Finite Group Databasebeta

← #358288

Subgroups — #358288

Order 512.

1822 conjugacy classes of subgroups.

The subgroup lattice has 637 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2317C211yesmaximal, characteristic
2C22×#343C211yesmaximal, characteristic
3#20347C211yesmaximal, characteristic
4#20347C211yesmaximal, characteristic
5C2×#1733C211yesmaximal, characteristic
6C2×#1733C211yesmaximal, characteristic
7#13795C211yesmaximal, characteristic
8#11276C218yesmaximal, complemented
9C23×M16C2211yescharacteristic
10C23×M16C2211yescharacteristic
11C22×#117C2211yescharacteristic
12C22×#117C2211yescharacteristic
13C22×#117C2211yescharacteristic
14C22×#117C2211yescharacteristic
15C23×C4⋊C4C2211yescharacteristic
16#2317—22no—
17#2317—22no—
18#2317—22no—
19#2317C2214yes—
20C2×#389—22no—
21C2×#385—22no—
22C2×#384—21no—
23C2×#382—21no—
24C2×#343—24no—
25C2×#343C2214yes—
26#1736—21no—
27#1736—21no—
28#1736—22no—
29#1736C2214yes—
30#1736C2214yes—
31#1735—22no—
32#1735—22no—
33#1733—24no—
34#1733C418yescomplemented
35C2×#339C2214yes—
36C2×#339C2214yes—
37C4⋊#95—22no—
38C4⋊#95C2214yes—
39C22⋊#95—22no—
40C24×C4D412yes—
41C24×C4C2311yescharacteristic
42C22×M16—21no—
43C22×M16—21no—
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×M16C2312yes—
57C22×M16C2312yes—
58C23×C8—21no—
59C23×C8—21no—
60C2×#117—21no—
61C2×#117—21no—
62C2×#117—22no—
63C2×#117—22no—
64C2×#117—21no—
65C2×#117—21no—
66C2×#117—22no—
67C2×#117—22no—
68C2×#117—22no—
69C2×#117—22no—
70C2×#117—22no—
71C2×#117—22no—
72C2×#117—22no—
73C2×#117—21no—
74C2×#117—21no—
75C2×#117—21no—
76C2×#117—21no—
77C2×#117—22no—
78C2×#117—22no—
79C2×#117—22no—
80C2×#117—22no—
81C2×#117—22no—
82C2×#117C2×C414yes—
83C2×#117C2×C414yes—
84C2×#117D412yes—
85C2×#117D412yes—
86C2×#117Q814yes—
87C2×#117Q814yes—
88C2×#117C2312yes—
89C2×#117C2312yes—
90C22×C4⋊C4—21no—
91C22×C4⋊C4—21no—
92C22×C4⋊C4—22no—
93C22×C4⋊C4—21no—
94C22×C4⋊C4—21no—
95C22×C4⋊C4—21no—
96C22×C4⋊C4—22no—
97C22×C4⋊C4—22no—
98C22×C4⋊C4—22no—
99C22×C4⋊C4—24no—
100C22×C4⋊C4D412yes—
101C22×C4⋊C4D411yescharacteristic
102C22×C4⋊C4D411yescharacteristic
103C22×C4⋊C4D412yes—
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2312yes—
106C22×C22⋊C4—21no—
107C22×C22⋊C4—21no—
108C22×C22⋊C4—21no—
109C22×C22⋊C4—21no—
110C22×C42—21no—
111C22×C42—21no—
112C22×C42—21no—
113C22×C42C2312yes—
114#389—28no—
115#389—28no—
116#385—28no—
117#385—28no—
118#384—24no—
119#384—24no—
120#382—24no—
121#382—24no—
122C2×#95—24no—
123C2×#95—28no—
124#343—28no—
125#340—24no—
126#340—24no—
127#340—28no—
128#339—28no—
129#339C2×C4116yescomplemented
130C25C2×D411yescharacteristic
131C23×C4—22no—
132C23×C4—21no—
133C23×C4—21no—
134C23×C4—22no—
135C23×C4—22no—
136C23×C4—22no—
137C23×C4—22no—
138C23×C4—24no—
139C23×C4—24no—
140C23×C4—84no—
141C23×C4C22×C412yes—
142C23×C4C2×D414yes—
143C23×C4C2×D411yescharacteristic
144C23×C4C2×D411yescharacteristic
145C23×C4C2×D411yescharacteristic
146C23×C4C2×D411yescharacteristic
147C23×C4C2×D411yescharacteristic
148C23×C4C2×D412yes—
149C23×C4C2×Q812yes—
150C23×C4C2×Q812yes—
