Finite Group Databasebeta

← #410010

Subgroups — #410010

Order 512.

2344 conjugacy classes of subgroups.

The subgroup lattice has 636 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#480C211yesmaximal, characteristic
2C2×#2854C211yesmaximal, characteristic
3C22×#389C211yesmaximal, characteristic, complemented
4C2×#1874C211yesmaximal, characteristic
5C2×#1823C211yesmaximal, characteristic, complemented
6C2×#1817C211yesmaximal, characteristic, complemented
7#11328C214yesmaximal, complemented
8#11066C214yesmaximal, complemented
9#10730C211yesmaximal, characteristic, complemented
10C22×#123—21no—
11C22×#121C2211yescharacteristic
12C23×C4⋊C4—21no—
13C23×C4⋊C4C2211yescharacteristic
14C23×C4⋊C4C2211yescharacteristic
15C23×C42C2211yescharacteristic
16C2×#480—24no—
17C2×#480—24no—
18C2×#480—24no—
19C22×#107C2212yes—
20C22×#102C2212yes—
21C22×#102C2212yes—
22C22×C22⋊C8C2211yescharacteristic
23C22×C22⋊C8C2211yescharacteristic
24#2854—24no—
25#2854—24no—
26C2×#390C2212yes—
27C2×#389—22no—
28C2×#389—24no—
29C2×#389—24no—
30C2×#389C2212yes—
31C2×#389C2214yes—
32C2×#388—22no—
33C2×#386—21no—
34C2×#380C2211yescharacteristic
35C22×#95—21no—
36#1874—24no—
37#1823—24no—
38#1817—24no—
39C2×#340C2214yes—
40C2×#340C2214yes—
41C2×#340C2212yes—
42C2×#336C2214yes—
43C23×D4—21no—
44C23×D4C2311yescharacteristic
45C24×C4—21no—
46C24×C4—21no—
47C24×C4D411yescharacteristic
48C24×C4D411yescharacteristic
49C24×C4C2311yescharacteristic
50C23×C8C2312yes—
51C23×C8C2312yes—
52C2×#123—28no—
53C2×#123—44no—
54C2×#123—44no—
55C2×#121—22no—
56C2×#121—22no—
57C2×#121—22no—
58C2×#121—22no—
59C2×#121—22no—
60C2×#121D414yes—
61C22×C4⋊C4—21no—
62C22×C4⋊C4—22no—
63C22×C4⋊C4—28no—
64C22×C4⋊C4—22no—
65C22×C4⋊C4—22no—
66C22×C4⋊C4—22no—
67C22×C4⋊C4—22no—
68C22×C4⋊C4—28no—
69C22×C4⋊C4—24no—
70C22×C4⋊C4—21no—
71C22×C4⋊C4—22no—
72C22×C4⋊C4—22no—
73C22×C4⋊C4—44no—
74C22×C4⋊C4—44no—
75C22×C4⋊C4D414yes—
76C22×C4⋊C4D414yes—
77C22×C4⋊C4C2312yes—
78C22×C4⋊C4C2311yescharacteristic
79C22×C4⋊C4C2312yes—
80C22×C4⋊C4C2312yes—
81C22×C22⋊C4—21no—
82C22×C22⋊C4—21no—
83C22×C22⋊C4—22no—
84C22×C22⋊C4—22no—
85C22×C42—22no—
86C22×C42—22no—
87C22×C42—22no—
88C22×C42—22no—
89C22×C42—22no—
90C22×C42—22no—
91C22×C42D412yes—
92C22×C42C2312yes—
93#480—232no—
94C2×#107—24no—
95C2×#107—24no—
96C2×#107—24no—
97C2×#102—24no—
98C2×#102—24no—
99C2×#102—24no—
100C2×#102—24no—
101C2×#102—28no—
102C2×C22⋊C8—22no—
103C2×C22⋊C8—22no—
104C2×C22⋊C8—22no—
105C2×C22⋊C8—22no—
106C2×C22⋊C8—22no—
107C2×C22⋊C8D414yes—
108#390—24no—
109#390—24no—
110#389—216no—
111#389—24no—
112#389—24no—
113#389—28no—
114#389—28no—
115#388—216no—
116#386—28no—
117#380—22no—
118#380D414yes—
119C2×#95—28no—
120C2×#95—24no—
121C2×#95—21no—
122C2×#95—22no—
123#340—28no—
124#340—24no—
125#340—28no—
126#336—28no—
127C25—21no—
128C25C2×D411yescharacteristic
129C22×D4—24no—
130C22×D4—21no—
131C22×D4—21no—
132C22×D4—21no—
133C22×D4—21no—
134C22×D4—21no—
135C22×D4—24no—
136C22×D4—24no—
137C22×D4—42no—
138C22×D4—42no—
