Finite Group Databasebeta

← #531048

Subgroups — #531048

Order 512.

2874 conjugacy classes of subgroups.

The subgroup lattice has 654 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2328C211yesmaximal, characteristic
2#14713C211yesmaximal, characteristic
3#14428C214yesmaximal, complemented
4#14376C214yesmaximal, complemented
5#14348C211yesmaximal, characteristic, complemented
6#14272C211yesmaximal, characteristic
7#14076C214yesmaximal, complemented
8#14022C212yesmaximal, complemented
9#14022C212yesmaximal, complemented
10#14007C214yesmaximal, complemented
11#13916C212yesmaximal, complemented
12#13820C212yesmaximal, complemented
13#13779C211yesmaximal, characteristic, complemented
14C4×#382C211yesmaximal, characteristic, complemented
15#13759C211yesmaximal, characteristic, complemented
16C22×#127C2211yescharacteristic
17C22×C4×D4C2212yes—
18#2328C4116yescomplemented
19C2×#390C2212yes—
20C2×#390C2212yes—
21C2×#390C2214yes—
22C2×#390C2212yes—
23C2×#390C2212yes—
24C2×#390C2214yes—
25C2×#390C2212yes—
26C2×#388C2214yes—
27C2×#388C2214yes—
28C2×#386C2214yes—
29C2×#386C2214yes—
30C2×#384C2211yescharacteristic
31C2×#382C2211yescharacteristic
32C2×#382C2211yescharacteristic
33C2×#381C2214yes—
34C2×C4×C4⋊C4C2211yescharacteristic
35C2×C4×C4⋊C4C2211yescharacteristic
36C2×C4×C4⋊C4C2212yes—
37C2×C4×C4⋊C4C2212yes—
38C2×C4×C22⋊C4C2214yes—
39C2×#376C2211yescharacteristic
40C2×#376C2211yescharacteristic
41C2×C43C2211yescharacteristic
42#1442C2214yes—
43#1442C2214yes—
44#1440C2214yes—
45#1440C2214yes—
46#1440C2214yes—
47#1440C2214yes—
48#1439C2214yes—
49#1439C2214yes—
50#1438C2214yes—
51#1438C2214yes—
52#1437C2214yes—
53#1437C2214yes—
54#1437C2214yes—
55#1437C2214yes—
56#1436C2212yes—
57#1436C2214yes—
58#1436C2212yes—
59#1436C2214yes—
60#1436C2212yes—
61#1436C2212yes—
62C4⋊#95C2212yes—
63C4⋊#95C2212yes—
64C4⋊#95C2212yes—
65C4⋊#95C2212yes—
66#1434C2214yes—
67C22⋊#95C2214yes—
68C22⋊#95C2214yes—
69C4×#95C2211yescharacteristic
70C4×#95C2214yes—
71C4×#95C2211yescharacteristic
72C23×D4C2312yes—
73C23×D4C2314yes—
74C24×C4C2314yes—
75C2×#127—24no—
76C2×#127—24no—
77C2×#127C2×C418yes—
78C2×C4×D4—44no—
79C2×C4×D4—44no—
80C2×C4×D4C2×C4116yes—
81C22×C4⋊C4C2312yes—
82C22×C4⋊C4C2312yes—
83C22×C4⋊C4C2311yescharacteristic
84C22×C4⋊C4C2312yes—
85C22×C4⋊C4C2312yes—
86C22×C4⋊C4C2311yescharacteristic
87C22×C4⋊C4C2314yes—
88C22×C22⋊C4C2314yes—
89C22×C22⋊C4C2314yes—
90C22×C22⋊C4C2314yes—
91C22×C22⋊C4C2314yes—
92C22×C22⋊C4C2314yes—
93C22×C22⋊C4C2314yes—
94C22×C22⋊C4C2314yes—
95C22×C22⋊C4C2314yes—
96C22×C22⋊C4C2314yes—
97C22×C22⋊C4C2314yes—
98C22×C22⋊C4C2314yes—
99C22×C42C2311yescharacteristic
100C22×C42C2311yescharacteristic
101C22×C42C2312yes—
102C22×C42C2312yes—
103C22×C42C2311yescharacteristic
104C22×C42C2311yescharacteristic
105C22×C42C2311yescharacteristic
106C22×C42C2311yescharacteristic
107C22×C42C2314yes—
108C22×C42C2314yes—
109C22×C42C2311yescharacteristic
110#390—28no—
111#390—28no—
112#390—48no—
113#390—48no—
114#390—44no—
115#390—44no—
116#390C2×C4116yes—
117#388—48no—
118#388—48no—
119#386C2×C4132yes—
120#386C2×C4132yes—
121#384—24no—
122#382—24no—
123#382C2×C418yes—
124#381—48no—
