
308C24—22no—
309C24—24no—
310C24—24no—
311C24—24no—
312C24—24no—
313C24—22no—
314C24H(5,𝔽2)12yes—
315C24H(5,𝔽2)14yes—
316C24C2511yescharacteristic
317#41—4128no—
318C2×Q8—44no—
319C2×Q8—44no—
320C2×Q8—48no—
321C2×Q8—48no—
322C2×Q8—44no—
323C2×Q8—44no—
324C2×Q8—44no—
325C2×Q8—84no—
326C2×Q8—84no—
327C2×Q8#124116yes—
328C2×D4—28no—
329C2×D4—216no—
330C2×D4—48no—
331C2×D4—48no—
332C2×D4—48no—
333C2×D4—48no—
334C2×D4—48no—
335C2×D4—48no—
336C2×D4—44no—
337C2×D4—48no—
338C2×D4—44no—
339C2×D4—48no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—21no—
346C22×C4—21no—
347C22×C4—28no—
348C22×C4—216no—
349C22×C4—21no—
350C22×C4—22no—
351C22×C4—24no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—24no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—21no—
358C22×C4—24no—
359C22×C4—22no—
360C22×C4—24no—
361C22×C4—22no—
362C22×C4—24no—
363C22×C4—22no—
364C22×C4—24no—
365C22×C4—24no—
366C22×C4—24no—
367C22×C4—22no—
368C22×C4—24no—
369C22×C4—24no—
370C22×C4—24no—
371C22×C4—24no—
372C22×C4—24no—
373C22×C4—24no—
374C22×C4—24no—
375C22×C4—24no—
376C22×C4—24no—
377C22×C4—24no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—24no—
381C22×C4—24no—
382C22×C4—24no—
383C22×C4—24no—
384C22×C4—24no—
385C22×C4—24no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—24no—
392C22×C4—24no—
393C22×C4—24no—
394C22×C4—24no—
395C22×C4—24no—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—48no—
399C22×C4—48no—
400C22×C4—48no—
401C22×C4—48no—
402C22×C4—48no—
403C22×C4—48no—
404C22×C4—44no—
405C22×C4—48no—
406C22×C4—44no—
407C22×C4—48no—
408C22×C4C22×D414yes—
409C22×C4C22×D414yes—
410C22×C4C22×D412yes—
411C22×C4C22×D414yes—
412C22×C4C2×#4114yes—
413C22×C4C2×#4114yes—
414C22×C4C2×#4114yes—
415C22×C4H(5,𝔽2)12yes—
416C22×C42−1+414yes—
417C4⋊C4—44no—
418C4⋊C4—48no—
419C4⋊C4—416no—
420C4⋊C4—48no—
421C4⋊C4—416no—
422C4⋊C4—84no—
423C4⋊C4—84no—
424C4⋊C4—84no—
425C4⋊C4—84no—
426C22⋊C4—48no—
427C22⋊C4—416no—
428C22⋊C4—48no—
429C22⋊C4—416no—
430C22⋊C4—416no—
431C22⋊C4—88no—
432C22⋊C4—88no—
433C22⋊C4—88no—
434C22⋊C4—88no—
435C22⋊C4—88no—
436C42—44no—
437C42—48no—
438C42—44no—
439C42—84no—
440C42—84no—
441C23—22no—
442C23—24no—
443C23—24no—
444C23—24no—
445C23—24no—
446C23—28no—
447C23—24no—
448C23—28no—
449C23—24no—
450C23—41no—
451C23—42no—
452C23—42no—
453C23—41no—
454C23—42no—
455C23—44no—
456C23—42no—
457C23—44no—
458C23—42no—
459C23—42no—
460C23—42no—
461C23—44no—
462C23—44no—
463C23—44no—
464C23—44no—
465C23—44no—
466C23—44no—
467C23—41no—
468C23—42no—
469C23—44no—
470C23—44no—
471C23—44no—
472C23—42no—
473C23—41no—
474C23—42no—
475C23—42no—
476C23—42no—
477C23—44no—
478C23—44no—
479C23—44no—
480C23C23×D411yescharacteristic
481C23C22×#4112yes—
482C23C2×H(5,𝔽2)11yescharacteristic
483C23C2×H(5,𝔽2)12yes—
484C23C2×H(5,𝔽2)11yescharacteristic
485C23C2×2−1+412yes—
486C23#58514yes—
487C23#58512yes—
488Q8—416no—
489Q8—88no—
490D4—432no—
491D4—416no—
492C2×C4—22no—
493C2×C4—24no—
494C2×C4—22no—
495C2×C4—24no—
496C2×C4—24no—
497C2×C4—24no—
498C2×C4—24no—
499C2×C4—24no—
500C2×C4—24no—
501C2×C4—24no—
502C2×C4—24no—
503C2×C4—24no—
504C2×C4—22no—
505C2×C4—24no—
506C2×C4—24no—
507C2×C4—28no—
508C2×C4—24no—
509C2×C4—24no—
510C2×C4—28no—
511C2×C4—432no—
512C2×C4—416no—
513C2×C4—42no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—44no—
519C2×C4—42no—
520C2×C4—44no—
521C2×C4—42no—
522C2×C4—44no—
523C2×C4—44no—
524C2×C4—42no—
525C2×C4—44no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—44no—
529C2×C4—44no—
530C2×C4—44no—
531C2×C4—44no—
532C2×C4—44no—
533C2×C4—44no—
534C2×C4—44no—
535C2×C4—44no—
536C2×C4—44no—
537C2×C4—44no—
538C2×C4—44no—
539C2×C4—44no—
540C2×C4—44no—
541C2×C4—44no—
542C2×C4—44no—
543C2×C4—44no—
544C2×C4—44no—
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×C4C2×#12414yes—
558C2×C4C2×#12418yes—
559C22—44no—
560C22—42no—
561C22—44no—
562C22—48no—
563C22—48no—
564C22—42no—
565C22—42no—
566C22—44no—
567C22—48no—
568C22—84no—
569C22—82no—
570C22—82no—
571C22—82no—
572C22—84no—
573C22—82no—
574C22—82no—
575C22—82no—
576C22—82no—
577C22—82no—
578C22C22×#12411yescharacteristic
579C22#348411yescharacteristic
580C22#348411yescharacteristic
581C22#348712yes—
582C22#348912yes—
583C22#349012yes—
584C22#349012yes—
585C22#351614yes—
586C22#352414yes—
587C22#352911yescharacteristic
588C22#353314yes—
589C22#354512yes—
590C22#355011yescharacteristic
591C22#355112yes—
592C22#355914yes—
593C22#355912yes—
594C4—28no—
595C4—24no—
596C4—44no—
597C4—42no—
598C4—44no—
599C4—44no—
600C4—82no—
601C4—84no—
602C4—84no—
603C4—84no—
604C4—84no—
605C2—84no—
606C2—82no—
607C2#3536711yescharacteristic
608C2#3585412yes—
609C2#4622811yescharacteristic
610C2#4627512yes—
611C2#4783612yes—
612C2#5000914yes—
613C2#5198812yes—
614C2#5208311yescharacteristic
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×#2809C2semidirect
3. #16558C2semidirect
4. #16371C2semidirect
5. #15820C2semidirect
6. #15670C2semidirect
7. #15639C2semidirect
8. #15431C2semidirect
9. #15045C2semidirect
10. #14803C2semidirect
11. #14450C2semidirect
13. #13834C2semidirect

Maximal subgroups

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

subgroupindex
C2×#28092
C2×#24322
#165582
#163712
#158202
#156702
#156392
#154312
#150452
#148032
#144502
#139372
#138342