Finite Group Databasebeta

← #900727

Subgroups — #900727

Order 512.

2810 conjugacy classes of subgroups.

The subgroup lattice has 660 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2811C211yesmaximal, characteristic, complemented
2C2×#2733C211yesmaximal, characteristic
3#17077C212yesmaximal, complemented
4#16878C212yesmaximal, complemented
5#15772C214yesmaximal, complemented
6#15317C214yesmaximal, complemented
7#15070C212yesmaximal, complemented
8#14928C214yesmaximal, complemented
9#14708C212yesmaximal, complemented
10#14492C212yesmaximal, complemented
11#14232C214yesmaximal, complemented
12#14026C212yesmaximal, complemented
13C4×#391C211yesmaximal, characteristic, complemented
14C22×#124C2212yes—
15C22×#124C2212yes—
16C22×#124C2212yes—
17C22×C4×Q8C2211yescharacteristic
18#2811—28no—
19#2733—28no—
20C2×#395C2214yes—
21C2×#395C2212yes—
22C2×#394C2212yes—
23C2×#394C2214yes—
24C2×#393C2214yes—
25C2×#393C2214yes—
26C2×#392C2214yes—
27C2×#391C2212yes—
28C2×#391C2214yes—
29C2×#391C2212yes—
30C2×#391C2211yescharacteristic
31C2×#391C2212yes—
32C2×#391C2211yescharacteristic
33C2×#390C2212yes—
34C2×#390C2212yes—
35C2×#390C2212yes—
36C2×#390C2212yes—
37C2×#388C2214yes—
38C2×#388C2214yes—
39C2×#382C2212yes—
40C2×C4×C4⋊C4C2211yescharacteristic
41C2×C43C2211yescharacteristic
42#1446C2212yes—
43#1442C2214yes—
44#1442C2214yes—
45#1440C2214yes—
46#1439C2214yes—
47#1439C2214yes—
48#1439C2214yes—
49#1439C2214yes—
50#1439C2214yes—
51#1439C2214yes—
52#1438C2214yes—
53#1438C2214yes—
54#1438C2212yes—
55#1437C2214yes—
56#1437C2214yes—
57#1437C2214yes—
58#1437C2214yes—
59#1437C2214yes—
60#1437C2214yes—
61#1436C2214yes—
62C4⋊#95C2212yes—
63C4⋊#95C2211yescharacteristic
64C4⋊#95C2211yescharacteristic
65#1434C2212yes—
66#1434C2214yes—
67C4×#95C2214yes—
68C4×#95C2212yes—
69C23×Q8C2312yes—
70C23×Q8C2312yes—
71C23×Q8C2311yescharacteristic
72C23×D4C2312yes—
73C23×D4C2312yes—
74C23×D4C2312yes—
75C2×#124—28no—
76C2×#124—28no—
77C2×#124—28no—
78C2×#124—28no—
79C2×#124—44no—
80C2×#124—44no—
81C2×#124—44no—
82C2×#124—48no—
83C2×C4×Q8—24no—
84C2×C4×Q8—42no—
85C2×C4×Q8—42no—
86C22×C4⋊C4C2312yes—
87C22×C4⋊C4C2314yes—
88C22×C4⋊C4C2312yes—
89C22×C4⋊C4C2312yes—
90C22×C4⋊C4C2311yescharacteristic
91C22×C4⋊C4C2311yescharacteristic
92C22×C4⋊C4C2312yes—
93C22×C4⋊C4C2311yescharacteristic
94C22×C22⋊C4C2314yes—
95C22×C22⋊C4C2314yes—
96C22×C22⋊C4C2314yes—
97C22×C22⋊C4C2314yes—
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×C42C2312yes—
111C22×C42C2312yes—
112C22×C42C2312yes—
113C22×C42C2311yescharacteristic
114C22×C42C2311yescharacteristic
115#395—28no—
116#395—216no—
117#394—28no—
118#394—48no—
119#393—216no—
120#393—48no—
121#392—216no—
122#391—28no—
123#391—24no—
124#391—28no—
125#391—216no—
126#391—44no—
127#391D418yes—
128#390—28no—
129#390—44no—
130#390—44no—
131#390—44no—
132#388—216no—
133#388—48no—
134#382—44no—
135C4×C4⋊C4—42no—
136C2×#95C2314yes—
137C2×#95C2314yes—
138C2×#95C2314yes—
139C2×#95C2314yes—
140C2×#95C2314yes—
141C2×#95C2312yes—
142C2×#95C2314yes—
