Finite Group Databasebeta

← #470286

Subgroups — #470286

Order 512.

1508 conjugacy classes of subgroups.

The subgroup lattice has 623 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2547C211yesmaximal, characteristic
2C2×#2057C211yesmaximal, characteristic, complemented
3C2×#1918C211yesmaximal, characteristic, complemented
4#15129C211yesmaximal, characteristic
5#11384C211yesmaximal, characteristic, complemented
6#11199C212yesmaximal, complemented
7#11096C212yesmaximal, complemented
8#11096C212yesmaximal, complemented
9#11096C212yesmaximal, complemented
10#10964C211yesmaximal, characteristic
11#10806C211yesmaximal, characteristic, complemented
12C22×C4⋊Q8C2211yescharacteristic
13C22×#124—21no—
14C22×C4×D4C2211yescharacteristic
15C22×#107C2211yescharacteristic
16C22×#107C2212yes—
17C22×C22⋊C8C2212yes—
18C22×C22⋊C8C2212yes—
19#2547—24no—
20#2547—24no—
21C2×#397—22no—
22C2×#394—21no—
23C2×#391—21no—
24C2×#389C2211yescharacteristic
25C2×#380C2212yes—
26C2×#380C2212yes—
27C2×#380C2212yes—
28C2×#380C2212yes—
29C2×C4×C4⋊C4C2211yescharacteristic
30#2057—24no—
31#1918—24no—
32C2×#340C2211yescharacteristic
33C2×#340C2211yescharacteristic
34C2×#340C2211yescharacteristic
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#340C2212yes—
40C2×#336C2211yescharacteristic
41C2×#336C2211yescharacteristic
42#1441—21no—
43#1440—22no—
44C4⋊#95C2211yescharacteristic
45C23×Q8—21no—
46C23×D4D411yescharacteristic
47C24×C4C2312yes—
48C23×C8C2311yescharacteristic
49C23×C8C2311yescharacteristic
50C23×C8C2312yes—
51C2×C4⋊Q8—22no—
52C2×C4⋊Q8—42no—
53C2×C4⋊Q8—42no—
54C2×C4⋊Q8D414yes—
55C2×#124—28no—
56C2×#124—44no—
57C2×#124—44no—
58C2×C4×D4—22no—
59C2×C4×D4—22no—
60C2×C4×D4—42no—
61C2×C4×D4—42no—
62C22×C4⋊C4—21no—
63C22×C4⋊C4—21no—
64C22×C4⋊C4D411yescharacteristic
65C22×C4⋊C4C2311yescharacteristic
66C22×C4⋊C4C2312yes—
67C22×C4⋊C4C2312yes—
68C22×C4⋊C4C2311yescharacteristic
69C22×C4⋊C4C2311yescharacteristic
70C22×C4⋊C4C2311yescharacteristic
71C22×C22⋊C4—22no—
72C22×C22⋊C4—22no—
73C22×C22⋊C4—22no—
74C22×C22⋊C4—22no—
75C22×C22⋊C4D412yes—
76C22×C42—21no—
77C22×C42C2311yescharacteristic
78C2×#107—22no—
79C2×#107—22no—
80C2×#107—22no—
81C2×#107—24no—
82C2×#107—42no—
83C2×#107—42no—
84C2×C22⋊C8—24no—
85C2×C22⋊C8—24no—
86C2×C22⋊C8—24no—
87C2×C22⋊C8—42no—
88C2×C22⋊C8—42no—
89C2×C22⋊C8—42no—
90#397—216no—
91#394—44no—
92#391—28no—
93#389—22no—
94#389—22no—
95#380—24no—
96#380—24no—
97#380—24no—
98#380—24no—
99#380—24no—
100#380—42no—
101#380—42no—
102#380—42no—
103C4×C4⋊C4—22no—
104C4×C4⋊C4D414yes—
105C2×#95—21no—
106C2×#95—22no—
107C2×#95—22no—
108C2×#95—21no—
109C2×#95—21no—
110C2×#95—22no—
111C2×#95—22no—
112C2×#95—21no—
113#340—24no—
114#340—24no—
115#340—22no—
