Finite Group Databasebeta

← #379792

Subgroups — #379792

Order 512.

2626 conjugacy classes of subgroups.

The subgroup lattice has 651 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#448C211yesmaximal, characteristic
2C22×#391C211yesmaximal, characteristic, complemented
3C2×#1874C211yesmaximal, characteristic
4C2×#1787C211yesmaximal, characteristic, complemented
5#14698C211yesmaximal, characteristic
6#11190C214yesmaximal, complemented
7#11023C214yesmaximal, complemented
8#10730C211yesmaximal, characteristic, complemented
9#10712C211yesmaximal, characteristic, complemented
10C24×Q8C2211yescharacteristic
11C23×SD16C2212yes—
12C22×#121C2211yescharacteristic
13C23×C4⋊C4C2211yescharacteristic
14C23×C42C2211yescharacteristic
15C2×#448—24no—
16C2×#448—24no—
17C2×#448C418yescomplemented
18C22×#103C2212yes—
19C22×#103C2214yes—
20C22×#102C2212yes—
21C22×C22⋊C8C2211yescharacteristic
22C22×C22⋊C8C2211yescharacteristic
23C2×#391—22no—
24C2×#391—24no—
25C2×#391—24no—
26C2×#391C2214yes—
27C2×#390C2212yes—
28C2×#386—21no—
29C2×#386—21no—
30C2×#380C2211yescharacteristic
31C22×#95—21no—
32C22×#95—21no—
33#1874—24no—
34#1787—24no—
35C2×#340C2212yes—
36C2×#340C2214yes—
37C2×#336C2214yes—
38#1437—22no—
39C4⋊#95C2212yes—
40C23×Q8—22no—
41C23×Q8—21no—
42C23×Q8—22no—
43C23×Q8—22no—
44C23×Q8—44no—
45C23×Q8—44no—
46C23×Q8C2×C414yes—
47C23×Q8D414yes—
48C23×Q8D414yes—
49C23×Q8C2312yes—
50C23×D4—21no—
51C23×D4C2311yescharacteristic
52C24×C4—21no—
53C24×C4—21no—
54C24×C4D411yescharacteristic
55C24×C4D411yescharacteristic
56C24×C4C2311yescharacteristic
57C22×SD16—24no—
58C22×SD16—24no—
59C22×SD16—44no—
60C22×SD16—44no—
61C22×SD16C2×C418yescomplemented
62C23×C8C2312yes—
63C23×C8C2312yes—
64C2×#121—22no—
65C2×#121—22no—
66C2×#121—22no—
67C2×#121—22no—
68C2×#121—22no—
69C2×#121C2×C414yes—
70C22×C4⋊C4—21no—
71C22×C4⋊C4—22no—
72C22×C4⋊C4—44no—
73C22×C4⋊C4D414yes—
74C22×C4⋊C4C2312yes—
75C22×C4⋊C4C2311yescharacteristic
76C22×C4⋊C4C2312yes—
77C22×C22⋊C4—21no—
78C22×C22⋊C4—21no—
79C22×C22⋊C4—22no—
80C22×C22⋊C4—22no—
81C22×C42—22no—
82C22×C42—22no—
83C22×C42—44no—
84C22×C42D412yes—
85C22×C42C2312yes—
86#448—232no—
87C2×#103—24no—
88C2×#103—24no—
89C2×#103—28no—
90C2×#103—44no—
91C2×#103—44no—
92C2×#103C2×C418yescomplemented
93C2×#102—24no—
94C2×#102—44no—
95C2×C22⋊C8—22no—
96C2×C22⋊C8—22no—
97C2×C22⋊C8—22no—
98C2×C22⋊C8—22no—
99C2×C22⋊C8—22no—
100C2×C22⋊C8C2×C414yes—
101#391—216no—
102#391—28no—
103#391—28no—
104#390—44no—
105#386—44no—
106#386—44no—
107#380—22no—
108#380—22no—
109C2×#95—28no—
110C2×#95—28no—
111C2×#95—24no—
112C2×#95—24no—
113C2×#95—22no—
114C2×#95—22no—
115C2×#95—21no—
116#340—24no—
117#340—44no—
118#336—44no—
119C25—21no—
120C25C2×D411yescharacteristic
121C22×Q8—24no—
122C22×Q8—28no—
123C22×Q8—24no—
124C22×Q8—24no—
125C22×Q8—28no—
126C22×Q8—28no—
127C22×Q8—22no—
128C22×Q8—22no—
129C22×Q8—22no—
130C22×Q8—24no—
131C22×Q8—24no—
132C22×Q8—24no—
133C22×Q8—24no—
134C22×Q8—416no—
135C22×Q8—48no—
136C22×Q8—48no—
137C22×Q8—416no—
138C22×Q8—44no—
