Finite Group Databasebeta

← #366550

Subgroups — #366550

Order 512.

2486 conjugacy classes of subgroups.

The subgroup lattice has 659 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#451C211yesmaximal, characteristic
2C23×#103C211yesmaximal, characteristic
3C2×#1971C212yesmaximal, complemented
4C2×#1892C211yesmaximal, characteristic
5C2×#1853C211yesmaximal, characteristic
6C2×#1752C211yesmaximal, characteristic
7#14139C211yesmaximal, characteristic
8#13922C211yesmaximal, characteristic
9#11122C212yesmaximal, complemented
10#11024C212yesmaximal, complemented
11#10928C212yesmaximal, complemented
12C24×Q8C2211yescharacteristic
13C23×Q16C2212yes—
14C24×C8C2211yescharacteristic
15C22×C22⋊Q8C2211yescharacteristic
16C23×C4⋊C4C2211yescharacteristic
17C2×#451—24no—
18C2×#451—24no—
19C2×#451C418yescomplemented
20C22×#106C2212yes—
21C22×#103—28no—
22C22×#103—24no—
23C22×#103C2212yes—
24C22×#103C2211yescharacteristic
25C22×#103C2212yes—
26C22×#103C2212yes—
27C22×#103C2211yescharacteristic
28C22×C22⋊C8C2211yescharacteristic
29C22×#97C2212yes—
30C2×#391C2212yes—
31C2×#391C2212yes—
32C2×#391C2212yes—
33C2×#380C2211yescharacteristic
34C2×C4×C22⋊C4C2211yescharacteristic
35#1971—28no—
36#1892—24no—
37#1853—24no—
38#1752—24no—
39C2×#340C2212yes—
40C2×#340C2211yescharacteristic
41C2×#340C2211yescharacteristic
42C2×#336C2212yes—
43#1446—22no—
44#1441—22no—
45C4⋊#95C2212yes—
46C22⋊#95—22no—
47C22⋊#95—22no—
48C23×Q8—21no—
49C23×Q8—22no—
50C23×Q8—24no—
51C23×Q8—24no—
52C23×Q8—22no—
53C23×Q8—22no—
54C23×Q8—22no—
55C23×Q8—22no—
56C23×Q8—42no—
57C23×Q8—42no—
58C23×Q8C2×C414yes—
59C23×Q8C2311yescharacteristic
60C23×Q8C2312yes—
61C24×C4—21no—
62C24×C4—21no—
63C24×C4C2311yescharacteristic
64C22×Q16—24no—
65C22×Q16—24no—
66C22×Q16—24no—
67C22×Q16—24no—
68C22×Q16—42no—
69C22×Q16—42no—
70C22×Q16C2×C418yescomplemented
71C23×C8—22no—
72C23×C8—22no—
73C23×C8—24no—
74C23×C8—42no—
75C23×C8D414yes—
76C23×C8C2311yescharacteristic
77C23×C8C2312yes—
78C23×C8C2311yescharacteristic
79C2×C22⋊Q8—22no—
80C2×C22⋊Q8—22no—
81C2×C22⋊Q8—22no—
82C2×C22⋊Q8—22no—
83C2×C22⋊Q8—22no—
84C2×C22⋊Q8C2×C414yes—
85C22×C4⋊C4—22no—
86C22×C4⋊C4—24no—
87C22×C4⋊C4—21no—
88C22×C4⋊C4—22no—
89C22×C4⋊C4—21no—
90C22×C4⋊C4—42no—
91C22×C4⋊C4C2311yescharacteristic
92C22×C4⋊C4C2311yescharacteristic
93C22×C4⋊C4C2311yescharacteristic
94C22×C4⋊C4C2312yes—
95C22×C22⋊C4—21no—
96C22×C22⋊C4—21no—
97C22×C22⋊C4D412yes—
98C22×C42C2312yes—
99#451—232no—
100C2×#106—24no—
101C2×#106—24no—
102C2×#106—42no—
