Finite Group Databasebeta

← #416179

Subgroups — #416179

Order 512.

1986 conjugacy classes of subgroups.

The subgroup lattice has 627 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2537C211yesmaximal, characteristic
2C2×#1954C211yesmaximal, characteristic, complemented
3C2×#1834C211yesmaximal, characteristic, complemented
4C2×#1802C211yesmaximal, characteristic, complemented
5C2×C4×#102C211yesmaximal, characteristic, complemented
6#13906C211yesmaximal, characteristic, complemented
7#11199C214yesmaximal, complemented
8#10969C214yesmaximal, complemented
9#10849C211yesmaximal, characteristic, complemented
10C22×#121—21no—
11C22×#121C2212yes—
12C22×C4×D4C2211yescharacteristic
13C22×C4×D4C2211yescharacteristic
14C22×C4⋊C8C2211yescharacteristic
15C22×#102C2212yes—
16C22×#102C2212yes—
17C22×C22⋊C8C2214yes—
18C22×C4×C8C2211yescharacteristic
19#2537—24no—
20#2537—24no—
21C2×#394—22no—
22C2×#390—21no—
23C2×#384C2211yescharacteristic
24C2×#384C2211yescharacteristic
25C2×#380—21no—
26C2×#380C2212yes—
27C2×#380C2214yes—
28C2×C4×C4⋊C4C2211yescharacteristic
29#1954—24no—
30#1834—24no—
31#1802—24no—
32C4×#102—24no—
33C2×#340C2214yes—
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#340C2214yes—
37#1439—22no—
38C4⋊#95—21no—
39C23×D4—22no—
40C23×D4D411yescharacteristic
41C23×D4C2311yescharacteristic
42C24×C4—21no—
43C24×C4C2312yes—
44C23×C8C2312yes—
45C23×C8C2312yes—
46C2×#121—28no—
47C2×#121—24no—
48C2×#121—24no—
49C2×#121—28no—
50C2×#121—44no—
51C2×#121—44no—
52C2×#121—44no—
53C2×C4×D4—22no—
54C2×C4×D4—22no—
55C2×C4×D4—22no—
56C2×C4×D4—22no—
57C2×C4×D4—22no—
58C2×C4×D4—22no—
59C2×C4×D4—22no—
60C2×C4×D4—22no—
61C2×C4×D4—22no—
62C2×C4×D4—22no—
63C2×C4×D4—22no—
64C2×C4×D4D414yes—
65C22×C4⋊C4—21no—
66C22×C4⋊C4—21no—
67C22×C4⋊C4—21no—
68C22×C4⋊C4—21no—
69C22×C4⋊C4D411yescharacteristic
70C22×C4⋊C4C2312yes—
71C22×C4⋊C4C2311yescharacteristic
72C22×C4⋊C4C2312yes—
73C22×C4⋊C4C2312yes—
74C22×C22⋊C4—22no—
75C22×C22⋊C4—21no—
76C22×C22⋊C4—22no—
77C22×C22⋊C4—24no—
78C22×C22⋊C4D412yes—
79C22×C42—21no—
80C22×C42C2311yescharacteristic
81C2×C4⋊C8—22no—
82C2×C4⋊C8—22no—
83C2×C4⋊C8—22no—
84C2×#102—24no—
85C2×#102—24no—
86C2×#102—24no—
87C2×#102—24no—
88C2×#102—44no—
89C2×C22⋊C8—28no—
90C2×C22⋊C8—44no—
91C2×C22⋊C8—44no—
92C2×C4×C8—22no—
93C2×C4×C8—22no—
94C2×C4×C8D414yescomplemented
95#394—216no—
96#390—44no—
97#384—22no—
98#384—22no—
99#384—22no—
100#384D414yes—
101#380—28no—
102#380—24no—
103#380—24no—
104#380—28no—
105#380—44no—
106C4×C4⋊C4—22no—
107C4×C4⋊C4—22no—
108C2×#95—22no—
109C2×#95—24no—
110C2×#95—21no—
111#340—24no—
112#340—24no—
113#340—28no—
114#340—44no—
115C25—21no—
116C25C2×D412yes—
117C22×D4—28no—
118C22×D4—22no—
119C22×D4—22no—
120C22×D4—22no—
121C22×D4—22no—
122C22×D4—22no—
123C22×D4—22no—
124C22×D4—21no—
125C22×D4—21no—
126C22×D4—21no—
127C22×D4—21no—
128C22×D4—21no—
129C22×D4—22no—
130C22×D4—22no—
131C22×D4—22no—
132C22×D4—48no—
133C22×D4—44no—
134C22×D4—44no—
135C22×D4—84no—
