Finite Group Databasebeta

← #380573

Subgroups — #380573

Order 512.

1988 conjugacy classes of subgroups.

The subgroup lattice has 634 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#465C211yesmaximal, characteristic
2C2×#1893C211yesmaximal, characteristic
3C2×#1892C211yesmaximal, characteristic, complemented
4C2×#1808C211yesmaximal, characteristic, complemented
5C22×#336C211yesmaximal, characteristic, complemented
6#14702C211yesmaximal, characteristic, complemented
7#14698C211yesmaximal, characteristic
8#11378C212yesmaximal, complemented
9#11124C212yesmaximal, complemented
10#11123C212yesmaximal, complemented
11#11023C212yesmaximal, complemented
12C23×SD16C2211yescharacteristic
13C24×C8C2211yescharacteristic
14C24×C8C2211yescharacteristic
15C22×C22⋊Q8C2211yescharacteristic
16C22×#121C2211yescharacteristic
17C23×C42C2211yescharacteristic
18C2×#465—24no—
19C2×#465—24no—
20C2×#465C418yescomplemented
21C22×#107C2212yes—
22C22×#106C2211yescharacteristic
23C22×#103C2211yescharacteristic
24C22×#103C2212yes—
25C22×#102C2211yescharacteristic
26C22×#102C2212yes—
27C2×#391C2212yes—
28C2×#390C2212yes—
29C2×#386—22no—
30C2×#380C2211yescharacteristic
31C2×#380C2211yescharacteristic
32#1893—24no—
33#1892—24no—
34#1808—24no—
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#336—24no—
39C2×#336C2212yes—
40C2×#336C2212yes—
41#1437—22no—
42#1436—22no—
43C4⋊#95C2212yes—
44C4⋊#95C2212yes—
45C22⋊#95—22no—
46C23×Q8C2311yescharacteristic
47C23×D4—21no—
48C23×D4C2311yescharacteristic
49C24×C4D412yes—
50C24×C4C2311yescharacteristic
51C22×SD16—22no—
52C22×SD16—22no—
53C22×SD16—42no—
54C22×SD16—42no—
55C22×SD16C2×C414yescomplemented
56C23×C8—22no—
57C23×C8—22no—
58C23×C8—42no—
59C23×C8—42no—
60C23×C8—42no—
61C23×C8—42no—
62C23×C8C2×C414yes—
63C23×C8D414yes—
64C23×C8D414yes—
65C23×C8D414yes—
66C23×C8C2312yes—
67C23×C8C2311yescharacteristic
68C23×C8C2311yescharacteristic
69C2×C22⋊Q8—22no—
70C2×C22⋊Q8—22no—
71C2×C22⋊Q8—22no—
72C2×C22⋊Q8—22no—
73C2×C22⋊Q8—22no—
74C2×C22⋊Q8C2×C414yes—
75C2×#121—22no—
76C2×#121—22no—
77C2×#121—22no—
78C2×#121—22no—
79C2×#121—22no—
80C2×#121C2×C414yes—
81C22×C4⋊C4—21no—
82C22×C4⋊C4C2312yes—
83C22×C4⋊C4C2312yes—
84C22×C4⋊C4C2311yescharacteristic
85C22×C4⋊C4C2311yescharacteristic
86C22×C22⋊C4—21no—
87C22×C22⋊C4—21no—
88C22×C22⋊C4—21no—
89C22×C22⋊C4—21no—
90C22×C22⋊C4—24no—
91C22×C42—22no—
92C22×C42—22no—
93C22×C42—42no—
94C22×C42—42no—
95C22×C42D412yes—
96C22×C42C2312yes—
97#465—232no—
98C2×#107—24no—
99C2×#107—42no—
100C2×#107—42no—
101C2×#106—22no—
102C2×#106—22no—
103C2×#106C2×C414yescomplemented
104C2×#103—22no—
105C2×#103—22no—
106C2×#103—24no—
107C2×#103—42no—
108C2×#103—42no—
109C2×#103C2×C414yescomplemented
110C2×#102—22no—
111C2×#102—22no—
112C2×#102—24no—
113C2×#102—42no—
114C2×#102—42no—
115C2×#102C2×C414yescomplemented
116#391—42no—
117#391—42no—
118#390—42no—
119#390—42no—
120#386—48no—
121#380—22no—
122#380—22no—
123#380—22no—
124#380—22no—
125C2×#95—24no—
126C2×#95—24no—
127C2×#95—24no—
128C2×#95—21no—
129C2×#95—21no—
130#340—24no—
