Finite Group Databasebeta

← #4941961

Subgroups — #4941961

Order 512.

2888 conjugacy classes of subgroups.

The subgroup lattice has 624 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2432C211yesmaximal, characteristic
2C2×#2379C211yesmaximal, characteristic, complemented
3#16695C212yesmaximal, complemented
4#16267C214yesmaximal, complemented
5#16093C212yesmaximal, complemented
6#15955C214yesmaximal, complemented
7#15465C212yesmaximal, complemented
8#15372C214yesmaximal, complemented
9#15017C212yesmaximal, complemented
10#14738C214yesmaximal, complemented
11#14513C212yesmaximal, complemented
12#14473C211yesmaximal, characteristic, complemented
13#13956C212yesmaximal
14C23×#41C2211yescharacteristic
15C22×C4⋊Q8C2211yescharacteristic
16C22×#124C2211yescharacteristic
17C22×C22⋊Q8C2211yescharacteristic
18#2432—28no—
19C2×#400C2212yes—
20C2×#397C2212yes—
21C2×#394C2214yes—
22#2379—28no—
23C2×#391C2212yes—
24C2×#391C2211yescharacteristic
25C2×#390C2211yescharacteristic
26C2×#388C2214yes—
27C2×#387C2214yes—
28C2×#386C2212yes—
29C2×#386C2212yes—
30C2×#386C2212yes—
31C2×#386C2212yes—
32C2×#385C2214yes—
33C2×#385C2214yes—
34C2×#382C2212yes—
35C2×#381C2212yes—
36C2×C4×C22⋊C4C2211yescharacteristic
37C22×#95C2212yes—
38#1446C2214yes—
39#1446C2212yes—
40#1445C2214yes—
41#1445C2214yes—
42#1444C2212yes—
43#1443C2214yes—
44#1443C2214yes—
45#1443C2214yes—
46#1442C2214yes—
47#1442C2214yes—
48#1442C2214yes—
49#1442C2212yes—
50#1441C2214yes—
51#1441C2214yes—
52#1440C2214yes—
53#1440C2214yes—
54#1439C2212yes—
55#1439C2214yes—
56#1439C2214yes—
57#1439C2214yes—
58#1439C2212yes—
59#1438C2214yes—
60#1437C2212yes—
61#1436C2212yes—
62#1436C2214yes—
63#1434C2212yes—
64C22⋊#95C2214yes—
65C22⋊#95C2214yes—
66C22⋊#95C2214yes—
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—
80C24×C4C2312yes—
81C24×C4C2312yes—
82C2×C4⋊Q8—24no—
83C2×C4⋊Q8—44no—
84C2×#124—28no—
85C2×#124D418yes—
86C2×C22⋊Q8—24no—
87C2×C22⋊Q8—42no—
88C2×C22⋊Q8—42no—
89C22×C4⋊C4C2314yes—
90C22×C4⋊C4C2312yes—
91C22×C4⋊C4C2312yes—
92C22×C4⋊C4C2311yescharacteristic
93C22×C4⋊C4C2314yes—
94C22×C4⋊C4C2312yes—
95C22×C4⋊C4C2312yes—
96C22×C22⋊C4C2312yes—
97C22×C22⋊C4C2312yes—
98C22×C22⋊C4C2314yes—
99C22×C22⋊C4C2314yes—
100C22×C22⋊C4C2314yes—
101C22×C22⋊C4C2314yes—
102C22×C22⋊C4C2314yes—
103C22×C22⋊C4C2312yes—
104C22×C22⋊C4C2312yes—
105C22×C22⋊C4C2314yes—
106C22×C42C2311yescharacteristic
107C22×C42C2311yescharacteristic
108C22×C42C2312yes—
109#400—44no—
110#397—44no—
111#394—216no—
112#391—28no—
113#391—24no—
114#390—24no—
115#388—216no—
116#387—48no—
117#386—28no—
118#386—28no—
119#386—44no—
120#386—44no—
121#385—216no—
122#385—216no—
123#382—28no—
124#381—44no—
125C4×C22⋊C4—24no—