125C4×C4⋊C4—24no—
126C4×C4⋊C4—24no—
127C4×C4⋊C4—44no—
128C4×C4⋊C4—44no—
129C4×C22⋊C4—48no—
130#376—24no—
131#376C2×C418yes—
132C2×#95C2314yes—
133C2×#95C2314yes—
134C2×#95C2312yes—
135C2×#95C2312yes—
136C2×#95C2312yes—
137C2×#95C2312yes—
138C2×#95C2311yescharacteristic
139C2×#95C2312yes—
140C2×#95C2311yescharacteristic
141C2×#95C2312yes—
142C2×#95C2312yes—
143C2×#95C2314yes—
144C2×#95C2314yes—
145C2×#95C2314yes—
146C2×#95C2314yes—
147C2×#95C2314yes—
148C2×#95C2314yes—
149C2×#95C2314yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2312yes—
154C2×#95C2312yes—
155C43—24no—
156C25C2414yes—
157C22×D4—24no—
158C22×D4—24no—
159C22×D4—24no—
160C22×D4—24no—
161C22×D4—28no—
162C22×D4—28no—
163C22×D4—24no—
164C22×D4—24no—
165C22×D4—44no—
166C22×D4—44no—
167C22×D4—44no—
168C22×D4—44no—
169C22×D4C22×C418yes—
170C22×D4C22×C4116yes—
171C23×C4—44no—
172C23×C4—44no—
173C23×C4—44no—
174C23×C4—44no—
175C23×C4—44no—
176C23×C4—44no—
177C23×C4C22×C4116yes—
178C23×C4C2414yes—
179C23×C4C2411yescharacteristic
180C23×C4C2411yescharacteristic
181C23×C4C2414yes—
182C23×C4C2411yescharacteristic
183C23×C4C2411yescharacteristic
184C23×C4C2414yes—
185C23×C4C2412yes—
186C23×C4C2412yes—
187C23×C4C2411yescharacteristic
188C23×C4C2412yes—
189C23×C4C2412yes—
190C23×C4C2411yescharacteristic
191C23×C4C2411yescharacteristic
192#127—232no—
193C4×D4—432no—
194C2×C4⋊C4—24no—
195C2×C4⋊C4—24no—
196C2×C4⋊C4—24no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—24no—
200C2×C4⋊C4—24no—
201C2×C4⋊C4—24no—
202C2×C4⋊C4—24no—
203C2×C4⋊C4—24no—
204C2×C4⋊C4—24no—
205C2×C4⋊C4—24no—
206C2×C4⋊C4—24no—
207C2×C4⋊C4—28no—
208C2×C4⋊C4—42no—
209C2×C4⋊C4—42no—
210C2×C4⋊C4—44no—
211C2×C4⋊C4—44no—
212C2×C4⋊C4C22×C418yes—
213C2×C22⋊C4—28no—
214C2×C22⋊C4—28no—
215C2×C22⋊C4—28no—
216C2×C22⋊C4—28no—
217C2×C22⋊C4—28no—
218C2×C22⋊C4—28no—
219C2×C22⋊C4—28no—
220C2×C22⋊C4—28no—
221C2×C22⋊C4—44no—
222C2×C22⋊C4—44no—
223C2×C22⋊C4—44no—
224C2×C22⋊C4—44no—
225C2×C22⋊C4—44no—
226C2×C22⋊C4—44no—
227C2×C22⋊C4—44no—
228C2×C22⋊C4—44no—
229C2×C22⋊C4—44no—
230C2×C22⋊C4—44no—
231C2×C22⋊C4—44no—
232C2×C22⋊C4—44no—
233C2×C22⋊C4—44no—
234C2×C22⋊C4—44no—
235C2×C22⋊C4—44no—
236C2×C22⋊C4—44no—
237C2×C22⋊C4—44no—
238C2×C22⋊C4—44no—
239C2×C22⋊C4—44no—
240C2×C22⋊C4—44no—
241C2×C22⋊C4—44no—
242C2×C22⋊C4—44no—
243C2×C22⋊C4C22×C4116yes—
244C2×C22⋊C4C22×C4116yes—
245C2×C22⋊C4C22×C4116yes—
246C2×C42—22no—
247C2×C42—22no—
248C2×C42—22no—
249C2×C42—22no—
250C2×C42—22no—
251C2×C42—22no—
252C2×C42—22no—
253C2×C42—22no—
254C2×C42—24no—
255C2×C42—24no—
256C2×C42—22no—
257C2×C42—22no—
258C2×C42—22no—
259C2×C42—22no—
260C2×C42—24no—
261C2×C42—24no—
262C2×C42—22no—
263C2×C42—22no—
264C2×C42—28no—
265C2×C42—42no—
266C2×C42—42no—
267C2×C42—44no—
268C2×C42—44no—
269C2×C42—44no—
270C2×C42—44no—
271C2×C42—44no—
272C2×C42C22×C414yes—
273C2×C42C22×C414yes—
274C2×C42C22×C414yes—
275C2×C42#4114yes—
276C2×C42#4114yes—
277C2×C42#4114yes—
278C2×C42#4114yes—
279#95—24no—
280#95—24no—
281#95—24no—
282#95—22no—
283#95—22no—
284#95—28no—
285#95—24no—
286#95—24no—
287#95—28no—
288#95—44no—
289#95—42no—
290#95—42no—
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#95C22×C418yes—
