Finite Group Databasebeta

← #392271

Subgroups — #392271

Order 512.

3730 conjugacy classes of subgroups.

The subgroup lattice has 600 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#520C211yesmaximal, characteristic
2C2×#1824C211yesmaximal, characteristic, complemented
3C2×#1823C211yesmaximal, characteristic, complemented
4C2×#1818C211yesmaximal, characteristic, complemented
5C2×#1788C211yesmaximal, characteristic
6C2×#1788C211yesmaximal, characteristic
7#14343C211yesmaximal, characteristic, complemented
8#10789C214yesmaximal, complemented
9#10788C214yesmaximal, complemented
10C23×#41C2211yescharacteristic
11C2×#520—24no—
12C2×#520—24no—
13C2×#520—24no—
14C2×#520—24no—
15C2×#520—24no—
16C2×#520C418yescomplemented
17C22×C4×Q8C2212yes—
18C22×C4×D4—22no—
19C22×C4×D4C2212yes—
20C22×#117—21no—
21C22×#117C2211yescharacteristic
22C23×C4⋊C4—21no—
23C23×C4⋊C4C2211yescharacteristic
24C22×#103C2212yes—
25C22×#103C2212yes—
26C22×#102C2212yes—
27C22×#102C2212yes—
28C22×C22⋊C8C2211yescharacteristic
29C22×C22⋊C8C2211yescharacteristic
30C2×#380C2211yescharacteristic
31C2×#380C2211yescharacteristic
32#1824—24no—
33#1823—24no—
34#1818—24no—
35#1788—24no—
36#1788—24no—
37C2×#340C2214yes—
38C2×#340C2214yes—
39C2×#336C2214yes—
40#1445—21no—
41#1445—21no—
42#1440—22no—
43C4⋊#95C2214yes—
44C22×#41—28no—
45C22×#41—28no—
46C22×#41—22no—
47C22×#41—22no—
48C22×#41—22no—
49C22×#41C2×C414yes—
50C22×#41D414yes—
51C22×#41D414yes—
52C23×Q8C2311yescharacteristic
53C23×D4—21no—
54C23×D4C2311yescharacteristic
55C24×C4—21no—
56C24×C4—21no—
57C24×C4—22no—
58C24×C4C2311yescharacteristic
59C23×C8C2312yes—
60C23×C8C2312yes—
61#520—232no—
62#520—232no—
63#520—232no—
64#520—232no—
65C2×C4×Q8—24no—
66C2×C4×Q8—24no—
67C2×C4×Q8—24no—
68C2×C4×Q8—24no—
69C2×C4×Q8—24no—
70C2×C4×Q8C2×C418yes—
71C2×C4×D4—216no—
72C2×C4×D4—216no—
73C2×C4×D4—24no—
74C2×C4×D4—24no—
75C2×C4×D4—24no—
76C2×C4×D4—24no—
77C2×C4×D4—216no—
78C2×C4×D4—24no—
79C2×C4×D4C2×C418yes—
80C2×#117—28no—
81C2×#117—28no—
82C2×#117—22no—
83C2×#117—22no—
84C2×#117—22no—
85C2×#117—22no—
86C2×#117—28no—
87C2×#117C2×C414yes—
88C2×#117D414yes—
89C22×C4⋊C4—28no—
90C22×C4⋊C4—28no—
91C22×C4⋊C4—21no—
92C22×C4⋊C4—22no—
93C22×C4⋊C4—22no—
94C22×C4⋊C4—22no—
95C22×C4⋊C4—22no—
96C22×C4⋊C4—28no—
97C22×C4⋊C4C2×C414yes—
98C22×C4⋊C4D411yescharacteristic
99C22×C4⋊C4D411yescharacteristic
100C22×C4⋊C4D414yes—
101C22×C4⋊C4D414yes—
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2312yes—
105C22×C22⋊C4—22no—
106C22×C22⋊C4—21no—
107C22×C22⋊C4—21no—
108C22×C22⋊C4D412yes—
109C22×C42—22no—
110C22×C42C2312yes—
111C2×#103—24no—
112C2×#103—24no—
113C2×#103—28no—
114C2×#103—28no—
115C2×#102—24no—
116C2×#102—24no—
117C2×#102—28no—
118C2×#102—28no—
119C2×C22⋊C8—22no—
120C2×C22⋊C8—22no—
121C2×C22⋊C8—22no—
122C2×C22⋊C8—22no—
123C2×C22⋊C8—22no—
124C2×C22⋊C8—22no—