151C23×C4#4112yes—
152C23×C4#4112yes—
153C23×C4#4112yes—
154C23×C4#4112yes—
155C23×C4#4112yes—
156C23×C4#4112yes—
157C23×C4#4112yes—
158C23×C4#4112yes—
159C23×C4C2411yescharacteristic
160C2×M16—24no—
161C2×M16—24no—
162C2×M16—22no—
163C2×M16—22no—
164C2×M16—28no—
165C2×M16—28no—
166C2×M16—28no—
167C2×M16—28no—
168C2×M16—22no—
169C2×M16—22no—
170C2×M16—22no—
171C2×M16—22no—
172C2×M16—22no—
173C2×M16—22no—
174C2×M16—22no—
175C2×M16—22no—
176C2×M16—42no—
177C2×M16—42no—
178C2×M16—44no—
179C2×M16—42no—
180C2×M16—42no—
181C2×M16—44no—
182C2×M16—44no—
183C2×M16—44no—
184C2×M16C22×C414yes—
185C2×M16C2×D414yes—
186C22×C8—24no—
187C22×C8—24no—
188C22×C8—22no—
189C22×C8—22no—
190C22×C8—42no—
191C22×C8—42no—
192C22×C8—42no—
193C22×C8—42no—
194#117—24no—
195#117—24no—
196#117—24no—
197#117—24no—
198#117—24no—
199#117—22no—
200#117—22no—
201#117—28no—
202#117—28no—
203#117—24no—
204#117—24no—
205#117—24no—
206#117—24no—
207#117—24no—
208#117—24no—
209#117—22no—
210#117—22no—
211#117—22no—
212#117—22no—
213#117—42no—
214#117—42no—
215#117—44no—
216#117—42no—
217#117—42no—
218#117—42no—
219#117—42no—
220#117—42no—
221#117—42no—
222#117—44no—
223#117C4⋊C418yes—
224#117C22×C414yes—
225#117C2×Q814yes—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—21no—
240C2×C4⋊C4—21no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—216no—
251C2×C4⋊C4—42no—
252C2×C4⋊C4—42no—
253C2×C4⋊C4—42no—
254C2×C4⋊C4—42no—
255C2×C4⋊C4—42no—
256C2×C4⋊C4—42no—
257C2×C4⋊C4—42no—
258C2×C4⋊C4—42no—
259C2×C4⋊C4—42no—
260C2×C4⋊C4—42no—
261C2×C4⋊C4—88no—
262C2×C4⋊C4C4⋊C418yes—
263C2×C4⋊C4C2×D412yes—
264C2×C4⋊C4C2×D412yes—
265C2×C4⋊C4C2×D414yes—
266C2×C4⋊C4C2×Q814yes—
267C2×C22⋊C4—24no—
268C2×C22⋊C4—22no—
269C2×C22⋊C4—24no—
270C2×C22⋊C4—22no—
271C2×C22⋊C4—24no—
272C2×C22⋊C4—22no—
273C2×C22⋊C4—24no—
274C2×C22⋊C4—22no—
275C2×C22⋊C4—41no—
276C2×C22⋊C4—41no—
277C2×C22⋊C4—42no—
278C2×C22⋊C4—42no—
279C2×C22⋊C4—42no—
280C2×C22⋊C4—42no—
281C2×C22⋊C4—42no—
282C2×C42—24no—
283C2×C42—22no—
284C2×C42—24no—
285C2×C42—22no—
286C2×C42—22no—
287C2×C42—22no—
288C2×C42—22no—
289C2×C42—22no—
290C2×C42—22no—
291C2×C42—24no—
292C2×C42—22no—
293C2×C42—22no—
294C2×C42—41no—
295C2×C42—41no—
296C2×C42—42no—
297C2×C42—42no—
298C2×C42—42no—
299C2×C42C22×C414yes—
300#95—216no—
301#95—88no—
302C24—42no—
303C24—42no—
304C24C4×D412yes—
305C24#12011yescharacteristic
306C24#12011yescharacteristic
307C24#12112yes—
308C24C22⋊Q812yes—
309C24C22⋊Q812yes—
310C24#12312yes—
311C24#12412yes—
312C24C22×D411yescharacteristic
313C22×C4—24no—
314C22×C4—28no—
315C22×C4—24no—
316C22×C4—22no—
317C22×C4—24no—
318C22×C4—22no—
319C22×C4—24no—
320C22×C4—24no—
321C22×C4—24no—
322C22×C4—44no—
323C22×C4—44no—
324C22×C4—48no—
325C22×C4—44no—
326C22×C4—44no—
327C22×C4—44no—
328C22×C4—48no—
329C22×C4—44no—
330C22×C4—44no—
331C22×C4—44no—
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—84no—
341C22×C4—88no—
342C22×C4—88no—
343C22×C4C2×C4⋊C412yes—
344C22×C4C2×C4⋊C412yes—
345C22×C4#11714yes—
346C22×C4#11714yes—
347C22×C4C4×D412yes—
348C22×C4C4×D412yes—
349C22×C4C4×D414yes—
