Finite Group Databasebeta

← #674791

Subgroups — #674791

Order 512.

2278 conjugacy classes of subgroups.

The subgroup lattice has 645 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2494C212yesmaximal, complemented
2#15694C214yesmaximal
3#15604C214yesmaximal, complemented
4#15220C212yesmaximal, complemented
5#15132C212yesmaximal, complemented
6#15038C214yesmaximal, complemented
7#14707C211yesmaximal, characteristic, complemented
8#14233C214yesmaximal, complemented
9#13973C212yesmaximal, complemented
10#13954C211yesmaximal, characteristic, complemented
11#13905C211yesmaximal, characteristic
12#13905C211yesmaximal, characteristic
13#13804C211yesmaximal, characteristic, complemented
14C4×#382C211yesmaximal, characteristic, complemented
15C4×#390C211yesmaximal, characteristic
16C22×C4×D4C2211yescharacteristic
17#2494—216no—
18C2×#401C2214yes—
19C2×#397C2214yes—
20C2×#397C2214yes—
21C2×#390C2212yes—
22C2×#389C2212yes—
23C2×#389C2212yes—
24C2×#389C2212yes—
25C2×#388C2214yes—
26C2×#387C2214yes—
27C2×#382C2211yescharacteristic
28C2×#382C2211yescharacteristic
29C2×#382C2211yescharacteristic
30C2×#382C2211yescharacteristic
31C2×#380C2212yes—
32C2×#380C2212yes—
33C2×#380C2212yes—
34C2×#380C2212yes—
35C2×C4×C4⋊C4C2211yescharacteristic
36C2×C4×C4⋊C4C2211yescharacteristic
37C2×C4×C4⋊C4C2211yescharacteristic
38C2×C4×C4⋊C4C2212yes—
39C2×C4×C22⋊C4C2212yes—
40C2×C4×C22⋊C4C2212yes—
41C2×C43C2211yescharacteristic
42#1445C2212yes—
43#1445C2212yes—
44#1444C2214yes—
45#1443C2214yes—
46#1443C2214yes—
47#1442C2214yes—
48#1442C2214yes—
49#1441C2214yes—
50#1440C2214yes—
51#1439C2214yes—
52#1439C2214yes—
53#1439C2214yes—
54#1438C2212yes—
55#1438C2212yes—
56#1437C2214yes—
57#1437C2214yes—
58#1437C2214yes—
59#1437C2212yes—
60#1437C2212yes—
61#1436C2214yes—
62#1436C2214yes—
63#1436C2214yes—
64C4⋊#95C2211yescharacteristic
65C4⋊#95C2212yes—
66C4⋊#95C2212yes—
67C4⋊#95C2212yes—
68C4⋊#95C2212yes—
69C4⋊#95C2211yescharacteristic
70C4⋊#95C2212yes—
71#1434C2214yes—
72#1434C2214yes—
73#1434C2212yes—
74C4×#95C2212yes—
75C4×#95C2214yes—
76C23×D4C2311yescharacteristic
77C24×C4C2312yes—
78C2×C4×D4—24no—
79C2×C4×D4—24no—
80C2×C4×D4—24no—
81C22×C4⋊C4C2311yescharacteristic
82C22×C4⋊C4C2311yescharacteristic
83C22×C4⋊C4C2312yes—
84C22×C4⋊C4C2312yes—
85C22×C4⋊C4C2314yes—
86C22×C4⋊C4C2312yes—
87C22×C4⋊C4C2311yescharacteristic
88C22×C4⋊C4C2311yescharacteristic
89C22×C4⋊C4C2311yescharacteristic
90C22×C4⋊C4C2314yes—
91C22×C4⋊C4C2314yes—
92C22×C22⋊C4C2312yes—
93C22×C22⋊C4C2314yes—
94C22×C22⋊C4C2312yes—
95C22×C22⋊C4C2312yes—
96C22×C22⋊C4C2314yes—
97C22×C22⋊C4C2314yes—
98C22×C22⋊C4C2314yes—
99C22×C22⋊C4C2312yes—
100C22×C22⋊C4C2312yes—
101C22×C42C2311yescharacteristic
102C22×C42C2312yes—
103C22×C42C2312yes—
104C22×C42C2311yescharacteristic
105C22×C42C2312yes—
106C22×C42C2312yes—
107C22×C42C2312yes—
108C22×C42C2314yes—
109C22×C42C2311yescharacteristic
110#401—216no—
111#397—216no—
112#397—216no—
113#390—28no—
114#389—28no—
