Finite Group Databasebeta

← #723044

Subgroups — #723044

Order 512.

2154 conjugacy classes of subgroups.

The subgroup lattice has 627 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#16419C214yesmaximal, complemented
2#15491C214yesmaximal, complemented
3#15488C214yesmaximal, complemented
4#15152C212yesmaximal, complemented
5#15134C212yesmaximal
6#14959C214yesmaximal, complemented
7#14276C211yesmaximal, characteristic, complemented
8#14041C212yesmaximal, complemented
9#14022C212yesmaximal
10#13941C211yesmaximal, characteristic
11#13910C211yesmaximal, characteristic, complemented
12#13906C211yesmaximal, characteristic
13#13856C211yesmaximal, characteristic, complemented
14#13804C211yesmaximal, characteristic, complemented
15#13785C211yesmaximal, characteristic
16C22×C4×D4C2211yescharacteristic
17C2×#400C2214yes—
18C2×#396C2214yes—
19C2×#396C2214yes—
20C2×#390C2211yescharacteristic
21C2×#390C2212yes—
22C2×#390C2212yes—
23C2×#390C2211yescharacteristic
24C2×#389C2212yes—
25C2×#389C2214yes—
26C2×#388C2214yes—
27C2×#384C2211yescharacteristic
28C2×#384C2212yes—
29C2×#382C2211yescharacteristic
30C2×#380C2214yes—
31C2×C4×C4⋊C4C2211yescharacteristic
32C2×C4×C4⋊C4C2211yescharacteristic
33C2×C4×C4⋊C4C2211yescharacteristic
34C2×C4×C4⋊C4C2212yes—
35C2×C4×C4⋊C4C2212yes—
36C2×C4×C4⋊C4C2212yes—
37C2×#376C2211yescharacteristic
38#1445C2214yes—
39#1445C2214yes—
40#1444C2212yes—
41#1444C2212yes—
42#1444C2212yes—
43#1444C2212yes—
44#1444C2212yes—
45#1444C2212yes—
46#1442C2214yes—
47#1442C2214yes—
48#1441C2214yes—
49#1441C2214yes—
50#1441C2214yes—
51#1440C2214yes—
52#1439C2214yes—
53#1439C2214yes—
54#1439C2212yes—
55#1439C2214yes—
56#1439C2214yes—
57#1439C2212yes—
58#1437C2214yes—
59#1437C2214yes—
60#1436C2214yes—
61#1436C2214yes—
62#1436C2214yes—
63C4⋊#95C2212yes—
64C4⋊#95C2211yescharacteristic
65C4⋊#95C2212yes—
66C4⋊#95C2212yes—
67C4⋊#95C2212yes—
68C4⋊#95C2212yes—
69C4⋊#95C2211yescharacteristic
70#1434C2212yes—
71C22⋊#95C2212yes—
72C22⋊#95C2212yes—
73C22⋊#95C2214yes—
74C4×#95C2212yes—
75C23×D4C2311yescharacteristic
76C24×C4C2312yes—
77C2×C4×D4—24no—
78C2×C4×D4—24no—
79C2×C4×D4—24no—
80C22×C4⋊C4C2311yescharacteristic
81C22×C4⋊C4C2312yes—
82C22×C4⋊C4C2311yescharacteristic
83C22×C4⋊C4C2312yes—
84C22×C4⋊C4C2312yes—
85C22×C4⋊C4C2312yes—
86C22×C4⋊C4C2312yes—
87C22×C4⋊C4C2312yes—
88C22×C4⋊C4C2311yescharacteristic
89C22×C4⋊C4C2314yes—
90C22×C4⋊C4C2312yes—
91C22×C4⋊C4C2314yes—
92C22×C4⋊C4C2312yes—
93C22×C4⋊C4C2312yes—
94C22×C4⋊C4C2312yes—
95C22×C4⋊C4C2314yes—
96C22×C22⋊C4C2312yes—
97C22×C22⋊C4C2314yes—
98C22×C22⋊C4C2314yes—
99C22×C22⋊C4C2314yes—
100C22×C22⋊C4C2314yes—
101C22×C22⋊C4C2314yes—
102C22×C22⋊C4C2314yes—
103C22×C42C2312yes—
104C22×C42C2312yes—
105C22×C42C2311yescharacteristic
106C22×C42C2311yescharacteristic
107C22×C42C2312yes—
108C22×C42C2314yes—
109C22×C42C2311yescharacteristic
110#400—48no—
111#396—48no—
112#396—48no—
113#390—24no—
114#390—44no—
