Finite Group Databasebeta

← #404173

Subgroups — #404173

Order 512.

1590 conjugacy classes of subgroups.

The subgroup lattice has 633 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2354C211yesmaximal, characteristic
2C2×#2023C211yesmaximal, characteristic
3C2×#1988C211yesmaximal, characteristic, complemented
4#15128C211yesmaximal, characteristic, complemented
5#11267C212yesmaximal, complemented
6#11202C211yesmaximal, characteristic, complemented
7#11076C211yesmaximal, characteristic
8#11066C212yesmaximal
9#10884C212yesmaximal, complemented
10#10806C211yesmaximal, characteristic, complemented
11#10788C212yesmaximal
12C22×C4⋊Q8C2211yescharacteristic
13C22×C4×D4C2211yescharacteristic
14C22×#117—21no—
15C22×C4⋊C8C2211yescharacteristic
16C22×#102C2212yes—
17C22×#102C2212yes—
18C2×#395—21no—
19#2354—24no—
20#2354—24no—
21C2×#390C2211yescharacteristic
22C2×#390C2211yescharacteristic
23C2×#389C2212yes—
24C2×#388—21no—
25C2×#388—21no—
26C2×#384C2211yescharacteristic
27C2×#380—21no—
28C2×C4×C4⋊C4C2211yescharacteristic
29C2×C4×C4⋊C4C2211yescharacteristic
30C2×C4×C4⋊C4C2211yescharacteristic
31#2023—24no—
32#1988—24no—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#336C2212yes—
40C2×#336C2211yescharacteristic
41C2×#336C2212yes—
42#1446—21no—
43#1444—21no—
44#1441—21no—
45C4⋊#95C2212yes—
46C4⋊#95C2211yescharacteristic
47C23×Q8—21no—
48C23×D4C2311yescharacteristic
49C24×C4—21no—
50C23×C8C2312yes—
51C23×C8C2312yes—
52C2×C4⋊Q8—24no—
53C2×C4⋊Q8—22no—
54C2×C4⋊Q8—22no—
55C2×C4⋊Q8—42no—
56C2×C4×D4—22no—
57C2×C4×D4—22no—
58C2×C4×D4—22no—
59C2×C4×D4—22no—
60C2×C4×D4—22no—
61C2×C4×D4Q814yescomplemented
62C2×#117—28no—
63C2×#117—44no—
64C2×#117—44no—
65C22×C4⋊C4—21no—
66C22×C4⋊C4—21no—
67C22×C4⋊C4—21no—
68C22×C4⋊C4—22no—
69C22×C4⋊C4D411yescharacteristic
70C22×C4⋊C4D411yescharacteristic
71C22×C4⋊C4C2311yescharacteristic
72C22×C4⋊C4C2311yescharacteristic
73C22×C4⋊C4C2312yes—
74C22×C4⋊C4C2312yes—
75C22×C4⋊C4C2311yescharacteristic
76C22×C22⋊C4—21no—
77C22×C22⋊C4—21no—
78C22×C22⋊C4—21no—
79C22×C22⋊C4—21no—
80C22×C22⋊C4—21no—
81C22×C42—21no—
82C22×C42—21no—
83C22×C42C2311yescharacteristic
84C22×C42C2311yescharacteristic
85C22×C42C2311yescharacteristic
86C2×C4⋊C8—22no—
87C2×C4⋊C8—42no—
88C2×#102—24no—
89C2×#102—24no—
90C2×#102—24no—
91C2×#102—24no—
92C2×#102—42no—
93C2×#102—42no—
94#395—44no—
95#390—22no—
96#390—22no—
97#390—22no—
98#390D414yes—
99#389—24no—
100#389—42no—
101#388—28no—
102#388—28no—
103#384—22no—
104#384—22no—
105#380—28no—
106C4×C4⋊C4—22no—
107C4×C4⋊C4—22no—
108C4×C4⋊C4—22no—
109C4×C4⋊C4—22no—
110C4×C4⋊C4—22no—
111C4×C4⋊C4D414yes—
112C2×#95—21no—
113C2×#95—21no—
114C2×#95—22no—
115C2×#95—21no—
116C2×#95—21no—
117C2×#95—22no—
118C2×#95—22no—
119C2×#95D411yescharacteristic
120C2×#95D411yescharacteristic
121#340—24no—
122#340—24no—
123#340—24no—
124#340—24no—
125#340—24no—
126#340—24no—
127#336—22no—
128#336—24no—
129#336—42no—
130C25—21no—
131C22×Q8—44no—