115#389—28no—
116#389—44no—
117#388—216no—
118#387—216no—
119#382—24no—
120#382—24no—
121#382—24no—
122#382D418yescomplemented
123#380—28no—
124#380—28no—
125#380—28no—
126#380—44no—
127C4×C4⋊C4—28no—
128C4×C4⋊C4—24no—
129C4×C4⋊C4—24no—
130C4×C4⋊C4—24no—
131C4×C22⋊C4—28no—
132C4×C22⋊C4—44no—
133C2×#95C2314yes—
134C2×#95C2314yes—
135C2×#95C2314yes—
136C2×#95C2314yes—
137C2×#95C2314yes—
138C2×#95C2312yes—
139C2×#95C2314yes—
140C2×#95C2314yes—
141C2×#95C2314yes—
142C2×#95C2311yescharacteristic
143C2×#95C2312yes—
144C2×#95C2314yes—
145C2×#95C2311yescharacteristic
146C2×#95C2312yes—
147C2×#95C2312yes—
148C2×#95C2314yes—
149C2×#95C2312yes—
150C2×#95C2312yes—
151C2×#95C2312yes—
152C2×#95C2312yes—
153C2×#95C2314yes—
154C2×#95C2312yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2312yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C43—24no—
162C25C2412yes—
163C22×D4—24no—
164C22×D4—22no—
165C22×D4—22no—
166C22×D4—22no—
167C22×D4—42no—
168C23×C4—24no—
169C23×C4—24no—
170C23×C4—24no—
171C23×C4—24no—
172C23×C4—24no—
173C23×C4—24no—
174C23×C4—24no—
175C23×C4C2414yes—
176C23×C4C2412yes—
177C23×C4C2414yes—
178C23×C4C2412yes—
179C23×C4C2412yes—
180C23×C4C2412yes—
181C23×C4C2412yes—
182C23×C4C2412yes—
183C23×C4C2412yes—
184C23×C4C2412yes—
185C23×C4C2412yes—
186C23×C4C2411yescharacteristic
187C23×C4C2411yescharacteristic
188C23×C4C2411yescharacteristic
189C4×D4—416no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—24no—
197C2×C4⋊C4—24no—
198C2×C4⋊C4—28no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—24no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—24no—
205C2×C4⋊C4—24no—
206C2×C4⋊C4—24no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—24no—
209C2×C4⋊C4—28no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—24no—
212C2×C4⋊C4—28no—
213C2×C4⋊C4—42no—
214C2×C4⋊C4—44no—
215C2×C4⋊C4—44no—
216C2×C4⋊C4—44no—
217C2×C4⋊C4—44no—
218C2×C4⋊C4—44no—
219C2×C4⋊C4—44no—
220C2×C4⋊C4C2×D414yes—
221C2×C4⋊C4C2×D414yes—
222C2×C4⋊C4#4114yes—
223C2×C22⋊C4—24no—
224C2×C22⋊C4—28no—
225C2×C22⋊C4—24no—
226C2×C22⋊C4—24no—
227C2×C22⋊C4—28no—
228C2×C22⋊C4—24no—
229C2×C22⋊C4—24no—
230C2×C22⋊C4—24no—
231C2×C22⋊C4—24no—
232C2×C22⋊C4—24no—
233C2×C22⋊C4—28no—
234C2×C22⋊C4—28no—
235C2×C22⋊C4—28no—
236C2×C22⋊C4—42no—
237C2×C22⋊C4—42no—
238C2×C22⋊C4—42no—
239C2×C22⋊C4—44no—
240C2×C22⋊C4—42no—
241C2×C22⋊C4—42no—
242C2×C22⋊C4—42no—
243C2×C22⋊C4—44no—
244C2×C22⋊C4—44no—
245C2×C22⋊C4—44no—
246C2×C22⋊C4—44no—
247C2×C22⋊C4—44no—
248C2×C22⋊C4—44no—
249C2×C22⋊C4#4118yes—
250C2×C42—22no—
251C2×C42—22no—
252C2×C42—24no—
253C2×C42—24no—
254C2×C42—24no—
255C2×C42—22no—
256C2×C42—24no—
257C2×C42—24no—
258C2×C42—22no—
259C2×C42—24no—
260C2×C42—22no—
261C2×C42—28no—
262C2×C42—24no—
263C2×C42—24no—
264C2×C42—42no—
265C2×C42—42no—
266C2×C42—42no—
267C2×C42—44no—
268C2×C42—44no—
269C2×C42—44no—
270C2×C42—44no—
271C2×C42C2×D414yes—
272C2×C42#4114yes—
273C2×C42#4114yes—
274C2×C42#4114yes—
275#95—24no—
276#95—24no—