143C2×#95C2312yes—
144C2×#95C2314yes—
145C2×#95C2314yes—
146C2×#95C2314yes—
147C2×#95C2314yes—
148C2×#95C2314yes—
149C2×#95C2314yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2312yes—
153C2×#95C2314yes—
154C43—42no—
155C25C2414yes—
156C22×Q8—24no—
157C22×Q8—24no—
158C22×Q8—24no—
159C22×Q8—41no—
160C22×Q8—41no—
161C22×Q8—41no—
162C22×Q8—41no—
163C22×Q8—42no—
164C22×Q8—42no—
165C22×Q8—42no—
166C22×Q8—44no—
167C22×Q8—42no—
168C22×Q8—42no—
169C22×Q8—42no—
170C22×Q8—44no—
171C22×Q8C2×D414yes—
172C22×Q8#4118yes—
173C22×Q8#4114yes—
174C22×Q8#4114yes—
175C22×D4—24no—
176C22×D4—24no—
177C22×D4—24no—
178C22×D4—24no—
179C22×D4—24no—
180C22×D4—24no—
181C22×D4—24no—
182C22×D4—42no—
183C22×D4—42no—
184C22×D4—42no—
185C22×D4—44no—
186C22×D4—42no—
187C22×D4—42no—
188C22×D4—42no—
189C22×D4—42no—
190C22×D4—42no—
191C22×D4—42no—
192C22×D4—42no—
193C22×D4—44no—
194C23×C4C2414yes—
195C23×C4C2412yes—
196C23×C4C2414yes—
197C23×C4C2412yes—
198C23×C4C2412yes—
199C23×C4C2414yes—
200C23×C4C2411yescharacteristic
201C23×C4C2412yes—
202C23×C4C2411yescharacteristic
203C23×C4C2412yes—
204C23×C4C2412yes—
205C23×C4C2411yescharacteristic
206#124—432no—
207#124—432no—
208#124—432no—
209C4×Q8—416no—
210C2×C4⋊C4—24no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—24no—
214C2×C4⋊C4—24no—
215C2×C4⋊C4—28no—
216C2×C4⋊C4—24no—
217C2×C4⋊C4—24no—
218C2×C4⋊C4—41no—
219C2×C4⋊C4—41no—
220C2×C4⋊C4—42no—
221C2×C4⋊C4—41no—
222C2×C4⋊C4—41no—
223C2×C4⋊C4—41no—
224C2×C4⋊C4—41no—
225C2×C4⋊C4—42no—
226C2×C4⋊C4—42no—
227C2×C4⋊C4—42no—
228C2×C4⋊C4—44no—
229C2×C4⋊C4—42no—
230C2×C4⋊C4—42no—
231C2×C4⋊C4—42no—
232C2×C4⋊C4—44no—
233C2×C4⋊C4C2×D414yes—
234C2×C22⋊C4—28no—
235C2×C22⋊C4—28no—
236C2×C22⋊C4—28no—
237C2×C22⋊C4—28no—
238C2×C22⋊C4—28no—
239C2×C22⋊C4—28no—
240C2×C22⋊C4—28no—
241C2×C22⋊C4—28no—
242C2×C22⋊C4—28no—
243C2×C22⋊C4—28no—
244C2×C22⋊C4—28no—
245C2×C22⋊C4—28no—
246C2×C22⋊C4—28no—
247C2×C22⋊C4—28no—
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—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×C22⋊C4—44no—
263C2×C22⋊C4—44no—
264C2×C22⋊C4—44no—
265C2×C22⋊C4—44no—
266C2×C22⋊C4—44no—
267C2×C22⋊C4—44no—
268C2×C22⋊C4—44no—
269C2×C22⋊C4—44no—
270C2×C42—22no—
271C2×C42—24no—
272C2×C42—22no—
273C2×C42—22no—
274C2×C42—24no—
275C2×C42—24no—
276C2×C42—24no—
277C2×C42—24no—
278C2×C42—24no—
279C2×C42—24no—
280C2×C42—22no—
281C2×C42—24no—
282C2×C42—24no—
283C2×C42—28no—
284C2×C42—24no—
285C2×C42—24no—
286C2×C42—24no—
287C2×C42—42no—
288C2×C42—41no—
289C2×C42—42no—
290C2×C42—41no—
291C2×C42—41no—
292C2×C42—41no—
293C2×C42—42no—
294C2×C42—42no—
295C2×C42C2×D414yes—
296#95—24no—
297#95—24no—
298#95—28no—
299#95—28no—
300#95—28no—
301#95—28no—
302#95—28no—
303#95—28no—
304#95—28no—
305#95—28no—
306#95—28no—
307#95—28no—
308#95—42no—
309#95—44no—
310#95—44no—
311#95—44no—
312#95—44no—
313#95C2×D4116yes—
314C24—24no—
315C24—24no—
316C24—24no—
317C24—24no—
318C24—24no—
319C24—24no—
320C24—24no—
321C24—24no—
322C24—24no—
323C24—24no—
324C24—24no—
325C24—24no—
326C24—24no—
327C24—24no—