125#380—22no—
126#380—22no—
127#380—22no—
128#380—22no—
129C2×#95—21no—
130C2×#95—21no—
131C2×#95—24no—
132C2×#95—24no—
133#340—28no—
134#340—28no—
135#336—28no—
136C25—21no—
137C25C2×D411yescharacteristic
138C2×#41—232no—
139C2×#41—232no—
140C2×#41—232no—
141C2×#41—216no—
142C2×#41—216no—
143C2×#41—216no—
144C2×#41—216no—
145C2×#41—216no—
146C2×#41—216no—
147C2×#41—28no—
148C2×#41—28no—
149C2×#41—28no—
150C2×#41C22⋊C4116yes—
151C22×Q8—21no—
152C22×Q8—21no—
153C22×Q8—21no—
154C22×Q8—24no—
155C22×Q8C22×C412yes—
156C22×Q8C2×D412yes—
157C22×Q8C2×D412yes—
158C22×Q8#4118yes—
159C22×D4—216no—
160C22×D4—24no—
161C22×D4—24no—
162C22×D4—24no—
163C22×D4—21no—
164C22×D4—21no—
165C22×D4—21no—
166C22×D4—24no—
167C22×D4C22×C412yes—
168C22×D4C2×D412yes—
169C22×D4C2×D412yes—
170C22×D4#4118yes—
171C23×C4—216no—
172C23×C4—21no—
173C23×C4—24no—
174C23×C4—24no—
175C23×C4—24no—
176C23×C4—24no—
177C23×C4—24no—
178C23×C4—22no—
179C23×C4—28no—
180C23×C4—28no—
181C23×C4—24no—
182C23×C4—24no—
183C23×C4—24no—
184C23×C4—28no—
185C23×C4—22no—
186C23×C4—22no—
187C23×C4—84no—
188C23×C4—84no—
189C23×C4—82no—
190C23×C4—82no—
191C23×C4C22×C412yes—
192C23×C4C2×D412yes—
193C23×C4C2×D412yes—
194C23×C4C2×D411yescharacteristic
195C23×C4C2×D412yes—
196C23×C4C2×D412yes—
197C23×C4C2×D412yes—
198C23×C4C2×D412yes—
199C23×C4C2×D412yes—
200C23×C4C2411yescharacteristic
201C22×C8—22no—
202C22×C8—22no—
203C22×C8—22no—
204C22×C8—22no—
205C22×C8—22no—
206C22×C8—22no—
207C22×C8—24no—
208C22×C8—24no—
209C22×C8—24no—
210C22×C8—24no—
211C4×Q8—216no—
212C4×Q8—216no—
213C4×Q8—216no—
214C4×Q8—216no—
215C4×D4—2128no—
216C4×D4—216no—
217C4×D4—216no—
218C4×D4—216no—
219C4×D4—216no—
220#117—264no—
221#117—28no—
222#117—28no—
223#117—28no—
224#117—28no—
225C2×C4⋊C4—28no—
226C2×C4⋊C4—28no—
227C2×C4⋊C4—264no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—28no—
230C2×C4⋊C4—28no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—28no—
236C2×C4⋊C4—28no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4C22⋊C414yes—
251C2×C4⋊C4C22⋊C414yes—
252C2×C4⋊C4C22⋊C4116yes—
253C2×C4⋊C4C22×C414yes—
254C2×C4⋊C4C2×D414yes—
255C2×C4⋊C4C2×D414yes—
256C2×C22⋊C4—28no—
257C2×C22⋊C4—28no—
258C2×C22⋊C4—24no—
259C2×C22⋊C4—24no—
260C2×C22⋊C4—28no—
261C2×C22⋊C4—42no—
262C2×C22⋊C4—42no—
263C2×C22⋊C4—82no—
264C2×C22⋊C4—82no—
265C2×C22⋊C4C22⋊C418yes—
266C2×C42—28no—
267C2×C42—28no—
268C2×C42—28no—
269C2×C42—22no—
270C2×C42—22no—
271C2×C42—22no—
272C2×C42—22no—
273C2×C42C22×C414yes—
274C2×C42C2×D414yes—
275#103—48no—
276#103—48no—
277#102—48no—
278#102—48no—
279C22⋊C8—44no—
280C22⋊C8—44no—
281#95—42no—
282#95—42no—
283#95—84no—
284#95—84no—
285C24—22no—
286C24—22no—
287C24—22no—
288C24—24no—
289C24—28no—
290C24—22no—
291C24—22no—
292C24—22no—
293C24—22no—
294C24—42no—
295C24—42no—
296C24—42no—
297C24—42no—
298C24C2×C22⋊C412yes—
