Finite Group Databasebeta

← #4353092

Subgroups — #4353092

Order 512.

2314 conjugacy classes of subgroups.

The subgroup lattice has 621 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2547C211yesmaximal, characteristic
2C2×#2564C211yesmaximal, characteristic
3#16831C212yesmaximal, complemented
4#16830C212yesmaximal, complemented
5#16560C212yesmaximal, complemented
6#16366C212yesmaximal, complemented
7#15884C212yesmaximal, complemented
8#15854C212yesmaximal, complemented
9#15487C214yesmaximal, complemented
10#15088C212yesmaximal, complemented
11#15015C212yesmaximal, complemented
12#14841C212yesmaximal, complemented
13#14826C212yesmaximal, complemented
14#14473C214yesmaximal
15#13784C211yesmaximal, characteristic
16C22×C4⋊Q8C2211yescharacteristic
17C22×#124C2212yes—
18C22×C22⋊Q8C2211yescharacteristic
19C22×C4×D4C2211yescharacteristic
20#2564—28no—
21#2547—28no—
22C2×#400C2211yescharacteristic
23C2×#400C2212yes—
24C2×#400C2212yes—
25C2×#399C2211yescharacteristic
26C2×#397C2212yes—
27C2×#397C2212yes—
28C2×#395C2211yescharacteristic
29C2×#393C2211yescharacteristic
30C2×#391C2212yes—
31C2×#387C2212yes—
32C2×#386C2212yes—
33C2×#386C2212yes—
34C2×#385C2212yes—
35C2×#380C2212yes—
36C2×#380C2212yes—
37C2×#380C2212yes—
38C2×C4×C4⋊C4C2211yescharacteristic
39C2×C4×C22⋊C4C2211yescharacteristic
40C22×#95C2211yescharacteristic
41C22×#95C2211yescharacteristic
42#1446C2214yes—
43#1446C2214yes—
44#1446C2214yes—
45#1445C2212yes—
46#1445C2212yes—
47#1443C2214yes—
48#1443C2214yes—
49#1443C2214yes—
50#1443C2214yes—
51#1442C2214yes—
52#1442C2214yes—
53#1442C2214yes—
54#1442C2214yes—
55#1442C2212yes—
56#1442C2212yes—
57#1441C2214yes—
58#1441C2212yes—
59#1441C2212yes—
60#1440C2214yes—
61#1440C2214yes—
62#1440C2214yes—
63#1440C2214yes—
64#1437C2212yes—
65#1436C2212yes—
66#1436C2214yes—
67#1434C2214yes—
68#1434C2214yes—
69#1434C2214yes—
70#1434C2212yes—
71C22⋊#95C2214yes—
72C22⋊#95C2214yes—
73C22⋊#95C2214yes—
74C22⋊#95C2214yes—
75C22⋊#95C2212yes—
76C22⋊#95C2212yes—
77C23×Q8C2312yes—
78C23×Q8C2311yescharacteristic
79C23×D4C2311yescharacteristic
80C24×C4C2312yes—
81C24×C4C2311yescharacteristic
82C24×C4C2311yescharacteristic
83C24×C4C2312yes—
84C2×C4⋊Q8—24no—
85C2×C4⋊Q8—42no—
86C2×C4⋊Q8—42no—
87C2×#124—28no—
88C2×#124—216no—
89C2×C22⋊Q8—24no—
90C2×C22⋊Q8—42no—
91C2×C22⋊Q8—42no—
92C2×C4×D4—42no—
93C2×C4×D4—42no—
94C2×C4×D4D418yes—
95C22×C4⋊C4C2312yes—
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2311yescharacteristic
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2311yescharacteristic
103C22×C22⋊C4C2314yes—
104C22×C22⋊C4C2314yes—
105C22×C22⋊C4C2312yes—
106C22×C22⋊C4C2312yes—
107C22×C22⋊C4C2312yes—
108C22×C22⋊C4C2314yes—
109C22×C22⋊C4C2312yes—
