Finite Group Databasebeta

← #362618

Subgroups — #362618

Order 512.

1764 conjugacy classes of subgroups.

The subgroup lattice has 635 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C8×D4C211yesmaximal, characteristic
2C2×#1949C212yesmaximal, complemented
3C2×#1808C211yesmaximal, characteristic, complemented
4C2×#1807C211yesmaximal, characteristic, complemented
5#13905C212yesmaximal
6#11096C212yesmaximal, complemented
7#11095C212yesmaximal, complemented
8#10842C211yesmaximal, characteristic, complemented
9#10721C211yesmaximal, characteristic, complemented
10#10721C211yesmaximal, characteristic, complemented
11C4×#340C211yesmaximal, characteristic, complemented
12C24×C8C2211yescharacteristic
13C24×C8C2211yescharacteristic
14C22×C4×D4C2211yescharacteristic
15C2×C8×D4—28no—
16C2×C8×D4C418yescomplemented
17C22×#107C2212yes—
18C22×#106C2212yes—
19C22×C4⋊C8C2211yescharacteristic
20C22×C22⋊C8C2211yescharacteristic
21C22×C22⋊C8C2211yescharacteristic
22C22×C4×C8C2211yescharacteristic
23C2×#382C2212yes—
24C2×#380C2212yes—
25C2×#380C2212yes—
26C2×#380C2212yes—
27C2×#380C2212yes—
28C2×C4×C4⋊C4C2212yes—
29#1949—28no—
30#1808—24no—
31#1807—24no—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#340C2211yescharacteristic
36C2×#340C2211yescharacteristic
37C2×#340C2211yescharacteristic
38C2×#340C2212yes—
39C2×#340C2211yescharacteristic
40#1445—22no—
41#1438—22no—
42#1437—22no—
43#1437—22no—
44C23×D4D411yescharacteristic
45C24×C4C2311yescharacteristic
46C24×C4C2311yescharacteristic
47C23×C8—22no—
48C23×C8—22no—
49C23×C8—24no—
50C23×C8—24no—
51C23×C8—22no—
52C23×C8—22no—
53C23×C8—42no—
54C23×C8—42no—
55C23×C8C2×C414yes—
56C23×C8C2×C414yes—
57C23×C8C2311yescharacteristic
58C23×C8C2311yescharacteristic
59C23×C8C2311yescharacteristic
60C23×C8C2311yescharacteristic
61C2×C4×D4—24no—
62C2×C4×D4—24no—
63C2×C4×D4C2×C414yes—
64C2×C4×D4Q814yes—
65C22×C4⋊C4D411yescharacteristic
66C22×C4⋊C4C2312yes—
67C22×C4⋊C4C2312yes—
68C22×C4⋊C4C2312yes—
69C22×C4⋊C4C2312yes—
70C22×C22⋊C4—22no—
71C22×C22⋊C4—22no—
72C22×C22⋊C4—22no—
73C22×C22⋊C4—22no—
74C22×C22⋊C4D411yescharacteristic
75C22×C22⋊C4D411yescharacteristic
76C22×C42—22no—
77C22×C42C2311yescharacteristic
78C8×D4—232no—
79C2×#107—24no—
80C2×#107—24no—
81C2×#107—42no—
82C2×#106—24no—
83C2×#106—24no—
84C2×#106—42no—
85C2×C4⋊C8C2×C414yes—
86C2×C4⋊C8D418yes—
87C2×C22⋊C8—24no—
88C2×C22⋊C8—24no—
89C2×C22⋊C8C2×C414yes—
90C2×C22⋊C8C2×C414yes—
91C2×C4×C8—24no—
92C2×C4×C8C2×C414yes—
93#382—24no—
94#382—24no—
95#380—24no—
96#380—24no—
97#380—24no—
98#380—24no—
99#380—42no—
100#380—42no—
101#380—42no—
102#380—42no—
103C4×C4⋊C4—24no—
104C4×C4⋊C4—24no—
105C2×#95—22no—
106C2×#95—22no—
107C2×#95—22no—
108C2×#95—22no—
109C2×#95—22no—
110C2×#95—22no—
111C2×#95—22no—
112#340—22no—
113#340—22no—