131#340—24no—
132#340—24no—
133#336—24no—
134#336—24no—
135C25—21no—
136C25C2×D411yescharacteristic
137C22×Q8—21no—
138C22×Q8—21no—
139C22×Q8—21no—
140C22×Q8—21no—
141C22×Q8—21no—
142C22×Q8—22no—
143C22×Q8—22no—
144C22×Q8—22no—
145C22×Q8—22no—
146C22×Q8C22×C412yes—
147C22×D4—24no—
148C22×D4—21no—
149C22×D4—21no—
150C22×D4—21no—
151C22×D4—21no—
152C22×D4—21no—
153C22×D4—22no—
154C22×D4—22no—
155C22×D4—22no—
156C22×D4—22no—
157C22×D4—42no—
158C22×D4—42no—
159C22×D4—44no—
160C22×D4—44no—
161C22×D4C22×C412yes—
162C23×C4—22no—
163C23×C4—22no—
164C23×C4—22no—
165C23×C4—22no—
166C23×C4—24no—
167C23×C4—28no—
168C23×C4—24no—
169C23×C4—24no—
170C23×C4—21no—
171C23×C4—21no—
172C23×C4—21no—
173C23×C4—22no—
174C23×C4—22no—
175C23×C4—44no—
176C23×C4C22×C412yes—
177C23×C4C2×D414yes—
178C23×C4C2×D412yes—
179C23×C4C2×D412yes—
180C23×C4C2×D411yescharacteristic
181C23×C4C2×D412yes—
182C23×C4#4112yes—
183C23×C4#4112yes—
184C23×C4C2411yescharacteristic
185C2×SD16—28no—
186C2×SD16—48no—
187C2×SD16—44no—
188C2×SD16—44no—
189C2×SD16—48no—
190C22×C8—22no—
191C22×C8—22no—
192C22×C8—22no—
193C22×C8—22no—
194C22×C8—28no—
195C22×C8—22no—
196C22×C8—22no—
197C22×C8—22no—
198C22×C8—22no—
199C22×C8—22no—
200C22×C8—22no—
201C22×C8—48no—
202C22×C8—48no—
203C22×C8—44no—
204C22×C8—44no—
205C22×C8—48no—
206C22×C8—44no—
207C22×C8—44no—
208C22×C8—48no—
209C22×C8C22⋊C4116yescomplemented
210C22×C8C22×C412yes—
211C22×C8C22×C412yes—
212C22×C8C2×D414yes—
213C22×C8C2×D412yes—
214C22×C8C2×D412yes—
215C22×C8C2×D412yes—
216C22×C8C2×D412yes—
217C22⋊Q8—28no—
218C22⋊Q8—28no—
219C22⋊Q8—44no—
220C22⋊Q8—44no—
221#121—28no—
222#121—28no—
223#121—44no—
224#121—44no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4—21no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4—22no—
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—22no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—42no—
249C2×C4⋊C4—42no—
250C2×C4⋊C4C22×C412yes—
251C2×C4⋊C4C22×C412yes—
252C2×C22⋊C4—24no—
253C2×C22⋊C4—24no—
254C2×C22⋊C4—42no—
255C2×C22⋊C4—42no—
256C2×C22⋊C4—42no—
257C2×C22⋊C4—42no—
258C2×C22⋊C4—48no—
259C2×C22⋊C4—48no—
260C2×C22⋊C4—42no—
261C2×C22⋊C4—42no—
262C2×C22⋊C4—44no—
263C2×C22⋊C4—44no—
264C2×C22⋊C4—48no—
265C2×C42—24no—
266C2×C42—22no—
267C2×C42—22no—
268C2×C42—22no—
269C2×C42—22no—
270C2×C42—22no—
271C2×C42—22no—
272C2×C42—44no—
273C2×C42—44no—
274C2×C42—44no—
275C2×C42—48no—
276#107—48no—
277#106—28no—
278#103—28no—
279#103—48no—
280#102—28no—
281#102—48no—
282#95—48no—
283#95—48no—
284#95—42no—
285#95—42no—
286#95—48no—
287C24—22no—
288C24—22no—
289C24—24no—
290C24—22no—
291C24—41no—
292C24—41no—
293C24—42no—
294C24—42no—
295C24—44no—
296C24—42no—
297C24—42no—
298C24C4×D412yes—
299C24#12112yes—
300C24#12712yes—
301C24AGL(1,ℤ/8)14yes—
302C24#13714yes—
303C24C22×D411yescharacteristic
304C2×Q8—22no—
305C2×Q8—22no—
306C2×Q8—24no—
307C2×Q8—24no—
308C2×Q8—41no—
309C2×Q8—41no—
310C2×Q8—42no—
311C2×Q8—42no—
312C2×Q8—42no—
313C2×Q8—41no—
314C2×Q8—41no—
315C2×Q8—42no—
316C2×Q8—41no—
317C2×Q8—41no—
318C2×Q8—42no—
319C2×Q8—42no—