139C22×Q8—44no—
140C22×Q8—48no—
141C22×Q8—48no—
142C22×Q8—416no—
143C22×Q8—416no—
144C22×Q8C22⋊C4116yescomplemented
145C22×Q8C22×C414yes—
146C22×Q8C2×D414yes—
147C22×Q8C2×D414yes—
148C22×D4—24no—
149C22×D4—21no—
150C22×D4—21no—
151C22×D4—21no—
152C22×D4—21no—
153C22×D4—21no—
154C22×D4—24no—
155C22×D4—24no—
156C22×D4—42no—
157C22×D4—42no—
158C22×D4—42no—
159C22×D4—42no—
160C22×D4—42no—
161C22×D4—42no—
162C22×D4C22×C412yes—
163C23×C4—24no—
164C23×C4—24no—
165C23×C4—24no—
166C23×C4—24no—
167C23×C4—22no—
168C23×C4—22no—
169C23×C4—24no—
170C23×C4—22no—
171C23×C4—22no—
172C23×C4—24no—
173C23×C4—24no—
174C23×C4—22no—
175C23×C4—22no—
176C23×C4—21no—
177C23×C4—22no—
178C23×C4—22no—
179C23×C4—42no—
180C23×C4—42no—
181C23×C4—42no—
182C23×C4—44no—
183C23×C4—44no—
184C23×C4—44no—
185C23×C4—44no—
186C23×C4—42no—
187C23×C4—42no—
188C23×C4SD1614yescomplemented
189C23×C4C22×C412yes—
190C23×C4C2×D412yes—
191C23×C4C2×D412yes—
192C23×C4C2×D412yes—
193C23×C4C2×D412yes—
194C23×C4C2×D411yescharacteristic
195C23×C4C2×D412yes—
196C23×C4#4112yes—
197C23×C4#4112yes—
198C23×C4C2411yescharacteristic
199C2×SD16—216no—
200C2×SD16—416no—
201C2×SD16—416no—
202C2×SD16—48no—
203C2×SD16—48no—
204C22×C8—22no—
205C22×C8—22no—
206C22×C8—24no—
207C22×C8—24no—
208C22×C8—22no—
209C22×C8—22no—
210C22×C8—22no—
211C22×C8—24no—
212C22×C8—24no—
213C22×C8C22×C414yes—
214#121—28no—
215#121—28no—
216#121—44no—
217#121—44no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—48no—
232C2×C4⋊C4—44no—
233C2×C4⋊C4—48no—
234C2×C4⋊C4C22×C412yes—
235C2×C4⋊C4C2×D414yes—
236C2×C22⋊C4—24no—
237C2×C22⋊C4—42no—
238C2×C22⋊C4—42no—
239C2×C22⋊C4—44no—
240C2×C22⋊C4—44no—
241C2×C22⋊C4—44no—
242C2×C22⋊C4—44no—
243C2×C22⋊C4—42no—
244C2×C22⋊C4—42no—
245C2×C22⋊C4—42no—
246C2×C22⋊C4—44no—
247C2×C22⋊C4—44no—
248C2×C42—24no—
249C2×C42—22no—
250C2×C42—22no—
251C2×C42—24no—
252C2×C42—24no—
253C2×C42—42no—
254C2×C42—42no—
255C2×C42—48no—
256C2×C42—48no—
257#103—216no—
258#103—416no—
259#102—48no—
260C22⋊C8—28no—
261C22⋊C8—44no—
262#95—216no—
263#95—216no—
264#95—44no—
265#95—42no—
266#95—44no—
267C24—22no—
268C24—22no—
269C24—22no—
270C24—22no—
271C24—22no—
272C24—41no—
273C24—41no—
274C24—42no—
275C24—42no—
276C24—41no—
277C24—41no—
278C24—41no—
279C24—42no—
280C24—41no—
281C24—42no—
282C24C4×D412yes—
283C24#12012yes—
284C24#12112yes—
285C24C2×SD1612yes—
286C24#13512yes—
287C24AGL(1,ℤ/8)12yes—
288C24#13712yes—
289C24C22×D411yescharacteristic
290C2×Q8—24no—
291C2×Q8—24no—
292C2×Q8—24no—
293C2×Q8—216no—
294C2×Q8—28no—
295C2×Q8—28no—
296C2×Q8—28no—
297C2×Q8—24no—
298C2×Q8—24no—
299C2×Q8—28no—
300C2×Q8—24no—
301C2×Q8—24no—
302C2×Q8—432no—
303C2×Q8—416no—
304C2×Q8—48no—
305C2×Q8—48no—
306C2×Q8—416no—
307C2×Q8—432no—
308C2×Q8—432no—
309C2×Q8—432no—
310C2×Q8—44no—
311C2×Q8—44no—
312C2×Q8—44no—
313C2×Q8—44no—
314C2×Q8—44no—
315C2×Q8—44no—
316C2×Q8C2×C22⋊C418yes—
317C2×D4—22no—
318C2×D4—22no—
319C2×D4—28no—
320C2×D4—48no—
321C2×D4—48no—
322C2×D4—48no—
323C2×D4—41no—
324C2×D4—41no—
325C2×D4—42no—
326C2×D4—44no—