136C22×D4D814yes—
137C22×D4C2×D412yes—
138C22×D4#4114yes—
139C23×C4—21no—
140C23×C4—22no—
141C23×C4—22no—
142C23×C4—24no—
143C23×C4—22no—
144C23×C4—22no—
145C23×C4—22no—
146C23×C4—22no—
147C23×C4—22no—
148C23×C4—22no—
149C23×C4—24no—
150C23×C4—24no—
151C23×C4—24no—
152C23×C4—24no—
153C23×C4—22no—
154C23×C4—42no—
155C23×C4—42no—
156C23×C4—42no—
157C23×C4—42no—
158C23×C4—42no—
159C23×C4—42no—
160C23×C4C2×D412yes—
161C23×C4C2×D411yescharacteristic
162C23×C4C2×D411yescharacteristic
163C23×C4C2411yescharacteristic
164C22×C8—22no—
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—24no—
174C22×C8—24no—
175C22×C8C2×D414yes—
176#121—416no—
177#121—48no—
178#121—432no—
179#121—48no—
180C4×D4—28no—
181C4×D4—28no—
182C4×D4—44no—
183C4×D4—44no—
184C4×D4—44no—
185C4×D4—44no—
186C4×D4—44no—
187C4×D4—44no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—24no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—24no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—24no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—24no—
208C2×C4⋊C4—24no—
209C2×C4⋊C4—24no—
210C2×C4⋊C4—42no—
211C2×C4⋊C4—42no—
212C2×C4⋊C4—42no—
213C2×C4⋊C4—42no—
214C2×C4⋊C4—42no—
215C2×C4⋊C4—42no—
216C2×C4⋊C4—82no—
217C2×C4⋊C4—82no—
218C2×C4⋊C4C2×D412yes—
219C2×C4⋊C4C2×D414yes—
220C2×C4⋊C4#4114yes—
221C2×C22⋊C4—28no—
222C2×C22⋊C4—28no—
223C2×C22⋊C4—24no—
224C2×C22⋊C4—24no—
225C2×C22⋊C4—24no—
226C2×C22⋊C4—24no—
227C2×C22⋊C4—48no—
228C2×C22⋊C4—44no—
229C2×C22⋊C4—44no—
230C2×C22⋊C4—42no—
231C2×C22⋊C4—42no—
232C2×C22⋊C4—48no—
233C2×C22⋊C4—48no—
234C2×C22⋊C4—84no—
235C2×C42—42no—
236C2×C42—42no—
237C2×C42—42no—
238C2×C42C2×D412yes—
239C2×C42C2×D412yes—
240C2×C42C2×D412yes—
241C2×C42#4112yes—
242C2×C42#4112yes—
243C2×C42#4112yes—
244C4⋊C8—44no—
245#102—216no—
246#102—48no—
247C22⋊C8—416no—
248C4×C8—28no—
249#95—28no—
250#95—48no—
251#95—42no—
252C24—22no—
253C24—22no—
254C24—22no—
255C24—22no—
256C24—24no—
257C24—24no—
258C24—24no—
259C24—22no—
260C24—22no—
261C24—22no—
262C24—24no—
263C24—24no—
264C24—41no—
265C24—41no—
266C24—42no—
267C24—41no—
268C24—41no—
269C24—41no—
270C24—41no—
271C24—42no—
272C24—42no—
273C24C2×D814yes—
274C24C22×D411yescharacteristic
275C2×D4—22no—
276C2×D4—22no—
277C2×D4—24no—
278C2×D4—24no—
279C2×D4—22no—
280C2×D4—22no—
281C2×D4—22no—
282C2×D4—24no—
283C2×D4—24no—
284C2×D4—24no—
285C2×D4—28no—
286C2×D4—28no—
287C2×D4—28no—
288C2×D4—416no—
289C2×D4—416no—
290C2×D4—41no—
291C2×D4—41no—
292C2×D4—42no—
293C2×D4—42no—
294C2×D4—42no—
295C2×D4—42no—
296C2×D4—42no—
297C2×D4—44no—
298C2×D4—44no—
299C2×D4—44no—
300C2×D4—44no—
301C2×D4—88no—
302C2×D4—816no—
303C2×D4—816no—
304C2×D4#12414yes—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—24no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—24no—
313C22×C4—24no—
314C22×C4—24no—
315C22×C4—42no—
316C22×C4—41no—
317C22×C4—41no—
318C22×C4—44no—
319C22×C4—42no—
320C22×C4—42no—
321C22×C4—41no—
322C22×C4—42no—
323C22×C4—42no—
324C22×C4—41no—
325C22×C4—41no—