110C22×C22⋊C4C2311yescharacteristic
111C22×C22⋊C4C2311yescharacteristic
112C22×C42C2312yes—
113C22×C42C2311yescharacteristic
114C22×C42C2312yes—
115#400—24no—
116#400—44no—
117#400—44no—
118#399—24no—
119#397—28no—
120#397—28no—
121#395—24no—
122#393—24no—
123#391—28no—
124#387—28no—
125#386—28no—
126#386—28no—
127#385—28no—
128#380—28no—
129#380—28no—
130#380—28no—
131C4×C4⋊C4—24no—
132C4×C22⋊C4—42no—
133C2×#95—24no—
134C2×#95—42no—
135C2×#95C2311yescharacteristic
136C2×#95C2312yes—
137C2×#95C2311yescharacteristic
138C2×#95C2312yes—
139C2×#95C2312yes—
140C2×#95C2314yes—
141C2×#95C2314yes—
142C2×#95C2312yes—
143C2×#95C2312yes—
144C2×#95C2312yes—
145C2×#95C2312yes—
146C2×#95C2314yes—
147C2×#95C2314yes—
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×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2312yes—
167C25C2411yescharacteristic
168C25C2411yescharacteristic
169C22×Q8—24no—
170C22×Q8—22no—
171C22×Q8—28no—
172C22×Q8—41no—
173C22×Q8—41no—
174C22×Q8—42no—
175C22×Q8—42no—
176C22×Q8—44no—
177C22×Q8—44no—
178C22×D4—41no—
179C22×D4—41no—
180C22×D4—42no—
181C22×D4C2×D414yes—
182C22×D4#4118yes—
183C23×C4—22no—
184C23×C4—22no—
185C23×C4—22no—
186C23×C4—22no—
187C23×C4—24no—
188C23×C4—24no—
189C23×C4—24no—
190C23×C4—44no—
191C23×C4—44no—
192C23×C4—44no—
193C23×C4—42no—
194C23×C4—42no—
195C23×C4—42no—
196C23×C4—44no—
197C23×C4—44no—
198C23×C4—44no—
199C23×C4C2×D414yes—
200C23×C4C2×D414yes—
201C23×C4C2412yes—
202C23×C4C2414yes—
203C23×C4C2412yes—
204C23×C4C2414yes—
205C23×C4C2414yes—
206C23×C4C2412yes—
207C23×C4C2414yes—
208C23×C4C2411yescharacteristic
209C23×C4C2411yescharacteristic
210C23×C4C2411yescharacteristic
211C23×C4C2412yes—
212C23×C4C2411yescharacteristic
213C23×C4C2411yescharacteristic
214C4⋊Q8—416no—
215#124—432no—
216C22⋊Q8—416no—
217C4×D4—416no—
218C2×C4⋊C4—24no—
219C2×C4⋊C4—22no—
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—42no—
231C2×C4⋊C4—44no—
232C2×C4⋊C4—41no—
233C2×C4⋊C4—41no—
234C2×C4⋊C4—41no—
235C2×C4⋊C4—41no—
236C2×C4⋊C4—42no—
237C2×C4⋊C4—42no—
238C2×C4⋊C4—44no—
239C2×C4⋊C4C2×D414yes—
240C2×C22⋊C4—24no—
241C2×C22⋊C4—28no—
242C2×C22⋊C4—28no—
243C2×C22⋊C4—24no—
244C2×C22⋊C4—28no—
245C2×C22⋊C4—28no—
246C2×C22⋊C4—28no—
247C2×C22⋊C4—28no—
248C2×C22⋊C4—24no—
249C2×C22⋊C4—24no—
250C2×C22⋊C4—24no—
251C2×C22⋊C4—28no—
252C2×C22⋊C4—42no—
253C2×C22⋊C4—42no—
254C2×C22⋊C4—41no—
255C2×C22⋊C4—41no—
256C2×C22⋊C4—41no—
257C2×C22⋊C4—41no—
258C2×C22⋊C4—42no—
259C2×C22⋊C4—42no—
260C2×C22⋊C4—42no—
261C2×C22⋊C4—44no—
262C2×C22⋊C4—44no—
263C2×C22⋊C4—44no—
264C2×C22⋊C4C2×D414yes—
265C2×C22⋊C4C2×D414yes—
266C2×C42—22no—
267C2×C42—22no—
268C2×C42—24no—
269C2×C42—28no—
270C2×C42—42no—
271C2×C42—44no—
272C2×C42C2×D414yes—
273#95—24no—