300#95C22×C418yes—
301#95C22×C4116yes—
302C24—24no—
303C24—24no—
304C24—24no—
305C24—24no—
306C24—24no—
307C24—24no—
308C24—24no—
309C24—24no—
310C24—24no—
311C24—24no—
312C24—24no—
313C24—24no—
314C24—24no—
315C24—24no—
316C24C23×C418yes—
317C24C2511yescharacteristic
318C2×D4—216no—
319C2×D4—232no—
320C2×D4—216no—
321C2×D4—216no—
322C2×D4—416no—
323C2×D4—416no—
324C2×D4—48no—
325C2×D4—48no—
326C2×D4—48no—
327C2×D4—48no—
328C2×D4—88no—
329C2×D4—88no—
330C2×D4—88no—
331C2×D4—88no—
332C22×C4—21no—
333C22×C4—21no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—21no—
340C22×C4—21no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—21no—
344C22×C4—22no—
345C22×C4—21no—
346C22×C4—21no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—24no—
350C22×C4—22no—
351C22×C4—21no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—24no—
356C22×C4—24no—
357C22×C4—24no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—24no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—24no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—21no—
376C22×C4—21no—
377C22×C4—24no—
378C22×C4—21no—
379C22×C4—21no—
380C22×C4—24no—
381C22×C4—24no—
382C22×C4—24no—
383C22×C4—24no—
384C22×C4—22no—
385C22×C4—24no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—22no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—24no—
392C22×C4—24no—
393C22×C4—21no—
394C22×C4—21no—
395C22×C4—24no—
396C22×C4—22no—
397C22×C4—24no—
398C22×C4—22no—
399C22×C4—416no—
400C22×C4—416no—
401C22×C4—416no—
402C22×C4—88no—
403C22×C4—88no—
404C22×C4—88no—
405C22×C4—88no—
406C22×C4C23×C412yes—
407C22×C4C23×C412yes—
408C22×C4C23×C414yes—
409C22×C4C23×C418yes—
410C22×C4C23×C414yes—
411C22×C4C23×C412yes—
412C22×C4C2×#4112yes—
413C22×C4C2×#4114yes—
414C22×C4C2×#4114yes—
415C22×C4C2×#4114yes—
416C22×C4C2×#4112yes—
417C22×C4C2×#4112yes—
418C22×C4C2×#4112yes—
419C22×C4C2×#4112yes—
420C22×C4C2×#4112yes—
421C22×C4H(5,𝔽2)14yes—
422C22×C4H(5,𝔽2)14yes—
423C22×C4H(5,𝔽2)12yes—
424C22×C4H(5,𝔽2)12yes—
425C4⋊C4—216no—
426C4⋊C4—48no—
427C4⋊C4—44no—
428C4⋊C4—44no—
429C4⋊C4—48no—
430C4⋊C4—48no—
431C4⋊C4—88no—
432C22⋊C4—416no—
433C22⋊C4—416no—
434C22⋊C4—416no—
435C22⋊C4—416no—
436C22⋊C4—416no—
437C22⋊C4—88no—
438C22⋊C4—88no—
439C22⋊C4—88no—
440C22⋊C4—88no—
441C22⋊C4—88no—
442C22⋊C4—88no—
443C42—28no—
444C42—28no—
445C42—28no—
446C42—28no—
447C42—48no—
448C42—48no—
449C42—44no—
450C42—44no—
451C42—88no—
452C42—88no—
453C42#117116yes—
454C23—28no—
455C23—28no—
456C23—28no—
457C23—28no—
458C23—28no—
459C23—28no—
460C23—28no—
461C23—44no—
462C23—44no—
463C23—44no—
464C23—44no—
465C23—44no—
466C23—44no—
467C23—44no—
468C23—44no—
469C23—44no—
470C23—44no—
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—
489C23C24×C411yescharacteristic
490C23C22×#4111yescharacteristic
491C23C22×#4111yescharacteristic
492C23C22×#4111yescharacteristic
493C23C22×#4111yescharacteristic
494C23C2×H(5,𝔽2)11yescharacteristic
495C23C2×H(5,𝔽2)11yescharacteristic
496C23C2×H(5,𝔽2)12yes—
497C23C2×H(5,𝔽2)12yes—
498C23#58512yes—
499C23#58512yes—
500D4—432no—
501D4—832no—
502C2×C4—24no—
503C2×C4—24no—
504C2×C4—22no—
505C2×C4—24no—
