Finite Group Databasebeta

← #450941

Subgroups — #450941

Order 512.

1598 conjugacy classes of subgroups.

The subgroup lattice has 625 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2486C211yesmaximal, characteristic
2C2×#1918C212yesmaximal, complemented
3#13906C211yesmaximal, characteristic
4#11207C212yesmaximal, complemented
5#11199C212yesmaximal, complemented
6#11096C212yesmaximal, complemented
7#11096C212yesmaximal, complemented
8#10964C211yesmaximal, characteristic
9#10730C212yesmaximal, complemented
10C22×#123—21no—
11C22×C4×D4C2211yescharacteristic
12C23×C4⋊C4C2211yescharacteristic
13C22×#107C2212yes—
14C22×C22⋊C8C2211yescharacteristic
15C22×C22⋊C8C2212yes—
16C22×C22⋊C8C2211yescharacteristic
17#2486—28no—
18C2×#398—21no—
19C2×#397—21no—
20C2×#390—21no—
21C2×#389C2212yes—
22C2×#384C2211yescharacteristic
23C2×#381—21no—
24C2×#380C2211yescharacteristic
25C2×#380C2212yes—
26C2×#380C2212yes—
27C2×#380C2212yes—
28C2×C4×C4⋊C4C2211yescharacteristic
29#1918—28no—
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#336C2212yes—
38#1439—21no—
39#1439—21no—
40C4⋊#95—21no—
41C23×D4D411yescharacteristic
42C24×C4—21no—
43C24×C4C2311yescharacteristic
44C24×C4C2311yescharacteristic
45C23×C8C2312yes—
46C23×C8C2312yes—
47C2×#123—28no—
48C2×#123—44no—
49C2×#123—44no—
50C2×C4×D4—24no—
51C2×C4×D4—24no—
52C2×C4×D4—41no—
53C2×C4×D4—41no—
54C22×C4⋊C4—21no—
55C22×C4⋊C4—22no—
56C22×C4⋊C4—24no—
57C22×C4⋊C4—24no—
58C22×C4⋊C4—21no—
59C22×C4⋊C4—21no—
60C22×C4⋊C4—41no—
61C22×C4⋊C4—41no—
62C22×C4⋊C4D411yescharacteristic
63C22×C4⋊C4C2312yes—
64C22×C4⋊C4C2312yes—
65C22×C4⋊C4C2312yes—
66C22×C4⋊C4C2312yes—
67C22×C22⋊C4—22no—
68C22×C22⋊C4—21no—
69C22×C22⋊C4—22no—
70C22×C22⋊C4—22no—
71C22×C22⋊C4D411yescharacteristic
72C22×C22⋊C4D411yescharacteristic
73C22×C42—21no—
74C22×C42C2311yescharacteristic
75C2×#107—24no—
76C2×#107—24no—
77C2×#107—24no—
78C2×C22⋊C8—24no—
79C2×C22⋊C8—24no—
80C2×C22⋊C8—24no—
81C2×C22⋊C8—41no—
82C2×C22⋊C8—42no—
83C2×C22⋊C8—41no—
84C2×C22⋊C8—42no—
85#398—28no—
86#397—28no—
87#390—44no—
88#389—24no—
89#389D418yes—
90#384—41no—
91#384—41no—
92#381—28no—
93#380—24no—
94#380—24no—
95#380—24no—
96#380—24no—
97#380—24no—
98#380—42no—
99#380—42no—
100C4×C4⋊C4—24no—
101C2×#95—21no—
102C2×#95—22no—
103C2×#95—22no—
104C2×#95—22no—
105C2×#95—22no—
106C2×#95—21no—
107#340—24no—
108#340—24no—
109#340—24no—
110#340—42no—
111#340—42no—
112#340—42no—
113#340—42no—
114#336—24no—
115C25C2×D411yescharacteristic
116C25C2×D411yescharacteristic
117C22×D4—22no—
118C22×D4—22no—
119C22×D4—22no—
120C22×D4—22no—
121C22×D4—41no—
122C22×D4—41no—
123C22×D4D814yes—
124C23×C4—22no—
125C23×C4—22no—
126C23×C4—22no—
127C23×C4—21no—
128C23×C4—21no—
129C23×C4—22no—
130C23×C4—22no—
131C23×C4—24no—
132C23×C4—22no—
133C23×C4—22no—
134C23×C4—22no—
135C23×C4—22no—
136C23×C4—22no—
137C23×C4—22no—