115#390—44no—
116#390Q818yes—
117#389—28no—
118#389—48no—
119#388—48no—
120#384—24no—
121#384—44no—
122#382—24no—
123#380—48no—
124C4×C4⋊C4—24no—
125C4×C4⋊C4—24no—
126C4×C4⋊C4—24no—
127C4×C4⋊C4—44no—
128C4×C4⋊C4—44no—
129C4×C4⋊C4—44no—
130#376—24no—
131C2×#95C2314yes—
132C2×#95C2314yes—
133C2×#95C2311yescharacteristic
134C2×#95C2312yes—
135C2×#95C2312yes—
136C2×#95C2312yes—
137C2×#95C2312yes—
138C2×#95C2311yescharacteristic
139C2×#95C2311yescharacteristic
140C2×#95C2311yescharacteristic
141C2×#95C2314yes—
142C2×#95C2314yes—
143C2×#95C2312yes—
144C2×#95C2314yes—
145C2×#95C2314yes—
146C2×#95C2314yes—
147C2×#95C2314yes—
148C2×#95C2314yes—
149C2×#95C2314yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2312yes—
155C2×#95C2312yes—
156C2×#95C2312yes—
157C2×#95C2312yes—
158C25C2412yes—
159C22×D4—22no—
160C22×D4—22no—
161C22×D4—22no—
162C22×D4—22no—
163C22×D4—22no—
164C22×D4—22no—
165C22×D4C2×Q814yes—
166C23×C4—24no—
167C23×C4—24no—
168C23×C4—24no—
169C23×C4—44no—
170C23×C4—44no—
171C23×C4C2414yes—
172C23×C4C2412yes—
173C23×C4C2412yes—
174C23×C4C2414yes—
175C23×C4C2412yes—
176C23×C4C2412yes—
177C23×C4C2412yes—
178C23×C4C2412yes—
179C23×C4C2412yes—
180C23×C4C2412yes—
181C23×C4C2412yes—
182C23×C4C2411yescharacteristic
183C23×C4C2411yescharacteristic
184C23×C4C2411yescharacteristic
185C4×D4—416no—
186C2×C4⋊C4—24no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—24no—
190C2×C4⋊C4—24no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—24no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—24no—
197C2×C4⋊C4—24no—
198C2×C4⋊C4—24no—
199C2×C4⋊C4—24no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—24no—
202C2×C4⋊C4—24no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—24no—
205C2×C4⋊C4—24no—
206C2×C4⋊C4—24no—
207C2×C4⋊C4—24no—
208C2×C4⋊C4—24no—
209C2×C4⋊C4—24no—
210C2×C4⋊C4—24no—
211C2×C4⋊C4—24no—
212C2×C4⋊C4—24no—
213C2×C4⋊C4—28no—
214C2×C4⋊C4—28no—
215C2×C4⋊C4—42no—
216C2×C4⋊C4—42no—
217C2×C4⋊C4—42no—
218C2×C4⋊C4—42no—
219C2×C4⋊C4—42no—
220C2×C4⋊C4—42no—
221C2×C4⋊C4—42no—
222C2×C4⋊C4—44no—
223C2×C4⋊C4—44no—
224C2×C4⋊C4—44no—
225C2×C4⋊C4—44no—
226C2×C4⋊C4—44no—
227C2×C4⋊C4—44no—
228C2×C4⋊C4—44no—
229C2×C4⋊C4—44no—
230C2×C4⋊C4—44no—
231C2×C4⋊C4C2×Q814yes—
232C2×C22⋊C4—24no—
233C2×C22⋊C4—24no—
234C2×C22⋊C4—24no—
235C2×C22⋊C4—28no—
236C2×C22⋊C4—28no—
237C2×C22⋊C4—28no—
238C2×C22⋊C4—28no—
239C2×C22⋊C4—44no—
240C2×C22⋊C4—44no—
241C2×C22⋊C4—44no—
242C2×C22⋊C4—44no—
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—44no—
250C2×C22⋊C4—44no—
251C2×C22⋊C4—44no—
252C2×C22⋊C4C2×Q8116yes—
253C2×C42—22no—
254C2×C42—22no—
255C2×C42—22no—
256C2×C42—22no—
257C2×C42—24no—
258C2×C42—24no—
259C2×C42—24no—
260C2×C42—24no—
261C2×C42—24no—
262C2×C42—24no—
263C2×C42—24no—
264C2×C42—44no—
265C2×C42—44no—
266C2×C42—44no—
267C2×C42—44no—
268C2×C42C2×Q814yes—
269C2×C42#4114yes—
270C2×C42#4114yes—
271C2×C42#4114yes—
272C2×C42#4114yes—
273#95—24no—