139C22×D4—42no—
140C22×D4—42no—
141C22×D4—42no—
142C22×D4—42no—
143C22×D4C2×D412yes—
144C23×C4—24no—
145C23×C4—24no—
146C23×C4—28no—
147C23×C4—22no—
148C23×C4—24no—
149C23×C4—22no—
150C23×C4—22no—
151C23×C4—22no—
152C23×C4—22no—
153C23×C4—22no—
154C23×C4—22no—
155C23×C4—22no—
156C23×C4—24no—
157C23×C4—24no—
158C23×C4—22no—
159C23×C4—22no—
160C23×C4—22no—
161C23×C4—21no—
162C23×C4—22no—
163C23×C4—42no—
164C23×C4—44no—
165C23×C4—42no—
166C23×C4—42no—
167C23×C4—42no—
168C23×C4—42no—
169C23×C4—42no—
170C23×C4—42no—
171C23×C4—42no—
172C23×C4—42no—
173C23×C4D814yes—
174C23×C4C2×D412yes—
175C23×C4C2×D412yes—
176C23×C4C2×D411yescharacteristic
177C23×C4C2×D412yes—
178C23×C4C2×D412yes—
179C23×C4C2×D412yes—
180C23×C4C2×D412yes—
181C23×C4C2×D412yes—
182C23×C4#4112yes—
183C23×C4C2411yescharacteristic
184C22×C8—22no—
185C22×C8—22no—
186C22×C8—24no—
187C22×C8—24no—
188C22×C8—22no—
189C22×C8—22no—
190C22×C8—22no—
191C22×C8—24no—
192C22×C8—24no—
193C22×C8C2×D414yes—
194#123—432no—
195#121—28no—
196#121—28no—
197#121—44no—
198#121—44no—
199C2×C4⋊C4—28no—
200C2×C4⋊C4—216no—
201C2×C4⋊C4—24no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—24no—
204C2×C4⋊C4—216no—
205C2×C4⋊C4—28no—
206C2×C4⋊C4—28no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—24no—
216C2×C4⋊C4—24no—
217C2×C4⋊C4—216no—
218C2×C4⋊C4—28no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—48no—
224C2×C4⋊C4—432no—
225C2×C4⋊C4—44no—
226C2×C4⋊C4—42no—
227C2×C4⋊C4—44no—
228C2×C4⋊C4—42no—
229C2×C4⋊C4—44no—
230C2×C4⋊C4—48no—
231C2×C4⋊C4—44no—
232C2×C4⋊C4C2×D414yes—
233C2×C4⋊C4C2×D412yes—
234C2×C4⋊C4C2×D414yes—
235C2×C4⋊C4C2×D414yes—
236C2×C4⋊C4#4118yes—
237C2×C22⋊C4—28no—
238C2×C22⋊C4—28no—
239C2×C22⋊C4—24no—
240C2×C22⋊C4—24no—
241C2×C22⋊C4—44no—
242C2×C22⋊C4—44no—
243C2×C22⋊C4—42no—
244C2×C22⋊C4—42no—
245C2×C22⋊C4—42no—
246C2×C22⋊C4—42no—
247C2×C22⋊C4—44no—
248C2×C22⋊C4—44no—
249C2×C42—24no—
250C2×C42—24no—
251C2×C42—24no—
252C2×C42—22no—
253C2×C42—22no—
254C2×C42—22no—
255C2×C42—22no—
256C2×C42—22no—
257C2×C42—22no—
258C2×C42—44no—
259C2×C42—44no—
260C2×C42—44no—
261C2×C42—44no—
262#107—216no—
263#102—216no—
264#102—48no—
265C22⋊C8—28no—
266C22⋊C8—44no—
267#95—216no—
268#95—28no—
269#95—24no—
270C24—22no—
271C24—22no—
272C24—22no—
273C24—22no—
274C24—22no—
275C24—41no—
276C24—41no—
277C24—42no—
278C24—42no—
279C24—41no—
280C24—41no—
281C24—41no—
282C24—41no—
283C24—42no—
284C24—42no—
285C24#12012yes—
286C24#12012yes—
287C24#12112yes—
288C24C2×D812yes—
289C24#13512yes—
290C24AGL(1,ℤ/8)12yes—
291C24AGL(1,ℤ/8)12yes—
292C24C22×D411yescharacteristic
293C2×D4—22no—
294C2×D4—22no—
295C2×D4—28no—
296C2×D4—48no—
297C2×D4—48no—
298C2×D4—48no—
299C2×D4—41no—
300C2×D4—42no—
301C2×D4—42no—
302C2×D4—41no—
303C2×D4—42no—
304C2×D4—42no—
305C2×D4—44no—
306C2×D4—44no—
307C2×D4—44no—
308C2×D4—84no—
309C2×D4—84no—
310C2×D4—84no—
311C2×D4—84no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—24no—
315C22×C4—24no—
316C22×C4—24no—
317C22×C4—24no—
318C22×C4—28no—
319C22×C4—28no—