132C22×Q8—42no—
133C22×Q8—42no—
134C22×Q8—42no—
135C22×Q8—82no—
136C22×D4—21no—
137C22×D4—22no—
138C22×D4—22no—
139C22×D4—21no—
140C22×D4—21no—
141C22×D4—22no—
142C22×D4—22no—
143C22×D4C2×D412yes—
144C22×D4C2×Q812yes—
145C22×D4#4112yes—
146C23×C4—24no—
147C23×C4—24no—
148C23×C4—21no—
149C23×C4—24no—
150C23×C4—21no—
151C23×C4—21no—
152C23×C4—22no—
153C23×C4—22no—
154C23×C4—42no—
155C23×C4—42no—
156C23×C4—42no—
157C23×C4—42no—
158C23×C4C2×D411yescharacteristic
159C23×C4C2×D411yescharacteristic
160C23×C4C2×D411yescharacteristic
161C23×C4C2×D411yescharacteristic
162C23×C4C2×D411yescharacteristic
163C23×C4C2×D411yescharacteristic
164C23×C4C2411yescharacteristic
165C22×C8—22no—
166C22×C8—22no—
167C22×C8—22no—
168C22×C8—22no—
169C22×C8—22no—
170C22×C8—22no—
171C22×C8—22no—
172C22×C8—22no—
173C22×C8—22no—
174C22×C8—22no—
175C22×C8—22no—
176C22×C8—22no—
177C22×C8—22no—
178C22×C8—22no—
179C4⋊Q8—28no—
180C4⋊Q8—48no—
181C4⋊Q8—44no—
182C4×D4—28no—
183C4×D4—28no—
184C4×D4—28no—
185C4×D4—28no—
186#117—432no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—21no—
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—24no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—24no—
212C2×C4⋊C4—41no—
213C2×C4⋊C4—44no—
214C2×C4⋊C4—42no—
215C2×C4⋊C4—42no—
216C2×C4⋊C4—42no—
217C2×C4⋊C4—44no—
218C2×C4⋊C4—44no—
219C2×C4⋊C4—82no—
220C2×C4⋊C4—82no—
221C2×C4⋊C4D814yes—
222C2×C4⋊C4C2×D412yes—
223C2×C4⋊C4C2×D412yes—
224C2×C4⋊C4C2×D412yes—
225C2×C4⋊C4C2×Q812yes—
226C2×C4⋊C4#4114yes—
227C2×C4⋊C4#4114yes—
228C2×C4⋊C4#4112yes—
229C2×C22⋊C4—24no—
230C2×C22⋊C4—24no—
231C2×C22⋊C4—24no—
232C2×C22⋊C4—24no—
233C2×C22⋊C4—24no—
234C2×C22⋊C4—24no—
235C2×C22⋊C4—24no—
236C2×C22⋊C4—42no—
237C2×C22⋊C4—42no—
238C2×C22⋊C4—42no—
239C2×C22⋊C4—42no—
240C2×C22⋊C4—82no—
241C2×C22⋊C4—82no—
242C2×C42—21no—
243C2×C42—21no—
244C2×C42—21no—
245C2×C42—21no—
246C2×C42—22no—
247C2×C42—22no—
248C2×C42—24no—
249C2×C42—24no—
250C2×C42—21no—
251C2×C42—21no—
252C2×C42—22no—
253C2×C42—22no—
254C2×C42—42no—
255C2×C42—42no—
256C2×C42—42no—
257C2×C42—42no—
258C2×C42C2×D412yes—
259C2×C42C2×D412yes—
260C2×C42C2×Q812yes—
261C2×C42#4112yes—
262C4⋊C8—44no—
263#102—48no—
264#102—48no—
265#95—24no—
266#95—24no—
267#95—22no—
268#95—22no—
269#95—44no—
270#95—42no—
271#95—42no—
272#95—44no—
273#95—82no—
274C24—22no—
275C24—22no—
276C24—22no—
277C24—42no—
278C24—42no—
279C24—41no—
280C24—41no—
281C24—41no—
282C24—41no—
283C24—42no—
284C24—42no—
285C24C22×D411yescharacteristic
286C2×Q8—48no—
287C2×Q8—88no—
288C2×Q8—88no—
289C2×Q8—84no—
290C2×Q8—84no—
291C2×D4—22no—
292C2×D4—22no—
293C2×D4—22no—
294C2×D4—24no—
295C2×D4—24no—
296C2×D4—24no—
297C2×D4—24no—
298C2×D4—24no—
299C2×D4—24no—
300C2×D4—41no—
301C2×D4—42no—
302C2×D4—41no—
303C2×D4—41no—
304C2×D4—41no—
305C2×D4—42no—
306C2×D4—42no—
307C2×D4—42no—
308C2×D4C22⋊Q814yes—
309C22×C4—21no—
310C22×C4—21no—
311C22×C4—22no—
312C22×C4—21no—
313C22×C4—21no—
314C22×C4—22no—