103C2×#103—232no—
104C2×#103—24no—
105C2×#103—24no—
106C2×#103—24no—
107C2×#103—24no—
108C2×#103—24no—
109C2×#103—24no—
110C2×#103—24no—
111C2×#103—22no—
112C2×#103—22no—
113C2×#103—24no—
114C2×#103—42no—
115C2×#103C2×C418yescomplemented
116C2×#103D418yes—
117C2×C22⋊C8C2×C414yes—
118C2×C22⋊C8D414yes—
119C2×C22⋊C8D414yes—
120C2×#97—24no—
121C2×#97—24no—
122C2×#97—42no—
123#391—24no—
124#391—24no—
125#391—24no—
126#391—42no—
127#391—42no—
128#391—42no—
129#380—22no—
130#380—22no—
131C4×C22⋊C4—22no—
132C4×C22⋊C4—22no—
133C2×#95—24no—
134C2×#95—24no—
135C2×#95—24no—
136C2×#95—21no—
137C2×#95—24no—
138C2×#95D412yes—
139#340—24no—
140#340—22no—
141#340—22no—
142#336—24no—
143C25C2×D411yescharacteristic
144C22×Q8—24no—
145C22×Q8—22no—
146C22×Q8—21no—
147C22×Q8—22no—
148C22×Q8—22no—
149C22×Q8—28no—
150C22×Q8—216no—
151C22×Q8—28no—
152C22×Q8—216no—
153C22×Q8—28no—
154C22×Q8—28no—
155C22×Q8—22no—
156C22×Q8—22no—
157C22×Q8—22no—
158C22×Q8—22no—
159C22×Q8—21no—
160C22×Q8—21no—
161C22×Q8—21no—
162C22×Q8—22no—
163C22×Q8—22no—
164C22×Q8—22no—
165C22×Q8—21no—
166C22×Q8—22no—
167C22×Q8—22no—
168C22×Q8—22no—
169C22×Q8—22no—
170C22×Q8—22no—
171C22×Q8—44no—
172C22×Q8—44no—
173C22×Q8—48no—
174C22×Q8—48no—
175C22×Q8—48no—
176C22×Q8—48no—
177C22×Q8—48no—
178C22×Q8—48no—
179C22×Q8—48no—
180C22×Q8—44no—
181C22×Q8—44no—
182C22×Q8—48no—
183C22×Q8—42no—
184C22×Q8—44no—
185C22×Q8—44no—
186C22×Q8—42no—
187C22×Q8C22×C412yes—
188C22×Q8C22×C414yes—
189C22×Q8C2×D414yes—
190C22×Q8#4114yes—
191C23×C4—22no—
192C23×C4—22no—
193C23×C4—22no—
194C23×C4—22no—
195C23×C4—22no—
196C23×C4—24no—
197C23×C4—21no—
198C23×C4—21no—
199C23×C4—22no—
200C23×C4—21no—
201C23×C4—21no—
202C23×C4—44no—
203C23×C4—44no—
204C23×C4—44no—
205C23×C4—42no—
206C23×C4—42no—
207C23×C4—42no—
208C23×C4—44no—
209C23×C4—82no—
210C23×C4C22×C412yes—
211C23×C4C2×D414yes—
212C23×C4C2×D412yes—
213C23×C4C2×D412yes—
214C23×C4C2×D411yescharacteristic
215C23×C4C2×D412yes—
216C23×C4#4112yes—
217C23×C4#4112yes—
218C23×C4C2411yescharacteristic
219C2×Q16—216no—
220C2×Q16—216no—
221C2×Q16—48no—
222C2×Q16—48no—
223C2×Q16—48no—
224C2×Q16—48no—
225C2×Q16—48no—
226C22×C8—21no—
227C22×C8—22no—
228C22×C8—21no—
229C22×C8—22no—
230C22×C8—22no—
231C22×C8—22no—
232C22×C8—22no—
233C22×C8—28no—
234C22×C8—21no—
235C22×C8—22no—
236C22×C8—21no—
237C22×C8—22no—
238C22×C8—28no—
239C22×C8—44no—
240C22×C8—48no—
241C22×C8—48no—
242C22×C8C22×C414yes—
243C22×C8C2×D412yes—
244C22×C8C2×D414yes—
245C22×C8C2×D414yes—
246C22×C8C2×D414yes—
