Finite Group Databasebeta

← #366884

Subgroups — #366884

Order 512.

1654 conjugacy classes of subgroups.

The subgroup lattice has 620 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#476C211yesmaximal, characteristic
2C2×#1987C211yesmaximal, characteristic, complemented
3C2×#1947C211yesmaximal, characteristic, complemented
4C2×#1802C211yesmaximal, characteristic
5C2×#1767C211yesmaximal, characteristic, complemented
6#14714C211yesmaximal, characteristic, complemented
7C4×#390C211yesmaximal, characteristic
8#11378C212yesmaximal, complemented
9#11123C212yesmaximal
10#10927C212yesmaximal
11#10863C212yesmaximal, complemented
12C22×C4⋊Q8C2211yescharacteristic
13C22×C4×D4C2211yescharacteristic
14C2×#476—24no—
15C2×#476—24no—
16C2×#476C418yescomplemented
17C22×#107C2212yes—
18C22×#106C2212yes—
19C22×C4⋊C8C2211yescharacteristic
20C22×C4⋊C8C2211yescharacteristic
21C22×#102C2212yes—
22C22×#102C2212yes—
23C22×#97C2212yes—
24C2×#391—22no—
25C2×#390C2211yescharacteristic
26C2×#390C2211yescharacteristic
27C2×#384C2211yescharacteristic
28C2×C4×C4⋊C4C2211yescharacteristic
29C2×C4×C22⋊C4—22no—
30C2×C43C2211yescharacteristic
31#1987—24no—
32#1947—24no—
33#1802—24no—
34#1767—24no—
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#336C2212yes—
40#1437—22no—
41C4⋊#95—22no—
42C4⋊#95C2211yescharacteristic
43C4⋊#95C2212yes—
44C4⋊#95C2211yescharacteristic
45C4⋊#95C2212yes—
46C23×Q8—21no—
47C23×D4C2311yescharacteristic
48C24×C4—21no—
49C23×C8C2312yes—
50C23×C8C2312yes—
51C2×C4⋊Q8—22no—
52C2×C4⋊Q8—22no—
53C2×C4⋊Q8—22no—
54C2×C4⋊Q8—22no—
55C2×C4⋊Q8—22no—
56C2×C4⋊Q8C2×C414yes—
57C2×C4×D4—22no—
58C2×C4×D4—22no—
59C2×C4×D4—22no—
60C2×C4×D4—22no—
61C2×C4×D4—22no—
62C2×C4×D4C2×C414yes—
63C22×C4⋊C4—21no—
64C22×C4⋊C4—21no—
65C22×C4⋊C4—21no—
66C22×C4⋊C4C2312yes—
67C22×C4⋊C4C2312yes—
68C22×C4⋊C4C2311yescharacteristic
69C22×C4⋊C4C2311yescharacteristic
70C22×C4⋊C4C2311yescharacteristic
71C22×C22⋊C4—21no—
72C22×C22⋊C4—22no—
73C22×C22⋊C4—22no—
74C22×C42—21no—
75C22×C42D412yes—
76C22×C42D412yes—
77C22×C42C2311yescharacteristic
78C22×C42C2311yescharacteristic
79C22×C42C2311yescharacteristic
80#476—232no—
81C2×#107—24no—
82C2×#107—24no—
83C2×#107—42no—
84C2×#106—24no—
85C2×#106—24no—
86C2×#106C2×C418yes—
87C2×C4⋊C8—22no—
88C2×C4⋊C8—22no—
89C2×C4⋊C8C2×C414yes—
90C2×C4⋊C8D414yes—
91C2×C4⋊C8D414yes—
92C2×C4⋊C8D414yes—
93C2×#102—24no—
94C2×#102—24no—
95C2×#102—24no—
96C2×#102—24no—
97C2×#102—42no—
98C2×#102C2×C418yes—
99C2×#97—24no—
100C2×#97—24no—
101C2×#97—42no—
102#391—48no—
103#390—22no—
104#390—22no—
105#390—22no—
106#390Q814yescomplemented
107#384—22no—
108#384—22no—
109C4×C4⋊C4—22no—
110C4×C4⋊C4—22no—
111C4×C22⋊C4—48no—
112C2×#95—24no—
113C2×#95—24no—
114C2×#95—22no—
115C2×#95—22no—
116C2×#95—21no—
117C43—22no—
118C43—22no—
119#340—24no—
120#340—24no—
121#340—24no—
122#340—24no—
123#336—24no—
124C25—21no—
125C22×Q8—24no—
126C22×Q8—44no—
127C22×Q8—44no—
128C22×Q8—42no—
129C22×Q8—42no—
130C22×D4—21no—
131C22×D4—21no—
132C22×D4—21no—