138C23×C4—22no—
139C23×C4—21no—
140C23×C4—21no—
141C23×C4—22no—
142C23×C4—22no—
143C23×C4—22no—
144C23×C4—42no—
145C23×C4—42no—
146C23×C4—42no—
147C23×C4—42no—
148C23×C4—42no—
149C23×C4—42no—
150C23×C4C2×D411yescharacteristic
151C23×C4C2×D411yescharacteristic
152C23×C4C2×D411yescharacteristic
153C23×C4C2×D411yescharacteristic
154C23×C4#4114yes—
155C23×C4C2411yescharacteristic
156C22×C8—22no—
157C22×C8—22no—
158C22×C8—22no—
159C22×C8—22no—
160C22×C8—22no—
161C22×C8—22no—
162C22×C8—22no—
163C22×C8—22no—
164C22×C8—22no—
165C22×C8—22no—
166C22×C8—22no—
167C22×C8—22no—
168C22×C8—22no—
169C22×C8#4114yes—
170#123—432no—
171C4×D4—48no—
172C4×D4—48no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—24no—
186C2×C4⋊C4—28no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
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—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—48no—
199C2×C4⋊C4—48no—
200C2×C4⋊C4—41no—
201C2×C4⋊C4—42no—
202C2×C4⋊C4—42no—
203C2×C4⋊C4—44no—
204C2×C4⋊C4—44no—
205C2×C4⋊C4—42no—
206C2×C4⋊C4—42no—
207C2×C4⋊C4—42no—
208C2×C4⋊C4—42no—
209C2×C4⋊C4—82no—
210C2×C4⋊C4C2×D414yes—
211C2×C4⋊C4C2×D414yes—
212C2×C4⋊C4#4114yes—
213C2×C22⋊C4—24no—
214C2×C22⋊C4—28no—
215C2×C22⋊C4—22no—
216C2×C22⋊C4—22no—
217C2×C22⋊C4—22no—
218C2×C22⋊C4—22no—
219C2×C22⋊C4—41no—
220C2×C22⋊C4—41no—
221C2×C22⋊C4—42no—
222C2×C22⋊C4—44no—
223C2×C22⋊C4—42no—
224C2×C22⋊C4—44no—
225C2×C22⋊C4—44no—
226C2×C22⋊C4—44no—
227C2×C22⋊C4—44no—
228C2×C22⋊C4—44no—
229C2×C22⋊C4—82no—
230C2×C22⋊C4—82no—
231C2×C42—22no—
232C2×C42—21no—
233C2×C42—21no—
234C2×C42—42no—
235C2×C42—42no—
236C2×C42—42no—
237C2×C42C2×D414yes—
238#107—216no—
239C22⋊C8—44no—
240C22⋊C8—48no—
241C22⋊C8—44no—
242#95—28no—
243#95—24no—
244#95—28no—
245#95—44no—
246#95—44no—
247#95—42no—
248C24—21no—
249C24—21no—
250C24—22no—
251C24—22no—
252C24—21no—
253C24—21no—
254C24—22no—
255C24—22no—
256C24—21no—
257C24—21no—
258C24—21no—
259C24—22no—
260C24—22no—
261C24—21no—
262C24—21no—
263C24—21no—
264C24—22no—
265C24—22no—
266C24#12312yes—
267C24C2×D812yes—
268C24C2×D812yes—
269C24#13712yes—
270C24C22×D411yescharacteristic
271C2×D4—28no—
272C2×D4—28no—
273C2×D4—44no—
274C2×D4—44no—
275C2×D4—44no—
276C2×D4—44no—
277C2×D4—44no—
278C22×C4—21no—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—21no—
282C22×C4—21no—
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—24no—
291C22×C4—24no—
292C22×C4—24no—
293C22×C4—24no—
294C22×C4—24no—
295C22×C4—24no—
296C22×C4—21no—
297C22×C4—24no—
298C22×C4—24no—
299C22×C4—21no—
300C22×C4—21no—
301C22×C4—24no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—42no—
305C22×C4—42no—
306C22×C4—42no—
307C22×C4—42no—
308C22×C4—42no—
309C22×C4—42no—
310C22×C4—41no—
311C22×C4—41no—
312C22×C4—41no—
313C22×C4—41no—
314C22×C4—48no—
315C22×C4—48no—
316C22×C4—48no—
317C22×C4—42no—
318C22×C4—42no—