326C22×C4—42no—
327C22×C4—41no—
328C22×C4—42no—
329C22×C4—44no—
330C22×C4—42no—
331C22×C4—42no—
332C22×C4—42no—
333C22×C4—42no—
334C22×C4—44no—
335C22×C4—42no—
336C22×C4—42no—
337C22×C4—44no—
338C22×C4—42no—
339C22×C4—44no—
340C22×C4—42no—
341C22×C4—42no—
342C22×C4—44no—
343C22×C4—44no—
344C22×C4—44no—
345C22×C4—44no—
346C22×C4—44no—
347C22×C4—44no—
348C22×C4—44no—
349C22×C4—48no—
350C22×C4—48no—
351C22×C4—48no—
352C22×C4—42no—
353C22×C4—44no—
354C22×C4—42no—
355C22×C4—42no—
356C22×C4—44no—
357C22×C4—84no—
358C22×C4—84no—
359C22×C4—84no—
360C22×C4—84no—
361C22×C4#12012yes—
362C22×C4#12012yes—
363C22×C4#12112yes—
364C22×C4#12112yes—
365C22×C4#12312yes—
366C22×C4#12312yes—
367C22×C4C2×D812yes—
368C22×C4#13512yes—
369C22×C4AGL(1,ℤ/8)12yes—
370C22×C4#13712yes—
371C22×C4C22×D411yescharacteristic
372C22×C4C22×D411yescharacteristic
373C22×C4C22×D411yescharacteristic
374C22×C4C2×#4111yescharacteristic
375C22×C4C2×#4111yescharacteristic
376C22×C4C2×#4111yescharacteristic
377C22×C4C2×#4112yes—
378C22×C4C2×#4112yes—
379C22×C4H(5,𝔽2)12yes—
380C22×C4H(5,𝔽2)12yes—
381C2×C8—24no—
382C2×C8—24no—
383C2×C8—24no—
384C2×C8—42no—
385C2×C8—42no—
386C2×C8—42no—
387C2×C8—42no—
388C2×C8—42no—
389C2×C8—44no—
390C2×C8—42no—
391C2×C8—42no—
392C2×C8—44no—
393C4⋊C4—22no—
394C4⋊C4—22no—
395C4⋊C4—24no—
396C4⋊C4—28no—
397C4⋊C4—24no—
398C4⋊C4—41no—
399C4⋊C4—41no—
400C4⋊C4—42no—
401C4⋊C4—42no—
402C4⋊C4—42no—
403C4⋊C4—42no—
404C4⋊C4—44no—
405C4⋊C4—48no—
406C4⋊C4—42no—
407C4⋊C4—42no—
408C4⋊C4—44no—
409C4⋊C4—84no—
410C4⋊C4—84no—
411C4⋊C4—84no—
412C22⋊C4—416no—
413C22⋊C4—416no—
414C22⋊C4—48no—
415C22⋊C4—48no—
416C22⋊C4—816no—
417C42—84no—
418C42#12114yes—
419C42#12114yes—
420C42#12314yes—
421C42#12414yes—
422C23—24no—
423C23—24no—
424C23—24no—
425C23—28no—
426C23—28no—
427C23—24no—
428C23—44no—
429C23—44no—
430C23—44no—
431C23—44no—
432C23—42no—
433C23—44no—
434C23—42no—
435C23—42no—
436C23—44no—
437C23—44no—
438C23—42no—
439C23—42no—
440C23—42no—
441C23—42no—
442C23—42no—
443C23—42no—
444C23—42no—
445C23—42no—
446C23—42no—
447C23—42no—
448C23—44no—
449C23—44no—
450C23—44no—
451C23—44no—
452C23—44no—
453C23—44no—
454C23—42no—
455C23—44no—
456C23—44no—
457C23—44no—
458C23—82no—
459C23—82no—
460C23—81no—
461C23—81no—
462C23—82no—
463C23—82no—
464C23—82no—
465C23—81no—
466C23—81no—
467C23—82no—
468C23C2×#12011yescharacteristic
469C23C2×#12111yescharacteristic
470C23C2×#12311yescharacteristic
471C23#54612yes—
472C23#54612yes—
473C23C22×D811yescharacteristic
474C23C2×#13511yescharacteristic
475C23C2×AGL(1,ℤ/8)11yescharacteristic
476C23C2×#13711yescharacteristic
477C23#57712yes—
478C23#57712yes—
479D4—24no—
480D4—24no—
481D4—42no—
482D4—44no—
483D4—44no—
484D4—42no—
485D4—416no—
486D4—832no—
487C2×C4—24no—
488C2×C4—22no—
489C2×C4—22no—
490C2×C4—24no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—44no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—44no—
497C2×C4—42no—
498C2×C4—44no—
499C2×C4—42no—
500C2×C4—44no—
