Finite Group Databasebeta

← #358221

Subgroups — #358221

Order 512.

2078 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
1C22×#428C211yesmaximal, characteristic
2C2×#1736C211yesmaximal, characteristic
3C2×#1736C211yesmaximal, characteristic
4C2×#1736C211yesmaximal, characteristic
5C2×#1728C211yesmaximal, characteristic
6#13756C211yesmaximal, characteristic
7C2×C4⋊#95C211yesmaximal, characteristic
8#11257C214yesmaximal, complemented
9#11256C214yesmaximal, complemented
10C23×M16C2211yescharacteristic
11C23×M16C2211yescharacteristic
12C22×#117C2211yescharacteristic
13C22×#117C2211yescharacteristic
14C23×C4⋊C4C2211yescharacteristic
15C23×C4⋊C4C2211yescharacteristic
16C23×C42C2211yescharacteristic
17C2×#428—24no—
18C2×#428—24no—
19C2×#428—24no—
20C22×#107C2212yes—
21C22×#106C2212yes—
22C22×#95—22no—
23#1736—24no—
24#1736—24no—
25#1736C418yescomplemented
26C2×#340C2212yes—
27C2×#340C2214yes—
28C2×#340C2212yes—
29C2×#340C2214yes—
30#1728—24no—
31C2×#336C2214yes—
32C4⋊#95—21no—
33C4⋊#95—21no—
34C4⋊#95C2214yes—
35C4⋊#95C2212yes—
36C4⋊#95C2212yes—
37C22⋊#95—22no—
38C4×#95—21no—
39C4×#95—21no—
40C24×C4—21no—
41C24×C4—21no—
42C24×C4D412yes—
43C24×C4C2311yescharacteristic
44C22×M16—22no—
45C22×M16—22no—
46C22×M16—22no—
47C22×M16—22no—
48C22×M16—22no—
49C22×M16—22no—
50C22×M16—22no—
51C22×M16—22no—
52C22×M16—22no—
53C22×M16—22no—
54C22×M16C2×C414yes—
55C22×M16D414yes—
56C22×M16Q814yes—
57C22×M16Q814yes—
58C23×C8C2312yes—
59C23×C8C2312yes—
60C2×#117—22no—
61C2×#117—22no—
62C2×#117—22no—
63C2×#117—22no—
64C2×#117—22no—
65C2×#117—22no—
66C2×#117—22no—
67C2×#117—22no—
68C2×#117—22no—
69C2×#117—22no—
70C2×#117C2×C414yes—
71C2×#117D414yes—
72C22×C4⋊C4—22no—
73C22×C4⋊C4—22no—
74C22×C4⋊C4—22no—
75C22×C4⋊C4—22no—
76C22×C4⋊C4—22no—
77C22×C4⋊C4—22no—
78C22×C4⋊C4—22no—
79C22×C4⋊C4—22no—
80C22×C4⋊C4—21no—
81C22×C4⋊C4—21no—
82C22×C4⋊C4—22no—
83C22×C4⋊C4—22no—
84C22×C4⋊C4C2×C414yes—
85C22×C4⋊C4D414yes—
86C22×C4⋊C4C2312yes—
87C22×C4⋊C4C2312yes—
88C22×C4⋊C4C2311yescharacteristic
89C22×C4⋊C4C2312yes—
90C22×C4⋊C4C2311yescharacteristic
91C22×C22⋊C4—21no—
92C22×C22⋊C4—21no—
93C22×C42—22no—
94C22×C42—21no—
95C22×C42—21no—
96C22×C42—22no—
97C22×C42—22no—
98C22×C42—22no—
99C22×C42—22no—
100C22×C42—22no—
101C22×C42D411yescharacteristic
102C22×C42D411yescharacteristic
103C22×C42C2312yes—
104#428—232no—
105C2×#107—24no—
106C2×#107—24no—
107C2×#107—24no—
108C2×#106—24no—
109C2×#106—24no—
110C2×#106—24no—
111C2×#95—22no—
112C2×#95—22no—
113C2×#95—22no—
114C2×#95—22no—
115C2×#95—24no—
116C2×#95—24no—
117C2×#95—48no—
118#340—24no—
119#340—24no—
120#340—28no—
121#340C2×C4116yescomplemented
122#336—28no—
123C25C2×D411yescharacteristic
124C23×C4—24no—
125C23×C4—24no—
126C23×C4—24no—
127C23×C4—24no—
128C23×C4—24no—
129C23×C4—24no—
130C23×C4—24no—
131C23×C4—22no—
132C23×C4—21no—
133C23×C4—21no—
134C23×C4—22no—
135C23×C4—22no—
136C23×C4—42no—
137C23×C4—42no—
138C23×C4—44no—
139C23×C4—44no—
140C23×C4—44no—
141C23×C4—44no—
142C23×C4—42no—
143C23×C4—42no—