320C2×Q8—42no—
321C2×Q8—42no—
322C2×D4—22no—
323C2×D4—22no—
324C2×D4—24no—
325C2×D4—24no—
326C2×D4—48no—
327C2×D4—416no—
328C2×D4—41no—
329C2×D4—41no—
330C2×D4—42no—
331C2×D4—41no—
332C2×D4—41no—
333C2×D4—42no—
334C2×D4—41no—
335C2×D4—41no—
336C2×D4—42no—
337C2×D4—42no—
338C2×D4—42no—
339C2×D4—42no—
340C2×D4—42no—
341C2×D4—42no—
342C2×D4—88no—
343C2×D4—84no—
344C2×D4—84no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—24no—
348C22×C4—24no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—24no—
352C22×C4—24no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—24no—
356C22×C4—24no—
357C22×C4—24no—
358C22×C4—24no—
359C22×C4—24no—
360C22×C4—24no—
361C22×C4—216no—
362C22×C4—24no—
363C22×C4—24no—
364C22×C4—24no—
365C22×C4—24no—
366C22×C4—24no—
367C22×C4—24no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—416no—
372C22×C4—416no—
373C22×C4—48no—
374C22×C4—48no—
375C22×C4—41no—
376C22×C4—41no—
377C22×C4—41no—
378C22×C4—41no—
379C22×C4—41no—
380C22×C4—41no—
381C22×C4—42no—
382C22×C4—42no—
383C22×C4—44no—
384C22×C4—42no—
385C22×C4—42no—
386C22×C4—42no—
387C22×C4—42no—
388C22×C4—42no—
389C22×C4—42no—
390C22×C4—42no—
391C22×C4—42no—
392C22×C4—44no—
393C22×C4—44no—
394C22×C4—44no—
395C22×C4—44no—
396C22×C4C2×C22⋊C414yes—
397C22×C4C4×D414yes—
398C22×C4C4×D412yes—
399C22×C4#12114yes—
400C22×C4#12112yes—
401C22×C4#12414yes—
402C22×C4#12712yes—
403C22×C4#13518yes—
404C22×C4C23×C411yescharacteristic
405C22×C4C22×D411yescharacteristic
406C22×C4C22×D411yescharacteristic
407C22×C4C22×D411yescharacteristic
408C22×C4C2×#4111yescharacteristic
409C22×C4C2×#4111yescharacteristic
410C22×C4H(5,𝔽2)12yes—
411C22×C4H(5,𝔽2)12yes—
412C22×C4H(5,𝔽2)12yes—
413C22×C42−1+412yes—
414SD16—416no—
415C2×C8—24no—
416C2×C8—24no—
417C2×C8—24no—
418C2×C8—22no—
419C2×C8—22no—
420C2×C8—22no—
421C2×C8—22no—
422C2×C8—24no—
423C2×C8—24no—
424C2×C8—24no—
425C2×C8—24no—
426C2×C8—416no—
427C2×C8—416no—
428C2×C8—42no—
429C2×C8—42no—
430C2×C8—42no—
431C2×C8—42no—
432C2×C8C2×C22⋊C414yes—
433C2×C8C2×C22⋊C414yes—
434C4⋊C4—22no—
435C4⋊C4—22no—
436C4⋊C4—22no—
437C4⋊C4—22no—
438C4⋊C4—41no—
439C4⋊C4—41no—
440C4⋊C4—41no—
441C4⋊C4—41no—
442C4⋊C4—48no—
443C4⋊C4—42no—
444C4⋊C4—42no—
445C4⋊C4—42no—
446C4⋊C4—42no—
447C4⋊C4—42no—
448C4⋊C4—42no—
449C4⋊C4—42no—
450C4⋊C4—42no—
451C22⋊C4—48no—
452C22⋊C4—48no—
453C22⋊C4—816no—
454C22⋊C4—84no—
455C22⋊C4—84no—
456C42—48no—
457C42—42no—
458C42—42no—
459C42—42no—
460C42—42no—
461C23—24no—
462C23—22no—
463C23—22no—
464C23—24no—
465C23—24no—
466C23—28no—
467C23—28no—
468C23—24no—
469C23—28no—
470C23—28no—
471C23—24no—
472C23—44no—
473C23—44no—
474C23—44no—
475C23—42no—
476C23—81no—
477C23—81no—
478C23—82no—
479C23—81no—
480C23—81no—
481C23—82no—
482C23—84no—
483C23—84no—
484C23—84no—
485C23—84no—
486C23—82no—
487C23—82no—
488C23#39014yes—
489C23#44018yes—
490C23#46818yes—
491C23#49614yes—
492C23#49714yes—
493C23C2×C4×D411yescharacteristic
494C23C2×#12111yescharacteristic
495C23C2×#12711yescharacteristic
496C23#53412yes—
497C23#53712yes—
498C23C2×#13514yes—
499C23C2×AGL(1,ℤ/8)12yes—