320C22×C4—28no—
321C22×C4—28no—
322C22×C4—216no—
323C22×C4—28no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—24no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—28no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—28no—
339C22×C4—28no—
340C22×C4—28no—
341C22×C4—24no—
342C22×C4—22no—
343C22×C4—24no—
344C22×C4—48no—
345C22×C4—48no—
346C22×C4—416no—
347C22×C4—416no—
348C22×C4—44no—
349C22×C4—48no—
350C22×C4—42no—
351C22×C4—42no—
352C22×C4—44no—
353C22×C4—44no—
354C22×C4—44no—
355C22×C4—48no—
356C22×C4—44no—
357C22×C4—44no—
358C22×C4—44no—
359C22×C4—41no—
360C22×C4—42no—
361C22×C4—41no—
362C22×C4—42no—
363C22×C4—42no—
364C22×C4—42no—
365C22×C4—42no—
366C22×C4—42no—
367C22×C4—41no—
368C22×C4—42no—
369C22×C4—41no—
370C22×C4—42no—
371C22×C4—42no—
372C22×C4—41no—
373C22×C4—42no—
374C22×C4—41no—
375C22×C4—42no—
376C22×C4—42no—
377C22×C4—48no—
378C22×C4—48no—
379C22×C4—48no—
380C22×C4—84no—
381C22×C4—84no—
382C22×C4—84no—
383C22×C4—84no—
384C22×C4#12014yes—
385C22×C4#12012yes—
386C22×C4#12012yes—
387C22×C4#12114yes—
388C22×C4#12114yes—
389C22×C4#12112yes—
390C22×C4#12714yes—
391C22×C4C2×D814yes—
392C22×C4#13714yes—
393C22×C4C22×D411yescharacteristic
394C22×C4C22×D411yescharacteristic
395C22×C4C22×D411yescharacteristic
396C22×C4C22×D411yescharacteristic
397C22×C4C22×D411yescharacteristic
398C22×C4C2×#4111yescharacteristic
399C22×C4C2×#4114yes—
400C22×C42−1+414yes—
401C2×C8—24no—
402C2×C8—28no—
403C2×C8—42no—
404C2×C8—42no—
405C2×C8—44no—
406C2×C8—42no—
407C2×C8—42no—
408C2×C8—42no—
409C2×C8—44no—
410C2×C8—44no—
411C4⋊C4—216no—
412C4⋊C4—24no—
413C4⋊C4—28no—
414C4⋊C4—24no—
415C4⋊C4—22no—
416C4⋊C4—24no—
417C4⋊C4—22no—
418C4⋊C4—28no—
419C4⋊C4—432no—
420C4⋊C4—416no—
421C4⋊C4—42no—
422C4⋊C4—41no—
423C4⋊C4—42no—
424C4⋊C4—41no—
425C4⋊C4—44no—
426C4⋊C4—48no—
427C4⋊C4#12318yescomplemented
428C22⋊C4—416no—
429C22⋊C4—416no—
430C22⋊C4—48no—
431C22⋊C4—48no—
432C42—42no—
433C42—42no—
434C42—42no—
435C42—42no—
436C42—48no—
437C23—24no—
438C23—22no—
439C23—24no—
440C23—24no—
441C23—24no—
442C23—22no—
443C23—24no—
444C23—24no—
445C23—24no—
446C23—24no—
447C23—24no—
448C23—24no—
449C23—22no—
450C23—22no—
451C23—24no—
452C23—24no—
453C23—42no—
454C23—44no—
455C23—42no—
456C23—42no—
457C23—44no—
458C23—82no—
459C23—81no—
460C23—82no—
461C23—82no—
462C23—82no—
463C23—81no—
464C23—82no—
465C23—82no—
466C23—82no—
467C23—82no—
468C23—82no—
469C23—81no—
470C23—81no—
471C23—81no—
472C23—81no—
473C23—81no—
474C23—81no—
475C23—82no—
476C23#39214yes—
477C23#44714yes—
478C23#44714yes—
479C23#44914yes—
480C23#44914yes—
481C23#46614yes—
482C23#46914yes—
483C23C2×#12011yescharacteristic
484C23C2×#12011yescharacteristic
485C23C2×#12111yescharacteristic
486C23#52512yes—
487C23#53612yes—
488C23C22×D811yescharacteristic
489C23C2×#13511yescharacteristic
490C23C2×AGL(1,ℤ/8)11yescharacteristic
491C23C2×AGL(1,ℤ/8)11yescharacteristic
492C23C2×#13712yes—
493C23#57512yes—
494D4—44no—
495D4—44no—
496D4—816no—
497D4—82no—