114#340—24no—
115#340—24no—
116#340—22no—
117#340—24no—
118#340—42no—
119#340D414yescomplemented
120C25C2×D411yescharacteristic
121C25C2×D411yescharacteristic
122C22×D4—22no—
123C22×D4—24no—
124C22×D4—22no—
125C22×D4—24no—
126C22×D4C4⋊C414yes—
127C23×C4—22no—
128C23×C4—22no—
129C23×C4—21no—
130C23×C4—21no—
131C23×C4—22no—
132C23×C4—22no—
133C23×C4—21no—
134C23×C4—21no—
135C23×C4—22no—
136C23×C4—22no—
137C23×C4—22no—
138C23×C4—22no—
139C23×C4—21no—
140C23×C4—21no—
141C23×C4—21no—
142C23×C4—21no—
143C23×C4—22no—
144C23×C4—22no—
145C23×C4—22no—
146C23×C4—22no—
147C23×C4C22×C412yes—
148C23×C4C22×C412yes—
149C23×C4C2×D411yescharacteristic
150C23×C4C2×D411yescharacteristic
151C23×C4C2×D411yescharacteristic
152C23×C4C2×D411yescharacteristic
153C23×C4C2×Q812yes—
154C23×C4C2×Q812yes—
155C23×C4C2411yescharacteristic
156C22×C8—22no—
157C22×C8—21no—
158C22×C8—21no—
159C22×C8—21no—
160C22×C8—21no—
161C22×C8—22no—
162C22×C8—22no—
163C22×C8—22no—
164C22×C8—22no—
165C22×C8—22no—
166C22×C8—21no—
167C22×C8—21no—
168C22×C8—28no—
169C22×C8—28no—
170C22×C8—28no—
171C22×C8—28no—
172C22×C8—44no—
173C22×C8—44no—
174C22×C8—48no—
175C22×C8—48no—
176C22×C8—48no—
177C22×C8—48no—
178C22×C8C22×C412yes—
179C22×C8C22×C412yes—
180C22×C8C22×C412yes—
181C22×C8C22×C412yes—
182C22×C8C2×D414yes—
183C22×C8C2×D414yes—
184C22×C8C2×D412yes—
185C22×C8#4112yes—
186C4×D4—28no—
187C4×D4—28no—
188C4×D4—216no—
189C2×C4⋊C4—24no—
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—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4C4⋊C414yes—
213C2×C4⋊C4C2×D414yes—
214C2×C4⋊C4#4114yes—
215C2×C22⋊C4—24no—
216C2×C22⋊C4—24no—
217C2×C22⋊C4—44no—
218C2×C22⋊C4—44no—
219C2×C22⋊C4—44no—
220C2×C22⋊C4—44no—
221C2×C22⋊C4—44no—
222C2×C22⋊C4—44no—
223C2×C22⋊C4—44no—
224C2×C22⋊C4—44no—
225C2×C22⋊C4—82no—
226C2×C22⋊C4—82no—
227C2×C22⋊C4—82no—
228C2×C22⋊C4—82no—
229C2×C22⋊C4C4⋊C414yes—
230C2×C22⋊C4C4⋊C414yes—
231C2×C42—44no—
232C2×C42—82no—
233C2×C42—82no—
234C2×C42C22×C412yes—
235C2×C42C2×D414yes—
236C2×C42C2×Q812yes—
237C2×C42#4114yes—
238#107—48no—
239#106—48no—
240C4⋊C8C22⋊C4116yescomplemented
241C22⋊C8—28no—
242C22⋊C8—28no—
243C4×C8—28no—
244#95—44no—
245#95—44no—
246#95—44no—
247#95—44no—
248#95—44no—
249#95—44no—
250#95—44no—
251C24—22no—
252C24—22no—
253C24—21no—
254C24—21no—
255C24—22no—
256C24—22no—
257C24—21no—
258C24—21no—
259C24—21no—
260C24—22no—
261C24—21no—
262C24—21no—
263C24—21no—
264C24—22no—
265C24—22no—
266C24—21no—
267C24—21no—
268C24C2×C4⋊C412yes—
269C24C2×C4⋊C412yes—
270C24#13512yes—
271C24#13514yes—
272C24#13512yes—
273C24C22×D411yescharacteristic
274C2×D4—216no—
275C2×D4—216no—
276C2×D4—28no—
277C2×D4—28no—
278C2×D4—28no—
279C22×C4—21no—
280C22×C4—22no—
281C22×C4—21no—
282C22×C4—21no—
283C22×C4—22no—
284C22×C4—21no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—24no—