116#340—22no—
117#340—22no—
118#340—42no—
119#340—42no—
120#340—42no—
121#336—22no—
122#336—22no—
123C25C2×D412yes—
124C22×Q8—24no—
125C22×Q8—44no—
126C22×Q8—44no—
127C22×Q8—42no—
128C22×Q8—42no—
129C22×D4—22no—
130C22×D4—22no—
131C22×D4—42no—
132C22×D4—42no—
133C22×D4D814yes—
134C23×C4—22no—
135C23×C4—21no—
136C23×C4—21no—
137C23×C4—21no—
138C23×C4—22no—
139C23×C4—22no—
140C23×C4—22no—
141C23×C4—22no—
142C23×C4—22no—
143C23×C4—22no—
144C23×C4—22no—
145C23×C4—22no—
146C23×C4—22no—
147C23×C4—22no—
148C23×C4—22no—
149C23×C4—22no—
150C23×C4—22no—
151C23×C4—22no—
152C23×C4—21no—
153C23×C4—22no—
154C23×C4C2×D412yes—
155C23×C4C2×D411yescharacteristic
156C23×C4C2×D411yescharacteristic
157C23×C4C2411yescharacteristic
158C22×C8—21no—
159C22×C8—22no—
160C22×C8—22no—
161C22×C8—22no—
162C22×C8—21no—
163C22×C8—22no—
164C22×C8—22no—
165C22×C8—22no—
166C22×C8—22no—
167C22×C8—22no—
168C22×C8—21no—
169C22×C8—22no—
170C22×C8—22no—
171C22×C8—21no—
172C22×C8—21no—
173C22×C8#4114yes—
174C22×C8#4112yes—
175C4⋊Q8—48no—
176C4⋊Q8—44no—
177C4⋊Q8—44no—
178#124—432no—
179C4×D4—48no—
180C4×D4—44no—
181C4×D4—44no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—24no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
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⋊C4—42no—
213C2×C4⋊C4—44no—
214C2×C4⋊C4—42no—
215C2×C4⋊C4—42no—
216C2×C4⋊C4—42no—
217C2×C4⋊C4C2×D414yes—
218C2×C4⋊C4C2×D412yes—
219C2×C4⋊C4C2×D412yes—
220C2×C4⋊C4#4112yes—
221C2×C22⋊C4—28no—
222C2×C22⋊C4—28no—
223C2×C22⋊C4—24no—
224C2×C22⋊C4—42no—
225C2×C22⋊C4—42no—
226C2×C22⋊C4—44no—
227C2×C22⋊C4—44no—
228C2×C22⋊C4—44no—
229C2×C22⋊C4—44no—
230C2×C22⋊C4—44no—
231C2×C22⋊C4—44no—
232C2×C22⋊C4—44no—
233C2×C22⋊C4—44no—
234C2×C22⋊C4—44no—
235C2×C22⋊C4—82no—
236C2×C42—22no—
237C2×C42—22no—
238C2×C42—24no—
239C2×C42—42no—
240C2×C42—42no—
241C2×C42C2×D412yes—
242C2×C42C2×D412yes—
243#107—28no—
244#107—48no—
245C22⋊C8—48no—
246C22⋊C8—48no—
247#95—28no—
248#95—28no—
249#95—24no—
250#95—44no—
251#95—44no—
252#95—42no—
253#95—42no—
254#95—81no—
255C24—22no—
256C24—22no—
257C24—22no—
258C24—22no—
259C24—22no—
260C24—22no—
261C24—22no—
262C24—22no—
263C24—22no—
264C24—22no—
265C24—22no—
266C24—22no—
267C24—22no—
268C24—22no—
269C24C2×D814yes—
270C24C22×D411yescharacteristic
271C2×Q8—48no—
272C2×Q8—48no—
273C2×Q8—48no—
274C2×Q8—88no—
275C2×Q8—88no—
276C2×D4—28no—
277C2×D4—48no—
278C2×D4—44no—
279C2×D4—48no—
280C2×D4—44no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—21no—
285C22×C4—22no—
286C22×C4—21no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—24no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—41no—
299C22×C4—42no—
300C22×C4—42no—
301C22×C4—41no—
302C22×C4—41no—
303C22×C4—41no—