506C2×C4—24no—
507C2×C4—24no—
508C2×C4—24no—
509C2×C4—24no—
510C2×C4—22no—
511C2×C4—24no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—24no—
515C2×C4—28no—
516C2×C4—24no—
517C2×C4—24no—
518C2×C4—28no—
519C2×C4—28no—
520C2×C4—24no—
521C2×C4—24no—
522C2×C4—22no—
523C2×C4—24no—
524C2×C4—22no—
525C2×C4—24no—
526C2×C4—24no—
527C2×C4—24no—
528C2×C4—24no—
529C2×C4—22no—
530C2×C4—24no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—42no—
544C2×C4—44no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—44no—
548C2×C4—42no—
549C2×C4—44no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—44no—
554C2×C4—42no—
555C2×C4—44no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—44no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—44no—
562C2×C4—44no—
563C2×C4—44no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—44no—
567C2×C4—44no—
568C2×C4—44no—
569C2×C4—41no—
570C2×C4—44no—
571C2×C4—44no—
572C2×C4—41no—
573C2×C4—44no—
574C2×C4—44no—
575C2×C4—42no—
576C2×C4—44no—
577C2×C4—44no—
578C2×C4—42no—
579C2×C4—832no—
580C2×C4C2×#11718yes—
581C2×C4C2×#11714yes—
582C2×C4C4×#4114yes—
583C2×C4#51814yes—
584C2×C4#54014yes—
585C2×C4#55314yes—
586C2×C4#55314yes—
587C2×C4#55914yes—
588C2×C4#55914yes—
589C22—48no—
590C22—48no—
591C22—48no—
592C22—48no—
593C22—48no—
594C22—48no—
595C22—48no—
596C22—84no—
597C22—84no—
598C22—84no—
599C22—84no—
600C22—84no—
601C22—84no—
602C22—84no—
603C22—84no—
604C22C22×#11711yescharacteristic
605C22C2×C4×#4111yescharacteristic
606C22C2×#51811yescharacteristic
607C22C4×H(5,𝔽2)12yes—
608C22C4×H(5,𝔽2)12yes—
609C22C2×#54011yescharacteristic
610C22#345511yescharacteristic
611C22#346012yes—
612C22C2×#55311yescharacteristic
613C22C2×#55311yescharacteristic
614C22C2×#55911yescharacteristic
615C22C2×#55911yescharacteristic
616C22#349412yes—
617C22#349412yes—
618C22#349412yes—
619C22#350811yescharacteristic
620C22#350812yes—
621C22#351012yes—
622C22#351211yescharacteristic
623C22#354312yes—
624C22#354612yes—
625C22#355812yes—
626C22#356212yes—
627C4—24no—
628C4—28no—
629C4—42no—
630C4—42no—
631C4—44no—
632C4—44no—
633C4—48no—
634C4—44no—
635C4—42no—
636C4—81no—
637C4—81no—
638C4—84no—
639C4—84no—
640C4#235718yes—
641C2—88no—
642C2C4×#54011yescharacteristic
643C2C2×#235711yescharacteristic
644C2#3425512yes—
645C2#3428011yescharacteristic
646C2#3443812yes—
647C2#3789311yescharacteristic
648C2#3789811yescharacteristic
649C2#3794612yes—
650C2#3976611yescharacteristic
651C2#3977011yescharacteristic
652C2#3981012yes—
subgroupG/H
center, Frattini, socle, cosocleC24C25
commutatorC23C24×C4
prime radicalC22×#127C22
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
3. #14428C2semidirect
4. #14376C2semidirect
5. #14348C2semidirect
7. #14076C2semidirect
8. #14022C2semidirect
9. #14022C2semidirect
10. #14007C2semidirect
11. #13916C2semidirect
12. #13820C2semidirect
13. #13779C2semidirect
14. C4×#382C2semidirect
15. #13759C2semidirect
18. #2328C4semidirect

Maximal subgroups

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

subgroupindex
C2×#23282
#147132
#144282
#143762
#143482
#142722
#140762
#140222
#140072
#139162
#138202
#137792
C4×#3822
#137592