144C23×C4—42no—
145C23×C4—42no—
146C23×C4C22×C412yes—
147C23×C4C2×D412yes—
148C23×C4C2×D412yes—
149C23×C4C2×D412yes—
150C23×C4C2×D412yes—
151C23×C4C2×D412yes—
152C23×C4C2×D411yescharacteristic
153C23×C4C2×Q812yes—
154C23×C4C2×Q812yes—
155C23×C4#4112yes—
156C23×C4#4112yes—
157C23×C4#4112yes—
158C23×C4#4112yes—
159C23×C4#4112yes—
160C23×C4#4112yes—
161C23×C4#4112yes—
162C23×C4#4112yes—
163C23×C4C2411yescharacteristic
164C2×M16—28no—
165C2×M16—28no—
166C2×M16—28no—
167C2×M16—28no—
168C2×M16—28no—
169C2×M16—28no—
170C2×M16—28no—
171C2×M16—28no—
172C2×M16—28no—
173C2×M16—44no—
174C2×M16—44no—
175C2×M16—44no—
176C2×M16—44no—
177C2×M16C4⋊C4116yes—
178C22×C8—22no—
179C22×C8—22no—
180C22×C8—22no—
181C22×C8—22no—
182C22×C8—22no—
183C22×C8—22no—
184C22×C8—22no—
185C22×C8—22no—
186C22×C8—22no—
187C22×C8—22no—
188C22×C8C22×C414yes—
189C22×C8C2×D414yes—
190C22×C8C2×Q814yes—
191C22×C8C2×Q814yes—
192#117—28no—
193#117—28no—
194#117—28no—
195#117—28no—
196#117—44no—
197#117—44no—
198#117—44no—
199#117—44no—
200C2×C4⋊C4—28no—
201C2×C4⋊C4—28no—
202C2×C4⋊C4—28no—
203C2×C4⋊C4—28no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—42no—
232C2×C4⋊C4—42no—
233C2×C4⋊C4—42no—
234C2×C4⋊C4—44no—
235C2×C4⋊C4—42no—
236C2×C4⋊C4—44no—
237C2×C4⋊C4—44no—
238C2×C4⋊C4—44no—
239C2×C4⋊C4C22×C414yes—
240C2×C4⋊C4C22×C414yes—
241C2×C4⋊C4C2×D414yes—
242C2×C4⋊C4C2×D412yes—
243C2×C4⋊C4C2×D412yes—
244C2×C22⋊C4—24no—
245C2×C22⋊C4—24no—
246C2×C22⋊C4—42no—
247C2×C22⋊C4—42no—
248C2×C22⋊C4—42no—
249C2×C22⋊C4—42no—
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—22no—
259C2×C42—22no—
260C2×C42—22no—
261C2×C42—22no—
262C2×C42—22no—
263C2×C42—22no—
264C2×C42—42no—
265C2×C42—42no—
266C2×C42—42no—
267C2×C42—42no—
268C2×C42—44no—
269C2×C42—44no—
270C2×C42—44no—
271C2×C42—44no—
272#107—216no—
273#106—216no—
274#95—44no—
275#95—44no—
276#95—48no—
277#95—44no—
278#95—48no—
279#95—44no—
280C24—22no—
281C24—22no—
282C24—22no—
283C24—22no—
284C24C4×D412yes—
285C24#12112yes—
286C24#12112yes—
287C24#12112yes—
288C24#12312yes—
289C24#12312yes—
290C24C4⋊Q812yes—
291C24C22×D411yescharacteristic
292C22×C4—216no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—24no—
306C22×C4—24no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—24no—
310C22×C4—48no—
311C22×C4—48no—
312C22×C4—48no—
313C22×C4—48no—
314C22×C4—48no—
315C22×C4—48no—
316C22×C4—416no—
317C22×C4—416no—
318C22×C4—41no—
319C22×C4—41no—
320C22×C4—41no—
321C22×C4—41no—
322C22×C4—42no—
323C22×C4—42no—
324C22×C4—42no—
325C22×C4—42no—
326C22×C4—42no—
327C22×C4—42no—
328C22×C4—44no—
329C22×C4—44no—
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—44no—
339C22×C4—44no—
340C22×C4—48no—
341C22×C4—48no—
342C22×C4—88no—
343C22×C4—84no—
344C22×C4—84no—
345C22×C4—84no—
346C22×C4—84no—
347C22×C4—88no—
348C22×C4C2×C4⋊C414yes—
349C22×C4#11714yes—
350C22×C4#11714yes—
351C22×C4C4×D414yes—
352C22×C4C4×D414yes—
353C22×C4C4×D412yes—
354C22×C4C4×Q814yes—
355C22×C4#12014yes—
356C22×C4#12112yes—
357C22×C4#12112yes—