126C2×#95—44no—
127C2×#95C2314yes—
128C2×#95C2314yes—
129C2×#95C2314yes—
130C2×#95C2314yes—
131C2×#95C2314yes—
132C2×#95C2314yes—
133C2×#95C2314yes—
134C2×#95C2314yes—
135C2×#95C2312yes—
136C2×#95C2314yes—
137C2×#95C2312yes—
138C2×#95C2312yes—
139C2×#95C2314yes—
140C2×#95C2312yes—
141C2×#95C2314yes—
142C2×#95C2314yes—
143C2×#95C2314yes—
144C2×#95C2314yes—
145C2×#95C2314yes—
146C2×#95C2314yes—
147C2×#95C2314yes—
148C2×#95C2314yes—
149C2×#95C2312yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C25C2411yescharacteristic
153C25C2412yes—
154C2×#41—416no—
155C2×#41—432no—
156C2×#41—432no—
157C2×#41—416no—
158C2×#41—416no—
159C22×Q8—22no—
160C22×Q8—24no—
161C22×Q8—24no—
162C22×Q8—24no—
163C22×Q8—41no—
164C22×Q8—41no—
165C22×Q8—41no—
166C22×Q8—41no—
167C22×Q8—42no—
168C22×Q8C2×D414yes—
169C22×D4—24no—
170C22×D4—24no—
171C22×D4—24no—
172C22×D4—24no—
173C22×D4—24no—
174C22×D4—42no—
175C22×D4—41no—
176C22×D4—41no—
177C22×D4—42no—
178C22×D4—42no—
179C22×D4C2×D414yes—
180C22×D4#4118yes—
181C23×C4—24no—
182C23×C4—22no—
183C23×C4—22no—
184C23×C4—22no—
185C23×C4—24no—
186C23×C4—24no—
187C23×C4—24no—
188C23×C4—24no—
189C23×C4—24no—
190C23×C4—24no—
191C23×C4—24no—
192C23×C4—42no—
193C23×C4—42no—
194C23×C4—42no—
195C23×C4—44no—
196C23×C4—44no—
197C23×C4—42no—
198C23×C4—42no—
199C23×C4—44no—
200C23×C4—44no—
201C23×C4—42no—
202C23×C4—42no—
203C23×C4C2412yes—
204C23×C4C2414yes—
205C23×C4C2414yes—
206C23×C4C2414yes—
207C23×C4C2414yes—
208C23×C4C2412yes—
209C23×C4C2412yes—
210C23×C4C2411yescharacteristic
211C23×C4C2412yes—
212C23×C4C2412yes—
213C23×C4C2411yescharacteristic
214C4⋊Q8—416no—
215#124—232no—
216C22⋊Q8—416no—
217C2×C4⋊C4—24no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—28no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—28no—
222C2×C4⋊C4—28no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—44no—
227C2×C4⋊C4—42no—
228C2×C4⋊C4—41no—
229C2×C4⋊C4—41no—
230C2×C4⋊C4—44no—
231C2×C4⋊C4—44no—
232C2×C4⋊C4—44no—
233C2×C4⋊C4—44no—
234C2×C4⋊C4—42no—
235C2×C4⋊C4—42no—
236C2×C22⋊C4—28no—
237C2×C22⋊C4—28no—
238C2×C22⋊C4—28no—
239C2×C22⋊C4—24no—
240C2×C22⋊C4—24no—
241C2×C22⋊C4—24no—
242C2×C22⋊C4—24no—
243C2×C22⋊C4—28no—
244C2×C22⋊C4—28no—
245C2×C22⋊C4—24no—
246C2×C22⋊C4—24no—
247C2×C22⋊C4—28no—
248C2×C22⋊C4—28no—
249C2×C22⋊C4—42no—
250C2×C22⋊C4—44no—
251C2×C22⋊C4—42no—
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—42no—
261C2×C22⋊C4—42no—
262C2×C22⋊C4—42no—
263C2×C22⋊C4—42no—
264C2×C22⋊C4—44no—
265C2×C22⋊C4C2×D4116yes—
266C2×C42—24no—
267C2×C42—22no—
268C2×C42—24no—
269C2×C42—42no—
270C2×C42—44no—
271C2×C42C2×D414yes—
272#95—24no—
273#95—28no—
274#95—28no—
275#95—28no—
276#95—28no—
277#95—28no—
278#95—28no—