274#95—22no—
275#95—22no—
276#95—24no—
277#95—24no—
278#95—24no—
279#95—28no—
280#95—28no—
281#95—24no—
282#95—24no—
283#95—28no—
284#95—24no—
285#95—24no—
286#95—28no—
287#95—42no—
288#95—44no—
289#95—44no—
290#95—44no—
291#95—44no—
292#95—44no—
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—
305C24—21no—
306C24—21no—
307C24—21no—
308C24—21no—
309C24—22no—
310C24—22no—
311C24—22no—
312C24—24no—
313C24—24no—
314C24—22no—
315C24—22no—
316C24—22no—
317C24C22×D412yes—
318C24C22×D412yes—
319C24C2×#4114yes—
320C24C2×#4114yes—
321C24C2511yescharacteristic
322C2×Q8—44no—
323C2×Q8—48no—
324C2×Q8—48no—
325C2×Q8—48no—
326C2×Q8—88no—
327C2×Q8—88no—
328C2×Q8—84no—
329C2×Q8—84no—
330C2×Q8—84no—
331C2×D4—44no—
332C2×D4—44no—
333C2×D4—48no—
334C2×D4—48no—
335C2×D4#124116yes—
336C22×C4—21no—
337C22×C4—21no—
338C22×C4—24no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—21no—
342C22×C4—21no—
343C22×C4—21no—
344C22×C4—21no—
345C22×C4—21no—
346C22×C4—21no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—24no—
350C22×C4—24no—
351C22×C4—24no—
352C22×C4—24no—
353C22×C4—22no—
354C22×C4—28no—
355C22×C4—28no—
356C22×C4—22no—
357C22×C4—24no—
358C22×C4—22no—
359C22×C4—24no—
360C22×C4—22no—
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—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—24no—
374C22×C4—24no—
375C22×C4—24no—
376C22×C4—24no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—24no—
381C22×C4—24no—
382C22×C4—22no—
383C22×C4—24no—
384C22×C4—24no—
385C22×C4—24no—
386C22×C4—24no—
387C22×C4—22no—
388C22×C4—24no—
389C22×C4—22no—
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—24no—
401C22×C4—48no—
402C22×C4—48no—
403C22×C4—48no—
404C22×C4—48no—
405C22×C4—48no—
406C22×C4—84no—
407C22×C4—84no—
408C22×C4—84no—
409C22×C4—84no—
410C22×C4—88no—
411C22×C4—84no—
412C22×C4—88no—
413C22×C4—88no—
414C22×C4—84no—
415C22×C4—84no—
416C22×C4—88no—
417C22×C4C22×D412yes—
418C22×C4C22×D412yes—
419C22×C4C22×D412yes—
420C22×C4C22×D412yes—
421C22×C4C22×D412yes—
422C22×C4C2×#4114yes—
423C22×C4H(5,𝔽2)12yes—
424C22×C4H(5,𝔽2)12yes—
425C22×C42−1+414yes—
426C4⋊C4—44no—
427C4⋊C4—48no—
428C4⋊C4—48no—
429C4⋊C4—44no—
430C4⋊C4—48no—
431C4⋊C4—48no—
432C4⋊C4—48no—
433C4⋊C4—84no—
434C22⋊C4—44no—
435C22⋊C4—44no—
436C22⋊C4—48no—
437C22⋊C4—48no—
438C22⋊C4—48no—
439C22⋊C4—416no—
440C22⋊C4—416no—
441C22⋊C4—84no—
442C22⋊C4—88no—
443C42—28no—
444C42—48no—
445C42—84no—
446C23—22no—
447C23—22no—
448C23—22no—
449C23—22no—
450C23—24no—
451C23—24no—
452C23—24no—
453C23—24no—
454C23—24no—
455C23—24no—
456C23—24no—
457C23—24no—
458C23—28no—
459C23—28no—
460C23—28no—
461C23—28no—
462C23—42no—
463C23—42no—
464C23—42no—
465C23—42no—
466C23—42no—
467C23—42no—
468C23—41no—