299C24#12012yes—
300C24#12012yes—
301C24C22×D411yescharacteristic
302#41—264no—
303#41—264no—
304#41—264no—
305#41—232no—
306#41—232no—
307C2×Q8—22no—
308C2×Q8—24no—
309C2×Q8—22no—
310C2×Q8—22no—
311C2×Q8—24no—
312C2×Q8—24no—
313C2×Q8—28no—
314C2×Q8—28no—
315C2×Q8—28no—
316C2×Q8C2×C22⋊C414yes—
317C2×Q8#11718yes—
318C2×Q8#12318yes—
319C2×Q8#12318yes—
320C2×D4—232no—
321C2×D4—232no—
322C2×D4—232no—
323C2×D4—216no—
324C2×D4—22no—
325C2×D4—24no—
326C2×D4—22no—
327C2×D4—22no—
328C2×D4—24no—
329C2×D4—24no—
330C2×D4—28no—
331C2×D4—28no—
332C2×D4—28no—
333C2×D4C2×C22⋊C414yes—
334C2×D4#11718yes—
335C2×D4#12318yes—
336C2×D4#12318yes—
337C22×C4—232no—
338C22×C4—232no—
339C22×C4—232no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—24no—
344C22×C4—28no—
345C22×C4—24no—
346C22×C4—24no—
347C22×C4—24no—
348C22×C4—24no—
349C22×C4—24no—
350C22×C4—24no—
351C22×C4—28no—
352C22×C4—28no—
353C22×C4—28no—
354C22×C4—216no—
355C22×C4—24no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—24no—
361C22×C4—24no—
362C22×C4—24no—
363C22×C4—24no—
364C22×C4—24no—
365C22×C4—24no—
366C22×C4—232no—
367C22×C4—216no—
368C22×C4—44no—
369C22×C4—44no—
370C22×C4—42no—
371C22×C4—42no—
372C22×C4—44no—
373C22×C4—44no—
374C22×C4—42no—
375C22×C4—44no—
376C22×C4—42no—
377C22×C4—44no—
378C22×C4—42no—
379C22×C4—42no—
380C22×C4—816no—
381C22×C4—88no—
382C22×C4—84no—
383C22×C4—84no—
384C22×C4—88no—
385C22×C4—88no—
386C22×C4—88no—
387C22×C4—816no—
388C22×C4C2×C22⋊C414yes—
389C22×C4C2×C22⋊C414yes—
390C22×C4C2×C22⋊C414yes—
391C22×C4C2×C22⋊C414yes—
392C22×C4C2×C22⋊C412yes—
393C22×C4#12014yes—
394C22×C4#12014yes—
395C22×C4#12012yes—
396C22×C4#12012yes—
397C22×C4C23×C411yescharacteristic
398C22×C4C22×D411yescharacteristic
399C22×C4C22×D411yescharacteristic
400C22×C4C22×D411yescharacteristic
401C22×C4C22×D411yescharacteristic
402C22×C4C22×D411yescharacteristic
403C22×C4C2×#4114yes—
404C22×C4C2×#4114yes—
405C2×C8—42no—
406C2×C8—42no—
407C2×C8—42no—
408C2×C8—42no—
409C2×C8—42no—
410C2×C8—42no—
411C2×C8—44no—
412C2×C8—44no—
413C2×C8—44no—
414C2×C8—44no—
415C4⋊C4—232no—
416C4⋊C4—24no—
417C4⋊C4—24no—
418C4⋊C4—24no—
419C4⋊C4—28no—
420C4⋊C4—216no—
421C4⋊C4—28no—
422C4⋊C4—42no—
423C4⋊C4—42no—
424C4⋊C4—42no—
425C4⋊C4—42no—
426C4⋊C4—44no—
427C4⋊C4—44no—
428C4⋊C4C2×C22⋊C418yes—
429C22⋊C4—232no—
430C22⋊C4—216no—
431C22⋊C4—84no—
432C22⋊C4—84no—
433C42—232no—
434C42—24no—
435C42—24no—
436C42—24no—
437C42—24no—
438C23—24no—
439C23—24no—
440C23—28no—
441C23—28no—
442C23—24no—
443C23—24no—
444C23—22no—
445C23—22no—
446C23—22no—
447C23—22no—
448C23—24no—
449C23—24no—
450C23—24no—
451C23—24no—
452C23—28no—
453C23—44no—
454C23—48no—
455C23—44no—
456C23—48no—
457C23—44no—
458C23—48no—
459C23—44no—
460C23—48no—
461C23—44no—
462C23—44no—
463C23—42no—
464C23—42no—
465C23—42no—
466C23—42no—
467C23—44no—
468C23—44no—