315C22×C4—216no—
316C22×C4—22no—
317C22×C4—21no—
318C22×C4—21no—
319C22×C4—22no—
320C22×C4—21no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—21no—
325C22×C4—21no—
326C22×C4—21no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—21no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—24no—
336C22×C4—24no—
337C22×C4—21no—
338C22×C4—21no—
339C22×C4—22no—
340C22×C4—48no—
341C22×C4—48no—
342C22×C4—48no—
343C22×C4—48no—
344C22×C4—48no—
345C22×C4—41no—
346C22×C4—41no—
347C22×C4—41no—
348C22×C4—41no—
349C22×C4—48no—
350C22×C4—42no—
351C22×C4—42no—
352C22×C4—42no—
353C22×C4—42no—
354C22×C4—42no—
355C22×C4—42no—
356C22×C4—42no—
357C22×C4—42no—
358C22×C4—41no—
359C22×C4—41no—
360C22×C4—41no—
361C22×C4—41no—
362C22×C4—42no—
363C22×C4—42no—
364C22×C4—42no—
365C22×C4—42no—
366C22×C4—42no—
367C22×C4—42no—
368C22×C4—42no—
369C22×C4—42no—
370C22×C4#12012yes—
371C22×C4#12012yes—
372C22×C4C22⋊Q812yes—
373C22×C4C22⋊Q812yes—
374C22×C4C2×D812yes—
375C22×C4C2×D812yes—
376C22×C4C2×D812yes—
377C22×C4AGL(1,ℤ/8)12yes—
378C22×C4#13712yes—
379C22×C4C22×D411yescharacteristic
380C22×C4C22×D411yescharacteristic
381C22×C4C22×Q811yescharacteristic
382C22×C4C2×#4111yescharacteristic
383C22×C4C2×#4111yescharacteristic
384C22×C4C2×#4111yescharacteristic
385C22×C4C2×#4112yes—
386C22×C4C2×#4112yes—
387C22×C4H(5,𝔽2)12yes—
388C22×C42−1+412yes—
389C2×C8—42no—
390C2×C8—42no—
391C2×C8—42no—
392C2×C8—42no—
393C2×C8—42no—
394C2×C8—42no—
395C2×C8—42no—
396C2×C8—42no—
397C2×C8—42no—
398C2×C8—42no—
399C2×C8—42no—
400C2×C8—42no—
401C2×C8—42no—
402C2×C8—42no—
403C4⋊C4—22no—
404C4⋊C4—22no—
405C4⋊C4—22no—
406C4⋊C4—28no—
407C4⋊C4—24no—
408C4⋊C4—24no—
409C4⋊C4—22no—
410C4⋊C4—22no—
411C4⋊C4—24no—
412C4⋊C4—24no—
413C4⋊C4—24no—
414C4⋊C4—44no—
415C4⋊C4—48no—
416C4⋊C4—48no—
417C4⋊C4—41no—
418C4⋊C4—42no—
419C4⋊C4—41no—
420C4⋊C4—42no—
421C4⋊C4—42no—
422C4⋊C4—42no—
423C4⋊C4—42no—
424C4⋊C4—88no—
425C4⋊C4—84no—
426C4⋊C4C22⋊Q814yes—
427C22⋊C4—216no—
428C22⋊C4—48no—
429C22⋊C4—48no—
430C22⋊C4—84no—
431C22⋊C4—84no—
432C42—22no—
433C42—22no—
434C42—22no—
435C42—22no—
436C42—22no—
437C42—22no—
438C42—24no—
439C42—48no—
440C42—41no—
441C42—41no—
442C42—42no—
443C42—48no—
444C23—24no—
445C23—42no—
446C23—44no—
447C23—42no—
448C23—42no—
449C23—44no—
450C23—42no—
451C23—42no—
452C23—44no—
453C23—44no—
454C23—42no—
455C23—42no—
456C23—42no—
457C23—42no—
458C23—42no—
459C23—82no—
460C23—82no—
461C23—82no—
462C23—82no—
463C23—82no—
464C23—82no—
465C23—82no—
466C23—82no—
467C23C2×#12011yescharacteristic
468C23C2×C22⋊Q811yescharacteristic
469C23#52511yescharacteristic
470C23#54612yes—
471C23#54812yes—
472C23C22×D811yescharacteristic
473C23C2×AGL(1,ℤ/8)11yescharacteristic
474C23C2×#13711yescharacteristic
475C23#57511yescharacteristic
476C23#57712yes—
477C23#57712yes—
478Q8—816no—
479D4—28no—
480D4—42no—
481D4—42no—
482D4—42no—
483D4—42no—
484D4—44no—
485C2×C4—22no—
486C2×C4—22no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—22no—