247C22×C8C2×D412yes—
248C22⋊Q8—28no—
249C22⋊Q8—28no—
250C22⋊Q8—44no—
251C22⋊Q8—44no—
252C2×C4⋊C4—28no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4—21no—
258C2×C4⋊C4—21no—
259C2×C4⋊C4—21no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—24no—
262C2×C4⋊C4—21no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—21no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—21no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—48no—
273C2×C4⋊C4—44no—
274C2×C4⋊C4—44no—
275C2×C4⋊C4—42no—
276C2×C4⋊C4—42no—
277C2×C4⋊C4—42no—
278C2×C4⋊C4C22×C412yes—
279C2×C4⋊C4C2×D414yes—
280C2×C22⋊C4—24no—
281C2×C22⋊C4—24no—
282C2×C22⋊C4—24no—
283C2×C22⋊C4—24no—
284C2×C22⋊C4—42no—
285C2×C22⋊C4—42no—
286C2×C22⋊C4—42no—
287C2×C22⋊C4—44no—
288C2×C42—22no—
289C2×C42—22no—
290C2×C42—22no—
291C2×C42—22no—
292C2×C42—22no—
293C2×C42—22no—
294#106—48no—
295#103—216no—
296#103—216no—
297#103—216no—
298#103—44no—
299#103—44no—
300C22⋊C8C22⋊C4116yes—
301#97—48no—
302#95—24no—
303#95—48no—
304#95—48no—
305#95—48no—
306#95—42no—
307#95—84no—
308C24—22no—
309C24—22no—
310C24—22no—
311C24—22no—
312C24C4×D412yes—
313C24#12112yes—
314C24#12712yes—
315C24#13714yes—
316C24#13714yes—
317C24C22×D411yescharacteristic
318C2×Q8—24no—
319C2×Q8—24no—
320C2×Q8—24no—
321C2×Q8—232no—
322C2×Q8—24no—
323C2×Q8—22no—
324C2×Q8—24no—
325C2×Q8—24no—
326C2×Q8—22no—
327C2×Q8—24no—
328C2×Q8—24no—
329C2×Q8—24no—
330C2×Q8—24no—
331C2×Q8—24no—
332C2×Q8—24no—
333C2×Q8—24no—
334C2×Q8—24no—
335C2×Q8—24no—
336C2×Q8—24no—
337C2×Q8—24no—
338C2×Q8—416no—
339C2×Q8—432no—
340C2×Q8—416no—
341C2×Q8—48no—
342C2×Q8—48no—
343C2×Q8—416no—
344C2×Q8—416no—
345C2×Q8—416no—
346C2×Q8—432no—
347C2×Q8—42no—
348C2×Q8—41no—
349C2×Q8—42no—
350C2×Q8—41no—
351C2×Q8—42no—
352C2×Q8—42no—
353C2×Q8—42no—
354C2×Q8—42no—
355C2×Q8—42no—
356C2×Q8—42no—
357C2×Q8—42no—
358C2×Q8—42no—
359C2×Q8—42no—
360C2×Q8—42no—
361C2×Q8—42no—
362C2×Q8—42no—
363C2×Q8—48no—
364C2×Q8—41no—
365C2×Q8—42no—
366C2×Q8—41no—
367C2×Q8—41no—
368C2×Q8—41no—
369C2×Q8—42no—
370C2×Q8—42no—
371C2×Q8—42no—
372C2×Q8—42no—
373C2×Q8—42no—
374C2×Q8C4×D418yes—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—24no—
378C22×C4—24no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—24no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—24no—
385C22×C4—24no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—24no—
392C22×C4—24no—
393C22×C4—24no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—416no—
405C22×C4—416no—
406C22×C4—48no—
407C22×C4—48no—