469C23#37914yes—
470C23C22×C22⋊C411yescharacteristic
471C23C2×#12011yescharacteristic
472C23C2×#12011yescharacteristic
473C23#52514yes—
474C23#57612yes—
475C23#57712yes—
476C23#57712yes—
477C23#57812yes—
478Q8—28no—
479Q8—28no—
480Q8—24no—
481Q8—28no—
482Q8#38118yes—
483D4—264no—
484D4—28no—
485D4—28no—
486D4—24no—
487D4—28no—
488D4#38118yes—
489C2×C4—264no—
490C2×C4—24no—
491C2×C4—24no—
492C2×C4—28no—
493C2×C4—28no—
494C2×C4—24no—
495C2×C4—24no—
496C2×C4—28no—
497C2×C4—28no—
498C2×C4—28no—
499C2×C4—24no—
500C2×C4—24no—
501C2×C4—28no—
502C2×C4—28no—
503C2×C4—28no—
504C2×C4—24no—
505C2×C4—24no—
506C2×C4—28no—
507C2×C4—48no—
508C2×C4—42no—
509C2×C4—44no—
510C2×C4—42no—
511C2×C4—44no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—44no—
515C2×C4—48no—
516C2×C4—44no—
517C2×C4—816no—
518C2×C4—832no—
519C2×C4—82no—
520C2×C4—82no—
521C2×C4—84no—
522C2×C4—84no—
523C2×C4—84no—
524C2×C4—84no—
525C2×C4—82no—
526C2×C4—82no—
527C2×C4—82no—
528C2×C4—82no—
529C2×C4#37914yes—
530C2×C4C22×C22⋊C411yescharacteristic
531C2×C4C22×C22⋊C411yescharacteristic
532C2×C4C2×#11712yes—
533C2×C4C2×#11712yes—
534C2×C4C2×#12011yescharacteristic
535C2×C4C2×#12011yescharacteristic
536C2×C4C2×#12312yes—
537C2×C4C2×#12312yes—
538C2×C4C2×#12312yes—
539C2×C4C2×#12312yes—
540C2×C4#52514yes—
541C2×C4#52514yes—
542C2×C4#52514yes—
543C8—82no—
544C8—82no—
545C22—24no—
546C22—28no—
547C22—24no—
548C22—24no—
549C22—48no—
550C22—416no—
551C22—44no—
552C22—44no—
553C22—44no—
554C22—48no—
555C22—44no—
556C22—48no—
557C22—44no—
558C22—48no—
559C22—44no—
560C22—48no—
561C22C2×#37911yescharacteristic
562C22#228012yes—
563C22#284912yes—
564C22#284912yes—
565C22#289712yes—
566C22#289812yes—
567C22#300811yescharacteristic
568C22#300911yescharacteristic
569C22#301011yescharacteristic
570C22#301111yescharacteristic
571C22#301212yes—
572C22#301312yes—
573C22#311914yes—
574C22#312014yes—
575C22#312414yes—
576C22#312514yes—
577C4—24no—
578C4—28no—
579C4—44no—
580C4—48no—
581C4—44no—
582C4—162no—
583C4—162no—
584C4C2×#38111yescharacteristic
585C4C2×#38111yescharacteristic
586C4#228012yes—
587C4#284912yes—
588C4#284912yes—
589C2—44no—
590C2—48no—
591C2#1468111yescharacteristic
592C2#2060211yescharacteristic
593C2#2060311yescharacteristic
594C2#2074614yes—
595C2#3301212yes—
596C2#3301312yes—
597C2#3301612yes—
598C2#3301712yes—
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×#41C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×#117 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#1824C2semidirect
3. C2×#1823C2semidirect
4. C2×#1818C2semidirect
7. #14343C2semidirect
8. #10789C2semidirect
9. #10788C2semidirect
16. C2×#520C4semidirect

Maximal subgroups

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

subgroupindex
C22×#5202
C2×#18242
C2×#18232
C2×#18182
C2×#17882
#143432
#107892
#107882