277#95—28no—
278#95—28no—
279#95—28no—
280#95—28no—
281#95—28no—
282#95—22no—
283#95—24no—
284#95—24no—
285#95—22no—
286#95—28no—
287#95—28no—
288#95—24no—
289#95—24no—
290#95—24no—
291#95—24no—
292#95—28no—
293#95—28no—
294#95—28no—
295#95—44no—
296#95—44no—
297#95—44no—
298#95—42no—
299#95—42no—
300#95—44no—
301#95—44no—
302#95C2×D4116yes—
303C24—22no—
304C24—22no—
305C24—22no—
306C24—22no—
307C24—22no—
308C24—22no—
309C24—22no—
310C24—22no—
311C24—22no—
312C24—22no—
313C24—22no—
314C24—22no—
315C24—22no—
316C24—22no—
317C24C2×#4114yes—
318C24C2511yescharacteristic
319C2×D4—28no—
320C2×D4—48no—
321C2×D4—48no—
322C2×D4—44no—
323C2×D4—44no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—24no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—24no—
347C22×C4—24no—
348C22×C4—24no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—24no—
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—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—24no—
371C22×C4—24no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—24no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—24no—
383C22×C4—24no—
384C22×C4—24no—
385C22×C4—24no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—22no—
389C22×C4—24no—
390C22×C4—22no—
391C22×C4—48no—
392C22×C4—48no—
393C22×C4—48no—
394C22×C4—48no—
395C22×C4—48no—
396C22×C4—48no—
397C22×C4—48no—
398C22×C4C22×D414yes—
399C22×C4C22×D414yes—
400C22×C4C22×D414yes—
401C22×C4C22×D412yes—
402C22×C4C2×#4114yes—
403C22×C4C2×#4114yes—
404C22×C4C2×#4114yes—
405C22×C4C2×#4112yes—
406C22×C4C2×#4112yes—
407C22×C4C2×#4112yes—
408C22×C4C2×#4112yes—
409C22×C4C2×#4112yes—
410C22×C4C2×#4112yes—
411C22×C4C2×#4112yes—
412C22×C4C2×#4112yes—
413C22×C4C2×#4112yes—
414C22×C4C2×#4112yes—
415C22×C4H(5,𝔽2)12yes—
416C22×C42−1+412yes—
417C4⋊C4—28no—
418C4⋊C4—28no—
419C4⋊C4—28no—
420C4⋊C4—48no—
421C4⋊C4—416no—
422C4⋊C4—44no—
423C4⋊C4—48no—
424C4⋊C4—48no—
425C4⋊C4—44no—
426C4⋊C4—88no—
427C4⋊C4—88no—
428C22⋊C4—48no—
429C22⋊C4—48no—
430C22⋊C4—48no—
431C22⋊C4—416no—
432C22⋊C4—48no—
433C22⋊C4—48no—
434C22⋊C4—88no—
435C22⋊C4—88no—
436C22⋊C4—88no—
437C42—28no—
438C42—48no—
439C42—48no—
440C42—44no—
441C42—48no—
442C42—84no—
443C42—84no—
444C42—88no—
445C42#126116yes—
446C23—24no—
447C23—24no—
448C23—24no—
449C23—24no—
450C23—24no—
451C23—24no—
452C23—24no—
453C23—42no—
454C23—42no—
455C23—42no—
456C23—42no—
457C23—42no—
458C23—42no—
459C23—42no—
460C23—42no—
461C23—42no—
462C23—42no—
463C23—42no—
464C23—42no—
465C23—42no—
466C23—42no—
467C23—42no—
468C23—42no—
469C23—42no—
470C23—42no—
471C23—42no—
472C23—42no—
473C23—42no—
474C23—42no—
475C23—42no—
476C23—42no—
477C23—42no—
478C23—42no—
479C23—42no—
480C23—42no—
481C23C23×D411yescharacteristic
482C23C22×#4111yescharacteristic
483C23C22×#4111yescharacteristic
484C23C22×#4111yescharacteristic
485C23C22×#4111yescharacteristic
486C23C22×#4112yes—
487C23C2×H(5,𝔽2)11yescharacteristic
488C23C2×2−1+411yescharacteristic
489C23#58512yes—