327C2×D4—42no—
328C2×D4—42no—
329C2×D4—42no—
330C2×D4—44no—
331C2×D4—44no—
332C2×D4—84no—
333C2×D4—84no—
334C2×D4—84no—
335C2×D4—84no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—28no—
339C22×C4—24no—
340C22×C4—24no—
341C22×C4—24no—
342C22×C4—28no—
343C22×C4—24no—
344C22×C4—28no—
345C22×C4—28no—
346C22×C4—28no—
347C22×C4—24no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—24no—
351C22×C4—24no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—216no—
363C22×C4—28no—
364C22×C4—22no—
365C22×C4—24no—
366C22×C4—48no—
367C22×C4—48no—
368C22×C4—48no—
369C22×C4—44no—
370C22×C4—44no—
371C22×C4—48no—
372C22×C4—44no—
373C22×C4—416no—
374C22×C4—48no—
375C22×C4—48no—
376C22×C4—416no—
377C22×C4—44no—
378C22×C4—44no—
379C22×C4—44no—
380C22×C4—48no—
381C22×C4—48no—
382C22×C4—41no—
383C22×C4—41no—
384C22×C4—44no—
385C22×C4—41no—
386C22×C4—42no—
387C22×C4—41no—
388C22×C4—41no—
389C22×C4—42no—
390C22×C4—41no—
391C22×C4—44no—
392C22×C4—48no—
393C22×C4—42no—
394C22×C4—42no—
395C22×C4—42no—
396C22×C4—42no—
397C22×C4—42no—
398C22×C4—42no—
399C22×C4—42no—
400C22×C4—88no—
401C22×C4—88no—
402C22×C4C2×C22⋊C414yes—
403C22×C4C4×D414yes—
404C22×C4C4×D412yes—
405C22×C4#12012yes—
406C22×C4#12114yes—
407C22×C4#12112yes—
408C22×C4#12314yes—
409C22×C4C2×SD1614yes—
410C22×C4#13714yes—
411C22×C4C23×C411yescharacteristic
412C22×C4C22×D411yescharacteristic
413C22×C4C22×D411yescharacteristic
414C22×C4C22×D411yescharacteristic
415C22×C4C2×#4111yescharacteristic
416C22×C4C2×#4111yescharacteristic
417C22×C4H(5,𝔽2)14yes—
418C22×C4H(5,𝔽2)14yes—
419SD16—432no—
420C2×C8—24no—
421C2×C8—28no—
422C2×C8—44no—
423C2×C8—42no—
424C2×C8—42no—
425C2×C8—42no—
426C2×C8—42no—
427C2×C8—44no—
428C2×C8—42no—
429C2×C8—44no—
430C4⋊C4—22no—
431C4⋊C4—22no—
432C4⋊C4—24no—
433C4⋊C4—24no—
434C4⋊C4—48no—
435C4⋊C4—41no—
436C4⋊C4—41no—
437C4⋊C4—44no—
438C4⋊C4—44no—
439C4⋊C4—44no—
440C22⋊C4—48no—
441C22⋊C4—88no—
442C22⋊C4—88no—
443C22⋊C4—84no—
444C42—48no—
445C42—44no—
446C42—44no—
447C23—24no—
448C23—22no—
449C23—22no—
450C23—24no—
451C23—24no—
452C23—24no—
453C23—24no—
454C23—24no—
455C23—24no—
456C23—24no—
457C23—24no—
458C23—24no—
459C23—24no—
460C23—22no—
461C23—22no—
462C23—24no—
463C23—44no—
464C23—42no—
465C23—42no—
466C23—44no—
467C23—42no—
468C23—81no—
469C23—81no—
470C23—82no—
471C23—82no—
472C23—82no—
473C23—82no—
474C23—82no—
475C23—82no—
476C23—82no—
477C23—82no—
478C23—82no—
479C23—81no—
480C23—81no—
481C23—81no—
482C23—81no—
483C23—81no—
484C23—81no—
485C23—82no—
486C23#38614yes—
487C23C4×SD1614yes—
488C23#44014yes—
489C23#44814yes—
490C23#44914yes—
491C23#46014yes—
492C23#46414yes—
493C23C2×C4×D411yescharacteristic
494C23C2×#12011yescharacteristic
495C23C2×#12111yescharacteristic
496C23#53412yes—
497C23#53512yes—
498C23C22×SD1611yescharacteristic
499C23C2×#13511yescharacteristic
500C23C2×AGL(1,ℤ/8)11yescharacteristic
501C23C2×#13712yes—
502C23C2×#13711yescharacteristic
503C23#57512yes—
504Q8—28no—
505Q8—28no—
506Q8—432no—
507Q8—48no—
508Q8—48no—
509Q8—44no—
510Q8—44no—