500C23C2×#13712yes—
501Q8—42no—
502Q8—42no—
503Q8—44no—
504Q8—81no—
505Q8—81no—
506Q8—82no—
507D4—42no—
508D4—42no—
509D4—44no—
510D4—816no—
511D4—81no—
512D4—81no—
513D4—82no—
514C2×C4—24no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—24no—
518C2×C4—24no—
519C2×C4—24no—
520C2×C4—28no—
521C2×C4—24no—
522C2×C4—24no—
523C2×C4—28no—
524C2×C4—28no—
525C2×C4—28no—
526C2×C4—28no—
527C2×C4—432no—
528C2×C4—44no—
529C2×C4—44no—
530C2×C4—44no—
531C2×C4—44no—
532C2×C4—44no—
533C2×C4—44no—
534C2×C4—44no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—81no—
539C2×C4—81no—
540C2×C4—82no—
541C2×C4—81no—
542C2×C4—81no—
543C2×C4—82no—
544C2×C4—82no—
545C2×C4—82no—
546C2×C4—82no—
547C2×C4—82no—
548C2×C4—82no—
549C2×C4—82no—
550C2×C4—82no—
551C2×C4—82no—
552C2×C4—84no—
553C2×C4—84no—
554C2×C4—84no—
555C2×C4—84no—
556C2×C4#39014yes—
557C2×C4C22×C22⋊C411yescharacteristic
558C2×C4C2×C4×D411yescharacteristic
559C2×C4#51812yes—
560C2×C4#52012yes—
561C2×C4C2×#12111yescharacteristic
562C2×C4C2×#12411yescharacteristic
563C2×C4#53411yescharacteristic
564C2×C4#53411yescharacteristic
565C2×C4#53412yes—
566C2×C4#53412yes—
567C2×C4#53511yescharacteristic
568C2×C4#53611yescharacteristic
569C2×C4#53712yes—
570C2×C4#53712yes—
571C2×C4#53812yes—
572C2×C4#53812yes—
573C2×C4#53912yes—
574C2×C4#53912yes—
575C8—24no—
576C8—24no—
577C8—44no—
578C22—28no—
579C22—28no—
580C22—28no—
581C22—28no—
582C22—216no—
583C22—24no—
584C22—24no—
585C22—88no—
586C22—82no—
587C22—82no—
588C22—82no—
589C22—162no—
590C22—162no—
591C22—162no—
592C22—162no—
593C22#197318yes—
594C22C2×#39011yescharacteristic
595C22#232512yes—
596C22#261711yescharacteristic
597C22#261811yescharacteristic
598C22#265612yes—
599C22C4×#13512yes—
600C22C2×#44012yes—
601C22#305012yes—
602C22C2×#46812yes—
603C22#314712yes—
604C22C2×#49611yescharacteristic
605C22C2×#49711yescharacteristic
606C22#318814yes—
607C22#319014yes—
608C22#324714yes—
609C22#324814yes—
610C4—24no—
611C4—48no—
612C4—82no—
613C4—82no—
614C4—82no—
615C4—162no—
616C4—162no—
617C4#230811yescharacteristic
618C4#231111yescharacteristic
619C4#232512yes—
620C4#261711yescharacteristic
621C4#261811yescharacteristic
622C4#263912yes—
623C2—28no—
624C2—162no—
625C2#1427111yescharacteristic
626C2#2122411yescharacteristic
627C2C2×#197311yescharacteristic
628C2#2218514yes—
629C2#2973512yes—
630C2#2977512yes—
631C2#3024312yes—
632C2#3024412yes—
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC22×#121C22
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
3. C2×#1892C2semidirect
4. C2×#1808C2semidirect
5. C22×#336C2semidirect
6. #14702C2semidirect
8. #11378C2semidirect
9. #11124C2semidirect
10. #11123C2semidirect
11. #11023C2semidirect
20. C2×#465C4semidirect
55. C22×SD16C2×C4semidirect
103. C2×#106C2×C4semidirect
109. C2×#103C2×C4semidirect
115. C2×#102C2×C4semidirect
209. C22×C8C22⋊C4semidirect

Maximal subgroups

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

subgroupindex
C22×#4652
C2×#18932
C2×#18922
C2×#18082
C22×#3362
#147022
#146982
#113782
#111242
#111232
#110232