133C22×D4—21no—
134C22×D4—21no—
135C22×D4—21no—
136C22×D4—21no—
137C22×D4—21no—
138C22×D4—21no—
139C22×D4—21no—
140C22×D4—21no—
141C22×D4C22×C412yes—
142C22×D4C2×Q812yes—
143C22×D4#4112yes—
144C23×C4—21no—
145C23×C4—21no—
146C23×C4—21no—
147C23×C4—22no—
148C23×C4—24no—
149C23×C4—22no—
150C23×C4—42no—
151C23×C4—42no—
152C23×C4—44no—
153C23×C4—44no—
154C23×C4C2×D412yes—
155C23×C4C2×D412yes—
156C23×C4C2×D411yescharacteristic
157C23×C4C2×D411yescharacteristic
158C23×C4C2411yescharacteristic
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×C8C22×C414yes—
170C22×C8C2×D414yes—
171C22×C8C2×D414yes—
172C22×C8C2×D414yes—
173C4⋊Q8—28no—
174C4⋊Q8—28no—
175C4⋊Q8—44no—
176C4⋊Q8—44no—
177C4×D4—28no—
178C4×D4—28no—
179C4×D4—44no—
180C4×D4—44no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—24no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—21no—
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—42no—
209C2×C4⋊C4—42no—
210C2×C4⋊C4—44no—
211C2×C4⋊C4—42no—
212C2×C4⋊C4—42no—
213C2×C4⋊C4—44no—
214C2×C4⋊C4C22×C412yes—
215C2×C4⋊C4C22×C414yes—
216C2×C4⋊C4C2×Q812yes—
217C2×C4⋊C4#4112yes—
218C2×C22⋊C4—28no—
219C2×C22⋊C4—24no—
220C2×C22⋊C4—28no—
221C2×C22⋊C4—44no—
222C2×C22⋊C4—44no—
223C2×C22⋊C4—44no—
224C2×C22⋊C4—42no—
225C2×C22⋊C4—42no—
226C2×C22⋊C4—44no—
227C2×C42—21no—
228C2×C42—21no—
229C2×C42—21no—
230C2×C42—21no—
231C2×C42—24no—
232C2×C42—24no—
233C2×C42—24no—
234C2×C42—24no—
235C2×C42—24no—
236C2×C42—21no—
237C2×C42—21no—
238C2×C42—21no—
239C2×C42—21no—
240C2×C42—21no—
241C2×C42—21no—
242C2×C42—42no—
243C2×C42—42no—
244C2×C42—44no—
245C2×C42C22×C412yes—
246C2×C42C2×D412yes—
247C2×C42C2×D412yes—
248C2×C42C2×D412yes—
249C2×C42C2×Q812yes—
250C2×C42#4112yes—
251C2×C42#4112yes—
252C2×C42#4112yes—
253#107—48no—
254#106—216no—
255C4⋊C8—28no—
256C4⋊C8C22⋊C4116yescomplemented
257#102—216no—
258#102—48no—
259#97—48no—
260#95—44no—
261#95—44no—
262#95—48no—
263#95—42no—
264#95—48no—
265C24—22no—
266C24—22no—
267C24—22no—
268C24—41no—
269C24—41no—
270C24—41no—
271C24—41no—
272C24—42no—
273C24—42no—
274C24—42no—
275C24—42no—
276C24C22×D411yescharacteristic
277C2×Q8—416no—
278C2×Q8—48no—
279C2×Q8—84no—
280C2×Q8—84no—
281C2×Q8—88no—
282C2×D4—22no—
283C2×D4—22no—
284C2×D4—22no—
285C2×D4—22no—
286C2×D4—22no—
287C2×D4—22no—
288C2×D4—22no—
289C2×D4—22no—
290C2×D4—22no—
291C2×D4—22no—
292C2×D4—22no—
293C2×D4—22no—
294C2×D4—22no—
295C2×D4—22no—
296C2×D4—22no—
297C2×D4—41no—
298C2×D4—41no—
299C2×D4—41no—
300C2×D4—41no—
301C2×D4—41no—
302C2×D4—41no—
303C2×D4—41no—
304C2×D4—41no—
305C2×D4—41no—
306C2×D4—41no—
307C2×D4—41no—
308C2×D4—41no—
309C2×D4C4×Q814yes—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—24no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—41no—
329C22×C4—42no—
330C22×C4—41no—
331C22×C4—41no—
332C22×C4—41no—
333C22×C4—42no—
334C22×C4—41no—
335C22×C4—41no—
336C22×C4—42no—
337C22×C4—44no—
338C22×C4—42no—
339C22×C4—42no—
340C22×C4—42no—
341C22×C4—42no—
342C22×C4—42no—
343C22×C4—44no—
344C22×C4—48no—