304C22×C4—42no—
305C22×C4—41no—
306C22×C4—41no—
307C22×C4—41no—
308C22×C4—42no—
309C22×C4—41no—
310C22×C4—41no—
311C22×C4—42no—
312C22×C4—41no—
313C22×C4—42no—
314C22×C4—42no—
315C22×C4—41no—
316C22×C4—41no—
317C22×C4—41no—
318C22×C4—42no—
319C22×C4—41no—
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—41no—
349C22×C4—42no—
350C22×C4—42no—
351C22×C4—42no—
352C22×C4—42no—
353C22×C4—42no—
354C22×C4—41no—
355C22×C4—41no—
356C22×C4—42no—
357C22×C4—42no—
358C22×C4—42no—
359C22×C4—42no—
360C22×C4#12012yes—
361C22×C4#12012yes—
362C22×C4C2×D812yes—
363C22×C4AGL(1,ℤ/8)12yes—
364C22×C4C22×D411yescharacteristic
365C22×C4C22×D411yescharacteristic
366C22×C4C2×#4111yescharacteristic
367C22×C4C2×#4111yescharacteristic
368C22×C4C2×#4112yes—
369C22×C4H(5,𝔽2)12yes—
370C22×C4H(5,𝔽2)12yes—
371C22×C4H(5,𝔽2)11yescharacteristic
372C22×C42−1+412yes—
373C22×C42−1+411yescharacteristic
374C2×C8—22no—
375C2×C8—24no—
376C2×C8—24no—
377C2×C8—24no—
378C2×C8—24no—
379C2×C8—41no—
380C2×C8—42no—
381C2×C8—41no—
382C2×C8—42no—
383C2×C8—42no—
384C2×C8—42no—
385C2×C8—42no—
386C2×C8—41no—
387C2×C8—41no—
388C2×C8—41no—
389C2×C8—42no—
390C2×C8—42no—
391C4⋊C4—22no—
392C4⋊C4—22no—
393C4⋊C4—22no—
394C4⋊C4—24no—
395C4⋊C4—24no—
396C4⋊C4—22no—
397C4⋊C4—22no—
398C4⋊C4—22no—
399C4⋊C4—44no—
400C4⋊C4—48no—
401C4⋊C4—42no—
402C4⋊C4—42no—
403C4⋊C4—41no—
404C4⋊C4—41no—
405C4⋊C4—42no—
406C4⋊C4—41no—
407C4⋊C4—41no—
408C4⋊C4—42no—
409C4⋊C4—41no—
410C4⋊C4—41no—
411C4⋊C4—42no—
412C4⋊C4—42no—
413C4⋊C4—42no—
414C4⋊C4—42no—
415C4⋊C4—84no—
416C22⋊C4—48no—
417C22⋊C4—416no—
418C22⋊C4—416no—
419C22⋊C4—88no—
420C22⋊C4—88no—
421C42—24no—
422C42—22no—
423C42—22no—
424C42—48no—
425C23—24no—
426C23—24no—
427C23—24no—
428C23—24no—
429C23—24no—
430C23—24no—
431C23—24no—
432C23—42no—
433C23—42no—
434C23—42no—
435C23—42no—
436C23—42no—
437C23—42no—
438C23—42no—
439C23—42no—
440C23—42no—
441C23—42no—
442C23—42no—
443C23—42no—
444C23—42no—
445C23—42no—
446C23—42no—
447C23—42no—
448C23—42no—
449C23—42no—
450C23—42no—
451C23—42no—
452C23—42no—
453C23—42no—
454C23—42no—
455C23—42no—
456C23—42no—
457C23—42no—
458C23—42no—
459C23—42no—
460C23C2×#12011yescharacteristic
461C23#53412yes—
462C23#54611yescharacteristic
463C23#54711yescharacteristic
464C23#54712yes—
465C23C22×D811yescharacteristic
466C23C2×AGL(1,ℤ/8)11yescharacteristic
467C23#57512yes—
468C23#57711yescharacteristic
469C23#57811yescharacteristic
470C23#57812yes—
471Q8—816no—
472D4—416no—
473C2×C4—22no—
474C2×C4—24no—
475C2×C4—24no—
476C2×C4—22no—
477C2×C4—22no—
478C2×C4—22no—
479C2×C4—22no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—44no—
485C2×C4—42no—
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—42no—
495C2×C4—42no—
496C2×C4—44no—