501C2×C4—44no—
502C2×C4—44no—
503C2×C4—44no—
504C2×C4—44no—
505C2×C4—81no—
506C2×C4—81no—
507C2×C4—82no—
508C2×C4—84no—
509C2×C4—82no—
510C2×C4—82no—
511C2×C4—81no—
512C2×C4—81no—
513C2×C4—82no—
514C2×C4—84no—
515C2×C4—82no—
516C2×C4—82no—
517C2×C4—84no—
518C2×C4—84no—
519C2×C4—82no—
520C2×C4—82no—
521C2×C4—816no—
522C2×C4—84no—
523C2×C4—82no—
524C2×C4—82no—
525C2×C4—82no—
526C2×C4—84no—
527C2×C4#39414yes—
528C2×C4#39414yes—
529C2×C4#44714yes—
530C2×C4#45214yes—
531C2×C4#46614yes—
532C2×C4#46814yes—
533C2×C4#48014yes—
534C2×C4#48214yes—
535C2×C4C2×#12111yescharacteristic
536C2×C4C2×#12111yescharacteristic
537C2×C4C2×#12311yescharacteristic
538C2×C4#52512yes—
539C2×C4C2×#12411yescharacteristic
540C2×C4C2×#12411yescharacteristic
541C2×C4#52912yes—
542C2×C4#53211yescharacteristic
543C2×C4#53512yes—
544C2×C4#53511yescharacteristic
545C2×C4#53511yescharacteristic
546C2×C4#54012yes—
547C2×C4#54612yes—
548C2×C4#55012yes—
549C2×C4#55312yes—
550C2×C4#55312yes—
551C2×C4#55312yes—
552C2×C4#55512yes—
553C8—44no—
554C8—82no—
555C22—44no—
556C22—44no—
557C22—44no—
558C22—44no—
559C22—48no—
560C22—44no—
561C22—44no—
562C22—48no—
563C22—82no—
564C22—82no—
565C22—84no—
566C22—84no—
567C22—84no—
568C22—82no—
569C22—82no—
570C22—82no—
571C22—82no—
572C22—82no—
573C22—82no—
574C22—84no—
575C22—84no—
576C22—84no—
577C22C2×#39411yescharacteristic
578C22#242012yes—
579C22#243111yescharacteristic
580C22#243511yescharacteristic
581C22#253512yes—
582C22C2×#44711yescharacteristic
583C22C2×#45211yescharacteristic
584C22#300911yescharacteristic
585C22#301011yescharacteristic
586C22C2×#46611yescharacteristic
587C22C2×#46811yescharacteristic
588C22#306012yes—
589C22C2×#48011yescharacteristic
590C22C2×#48211yescharacteristic
591C22#309912yes—
592C22#329412yes—
593C22#329412yes—
594C22#330312yes—
595C22#330312yes—
596C22#332412yes—
597C22#332912yes—
598C22#333112yes—
599C22#333212yes—
600C4—24no—
601C4—84no—
602C4—82no—
603C4—82no—
604C4—84no—
605C4—162no—
606C4#203318yes—
607C4#241412yes—
608C4#249012yes—
609C4#250011yescharacteristic
610C4#250411yescharacteristic
611C4#261911yescharacteristic
612C4#285011yescharacteristic
613C2—84no—
614C2—84no—
615C2#1521911yescharacteristic
616C2C2×#203311yescharacteristic
617C2#2348911yescharacteristic
618C2#2496512yes—
619C2#2496612yes—
620C2#2556711yescharacteristic
621C2#2556811yescharacteristic
622C2#2588511yescharacteristic
623C2#2588611yescharacteristic
624C2#2814012yes—
625C2#2815012yes—
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC2×#2537C2
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×#1954C2semidirect
3. C2×#1834C2semidirect
4. C2×#1802C2semidirect
5. C2×C4×#102C2semidirect
6. #13906C2semidirect
7. #11199C2semidirect
8. #10969C2semidirect
9. #10849C2semidirect
94. C2×C4×C8D4semidirect

Maximal subgroups

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

subgroupindex
C2×#25372
C2×#19542
C2×#18342
C2×#18022
C2×C4×#1022
#139062
#111992
#109692
#108492