328C24—24no—
329C24C2511yescharacteristic
330C2×Q8—216no—
331C2×Q8—44no—
332C2×Q8—44no—
333C2×Q8—44no—
334C2×Q8—44no—
335C2×Q8—44no—
336C2×Q8—44no—
337C2×Q8—48no—
338C2×Q8—48no—
339C2×Q8—48no—
340C2×Q8—84no—
341C2×Q8—84no—
342C2×Q8—84no—
343C2×Q8—84no—
344C2×Q8—84no—
345C2×Q8—88no—
346C2×Q8—84no—
347C2×Q8—88no—
348C2×Q8#124116yescomplemented
349C2×D4—48no—
350C2×D4—48no—
351C2×D4—48no—
352C2×D4—48no—
353C2×D4—48no—
354C2×D4—48no—
355C2×D4—48no—
356C2×D4—84no—
357C2×D4—84no—
358C2×D4—84no—
359C2×D4—84no—
360C2×D4—84no—
361C2×D4—84no—
362C2×D4—84no—
363C2×D4—84no—
364C2×D4—84no—
365C2×D4—88no—
366C2×D4—84no—
367C2×D4—88no—
368C22×C4—21no—
369C22×C4—21no—
370C22×C4—21no—
371C22×C4—22no—
372C22×C4—21no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—21no—
379C22×C4—21no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—24no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—21no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—21no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—24no—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—22no—
399C22×C4—24no—
400C22×C4—22no—
401C22×C4—24no—
402C22×C4—22no—
403C22×C4—24no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—22no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—22no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—22no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—22no—
426C22×C4—24no—
427C22×C4C22×D414yes—
428C22×C4C22×D412yes—
429C22×C4C22×D414yes—
430C22×C4C22×D414yes—
431C22×C4C2×#4118yes—
432C22×C4C2×#4114yes—
433C22×C4C2×#4114yes—
434C22×C4C2×#4112yes—
435C22×C4C2×#4112yes—
436C22×C4C2×#4114yes—
437C22×C4H(5,𝔽2)14yes—
438C22×C4H(5,𝔽2)12yes—
439C22×C4H(5,𝔽2)14yes—
440C22×C4H(5,𝔽2)12yes—
441C22×C42−1+414yes—
442C4⋊C4—44no—
443C4⋊C4—48no—
444C4⋊C4—44no—
445C4⋊C4—44no—
446C4⋊C4—84no—
447C4⋊C4—84no—
448C4⋊C4—84no—
449C4⋊C4—88no—
450C22⋊C4—416no—
451C22⋊C4—416no—
452C22⋊C4—416no—
453C22⋊C4—416no—
454C22⋊C4—416no—
455C22⋊C4—416no—
456C22⋊C4—416no—
457C22⋊C4—416no—
458C22⋊C4—88no—
459C22⋊C4—88no—
460C22⋊C4—88no—
461C22⋊C4—88no—
462C42—28no—
463C42—216no—
464C42—44no—
465C42—48no—
466C42—48no—
467C42—44no—
468C42—48no—
469C42—48no—
470C42—84no—
471C23—44no—
472C23—44no—
473C23—44no—
474C23—44no—
475C23—44no—
476C23—44no—
477C23—44no—
478C23—44no—
479C23—44no—
480C23—44no—
481C23—44no—
482C23—44no—
483C23—44no—
484C23—44no—
485C23—44no—
486C23—44no—
487C23—44no—
488C23—44no—
489C23—44no—
490C23—44no—
491C23—44no—
492C23—44no—
493C23—44no—
494C23—44no—
495C23—44no—
496C23—44no—
497C23—44no—
498C23—44no—
499C23—44no—
500C23—44no—
501C23—44no—
502C23—44no—
503C23—44no—
504C23—44no—
505C23—44no—
506C23C23×D411yescharacteristic
507C23C22×#4111yescharacteristic
508C23C22×#4111yescharacteristic
509C23C22×#4112yes—
510C23C2×H(5,𝔽2)11yescharacteristic
511C23C2×H(5,𝔽2)12yes—
512C23C2×H(5,𝔽2)11yescharacteristic
513C23C2×H(5,𝔽2)12yes—
514C23C2×2−1+412yes—
515C23#58511yescharacteristic
516C23#58511yescharacteristic
517Q8—416no—
518Q8—816no—
519Q8—816no—
520D4—816no—