408C22×C4—42no—
409C22×C4—42no—
410C22×C4—42no—
411C22×C4—42no—
412C22×C4—48no—
413C22×C4—48no—
414C22×C4—41no—
415C22×C4—41no—
416C22×C4—41no—
417C22×C4—41no—
418C22×C4—42no—
419C22×C4—42no—
420C22×C4—42no—
421C22×C4—42no—
422C22×C4—44no—
423C22×C4—42no—
424C22×C4—44no—
425C22×C4—42no—
426C22×C4—41no—
427C22×C4—41no—
428C22×C4—42no—
429C22×C4—42no—
430C22×C4—42no—
431C22×C4—42no—
432C22×C4—42no—
433C22×C4—42no—
434C22×C4—816no—
435C22×C4—84no—
436C22×C4—84no—
437C22×C4C2×C22⋊C414yes—
438C22×C4C4×D414yes—
439C22×C4C4×D412yes—
440C22×C4#12114yes—
441C22×C4#12112yes—
442C22×C4#12414yes—
443C22×C4#12712yes—
444C22×C4C23×C411yescharacteristic
445C22×C4C22×D411yescharacteristic
446C22×C4C22×D411yescharacteristic
447C22×C4C22×D411yescharacteristic
448C22×C4C22×D412yes—
449C22×C4C2×#4111yescharacteristic
450C22×C4C2×#4112yes—
451C22×C4C2×#4111yescharacteristic
452C22×C4H(5,𝔽2)12yes—
453C22×C4H(5,𝔽2)12yes—
454Q16—432no—
455C2×C8—24no—
456C2×C8—24no—
457C2×C8—22no—
458C2×C8—22no—
459C2×C8—24no—
460C2×C8—24no—
461C2×C8—24no—
462C2×C8—24no—
463C2×C8—24no—
464C2×C8—24no—
465C2×C8—22no—
466C2×C8—22no—
467C2×C8—22no—
468C2×C8—416no—
469C2×C8—42no—
470C2×C8—42no—
471C2×C8C2×C22⋊C418yes—
472C2×C8#12014yes—
473C4⋊C4—22no—
474C4⋊C4—22no—
475C4⋊C4—24no—
476C4⋊C4—24no—
477C4⋊C4—48no—
478C4⋊C4—41no—
479C4⋊C4—41no—
480C4⋊C4—42no—
481C4⋊C4—42no—
482C4⋊C4—48no—
483C4⋊C4—41no—
484C4⋊C4—41no—
485C4⋊C4—41no—
486C4⋊C4—41no—
487C4⋊C4—42no—
488C4⋊C4—42no—
489C22⋊C4—48no—
490C22⋊C4—48no—
491C22⋊C4—84no—
492C42—42no—
493C42—42no—
494C42—42no—
495C42—42no—
496C23—24no—
497C23—22no—
498C23—22no—
499C23—24no—
500C23—22no—
501C23—24no—
502C23—28no—
503C23—28no—
504C23—22no—
505C23—28no—
506C23—28no—
507C23—24no—
508C23#39014yes—
509C23#44118yes—
510C23#47018yes—
511C23#49714yes—
512C23#49714yes—
513C23C2×C4×D411yescharacteristic
514C23C2×#12111yescharacteristic
515C23C2×#12711yescharacteristic
516C23D4212yes—
517C23#54612yes—
518C23C2×#13712yes—
519C23C2×#13712yes—
520C23#57712yes—
521C23#57812yes—
522Q8—28no—
523Q8—28no—
524Q8—432no—
525Q8—44no—
526Q8—44no—
527Q8—44no—
528Q8—44no—
529Q8—44no—
530Q8—44no—
531Q8—42no—
532Q8—44no—
533Q8—42no—
534Q8—81no—
535Q8—81no—
536Q8—82no—
537C2×C4—24no—
538C2×C4—24no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—24no—
542C2×C4—28no—
543C2×C4—24no—
544C2×C4—24no—
545C2×C4—28no—
546C2×C4—24no—
547C2×C4—28no—
548C2×C4—432no—
549C2×C4—44no—
550C2×C4—44no—
551C2×C4—44no—
552C2×C4—44no—
553C2×C4—44no—
554C2×C4—44no—
555C2×C4—44no—