274#95—24no—
275#95—22no—
276#95—22no—
277#95—22no—
278#95—24no—
279#95—22no—
280#95—24no—
281#95—24no—
282#95—28no—
283#95—28no—
284#95—28no—
285#95—28no—
286#95—44no—
287#95—44no—
288#95—42no—
289#95—42no—
290#95—42no—
291#95—42no—
292#95—44no—
293#95—44no—
294#95—44no—
295#95—44no—
296#95—44no—
297#95—44no—
298#95—44no—
299#95—44no—
300C24—22no—
301C24—22no—
302C24—22no—
303C24—22no—
304C24—24no—
305C24—24no—
306C24—22no—
307C24—22no—
308C24—24no—
309C24—24no—
310C24C22×Q814yes—
311C24C2511yescharacteristic
312C2×D4—28no—
313C2×D4—28no—
314C2×D4—28no—
315C2×D4—44no—
316C2×D4—44no—
317C2×D4—44no—
318C2×D4—44no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—24no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—22no—
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—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—24no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—24no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—24no—
361C22×C4—24no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—24no—
367C22×C4—24no—
368C22×C4—24no—
369C22×C4—24no—
370C22×C4—24no—
371C22×C4—24no—
372C22×C4—22no—
373C22×C4—24no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—24no—
378C22×C4—24no—
379C22×C4—24no—
380C22×C4—24no—
381C22×C4—24no—
382C22×C4—22no—
383C22×C4—24no—
384C22×C4—22no—
385C22×C4—24no—
386C22×C4—24no—
387C22×C4—22no—
388C22×C4—416no—
389C22×C4—48no—
390C22×C4—88no—
391C22×C4—84no—
392C22×C4—84no—
393C22×C4C22×Q814yes—
394C22×C4C22×Q814yes—
395C22×C4C22×Q812yes—
396C22×C4C2×#4114yes—
397C22×C4C2×#4112yes—
398C22×C4C2×#4112yes—
399C22×C4C2×#4112yes—
400C22×C4C2×#4112yes—
401C22×C4C2×#4112yes—
402C22×C4C2×#4112yes—
403C22×C4C2×#4112yes—
404C22×C4C2×#4112yes—
405C22×C4C2×#4112yes—
406C22×C4C2×#4112yes—
407C22×C4H(5,𝔽2)14yes—
408C22×C4H(5,𝔽2)14yes—
409C22×C4H(5,𝔽2)12yes—
410C22×C4H(5,𝔽2)12yes—
411C22×C42−1+414yes—
412C4⋊C4—216no—
413C4⋊C4—28no—
414C4⋊C4—48no—
415C4⋊C4—48no—
416C4⋊C4—48no—
417C4⋊C4—48no—
418C4⋊C4—48no—
419C4⋊C4—44no—
420C4⋊C4—48no—
421C4⋊C4—44no—
422C4⋊C4—48no—
423C4⋊C4—84no—
424C4⋊C4—84no—
425C4⋊C4—88no—
426C4⋊C4—88no—
427C4⋊C4—88no—
428C22⋊C4—416no—
429C22⋊C4—48no—
430C22⋊C4—88no—
431C22⋊C4—88no—
432C22⋊C4—88no—
433C22⋊C4—88no—
434C22⋊C4—88no—
435C42—28no—
436C42—28no—
437C42—48no—
438C42—48no—
439C42—84no—
440C42—88no—
441C42#126116yes—
442C23—24no—
443C23—24no—
444C23—24no—
445C23—28no—
446C23—28no—
447C23—42no—
448C23—42no—
449C23—44no—
450C23—42no—
451C23—42no—
452C23—44no—
453C23—44no—
454C23—44no—
455C23—44no—
456C23—42no—
457C23—42no—
458C23—42no—
459C23—42no—
460C23—44no—
461C23—44no—
462C23—44no—
463C23—44no—
464C23—44no—
465C23C23×Q811yescharacteristic
466C23C22×#4111yescharacteristic
467C23C22×#4111yescharacteristic
468C23C22×#4111yescharacteristic
469C23C22×#4111yescharacteristic
470C23C2×H(5,𝔽2)11yescharacteristic
471C23C2×H(5,𝔽2)11yescharacteristic
472C23C2×H(5,𝔽2)12yes—