498D4—82no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—24no—
502C2×C4—24no—
503C2×C4—28no—
504C2×C4—28no—
505C2×C4—28no—
506C2×C4—28no—
507C2×C4—28no—
508C2×C4—24no—
509C2×C4—24no—
510C2×C4—24no—
511C2×C4—24no—
512C2×C4—24no—
513C2×C4—24no—
514C2×C4—416no—
515C2×C4—432no—
516C2×C4—48no—
517C2×C4—48no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—44no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—44no—
525C2×C4—42no—
526C2×C4—416no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—44no—
531C2×C4—44no—
532C2×C4—44no—
533C2×C4—44no—
534C2×C4—42no—
535C2×C4—44no—
536C2×C4—42no—
537C2×C4—44no—
538C2×C4—82no—
539C2×C4—82no—
540C2×C4—82no—
541C2×C4—82no—
542C2×C4—82no—
543C2×C4—81no—
544C2×C4—82no—
545C2×C4—81no—
546C2×C4—816no—
547C2×C4—82no—
548C2×C4—81no—
549C2×C4—82no—
550C2×C4—81no—
551C2×C4#39214yes—
552C2×C4C2×#12011yescharacteristic
553C2×C4C2×#12111yescharacteristic
554C2×C4C2×#12111yescharacteristic
555C2×C4C2×#12312yes—
556C2×C4#52512yes—
557C2×C4C2×#12711yescharacteristic
558C2×C4#53612yes—
559C2×C4#53612yes—
560C2×C4#54714yes—
561C2×C4#54714yes—
562C2×C4#54714yes—
563C2×C4#55214yes—
564C8—44no—
565C8—82no—
566C22—28no—
567C22—28no—
568C22—28no—
569C22—24no—
570C22—24no—
571C22—24no—
572C22—24no—
573C22—24no—
574C22—24no—
575C22—28no—
576C22—84no—
577C22—82no—
578C22—82no—
579C22—82no—
580C22—84no—
581C22—162no—
582C22—162no—
583C22—161no—
584C22—161no—
585C22—161no—
586C22—161no—
587C22#199918yes—
588C22C2×#39211yescharacteristic
589C22#240612yes—
590C22#241312yes—
591C22#261411yescharacteristic
592C22#284911yescharacteristic
593C22C2×#44711yescharacteristic
594C22C2×#44711yescharacteristic
595C22C2×#44911yescharacteristic
596C22C2×#44911yescharacteristic
597C22#300411yescharacteristic
598C22#300511yescharacteristic
599C22#300712yes—
600C22C2×#46611yescharacteristic
601C22C2×#46911yescharacteristic
602C22#305612yes—
603C22#310012yes—
604C22#310812yes—
605C22#311212yes—
606C22#311412yes—
607C22#322612yes—
608C22#323212yes—
609C22#323412yes—
610C22#323812yes—
611C4—24no—
612C4—44no—
613C4—44no—
614C4—48no—
615C4—84no—
616C4—82no—
617C4—84no—
618C4#240612yes—
619C4#241312yes—
620C4#248612yes—
621C4#250112yes—
622C2—28no—
623C2—162no—
624C2#1506911yescharacteristic
625C2C2×#199911yescharacteristic
626C2#2339211yescharacteristic
627C2#2406012yes—
628C2#2406612yes—
629C2#2460012yes—
630C2#2460812yes—
631C2#2963811yescharacteristic
632C2#2964011yescharacteristic
633C2#3300211yescharacteristic
634C2#3300311yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC22×#121C22
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C23×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C22×#389C2semidirect
5. C2×#1823C2semidirect
6. C2×#1817C2semidirect
7. #11328C2semidirect
8. #11066C2semidirect
9. #10730C2semidirect
427. C4⋊C4#123semidirect

Maximal subgroups

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

subgroupindex
C22×#4802
C2×#28542
C22×#3892
C2×#18742
C2×#18232
C2×#18172
#113282
#110662
#107302