350C22×C4C4×D412yes—
351C22×C4C4×Q814yes—
352C22×C4#12011yescharacteristic
353C22×C4#12011yescharacteristic
354C22×C4#12112yes—
355C22×C4#12112yes—
356C22×C4#12112yes—
357C22×C4#12112yes—
358C22×C4#12112yes—
359C22×C4C22⋊Q812yes—
360C22×C4C22⋊Q812yes—
361C22×C4C22⋊Q812yes—
362C22×C4C22⋊Q812yes—
363C22×C4C22⋊Q812yes—
364C22×C4C22⋊Q812yes—
365C22×C4C22⋊Q812yes—
366C22×C4C22⋊Q812yes—
367C22×C4#12312yes—
368C22×C4#12312yes—
369C22×C4#12312yes—
370C22×C4#12312yes—
371C22×C4#12314yes—
372C22×C4#12314yes—
373C22×C4#12312yes—
374C22×C4#12412yes—
375C22×C4#12412yes—
376C22×C4#12412yes—
377C22×C4#12412yes—
378C22×C4#12412yes—
379C22×C4#12412yes—
380C22×C4#12412yes—
381C22×C4#12514yes—
382C22×C4#12514yes—
383C22×C4#12514yes—
384C22×C4#12514yes—
385C22×C4#12514yes—
386C22×C4#12614yes—
387C22×C4#12614yes—
388C22×C4#12614yes—
389C22×C4#12614yes—
390C22×C4C4⋊Q814yes—
391C22×C4C23×C411yescharacteristic
392C22×C4C22×D411yescharacteristic
393C22×C4C22×D411yescharacteristic
394C22×C4C22×D411yescharacteristic
395C22×C4C22×Q811yescharacteristic
396C22×C4C22×Q811yescharacteristic
397C22×C4C2×#4111yescharacteristic
398C22×C4C2×#4111yescharacteristic
399C22×C4C2×#4111yescharacteristic
400C22×C4C2×#4111yescharacteristic
401C22×C4C2×#4111yescharacteristic
402C22×C4C2×#4111yescharacteristic
403C22×C4C2×#4111yescharacteristic
404C22×C4C2×#4111yescharacteristic
405M16—24no—
406M16—24no—
407M16—24no—
408M16—24no—
409M16—42no—
410M16—42no—
411M16—42no—
412M16—42no—
413M16—48no—
414M16—48no—
415C2×C8—24no—
416C2×C8—24no—
417C2×C8—42no—
418C2×C8—42no—
419C2×C8—42no—
420C2×C8—42no—
421C2×C8—48no—
422C2×C8—48no—
423C4⋊C4—24no—
424C4⋊C4—28no—
425C4⋊C4—22no—
426C4⋊C4—22no—
427C4⋊C4—24no—
428C4⋊C4—24no—
429C4⋊C4—22no—
430C4⋊C4—22no—
431C4⋊C4—42no—
432C4⋊C4—42no—
433C4⋊C4—42no—
434C4⋊C4—42no—
435C4⋊C4—42no—
436C4⋊C4—42no—
437C4⋊C4—48no—
438C4⋊C4—44no—
439C4⋊C4—44no—
440C4⋊C4—42no—
441C4⋊C4—42no—
442C4⋊C4—816no—
443C4⋊C4C2×C4⋊C418yes—
444C22⋊C4—24no—
445C22⋊C4—24no—
446C22⋊C4—28no—
447C22⋊C4—42no—
448C22⋊C4—42no—
449C22⋊C4—48no—
450C22⋊C4—44no—
451C22⋊C4—44no—
452C22⋊C4—44no—
453C42—24no—
454C42—24no—
455C42—28no—
456C42—24no—
457C42—24no—
458C42—42no—
459C42—42no—
460C42—42no—
461C42—42no—
462C42—44no—
463C42—44no—
464C42—44no—
465C23—22no—
466C23—22no—
467C23—22no—
468C23—22no—
469C23—22no—
470C23—22no—
471C23—42no—
472C23—42no—
473C23—44no—
474C23—42no—
475C23—44no—
476C23—44no—
477C23—42no—
478C23—44no—
479C23#38512yes—
480C23#38512yes—
481C23#38714yes—
482C23#38814yes—
483C23#39312yes—
484C23#39312yes—
485C23#39412yes—
486C23#39412yes—
487C23#39814yes—
488C23#40014yes—
489C23C2×C4×D411yescharacteristic
490C23C2×#12011yescharacteristic
491C23C2×#12111yescharacteristic
492C23C2×C22⋊Q811yescharacteristic
493C23C2×C22⋊Q811yescharacteristic
494C23C2×#12311yescharacteristic
495C23C2×#12411yescharacteristic
496C2×C4—28no—
497C2×C4—416no—
498C2×C4—44no—
499C2×C4—44no—
500C2×C4—48no—
501C2×C4—48no—
502C2×C4—48no—
503C2×C4—44no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—44no—
507C2×C4—44no—
508C2×C4—48no—