490C2×C4—22no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—24no—
494C2×C4—24no—
495C2×C4—24no—
496C2×C4—22no—
497C2×C4—24no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—24no—
503C2×C4—432no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—41no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—44no—
520C2×C4—44no—
521C2×C4—41no—
522C2×C4—44no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—44no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—44no—
533C2×C4—44no—
534C2×C4—42no—
535C2×C4—82no—
536C2×C4—82no—
537C2×C4—82no—
538C2×C4—82no—
539C2×C4—82no—
540C2×C4—82no—
541C2×C4—82no—
542C2×C4—82no—
543C2×C4—82no—
544C2×C4—82no—
545C2×C4—82no—
546C2×C4—82no—
547C2×C4#44714yes—
548C2×C4#47414yes—
549C2×C4C2×C22⋊Q811yescharacteristic
550C2×C4C2×C22⋊Q811yescharacteristic
551C2×C4#52511yescharacteristic
552C2×C4#52511yescharacteristic
553C2×C4#52511yescharacteristic
554C2×C4#52912yes—
555C2×C4#53211yescharacteristic
556C2×C4#53312yes—
557C2×C4#53511yescharacteristic
558C2×C4#53611yescharacteristic
559C2×C4#54212yes—
560C2×C4#54412yes—
561C2×C4#54612yes—
562C2×C4#54712yes—
563C2×C4#55112yes—
564C2×C4#55212yes—
565C2×C4#55512yes—
566C2×C4#55612yes—
567C8—82no—
568C8—82no—
569C22—44no—
570C22—44no—
571C22—44no—
572C22—84no—
573C22—82no—
574C22—84no—
575C22—82no—
576C22—82no—
577C22—84no—
578C22—82no—
579C22—84no—
580C22#240411yescharacteristic
581C22#243211yescharacteristic
582C22#243912yes—
583C22#247712yes—
584C22#284811yescharacteristic
585C22C2×#44711yescharacteristic
586C22#300711yescharacteristic
587C22#300911yescharacteristic
588C22#301011yescharacteristic
589C22C2×#47411yescharacteristic
590C22#307711yescharacteristic
591C22#308412yes—
592C22#310011yescharacteristic
593C22#311011yescharacteristic
594C22#312512yes—
595C22#329412yes—
596C22#330612yes—
597C22#332112yes—
598C22#332912yes—
599C22#335812yes—
600C22#336812yes—
601C22#336912yes—
602C22#337612yes—
603C4—24no—
604C4—24no—
605C4—42no—
606C4—42no—
607C4—44no—
608C4—42no—
609C4—84no—
610C4—82no—
611C4—82no—
612C4—84no—
613C4—81no—
614C4#239611yescharacteristic
615C4#243812yes—
616C4#247412yes—
617C4#259911yescharacteristic
618C4#261011yescharacteristic
619C4#262811yescharacteristic
620C2—84no—
621C2#1488711yescharacteristic
622C2#2396011yescharacteristic
623C2#2397711yescharacteristic
624C2#2564511yescharacteristic
625C2#2566211yescharacteristic
626C2#2602012yes—
627C2#2603212yes—
628C2#2742212yes—
629C2#2744512yes—
630C2#3295311yescharacteristic
631C2#3296511yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC23×D4C23
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C2×#1988C2semidirect
4. #15128C2semidirect
5. #11267C2semidirect
6. #11202C2semidirect
9. #10884C2semidirect
10. #10806C2semidirect
61. C2×C4×D4Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#23542
C2×#20232
C2×#19882
#151282
#112672
#112022
#110762
#110662
#108842
#108062
#107882