345C22×C4—416no—
346C22×C4—42no—
347C22×C4—42no—
348C22×C4—42no—
349C22×C4—44no—
350C22×C4—44no—
351C22×C4—88no—
352C22×C4—84no—
353C22×C4—84no—
354C22×C4C4×D412yes—
355C22×C4C4×D412yes—
356C22×C4#12112yes—
357C22×C4#12112yes—
358C22×C4C22⋊Q814yes—
359C22×C4#12712yes—
360C22×C4#12712yes—
361C22×C4AGL(1,ℤ/8)14yes—
362C22×C4AGL(1,ℤ/8)14yes—
363C22×C4#13714yes—
364C22×C4C23×C411yescharacteristic
365C22×C4C22×D411yescharacteristic
366C22×C4C22×D411yescharacteristic
367C22×C4C22×D411yescharacteristic
368C22×C4C22×Q811yescharacteristic
369C22×C4C2×#4111yescharacteristic
370C22×C4C2×#4111yescharacteristic
371C22×C4C2×#4111yescharacteristic
372C22×C4C2×#4111yescharacteristic
373C22×C4C2×#4111yescharacteristic
374C22×C4H(5,𝔽2)11yescharacteristic
375C22×C42−1+411yescharacteristic
376C22×C42−1+411yescharacteristic
377C22×C42−1+411yescharacteristic
378C2×C8—24no—
379C2×C8—24no—
380C2×C8—24no—
381C2×C8—24no—
382C2×C8—24no—
383C2×C8—24no—
384C2×C8—24no—
385C2×C8—24no—
386C2×C8—24no—
387C2×C8—42no—
388C2×C8—42no—
389C2×C8—42no—
390C2×C8—42no—
391C2×C8C2×C22⋊C418yes—
392C4⋊C4—24no—
393C4⋊C4—22no—
394C4⋊C4—22no—
395C4⋊C4—24no—
396C4⋊C4—22no—
397C4⋊C4—22no—
398C4⋊C4—22no—
399C4⋊C4—22no—
400C4⋊C4—41no—
401C4⋊C4—42no—
402C4⋊C4—41no—
403C4⋊C4—42no—
404C4⋊C4—41no—
405C4⋊C4—41no—
406C4⋊C4—41no—
407C4⋊C4—41no—
408C4⋊C4—48no—
409C4⋊C4—42no—
410C4⋊C4—42no—
411C4⋊C4—42no—
412C4⋊C4—42no—
413C4⋊C4—84no—
414C4⋊C4—84no—
415C22⋊C4—416no—
416C22⋊C4—416no—
417C22⋊C4—48no—
418C42—22no—
419C42—22no—
420C42—22no—
421C42—22no—
422C42—48no—
423C42—41no—
424C42—41no—
425C42—41no—
426C42—41no—
427C42—48no—
428C42—84no—
429C42C2×C22⋊C414yes—
430C42C4×D414yes—
431C42#12114yes—
432C42#12414yes—
433C23—24no—
434C23—44no—
435C23—44no—
436C23—42no—
437C23—44no—
438C23—44no—
439C23—42no—
440C23—42no—
441C23—44no—
442C23—44no—
443C23—42no—
444C23—42no—
445C23—42no—
446C23—81no—
447C23—81no—
448C23—81no—
449C23—81no—
450C23—82no—
451C23—82no—
452C23—82no—
453C23—82no—
454C23—82no—
455C23—82no—
456C23C2×C4×D411yescharacteristic
457C23C2×#12111yescharacteristic
458C23C2×C22⋊Q811yescharacteristic
459C23#52511yescharacteristic
460C23C2×#12711yescharacteristic
461C23#53611yescharacteristic
462C23#53711yescharacteristic
463C23C2×AGL(1,ℤ/8)12yes—
464C23C2×AGL(1,ℤ/8)12yes—
465C23C2×#13712yes—
466C23#57512yes—
467Q8—816no—
468D4—24no—
469D4—24no—
470D4—42no—
471D4—42no—
472D4—42no—
473D4—42no—
474D4—42no—
475D4—42no—
476C2×C4—22no—
477C2×C4—22no—
478C2×C4—24no—
479C2×C4—24no—
480C2×C4—24no—
481C2×C4—24no—
482C2×C4—24no—
483C2×C4—24no—
484C2×C4—24no—
485C2×C4—24no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—44no—
495C2×C4—44no—
496C2×C4—44no—
497C2×C4—48no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—44no—
502C2×C4—81no—
503C2×C4—81no—
504C2×C4—81no—
505C2×C4—81no—
506C2×C4—81no—
507C2×C4—82no—
508C2×C4—81no—
509C2×C4—82no—
510C2×C4—82no—
511C2×C4—82no—
512C2×C4—82no—
513C2×C4—84no—
514C2×C4—816no—
515C2×C4—84no—
516C2×C4—82no—
517C2×C4—82no—