497C2×C4—44no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—81no—
502C2×C4—81no—
503C2×C4—81no—
504C2×C4—82no—
505C2×C4—82no—
506C2×C4—81no—
507C2×C4—81no—
508C2×C4—82no—
509C2×C4—82no—
510C2×C4—82no—
511C2×C4—82no—
512C2×C4—81no—
513C2×C4—81no—
514C2×C4—81no—
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—81no—
526C2×C4—82no—
527C2×C4—82no—
528C2×C4—82no—
529C2×C4—82no—
530C2×C4—82no—
531C2×C4—81no—
532C2×C4—81no—
533C2×C4—82no—
534C2×C4—82no—
535C2×C4#44714yes—
536C2×C4#52511yescharacteristic
537C2×C4#53211yescharacteristic
538C2×C4#53412yes—
539C2×C4#53511yescharacteristic
540C2×C4#53511yescharacteristic
541C2×C4#53611yescharacteristic
542C2×C4#53711yescharacteristic
543C2×C4#53812yes—
544C2×C4#54212yes—
545C2×C4#54212yes—
546C2×C4#54212yes—
547C2×C4#54812yes—
548C2×C4#54811yescharacteristic
549C2×C4#55011yescharacteristic
550C2×C4#55112yes—
551C2×C4#55512yes—
552C2×C4#56212yes—
553C2×C4#56312yes—
554C8—42no—
555C8—44no—
556C8—81no—
557C22—44no—
558C22—44no—
559C22—44no—
560C22—44no—
561C22—44no—
562C22—44no—
563C22—44no—
564C22—82no—
565C22—82no—
566C22—82no—
567C22—82no—
568C22—82no—
569C22—82no—
570C22—82no—
571C22—82no—
572C22#240711yescharacteristic
573C22#243211yescharacteristic
574C22#261311yescharacteristic
575C22#268612yes—
576C22#271712yes—
577C22C2×#44711yescharacteristic
578C22#300511yescharacteristic
579C22#301111yescharacteristic
580C22#301311yescharacteristic
581C22#318612yes—
582C22#319412yes—
583C22#320112yes—
584C22#320212yes—
585C22#329411yescharacteristic
586C22#331312yes—
587C22#333011yescharacteristic
588C22#333411yescharacteristic
589C22#333412yes—
590C22#334912yes—
591C22#334912yes—
592C22#335012yes—
593C22#335312yes—
594C22#335411yescharacteristic
595C4—24no—
596C4—42no—
597C4—82no—
598C4—82no—
599C4—82no—
600C4—82no—
601C4—84no—
602C4—162no—
603C4—161no—
604C4#261011yescharacteristic
605C4#263211yescharacteristic
606C4#266211yescharacteristic
607C4#273012yes—
608C4#273112yes—
609C4#282011yescharacteristic
610C2—84no—
611C2#1668111yescharacteristic
612C2#2411611yescharacteristic
613C2#2413511yescharacteristic
614C2#2570311yescharacteristic
615C2#2571111yescharacteristic
616C2#2959211yescharacteristic
617C2#2960511yescharacteristic
618C2#3131812yes—
619C2#3134312yes—
620C2#3225812yes—
621C2#3227612yes—
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×#2057C2semidirect
3. C2×#1918C2semidirect
5. #11384C2semidirect
6. #11199C2semidirect
7. #11096C2semidirect
8. #11096C2semidirect
9. #11096C2semidirect
11. #10806C2semidirect

Maximal subgroups

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

subgroupindex
C2×#25472
C2×#20572
C2×#19182
#151292
#113842
#111992
#110962
#109642
#108062