469C23—41no—
470C23—41no—
471C23—41no—
472C23—44no—
473C23—44no—
474C23—44no—
475C23—44no—
476C23C2×#12414yes—
477C23C2×#12414yes—
478C23C23×D411yescharacteristic
479C23C22×#4112yes—
480C23C2×H(5,𝔽2)11yescharacteristic
481C23C2×H(5,𝔽2)11yescharacteristic
482C23C2×2−1+412yes—
483C23#58512yes—
484C23#58514yes—
485C23#58512yes—
486Q8—816no—
487Q8—88no—
488D4—416no—
489C2×C4—22no—
490C2×C4—22no—
491C2×C4—24no—
492C2×C4—24no—
493C2×C4—24no—
494C2×C4—24no—
495C2×C4—24no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—24no—
500C2×C4—24no—
501C2×C4—24no—
502C2×C4—24no—
503C2×C4—28no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—44no—
514C2×C4—44no—
515C2×C4—44no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—44no—
519C2×C4—42no—
520C2×C4—44no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—44no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—44no—
528C2×C4—42no—
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—42no—
548C2×C4—42no—
549C2×C4—44no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—44no—
553C2×C4—44no—
554C2×C4—44no—
555C2×C4—44no—
556C2×C4—44no—
557C2×C4—44no—
558C2×C4—44no—
559C2×C4—88no—
560C2×C4—88no—
561C2×C4—816no—
562C2×C4—816no—
563C2×C4C2×#12414yes—
564C2×C4#53414yes—
565C22—28no—
566C22—28no—
567C22—24no—
568C22—24no—
569C22—42no—
570C22—42no—
571C22—44no—
572C22—44no—
573C22—44no—
574C22—44no—
575C22—48no—
576C22—48no—
577C22—44no—
578C22—44no—
579C22—42no—
580C22—42no—
581C22C22×#12411yescharacteristic
582C22C2×#53411yescharacteristic
583C22#344911yescharacteristic
584C22#348512yes—
585C22#348912yes—
586C22#350512yes—
587C22#350512yes—
588C22#351514yes—
589C22#352114yes—
590C22#354512yes—
591C22#354512yes—
592C22#354712yes—
593C22#355011yescharacteristic
594C22#355111yescharacteristic
595C22#356414yes—
596C22#356814yes—
597C4—24no—
598C4—24no—
599C4—42no—
600C4—44no—
601C4—42no—
602C4—42no—
603C4—82no—
604C4—84no—
605C4—84no—
606C4—84no—
607C4—82no—
608C4—84no—
609C4—82no—
610C2—44no—
611C2—44no—
612C2#3501911yescharacteristic
613C2#3604212yes—
614C2#3711911yescharacteristic
615C2#3737611yescharacteristic
616C2#4141812yes—
617C2#4691212yes—
618C2#4745212yes—
619C2#4902514yes—
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
3. #16831C2semidirect
4. #16830C2semidirect
5. #16560C2semidirect
6. #16366C2semidirect
7. #15884C2semidirect
8. #15854C2semidirect
9. #15487C2semidirect
10. #15088C2semidirect
11. #15015C2semidirect
12. #14841C2semidirect
13. #14826C2semidirect

Maximal subgroups

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

subgroupindex
C2×#25472
C2×#25642
#168312
#168302
#165602
#163662
#158842
#158542
#154872
#150882
#150152
#148412
#148262
#144732
#137842