556C2×C4—44no—
557C2×C4—44no—
558C2×C4—44no—
559C2×C4—44no—
560C2×C4—48no—
561C2×C4—44no—
562C2×C4—44no—
563C2×C4—44no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—44no—
567C2×C4—44no—
568C2×C4—816no—
569C2×C4—82no—
570C2×C4—81no—
571C2×C4—82no—
572C2×C4—81no—
573C2×C4—82no—
574C2×C4—81no—
575C2×C4—84no—
576C2×C4—81no—
577C2×C4—82no—
578C2×C4—81no—
579C2×C4—81no—
580C2×C4—81no—
581C2×C4—81no—
582C2×C4—81no—
583C2×C4—84no—
584C2×C4—81no—
585C2×C4—84no—
586C2×C4#39014yes—
587C2×C4C22×C22⋊C411yescharacteristic
588C2×C4C2×C4×D411yescharacteristic
589C2×C4C2×C4×D412yes—
590C2×C4#51812yes—
591C2×C4C2×#12011yescharacteristic
592C2×C4C2×#12111yescharacteristic
593C2×C4#52511yescharacteristic
594C2×C4#52512yes—
595C2×C4C2×#12411yescharacteristic
596C2×C4#53411yescharacteristic
597C2×C4#53511yescharacteristic
598C2×C4#53812yes—
599C2×C4D4212yes—
600C2×C4D4212yes—
601C2×C4#54612yes—
602C2×C4#54612yes—
603C2×C4#54612yes—
604C2×C4#55112yes—
605C8—28no—
606C8—24no—
607C8—42no—
608C22—216no—
609C22—216no—
610C22—24no—
611C22—24no—
612C22—24no—
613C22—24no—
614C22—24no—
615C22—24no—
616C22#197118yescomplemented
617C22C2×#39011yescharacteristic
618C22#235812yes—
619C22#242412yes—
620C22#243311yescharacteristic
621C22#253911yescharacteristic
622C22C2×#44112yes—
623C22#294712yes—
624C22C2×#47012yes—
625C22#306312yes—
626C22C2×#49711yescharacteristic
627C22C2×#49711yescharacteristic
628C22#315411yescharacteristic
629C22#315611yescharacteristic
630C22#326614yes—
631C22#327014yes—
632C22#331214yes—
633C22#331614yes—
634C4—24no—
635C4—48no—
636C4—84no—
637C4—82no—
638C4—84no—
639C4—82no—
640C4—161no—
641C4—161no—
642C4C4×#12011yescharacteristic
643C4#232111yescharacteristic
644C4D4×C22⋊C412yes—
645C4#241011yescharacteristic
646C4#241111yescharacteristic
647C4#247212yes—
648C2—28no—
649C2#1399511yescharacteristic
650C2C2×#197111yescharacteristic
651C2#2123411yescharacteristic
652C2#2285914yes—
653C2#2512614yes—
654C2#2578411yescharacteristic
655C2#2578911yescharacteristic
656C2#2834511yescharacteristic
657C2#2834711yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C2×#1971C2semidirect
9. #11122C2semidirect
10. #11024C2semidirect
11. #10928C2semidirect
19. C2×#451C4semidirect
70. C22×Q16C2×C4semidirect
115. C2×#103C2×C4semidirect
616. C22#1971semidirect

Maximal subgroups

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

subgroupindex
C22×#4512
C23×#1032
C2×#19712
C2×#18922
C2×#18532
C2×#17522
#141392
#139222
#111222
#110242
#109282