319C22×C4—42no—
320C22×C4—42no—
321C22×C4—42no—
322C22×C4—42no—
323C22×C4—42no—
324C22×C4—42no—
325C22×C4—42no—
326C22×C4—42no—
327C22×C4—42no—
328C22×C4—42no—
329C22×C4—42no—
330C22×C4—42no—
331C22×C4—42no—
332C22×C4—42no—
333C22×C4—42no—
334C22×C4—42no—
335C22×C4—42no—
336C22×C4—42no—
337C22×C4—42no—
338C22×C4—42no—
339C22×C4—42no—
340C22×C4—42no—
341C22×C4—42no—
342C22×C4—42no—
343C22×C4—42no—
344C22×C4—42no—
345C22×C4—41no—
346C22×C4—42no—
347C22×C4—42no—
348C22×C4—41no—
349C22×C4—41no—
350C22×C4—41no—
351C22×C4—42no—
352C22×C4—42no—
353C22×C4—42no—
354C22×C4—42no—
355C22×C4—42no—
356C22×C4—42no—
357C22×C4—42no—
358C22×C4—84no—
359C22×C4—84no—
360C22×C4—84no—
361C22×C4—84no—
362C22×C4#12012yes—
363C22×C4#12012yes—
364C22×C4#12312yes—
365C22×C4C2×D812yes—
366C22×C4AGL(1,ℤ/8)12yes—
367C22×C4C22×D412yes—
368C22×C4C2×#4112yes—
369C22×C4C2×#4112yes—
370C22×C4C2×#4112yes—
371C22×C4H(5,𝔽2)12yes—
372C22×C4H(5,𝔽2)11yescharacteristic
373C22×C4H(5,𝔽2)11yescharacteristic
374C22×C42−1+412yes—
375C2×C8—24no—
376C2×C8—24no—
377C2×C8—24no—
378C2×C8—42no—
379C2×C8—42no—
380C2×C8—42no—
381C2×C8—42no—
382C2×C8—42no—
383C2×C8—42no—
384C2×C8—42no—
385C2×C8—42no—
386C2×C8—42no—
387C2×C8—42no—
388C2×C8—42no—
389C4⋊C4—24no—
390C4⋊C4—24no—
391C4⋊C4—24no—
392C4⋊C4—24no—
393C4⋊C4—24no—
394C4⋊C4—24no—
395C4⋊C4—44no—
396C4⋊C4—48no—
397C4⋊C4—42no—
398C4⋊C4—42no—
399C4⋊C4—42no—
400C4⋊C4—42no—
401C4⋊C4—42no—
402C4⋊C4—42no—
403C4⋊C4—416no—
404C4⋊C4—42no—
405C4⋊C4—42no—
406C4⋊C4—42no—
407C4⋊C4—42no—
408C4⋊C4—84no—
409C4⋊C4—84no—
410C22⋊C4—44no—
411C22⋊C4—44no—
412C22⋊C4—48no—
413C22⋊C4—416no—
414C22⋊C4—88no—
415C22⋊C4—88no—
416C42—24no—
417C42—24no—
418C42—84no—
419C23—24no—
420C23—24no—
421C23—22no—
422C23—24no—
423C23—22no—
424C23—22no—
425C23—24no—
426C23—22no—
427C23—22no—
428C23—22no—
429C23—24no—
430C23—24no—
431C23—24no—
432C23—22no—
433C23—22no—
434C23—22no—
435C23—24no—
436C23—42no—
437C23—42no—
438C23—42no—
439C23—42no—
440C23—41no—
441C23—41no—
442C23—41no—
443C23—41no—
444C23—42no—
445C23—42no—
446C23—42no—
447C23—42no—
448C23—42no—
449C23—41no—
450C23—42no—
451C23—42no—
452C23—41no—
453C23—42no—
454C23—42no—
455C23—41no—
456C23—41no—
457C23—41no—
458C23—41no—
459C23—41no—
460C23—41no—
461C23—42no—
462C23—42no—
463C23—42no—
464C23#48014yes—
465C23C2×#12011yescharacteristic
466C23C2×#12311yescharacteristic
467C23#53411yescharacteristic
468C23#54612yes—
469C23#54712yes—
470C23C22×D811yescharacteristic
471C23C2×AGL(1,ℤ/8)11yescharacteristic
472C23C2×#13711yescharacteristic
473C23#57511yescharacteristic
474C23#57712yes—
475C23#57812yes—
476D4—416no—
477C2×C4—22no—
478C2×C4—22no—
479C2×C4—22no—
480C2×C4—22no—
481C2×C4—22no—
482C2×C4—22no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—24no—
486C2×C4—24no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—41no—
490C2×C4—41no—