306C22×C4—24no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—24no—
312C22×C4—24no—
313C22×C4—24no—
314C22×C4—24no—
315C22×C4—42no—
316C22×C4—42no—
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—42no—
346C22×C4—42no—
347C22×C4—42no—
348C22×C4—42no—
349C22×C4—42no—
350C22×C4—42no—
351C22×C4—42no—
352C22×C4—42no—
353C22×C4—42no—
354C22×C4—42no—
355C22×C4—42no—
356C22×C4—42no—
357C22×C4C2×C4⋊C412yes—
358C22×C4C2×C4⋊C412yes—
359C22×C4C2×C4⋊C412yes—
360C22×C4C2×C4⋊C412yes—
361C22×C4#12114yes—
362C22×C4#12114yes—
363C22×C4C23×C411yescharacteristic
364C22×C4C22×D412yes—
365C22×C4C22×D411yescharacteristic
366C22×C4C22×Q811yescharacteristic
367C22×C4C2×#4112yes—
368C22×C4C2×#4111yescharacteristic
369C22×C4C2×#4112yes—
370C22×C4H(5,𝔽2)12yes—
371C22×C4H(5,𝔽2)11yescharacteristic
372C22×C42−1+411yescharacteristic
373C2×C8—22no—
374C2×C8—22no—
375C2×C8—22no—
376C2×C8—22no—
377C2×C8—22no—
378C2×C8—22no—
379C2×C8—24no—
380C2×C8—24no—
381C2×C8—24no—
382C2×C8—24no—
383C2×C8—24no—
384C2×C8—24no—
385C2×C8—22no—
386C2×C8—22no—
387C2×C8—22no—
388C2×C8—416no—
389C2×C8—416no—
390C2×C8—42no—
391C2×C8—42no—
392C2×C8C2×C22⋊C414yes—
393C2×C8C2×C22⋊C414yes—
394C2×C8C4×D414yes—
395C4⋊C4—28no—
396C4⋊C4—24no—
397C4⋊C4—24no—
398C4⋊C4—24no—
399C4⋊C4—24no—
400C4⋊C4—42no—
401C4⋊C4—42no—
402C4⋊C4—42no—
403C4⋊C4—42no—
404C4⋊C4—42no—
405C4⋊C4—42no—
406C4⋊C4—42no—
407C4⋊C4—42no—
408C4⋊C4—42no—
409C4⋊C4—42no—
410C4⋊C4—42no—
411C4⋊C4—42no—
412C22⋊C4—28no—
413C22⋊C4—28no—
414C22⋊C4—88no—
415C22⋊C4—88no—
416C22⋊C4—88no—
417C22⋊C4—88no—
418C42—88no—
419C42C2×C22⋊C414yes—
420C42#11714yes—
421C42C22⋊Q818yes—
422C23—22no—
423C23—24no—
424C23—22no—
425C23—22no—
426C23—22no—
427C23—22no—
428C23—22no—
429C23—22no—
430C23—22no—
431C23—22no—
432C23—22no—
433C23—22no—
434C23—22no—
435C23—22no—
436C23—24no—
437C23—22no—
438C23—24no—
439C23—24no—
440C23—24no—
441C23—24no—
442C23—22no—
443C23—22no—
444C23—24no—
445C23—24no—
446C23—24no—
447C23—24no—
448C23—41no—
449C23—42no—
450C23—41no—
451C23—42no—
452C23—42no—
453C23—42no—
454C23—42no—
455C23—41no—
456C23—42no—
457C23—41no—
458C23—42no—
459C23—42no—
460C23—42no—
461C23—42no—
462C23—42no—
463C23—42no—
464C23—42no—
465C23—42no—
466C23#42714yes—
467C23#42714yes—
468C23C22×C4⋊C411yescharacteristic
469C23C2×#12112yes—
470C23D4211yescharacteristic
471C23#54612yes—
472C23#54711yescharacteristic
473C23C2×#13511yescharacteristic
474C23C2×#13512yes—
475C23C2×#13511yescharacteristic
476C23#57611yescharacteristic
477C23#57712yes—
478C23#57811yescharacteristic
479D4—232no—
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—22no—
489C2×C4—22no—
490C2×C4—22no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—44no—
505C2×C4—44no—
506C2×C4—44no—