518C2×C4—84no—
519C2×C4—82no—
520C2×C4#39014yes—
521C2×C4#39014yes—
522C2×C4#44018yes—
523C2×C4#46818yes—
524C2×C4#49614yes—
525C2×C4#49714yes—
526C2×C4C22×C22⋊C411yescharacteristic
527C2×C4C2×C4×D411yescharacteristic
528C2×C4C2×C4×Q811yescharacteristic
529C2×C4C4×#4111yescharacteristic
530C2×C4#51911yescharacteristic
531C2×C4#52011yescharacteristic
532C2×C4C2×#12111yescharacteristic
533C2×C4C2×#12411yescharacteristic
534C2×C4#52911yescharacteristic
535C2×C4#53311yescharacteristic
536C2×C4#54111yescharacteristic
537C2×C4#54211yescharacteristic
538C2×C4#54611yescharacteristic
539C2×C4#54611yescharacteristic
540C2×C4#54611yescharacteristic
541C2×C4#54711yescharacteristic
542C2×C4#54711yescharacteristic
543C2×C4#54711yescharacteristic
544C2×C4#54811yescharacteristic
545C2×C4#54811yescharacteristic
546C2×C4#54811yescharacteristic
547C2×C4D4×Q811yescharacteristic
548C2×C4D4×Q811yescharacteristic
549C2×C4D4×Q811yescharacteristic
550C2×C4#55211yescharacteristic
551C2×C4#55211yescharacteristic
552C2×C4#55411yescharacteristic
553C2×C4#55611yescharacteristic
554C8—28no—
555C8—44no—
556C22—48no—
557C22—44no—
558C22—84no—
559C22—84no—
560C22—82no—
561C22—82no—
562C22—84no—
563C22—82no—
564C22—82no—
565C22—82no—
566C22—82no—
567C22C2×#39011yescharacteristic
568C22C4×C22⋊Q811yescharacteristic
569C22#232311yescharacteristic
570C22#241511yescharacteristic
571C22#241711yescharacteristic
572C22#254011yescharacteristic
573C22#254211yescharacteristic
574C22C2×#44012yes—
575C22C4×AGL(1,ℤ/8)12yes—
576C22C2×#46812yes—
577C22#305512yes—
578C22#307712yes—
579C22#307912yes—
580C22#310912yes—
581C22#311612yes—
582C22C2×#49611yescharacteristic
583C22C2×#49711yescharacteristic
584C22#315111yescharacteristic
585C22#315311yescharacteristic
586C22#323512yes—
587C22#323812yes—
588C22#325212yes—
589C22#325312yes—
590C4—24no—
591C4—44no—
592C4—44no—
593C4—82no—
594C4—82no—
595C4—82no—
596C4—84no—
597C4—162no—
598C4#197318yes—
599C4#233911yescharacteristic
600C4Q8×C22⋊C411yescharacteristic
601C4#236011yescharacteristic
602C4#236211yescharacteristic
603C4#242711yescharacteristic
604C4#242911yescharacteristic
605C4#248911yescharacteristic
606C4#249211yescharacteristic
607C2—84no—
608C2#1399711yescharacteristic
609C2C2×#197311yescharacteristic
610C2#2123611yescharacteristic
611C2#2161412yes—
612C2#2211712yes—
613C2#2470812yes—
614C2#2484312yes—
615C2#2838111yescharacteristic
616C2#2838411yescharacteristic
617C2#2851911yescharacteristic
618C2#2852111yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C43 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#1987C2semidirect
3. C2×#1947C2semidirect
5. C2×#1767C2semidirect
6. #14714C2semidirect
8. #11378C2semidirect
11. #10863C2semidirect
16. C2×#476C4semidirect
106. #390Q8semidirect
256. C4⋊C8C22⋊C4semidirect

Maximal subgroups

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

subgroupindex
C22×#4762
C2×#19872
C2×#19472
C2×#18022
C2×#17672
#147142
C4×#3902
#113782
#111232
#109272
#108632