473C23C2×H(5,𝔽2)12yes—
474C23C2×2−1+412yes—
475C23#58512yes—
476D4—416no—
477C2×C4—24no—
478C2×C4—24no—
479C2×C4—24no—
480C2×C4—22no—
481C2×C4—22no—
482C2×C4—22no—
483C2×C4—22no—
484C2×C4—24no—
485C2×C4—24no—
486C2×C4—24no—
487C2×C4—24no—
488C2×C4—24no—
489C2×C4—24no—
490C2×C4—24no—
491C2×C4—24no—
492C2×C4—28no—
493C2×C4—24no—
494C2×C4—24no—
495C2×C4—24no—
496C2×C4—24no—
497C2×C4—24no—
498C2×C4—24no—
499C2×C4—24no—
500C2×C4—24no—
501C2×C4—24no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—44no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—44no—
528C2×C4—44no—
529C2×C4—44no—
530C2×C4—42no—
531C2×C4—44no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—44no—
535C2×C4—44no—
536C2×C4—42no—
537C2×C4—44no—
538C2×C4—44no—
539C2×C4—44no—
540C2×C4—44no—
541C2×C4—44no—
542C2×C4—44no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—44no—
547C2×C4—44no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—44no—
551C2×C4—44no—
552C2×C4—44no—
553C2×C4—42no—
554C2×C4—44no—
555C2×C4—42no—
556C2×C4—816no—
557C2×C4C2×#12614yes—
558C2×C4C2×#12614yes—
559C2×C4C2×#12614yes—
560C2×C4#53314yes—
561C2×C4#54014yes—
562C2×C4#55314yes—
563C2×C4#55414yes—
564C2×C4#55414yes—
565C2×C4#55514yes—
566C22—48no—
567C22—48no—
568C22—44no—
569C22—44no—
570C22—44no—
571C22—84no—
572C22—82no—
573C22—82no—
574C22—84no—
575C22—82no—
576C22—82no—
577C22C22×#12611yescharacteristic
578C22C2×#53311yescharacteristic
579C22C2×#54011yescharacteristic
580C22#346012yes—
581C22C2×#55311yescharacteristic
582C22C2×#55411yescharacteristic
583C22C2×#55411yescharacteristic
584C22C2×#55511yescharacteristic
585C22#349212yes—
586C22#349412yes—
587C22#350112yes—
588C22#350312yes—
589C22#350712yes—
590C22#350812yes—
591C22#350811yescharacteristic
592C22#350811yescharacteristic
593C22#350911yescharacteristic
594C22#351011yescharacteristic
595C22#351112yes—
596C22#353612yes—
597C22#353712yes—
598C22#354112yes—
599C22#354212yes—
600C4—24no—
601C4—24no—
602C4—44no—
603C4—44no—
604C4—44no—
605C4—44no—
606C4—44no—
607C4—44no—
608C4—82no—
609C4—82no—
610C4—84no—
611C4—84no—
612C4—82no—
613C4#249718yes—
614C2—84no—
615C2C2×#249711yescharacteristic
616C2#3526611yescharacteristic
617C2#3555711yescharacteristic
618C2#3593412yes—
619C2#3968812yes—
620C2#3981712yes—
621C2#4250812yes—
622C2#4494711yescharacteristic
623C2#4515111yescharacteristic
624C2#4543411yescharacteristic
625C2#4543711yescharacteristic
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. #16419C2semidirect
2. #15491C2semidirect
3. #15488C2semidirect
4. #15152C2semidirect
6. #14959C2semidirect
7. #14276C2semidirect
8. #14041C2semidirect
11. #13910C2semidirect
13. #13856C2semidirect
14. #13804C2semidirect

Maximal subgroups

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

subgroupindex
#164192
#154912
#154882
#151522
#151342
#149592
#142762
#140412
#140222
#139412
#139102
#139062
#138562
#138042
#137852