490C23#58512yes—
491C23#58512yes—
492D4—416no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—24no—
496C2×C4—24no—
497C2×C4—24no—
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—28no—
507C2×C4—24no—
508C2×C4—24no—
509C2×C4—24no—
510C2×C4—28no—
511C2×C4—24no—
512C2×C4—24no—
513C2×C4—24no—
514C2×C4—24no—
515C2×C4—28no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—44no—
531C2×C4—42no—
532C2×C4—44no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—44no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—44no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—44no—
545C2×C4—42no—
546C2×C4—44no—
547C2×C4—42no—
548C2×C4—44no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—44no—
554C2×C4—44no—
555C2×C4—44no—
556C2×C4—44no—
557C2×C4—44no—
558C2×C4—44no—
559C2×C4—44no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—44no—
563C2×C4—44no—
564C2×C4—42no—
565C2×C4—44no—
566C2×C4—42no—
567C2×C4—44no—
568C2×C4—42no—
569C2×C4—44no—
570C2×C4—816no—
571C2×C4C2×#12614yes—
572C2×C4C2×#12614yes—
573C2×C4C2×#12614yes—
574C2×C4#53214yes—
575C2×C4#54214yes—
576C2×C4#54714yes—
577C2×C4#55014yes—
578C2×C4#55314yes—
579C2×C4#55614yes—
580C22—44no—
581C22—44no—
582C22—44no—
583C22—44no—
584C22—44no—
585C22—44no—
586C22—44no—
587C22—82no—
588C22—82no—
589C22—82no—
590C22—82no—
591C22—82no—
592C22—82no—
593C22—82no—
594C22—82no—
595C22C22×#12611yescharacteristic
596C22C2×#53211yescharacteristic
597C22#344411yescharacteristic
598C22C2×#54211yescharacteristic
599C22#345511yescharacteristic
600C22C2×#54711yescharacteristic
601C22C2×#55011yescharacteristic
602C22D4×#4112yes—
603C22C2×#55311yescharacteristic
604C22C2×#55611yescharacteristic
605C22#347512yes—
606C22#347512yes—
607C22#348012yes—
608C22#349312yes—
609C22#350012yes—
610C22#350512yes—
611C22#351011yescharacteristic
612C22#351011yescharacteristic
613C22#351212yes—
614C22#351512yes—
615C22#352112yes—
616C22#355112yes—
617C22#356512yes—
618C4—24no—
619C4—24no—
620C4—44no—
621C4—44no—
622C4—44no—
623C4—44no—
624C4—44no—
625C4—44no—
626C4—84no—
627C4—82no—
628C4—84no—
629C4—82no—
630C4—82no—
631C4#251218yes—
632C2—84no—
633C2C2×#251211yescharacteristic
634C2#3481411yescharacteristic
635C2#3562911yescharacteristic
636C2#3613511yescharacteristic
637C2#3647012yes—
638C2#3653911yescharacteristic
639C2#3918711yescharacteristic
640C2#3937411yescharacteristic
641C2#3993112yes—
642C2#5943312yes—
643C2#5982412yes—
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
1. C2×#2494C2semidirect
3. #15604C2semidirect
4. #15220C2semidirect
5. #15132C2semidirect
6. #15038C2semidirect
7. #14707C2semidirect
8. #14233C2semidirect
9. #13973C2semidirect
10. #13954C2semidirect
13. #13804C2semidirect
14. C4×#382C2semidirect
122. #382D4semidirect

Maximal subgroups

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

subgroupindex
C2×#24942
#156942
#156042
#152202
#151322
#150382
#147072
#142332
#139732
#139542
#139052
#138042
C4×#3822
C4×#3902