507C2×C4—44no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—44no—
511C2×C4—44no—
512C2×C4—44no—
513C2×C4—44no—
514C2×C4—44no—
515C2×C4—44no—
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—82no—
536C2×C4—82no—
537C2×C4—82no—
538C2×C4#38014yes—
539C2×C4#38014yes—
540C2×C4C22×C22⋊C411yescharacteristic
541C2×C4C2×#11711yescharacteristic
542C2×C4C2×C4×D411yescharacteristic
543C2×C4C4×#4111yescharacteristic
544C2×C4#51811yescharacteristic
545C2×C4#52011yescharacteristic
546C2×C4C2×C22⋊Q812yes—
547C2×C4#52912yes—
548C2×C4#53212yes—
549C2×C4#54012yes—
550C2×C4#54212yes—
551C2×C4D4212yes—
552C2×C4#54612yes—
553C2×C4#54812yes—
554C2×C4D4×Q811yescharacteristic
555C2×C4#55012yes—
556C2×C4#55412yes—
557C2×C4#55411yescharacteristic
558C8—24no—
559C8—24no—
560C8—24no—
561C8—42no—
562C22—24no—
563C22—24no—
564C22—24no—
565C22—24no—
566C22—24no—
567C22—24no—
568C22—44no—
569C22—44no—
570C22—44no—
571C22—44no—
572C22—42no—
573C22—42no—
574C22—42no—
575C22—42no—
576C22—42no—
577C22—42no—
578C22—42no—
579C22—42no—
580C22—44no—
581C22—44no—
582C22—44no—
583C22—44no—
584C22C2×#38011yescharacteristic
585C22D4×C4⋊C411yescharacteristic
586C22#234911yescharacteristic
587C22#242712yes—
588C22#242812yes—
589C22C2×#42711yescharacteristic
590C22C2×#42711yescharacteristic
591C22#291211yescharacteristic
592C22#291311yescharacteristic
593C22#305012yes—
594C22#305012yes—
595C22#306212yes—
596C22#306312yes—
597C22#328011yescharacteristic
598C22#328212yes—
599C22#328511yescharacteristic
600C22#329612yes—
601C22#329712yes—
602C22#330012yes—
603C22#330112yes—
604C22#333611yescharacteristic
605C22#333712yes—
606C22#333911yescharacteristic
607C4—24no—
608C4—24no—
609C4—42no—
610C4—42no—
611C4—84no—
612C4—84no—
613C4—162no—
614C4—162no—
615C4#229311yescharacteristic
616C4#231911yescharacteristic
617C4D4×C22⋊C411yescharacteristic
618C4#234111yescharacteristic
619C4#245212yes—
620C4#245412yes—
621C2—44no—
622C2—44no—
623C2#1392011yescharacteristic
624C2#2065811yescharacteristic
625C2#2066411yescharacteristic
626C2#2265411yescharacteristic
627C2#2265611yescharacteristic
628C2#2270411yescharacteristic
629C2#2270611yescharacteristic
630C2#2534212yes—
631C2#2534312yes—
632C2#2538812yes—
633C2#2538912yes—
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×D4D4
LCSG ▷ C22×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×#1949C2semidirect
3. C2×#1808C2semidirect
4. C2×#1807C2semidirect
6. #11096C2semidirect
7. #11095C2semidirect
8. #10842C2semidirect
9. #10721C2semidirect
10. #10721C2semidirect
11. C4×#340C2semidirect
16. C2×C8×D4C4semidirect
119. #340D4semidirect
240. C4⋊C8C22⋊C4semidirect

Maximal subgroups

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

subgroupindex
C22×C8×D42
C2×#19492
C2×#18082
C2×#18072
#139052
#110962
#110952
#108422
#107212
C4×#3402