509C2×C4—44no—
510C2×C4—44no—
511C2×C4—84no—
512C2×C4—82no—
513C2×C4—82no—
514C2×C4—84no—
515C2×C4—84no—
516C2×C4—84no—
517C2×C4—88no—
518C2×C4—88no—
519C2×C4—82no—
520C2×C4—82no—
521C2×C4—84no—
522C2×C4#38214yes—
523C2×C4#38214yes—
524C2×C4#38512yes—
525C2×C4#38512yes—
526C2×C4#38714yes—
527C2×C4#38714yes—
528C2×C4#38714yes—
529C2×C4#38814yes—
530C2×C4#38814yes—
531C2×C4#38814yes—
532C2×C4#38914yes—
533C2×C4#38914yes—
534C2×C4#39312yes—
535C2×C4#39312yes—
536C2×C4#39412yes—
537C2×C4#39412yes—
538C2×C4#39714yes—
539C2×C4#39714yes—
540C2×C4#39814yes—
541C2×C4#39814yes—
542C2×C4#39814yes—
543C2×C4#40014yes—
544C2×C4#40014yes—
545C2×C4#40014yes—
546C2×C4#40014yes—
547C2×C4#40014yes—
548C2×C4C22×C4⋊C411yescharacteristic
549C2×C4C2×#11711yescharacteristic
550C2×C4C2×#11711yescharacteristic
551C2×C4C2×C4×D411yescharacteristic
552C2×C4C2×C4×D411yescharacteristic
553C2×C4C2×C4×Q811yescharacteristic
554C2×C4C2×#12111yescharacteristic
555C2×C4C2×#12111yescharacteristic
556C2×C4C2×C22⋊Q811yescharacteristic
557C2×C4C2×C22⋊Q811yescharacteristic
558C2×C4C2×C22⋊Q811yescharacteristic
559C2×C4C2×#12311yescharacteristic
560C2×C4C2×#12311yescharacteristic
561C2×C4C2×#12311yescharacteristic
562C2×C4C2×#12311yescharacteristic
563C2×C4C2×#12411yescharacteristic
564C2×C4C2×#12411yescharacteristic
565C2×C4C2×#12411yescharacteristic
566C2×C4C2×#12511yescharacteristic
567C2×C4C2×#12511yescharacteristic
568C2×C4C2×#12511yescharacteristic
569C2×C4C2×#12511yescharacteristic
570C2×C4C2×#12511yescharacteristic
571C2×C4C2×#12611yescharacteristic
572C2×C4C2×#12611yescharacteristic
573C2×C4C2×#12611yescharacteristic
574C2×C4C2×#12611yescharacteristic
575C2×C4C2×C4⋊Q811yescharacteristic
576C8—44no—
577C8—44no—
578C22—22no—
579C22—24no—
580C22—22no—
581C22—24no—
582C22—22no—
583C22—22no—
584C22—24no—
585C22—24no—
586C22—21no—
587C22—21no—
588C22—22no—
589C22—22no—
590C22—21no—
591C22—22no—
592C22—21no—
593C22—22no—
594C22—44no—
595C22—44no—
596C22—44no—
597C22—44no—
598C22#144214yes—
599C22#144214yes—
600C22C2×#38511yescharacteristic
601C22C2×#38711yescharacteristic
602C22C2×#38811yescharacteristic
603C22C2×#39311yescharacteristic
604C22C2×#39411yescharacteristic
605C22C2×#39811yescharacteristic
606C22C2×#40011yescharacteristic
607C22#301912yes—
608C22#302012yes—
609C4—48no—
610C4—84no—
611C4—84no—
612C4—84no—
613C4—82no—
614C4—82no—
615C4#144214yes—
616C4#144214yes—
617C4C2×#38211yescharacteristic
618C4C2×#38711yescharacteristic
619C4C2×#38811yescharacteristic
620C4C2×#38911yescharacteristic
621C4C2×#39711yescharacteristic
622C4C2×#39811yescharacteristic
623C4C2×#40011yescharacteristic
624C4C2×#40011yescharacteristic
625C2—21no—
626C2—21no—
627C2—21no—
628C2—24no—
629C2—24no—
630C2—21no—
631C2C2×#144211yescharacteristic
632C2#2086412yes—
633C2#2346212yes—
634C2#2349311yescharacteristic
635C2#2349411yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
8. #11276C2semidirect
34. #1733C4semidirect
129. #339C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#23172
C22×#3432
#203472
C2×#17332
#137952
#112762