358C22×C4#12112yes—
359C22×C4C22⋊Q814yes—
360C22×C4C22⋊Q814yes—
361C22×C4C22⋊Q814yes—
362C22×C4C22⋊Q814yes—
363C22×C4C22⋊Q814yes—
364C22×C4#12314yes—
365C22×C4#12314yes—
366C22×C4#12314yes—
367C22×C4#12312yes—
368C22×C4#12312yes—
369C22×C4#12414yes—
370C22×C4#12414yes—
371C22×C4#12414yes—
372C22×C4#12414yes—
373C22×C4#12514yes—
374C22×C4#12514yes—
375C22×C4#12514yes—
376C22×C4#12514yes—
377C22×C4#12514yes—
378C22×C4#12614yes—
379C22×C4#12614yes—
380C22×C4#12614yes—
381C22×C4#12614yes—
382C22×C4C4⋊Q812yes—
383C22×C4AGL(1,ℤ/8)14yes—
384C22×C4#13714yes—
385C22×C4C23×C411yescharacteristic
386C22×C4C22×D411yescharacteristic
387C22×C4C22×D411yescharacteristic
388C22×C4C22×D411yescharacteristic
389C22×C4C22×Q811yescharacteristic
390C22×C4C22×Q811yescharacteristic
391C22×C4C2×#4111yescharacteristic
392C22×C4C2×#4111yescharacteristic
393C22×C4C2×#4111yescharacteristic
394C22×C4C2×#4111yescharacteristic
395C22×C4C2×#4111yescharacteristic
396C22×C4C2×#4111yescharacteristic
397C22×C4C2×#4111yescharacteristic
398C22×C4C2×#4111yescharacteristic
399M16—232no—
400M16—416no—
401C2×C8—24no—
402C2×C8—24no—
403C2×C8—24no—
404C2×C8—24no—
405C2×C8—24no—
406C2×C8—24no—
407C2×C8—24no—
408C2×C8—24no—
409C2×C8—24no—
410C2×C8—42no—
411C2×C8—42no—
412C2×C8—42no—
413C2×C8—42no—
414C2×C8C2×C4⋊C418yes—
415C4⋊C4—24no—
416C4⋊C4—24no—
417C4⋊C4—24no—
418C4⋊C4—24no—
419C4⋊C4—22no—
420C4⋊C4—24no—
421C4⋊C4—22no—
422C4⋊C4—24no—
423C4⋊C4—22no—
424C4⋊C4—22no—
425C4⋊C4—42no—
426C4⋊C4—41no—
427C4⋊C4—41no—
428C4⋊C4—42no—
429C4⋊C4—42no—
430C4⋊C4—41no—
431C4⋊C4—41no—
432C4⋊C4—42no—
433C4⋊C4—42no—
434C4⋊C4—42no—
435C4⋊C4—48no—
436C4⋊C4—48no—
437C22⋊C4—48no—
438C22⋊C4—48no—
439C42—42no—
440C42—42no—
441C42—42no—
442C42—42no—
443C42—48no—
444C42—48no—
445C42—44no—
446C42—44no—
447C23—24no—
448C23—24no—
449C23—24no—
450C23—24no—
451C23—24no—
452C23—24no—
453C23—24no—
454C23—24no—
455C23—24no—
456C23—24no—
457C23—22no—
458C23—22no—
459C23—22no—
460C23—22no—
461C23—24no—
462C23—24no—
463C23#38814yes—
464C23#38814yes—
465C23#38914yes—
466C23#39414yes—
467C23#39714yes—
468C23#39814yes—
469C23#39814yes—
470C23C2×C4×D411yescharacteristic
471C23C2×#12111yescharacteristic
472C23C2×#12111yescharacteristic
473C23C2×#12111yescharacteristic
474C23C2×#12311yescharacteristic
475C23C2×#12311yescharacteristic
476C23C2×C4⋊Q811yescharacteristic
477C23C2×AGL(1,ℤ/8)12yes—
478C23C2×#13712yes—
479C23#57512yes—
480C23#57512yes—
481C2×C4—24no—
482C2×C4—24no—
483C2×C4—24no—
484C2×C4—24no—
485C2×C4—24no—
486C2×C4—24no—
487C2×C4—432no—
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—42no—
497C2×C4—42no—
498C2×C4—44no—
499C2×C4—44no—
500C2×C4—48no—
501C2×C4—44no—
502C2×C4—48no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—44no—
508C2×C4—44no—
509C2×C4—816no—
510C2×C4—816no—
511C2×C4—82no—
512C2×C4—82no—
513C2×C4—84no—
514C2×C4—82no—
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—84no—
524C2×C4—82no—
525C2×C4#38218yes—
526C2×C4#38518yes—
527C2×C4#38718yes—
528C2×C4#38718yes—