491C2×C4—41no—
492C2×C4—41no—
493C2×C4—44no—
494C2×C4—44no—
495C2×C4—44no—
496C2×C4—44no—
497C2×C4—44no—
498C2×C4—41no—
499C2×C4—44no—
500C2×C4—44no—
501C2×C4—44no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—44no—
507C2×C4—44no—
508C2×C4—44no—
509C2×C4—44no—
510C2×C4—44no—
511C2×C4—816no—
512C2×C4—82no—
513C2×C4—82no—
514C2×C4—82no—
515C2×C4—82no—
516C2×C4—82no—
517C2×C4—82no—
518C2×C4—82no—
519C2×C4—82no—
520C2×C4—82no—
521C2×C4—82no—
522C2×C4—82no—
523C2×C4—82no—
524C2×C4—82no—
525C2×C4—82no—
526C2×C4—82no—
527C2×C4—82no—
528C2×C4—82no—
529C2×C4—82no—
530C2×C4—82no—
531C2×C4—82no—
532C2×C4—82no—
533C2×C4—82no—
534C2×C4—82no—
535C2×C4#44714yes—
536C2×C4#52512yes—
537C2×C4#52912yes—
538C2×C4#53712yes—
539C2×C4#53912yes—
540C2×C4#54012yes—
541C2×C4#54212yes—
542C2×C4#54612yes—
543C2×C4#54612yes—
544C2×C4#54812yes—
545C2×C4#55012yes—
546C2×C4#55112yes—
547C2×C4#55111yescharacteristic
548C2×C4#55212yes—
549C2×C4#55911yescharacteristic
550C2×C4#56011yescharacteristic
551C2×C4#56311yescharacteristic
552C8—44no—
553C8—82no—
554C22—28no—
555C22—24no—
556C22—42no—
557C22—42no—
558C22—42no—
559C22—42no—
560C22—44no—
561C22—44no—
562C22—42no—
563C22—44no—
564C22—42no—
565C22—42no—
566C22—42no—
567C22—44no—
568C22—44no—
569C22—42no—
570C22—81no—
571C22—81no—
572C22—82no—
573C22—81no—
574C22—82no—
575C22—81no—
576C22#240511yescharacteristic
577C22#240712yes—
578C22#241912yes—
579C22#270911yescharacteristic
580C22#271711yescharacteristic
581C22C2×#44711yescharacteristic
582C22#300711yescharacteristic
583C22#301312yes—
584C22C2×#48011yescharacteristic
585C22#309411yescharacteristic
586C22#309712yes—
587C22#318611yescharacteristic
588C22#319311yescharacteristic
589C22#320011yescharacteristic
590C22#320211yescharacteristic
591C22#329412yes—
592C22#331212yes—
593C22#331712yes—
594C22#333012yes—
595C22#333412yes—
596C22#334412yes—
597C22#334912yes—
598C22#335412yes—
599C4—24no—
600C4—42no—
601C4—42no—
602C4—84no—
603C4—84no—
604C4—81no—
605C4—84no—
606C4—162no—
607C4#245812yes—
608C4#247112yes—
609C4#270512yes—
610C4#270712yes—
611C2—44no—
612C2—82no—
613C2#1598311yescharacteristic
614C2#2400211yescharacteristic
615C2#2401611yescharacteristic
616C2#2411612yes—
617C2#2413912yes—
618C2#2493312yes—
619C2#2495812yes—
620C2#3202611yescharacteristic
621C2#3203911yescharacteristic
622C2#3225811yescharacteristic
623C2#3227011yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC23×D4D4
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×C4×D4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#1918C2semidirect
4. #11207C2semidirect
5. #11199C2semidirect
6. #11096C2semidirect
7. #11096C2semidirect
9. #10730C2semidirect

Maximal subgroups

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

subgroupindex
C2×#24862
C2×#19182
#139062
#112072
#111992
#110962
#109642
#107302