511D4—44no—
512D4—44no—
513D4—816no—
514D4—82no—
515D4—82no—
516C2×C4—24no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—24no—
520C2×C4—28no—
521C2×C4—28no—
522C2×C4—28no—
523C2×C4—28no—
524C2×C4—24no—
525C2×C4—24no—
526C2×C4—24no—
527C2×C4—24no—
528C2×C4—24no—
529C2×C4—24no—
530C2×C4—28no—
531C2×C4—416no—
532C2×C4—416no—
533C2×C4—432no—
534C2×C4—48no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—44no—
538C2×C4—44no—
539C2×C4—48no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—44no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—44no—
550C2×C4—44no—
551C2×C4—42no—
552C2×C4—816no—
553C2×C4—84no—
554C2×C4—84no—
555C2×C4—81no—
556C2×C4—81no—
557C2×C4—81no—
558C2×C4—81no—
559C2×C4—82no—
560C2×C4—82no—
561C2×C4—82no—
562C2×C4—82no—
563C2×C4—82no—
564C2×C4—82no—
565C2×C4—82no—
566C2×C4#38614yes—
567C2×C4C22×C22⋊C411yescharacteristic
568C2×C4C2×C4×D411yescharacteristic
569C2×C4#51814yes—
570C2×C4C2×#12111yescharacteristic
571C2×C4C2×#12311yescharacteristic
572C2×C4#53414yes—
573C2×C4#53412yes—
574C2×C4#53412yes—
575C2×C4#53412yes—
576C2×C4#53512yes—
577C2×C4#53814yes—
578C2×C4#53814yes—
579C8—44no—
580C8—82no—
581C22—28no—
582C22—28no—
583C22—28no—
584C22—28no—
585C22—24no—
586C22—24no—
587C22—24no—
588C22—24no—
589C22—24no—
590C22—24no—
591C22—84no—
592C22—84no—
593C22—82no—
594C22—82no—
595C22—82no—
596C22—162no—
597C22—162no—
598C22—161no—
599C22—161no—
600C22—161no—
601C22—161no—
602C22#188018yes—
603C22C2×#38611yescharacteristic
604C22#232112yes—
605C22#261311yescharacteristic
606C22#261311yescharacteristic
607C22#261712yes—
608C22C2×C4×SD1611yescharacteristic
609C22C2×#44011yescharacteristic
610C22C4×#13712yes—
611C22C2×#44811yescharacteristic
612C22C2×#44911yescharacteristic
613C22#300411yescharacteristic
614C22#300511yescharacteristic
615C22C2×#46011yescharacteristic
616C22C2×#46411yescharacteristic
617C22#303712yes—
618C22#318712yes—
619C22#319012yes—
620C22#319312yes—
621C22#319612yes—
622C22#320612yes—
623C22#321112yes—
624C22#321312yes—
625C22#321812yes—
626C4—24no—
627C4—44no—
628C4—44no—
629C4—48no—
630C4—84no—
631C4—82no—
632C4—162no—
633C4#230812yes—
634C4#232112yes—
635C4#261712yes—
636C4#262212yes—
637C2—28no—
638C2—162no—
639C2#1425511yescharacteristic
640C2C2×#188011yescharacteristic
641C2#2092611yescharacteristic
642C2#2204112yes—
643C2#2204712yes—
644C2#2959311yescharacteristic
645C2#2959511yescharacteristic
646C2#2962311yescharacteristic
647C2#2962411yescharacteristic
648C2#2971312yes—
649C2#2971912yes—
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
2. C22×#391C2semidirect
4. C2×#1787C2semidirect
6. #11190C2semidirect
7. #11023C2semidirect
8. #10730C2semidirect
9. #10712C2semidirect
17. C2×#448C4semidirect
61. C22×SD16C2×C4semidirect
92. C2×#103C2×C4semidirect
144. C22×Q8C22⋊C4semidirect
188. C23×C4SD16semidirect

Maximal subgroups

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

subgroupindex
C22×#4482
C22×#3912
C2×#18742
C2×#17872
#146982
#111902
#110232
#107302
#107122