279#95—28no—
280#95—28no—
281#95—44no—
282#95—42no—
283#95—42no—
284#95—44no—
285#95—44no—
286#95—44no—
287#95—44no—
288#95—44no—
289#95—44no—
290#95—44no—
291#95—44no—
292#95—44no—
293#95—42no—
294#95—44no—
295#95—42no—
296#95—44no—
297C24—22no—
298C24—22no—
299C24—22no—
300C24—22no—
301C24—24no—
302C24—24no—
303C24—24no—
304C24—24no—
305C24—21no—
306C24—22no—
307C24—21no—
308C24—22no—
309C24—22no—
310C24—24no—
311C24—22no—
312C24C22×D414yes—
313C24C2×#4114yes—
314C24C2×#4114yes—
315C24H(5,𝔽2)12yes—
316C24C2511yescharacteristic
317#41—4128no—
318C2×Q8—28no—
319C2×Q8—28no—
320C2×Q8—44no—
321C2×Q8—44no—
322C2×Q8—48no—
323C2×Q8—48no—
324C2×Q8—44no—
325C2×Q8—44no—
326C2×Q8—84no—
327C2×Q8—84no—
328C2×D4—216no—
329C2×D4—44no—
330C2×D4—44no—
331C2×D4—48no—
332C2×D4—48no—
333C2×D4—48no—
334C2×D4—48no—
335C2×D4—48no—
336C2×D4—48no—
337C2×D4—48no—
338C2×D4—48no—
339C2×D4#123116yes—
340C22×C4—22no—
341C22×C4—21no—
342C22×C4—21no—
343C22×C4—28no—
344C22×C4—216no—
345C22×C4—21no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—24no—
349C22×C4—24no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—24no—
357C22×C4—24no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—21no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—24no—
365C22×C4—24no—
366C22×C4—22no—
367C22×C4—24no—
368C22×C4—24no—
369C22×C4—22no—
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—24no—
379C22×C4—22no—
380C22×C4—24no—
381C22×C4—22no—
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—24no—
399C22×C4—24no—
400C22×C4—48no—
401C22×C4—48no—
402C22×C4—48no—
403C22×C4—48no—
404C22×C4—48no—
405C22×C4—48no—
406C22×C4—44no—
407C22×C4—48no—
408C22×C4—48no—
409C22×C4—44no—
410C22×C4—48no—
411C22×C4—48no—
412C22×C4—84no—
413C22×C4—84no—
414C22×C4—88no—
415C22×C4—88no—
416C22×C4—84no—
417C22×C4—84no—
418C22×C4—88no—
419C22×C4—88no—
420C22×C4C22×D414yes—
421C22×C4C22×D414yes—
422C22×C4C22×D412yes—
423C22×C4C2×#4114yes—
424C22×C4H(5,𝔽2)12yes—
425C22×C4H(5,𝔽2)14yes—
426C22×C42−1+414yes—
427C4⋊C4—48no—
428C4⋊C4—416no—
429C4⋊C4—416no—
430C4⋊C4—44no—
431C4⋊C4—84no—
432C4⋊C4—84no—
433C4⋊C4—84no—
434C22⋊C4—232no—
435C22⋊C4—48no—
436C22⋊C4—416no—
437C22⋊C4—48no—
438C22⋊C4—84no—
439C22⋊C4—84no—
440C22⋊C4—88no—
441C22⋊C4—88no—
442C22⋊C4—88no—
443C22⋊C4—88no—
444C42—28no—
445C42—44no—
446C42—84no—
447C23—24no—
448C23—24no—
449C23—28no—
450C23—24no—
451C23—24no—
452C23—24no—
453C23—24no—
454C23—28no—
455C23—28no—
456C23—28no—
457C23—28no—
458C23—28no—
459C23—22no—
460C23—24no—
461C23—24no—
462C23—24no—
463C23—42no—
464C23—42no—
465C23—42no—
466C23—44no—
467C23—44no—
468C23—44no—
469C23—42no—
470C23—44no—
471C23—44no—
472C23—44no—
473C23—41no—
474C23—41no—