521D4—816no—
522D4—816no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—24no—
526C2×C4—24no—
527C2×C4—24no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—24no—
531C2×C4—24no—
532C2×C4—24no—
533C2×C4—22no—
534C2×C4—24no—
535C2×C4—22no—
536C2×C4—24no—
537C2×C4—22no—
538C2×C4—24no—
539C2×C4—24no—
540C2×C4—24no—
541C2×C4—24no—
542C2×C4—24no—
543C2×C4—22no—
544C2×C4—24no—
545C2×C4—28no—
546C2×C4—28no—
547C2×C4—24no—
548C2×C4—24no—
549C2×C4—22no—
550C2×C4—28no—
551C2×C4—24no—
552C2×C4—28no—
553C2×C4—24no—
554C2×C4—24no—
555C2×C4—42no—
556C2×C4—41no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—44no—
560C2×C4—44no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—44no—
565C2×C4—41no—
566C2×C4—42no—
567C2×C4—44no—
568C2×C4—44no—
569C2×C4—44no—
570C2×C4—44no—
571C2×C4—42no—
572C2×C4—44no—
573C2×C4—42no—
574C2×C4—44no—
575C2×C4—44no—
576C2×C4—44no—
577C2×C4—42no—
578C2×C4—44no—
579C2×C4—42no—
580C2×C4—44no—
581C2×C4—44no—
582C2×C4—44no—
583C2×C4—44no—
584C2×C4—44no—
585C2×C4—44no—
586C2×C4—44no—
587C2×C4—42no—
588C2×C4—44no—
589C2×C4—42no—
590C2×C4—44no—
591C2×C4—44no—
592C2×C4—44no—
593C2×C4C2×#12418yes—
594C2×C4C2×#12414yes—
595C2×C4#54818yes—
596C2×C4H(3,𝔽4)14yes—
597C22—84no—
598C22—84no—
599C22—84no—
600C22—84no—
601C22—84no—
602C22—84no—
603C22—84no—
604C22—84no—
605C22—84no—
606C22—84no—
607C22—84no—
608C22—84no—
609C22—84no—
610C22—84no—
611C22—84no—
612C22C22×#12411yescharacteristic
613C22#345911yescharacteristic
614C22#345912yes—
615C22#345911yescharacteristic
616C22C2×#54812yes—
617C22#348311yescharacteristic
618C22#348312yes—
619C22#348312yes—
620C22#349011yescharacteristic
621C22#349012yes—
622C22#349111yescharacteristic
623C22#349611yescharacteristic
624C22#349611yescharacteristic
625C22#349812yes—
626C22#351812yes—
627C22#352212yes—
628C22#352212yes—
629C22C2×H(3,𝔽4)11yescharacteristic
630C22#353612yes—
631C22#354411yescharacteristic
632C22#354911yescharacteristic
633C22#355812yes—
634C22#356112yes—
635C4—24no—
636C4—28no—
637C4—24no—
638C4—28no—
639C4—42no—
640C4—44no—
641C4—44no—
642C4—48no—
643C4—82no—
644C4—84no—
645C4—82no—
646C4—84no—
647C2—164no—
648C2#3499311yescharacteristic
649C2#3645712yes—
650C2#3726711yescharacteristic
651C2#3882012yes—
652C2#3883012yes—
653C2#3942012yes—
654C2#4599611yescharacteristic
655C2#4609911yescharacteristic
656C2#5230711yescharacteristic
657C2#5965911yescharacteristic
658C2#6004811yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC2×#2733C2
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×#2811C2semidirect
3. #17077C2semidirect
4. #16878C2semidirect
5. #15772C2semidirect
6. #15317C2semidirect
7. #15070C2semidirect
8. #14928C2semidirect
9. #14708C2semidirect
10. #14492C2semidirect
11. #14232C2semidirect
12. #14026C2semidirect
13. C4×#391C2semidirect
348. C2×Q8#124semidirect

Maximal subgroups

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

subgroupindex
C2×#28112
C2×#27332
#170772
#168782
#157722
#153172
#150702
#149282
#147082
#144922
#142322
#140262
C4×#3912