529C2×C4#38814yes—
530C2×C4#38814yes—
531C2×C4#38914yes—
532C2×C4#39318yes—
533C2×C4#39414yes—
534C2×C4#39714yes—
535C2×C4#39814yes—
536C2×C4#39814yes—
537C2×C4#40018yes—
538C2×C4#40018yes—
539C2×C4#40018yes—
540C2×C4C22×C4⋊C411yescharacteristic
541C2×C4C2×#11711yescharacteristic
542C2×C4C2×#11711yescharacteristic
543C2×C4C2×C4×D411yescharacteristic
544C2×C4C2×C4×D411yescharacteristic
545C2×C4C2×C4×Q811yescharacteristic
546C2×C4C2×#12011yescharacteristic
547C2×C4C2×C22⋊Q811yescharacteristic
548C2×C4C2×C22⋊Q811yescharacteristic
549C2×C4C2×C22⋊Q811yescharacteristic
550C2×C4C2×C22⋊Q811yescharacteristic
551C2×C4C2×C22⋊Q811yescharacteristic
552C2×C4C2×#12311yescharacteristic
553C2×C4C2×#12311yescharacteristic
554C2×C4C2×#12311yescharacteristic
555C2×C4C2×#12411yescharacteristic
556C2×C4C2×#12411yescharacteristic
557C2×C4C2×#12411yescharacteristic
558C2×C4C2×#12411yescharacteristic
559C2×C4C2×#12511yescharacteristic
560C2×C4C2×#12511yescharacteristic
561C2×C4C2×#12511yescharacteristic
562C2×C4C2×#12511yescharacteristic
563C2×C4C2×#12511yescharacteristic
564C2×C4C2×#12611yescharacteristic
565C2×C4C2×#12611yescharacteristic
566C2×C4C2×#12611yescharacteristic
567C2×C4C2×#12611yescharacteristic
568C8—28no—
569C8—44no—
570C22—28no—
571C22—24no—
572C22—24no—
573C22—28no—
574C22—28no—
575C22—24no—
576C22—24no—
577C22—28no—
578C22—24no—
579C22—24no—
580C22#144218yes—
581C22C2×#38811yescharacteristic
582C22C2×#38811yescharacteristic
583C22C2×#38911yescharacteristic
584C22C2×#39411yescharacteristic
585C22C2×#39711yescharacteristic
586C22C2×#39811yescharacteristic
587C22C2×#39811yescharacteristic
588C22C4×AGL(1,ℤ/8)12yes—
589C22C4×#13712yes—
590C22#303612yes—
591C22#303712yes—
592C22#303811yescharacteristic
593C22#303911yescharacteristic
594C22#304011yescharacteristic
595C22#304011yescharacteristic
596C22#305512yes—
597C22#305612yes—
598C22#309212yes—
599C22#309312yes—
600C22#309412yes—
601C22#309512yes—
602C22#316512yes—
603C22#316612yes—
604C4—44no—
605C4—44no—
606C4—84no—
607C4—84no—
608C4—82no—
609C4—84no—
610C4—82no—
611C4#144218yes—
612C4C2×#38211yescharacteristic
613C4C2×#38511yescharacteristic
614C4C2×#38711yescharacteristic
615C4C2×#38711yescharacteristic
616C4C2×#39311yescharacteristic
617C4C2×#40011yescharacteristic
618C4C2×#40011yescharacteristic
619C4C2×#40011yescharacteristic
620C2—28no—
621C2C2×#144211yescharacteristic
622C2#2105912yes—
623C2#2106011yescharacteristic
624C2#2106111yescharacteristic
625C2#2115912yes—
626C2#2354611yescharacteristic
627C2#2354711yescharacteristic
628C2#2364212yes—
629C2#2365712yes—
630C2#2365811yescharacteristic
631C2#2365911yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C22×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
8. #11257C2semidirect
9. #11256C2semidirect
25. #1736C4semidirect
121. #340C2×C4semidirect

Maximal subgroups

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

subgroupindex
C22×#4282
C2×#17362
C2×#17282
#137562
C2×C4⋊#952
#112572
#112562