475C23—42no—
476C23—42no—
477C23—42no—
478C23—44no—
479C23—44no—
480C23—41no—
481C23—42no—
482C23—44no—
483C23—41no—
484C23—44no—
485C23C2×#12318yes—
486C23C23×D411yescharacteristic
487C23C22×#4112yes—
488C23C2×H(5,𝔽2)11yescharacteristic
489C23C2×H(5,𝔽2)12yes—
490C23C2×H(5,𝔽2)11yescharacteristic
491C23C2×2−1+412yes—
492C23#58512yes—
493C23#58514yes—
494Q8—416no—
495Q8—88no—
496D4—416no—
497D4—432no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—24no—
501C2×C4—24no—
502C2×C4—24no—
503C2×C4—24no—
504C2×C4—24no—
505C2×C4—24no—
506C2×C4—22no—
507C2×C4—24no—
508C2×C4—24no—
509C2×C4—24no—
510C2×C4—24no—
511C2×C4—24no—
512C2×C4—24no—
513C2×C4—24no—
514C2×C4—28no—
515C2×C4—416no—
516C2×C4—432no—
517C2×C4—41no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—41no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—44no—
526C2×C4—44no—
527C2×C4—42no—
528C2×C4—44no—
529C2×C4—44no—
530C2×C4—44no—
531C2×C4—44no—
532C2×C4—42no—
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—42no—
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—42no—
557C2×C4—44no—
558C2×C4—44no—
559C2×C4—44no—
560C2×C4—44no—
561C2×C4—44no—
562C2×C4—44no—
563C2×C4—44no—
564C2×C4—816no—
565C2×C4—816no—
566C2×C4C2×#12314yes—
567C22—28no—
568C22—216no—
569C22—48no—
570C22—44no—
571C22—44no—
572C22—44no—
573C22—48no—
574C22—48no—
575C22—44no—
576C22—44no—
577C22—44no—
578C22—42no—
579C22—42no—
580C22—48no—
581C22—42no—
582C22—82no—
583C22—82no—
584C22—82no—
585C22—82no—
586C22C22×#12311yescharacteristic
587C22#348411yescharacteristic
588C22#348411yescharacteristic
589C22#348912yes—
590C22#349012yes—
591C22#349012yes—
592C22#350012yes—
593C22#351514yes—
594C22#352114yes—
595C22#352911yescharacteristic
596C22#354012yes—
597C22#354114yes—
598C22#354512yes—
599C22#355011yescharacteristic
600C22#355912yes—
601C22#356214yes—
602C4—24no—
603C4—44no—
604C4—42no—
605C4—44no—
606C4—44no—
607C4—44no—
608C4—84no—
609C4—84no—
610C4—84no—
611C4—82no—
612C4—84no—
613C2—48no—
614C2—82no—
615C2#3535011yescharacteristic
616C2#3606712yes—
617C2#4573112yes—
618C2#4622811yescharacteristic
619C2#4928314yes—
620C2#5023712yes—
621C2#5121312yes—
622C2#5174911yescharacteristic
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
2. C2×#2379C2semidirect
3. #16695C2semidirect
4. #16267C2semidirect
5. #16093C2semidirect
6. #15955C2semidirect
7. #15465C2semidirect
8. #15372C2semidirect
9. #15017C2semidirect
10. #14738C2semidirect
11. #14513C2semidirect
12. #14473C2semidirect

Maximal subgroups

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

subgroupindex
C2×#24322
C2×#23792
#166952
#162672
#160932
#159552
#154652
#153722
#150172
#147382
#145132
#144732
#139562