Finite Group Databasebeta

← #379993

Subgroups — #379993

Order 512.

2240 conjugacy classes of subgroups.

The subgroup lattice has 648 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#459C211yesmaximal, characteristic
2C2×#2293C211yesmaximal, characteristic, complemented
3C2×#1982C211yesmaximal, characteristic, complemented
4C2×#1802C211yesmaximal, characteristic, complemented
5#14713C211yesmaximal, characteristic, complemented
6#11066C214yesmaximal, complemented
7#11022C214yesmaximal, complemented
8#10849C211yesmaximal, characteristic, complemented
9#10832C211yesmaximal, characteristic, complemented
10C23×D8C2212yes—
11C22×#127C2211yescharacteristic
12C22×C4×D4C2211yescharacteristic
13C22×#117—21no—
14C2×#459—24no—
15C2×#459—24no—
16C2×#459C418yescomplemented
17C22×C4⋊C8C2211yescharacteristic
18C22×C4⋊C8C2211yescharacteristic
19C22×#102C2212yes—
20C22×#102C2214yes—
21C22×#102C2212yes—
22#2293—24no—
23#2293—24no—
24C2×#390—21no—
25C2×#390—22no—
26C2×#390—21no—
27C2×#390C2212yes—
28C2×#390C2214yes—
29C2×#389C2212yes—
30C2×#388—22no—
31C2×#384C2211yescharacteristic
32C2×C4×C4⋊C4C2211yescharacteristic
33C2×C4×C22⋊C4—21no—
34C2×C43C2211yescharacteristic
35#1982—24no—
36#1802—24no—
37C2×#340C2212yes—
38C2×#340C2214yes—
39C2×#336C2214yes—
40C23×D4—21no—
41C23×D4—21no—
42C23×D4C2312yes—
43C23×D4C2311yescharacteristic
44C24×C4—21no—
45C22×D8—24no—
46C22×D8—24no—
47C22×D8—44no—
48C22×D8—44no—
49C22×D8C2×C418yes—
50C23×C8C2312yes—
51C23×C8C2312yes—
52C2×#127—22no—
53C2×#127—22no—
54C2×#127—22no—
55C2×#127—22no—
56C2×#127—22no—
57C2×#127C2×C414yes—
58C2×C4×D4—22no—
59C2×C4×D4—22no—
60C2×C4×D4—22no—
61C2×C4×D4C2×C414yes—
62C2×C4×D4D414yes—
63C2×C4×D4D414yes—
64C2×#117—28no—
65C2×#117—44no—
66C2×#117—44no—
67C22×C4⋊C4—22no—
68C22×C4⋊C4—21no—
69C22×C4⋊C4—21no—
70C22×C4⋊C4C2312yes—
71C22×C4⋊C4C2312yes—
72C22×C4⋊C4C2311yescharacteristic
73C22×C22⋊C4—21no—
74C22×C22⋊C4—22no—
75C22×C22⋊C4—22no—
76C22×C22⋊C4—24no—
77C22×C22⋊C4—24no—
78C22×C42—21no—
79C22×C42D411yescharacteristic
80C22×C42D412yes—
81C22×C42D411yescharacteristic
82C22×C42C2312yes—
83C22×C42C2311yescharacteristic
84#459—232no—
85C2×C4⋊C8—22no—
86C2×C4⋊C8—22no—
87C2×C4⋊C8—22no—
88C2×C4⋊C8—22no—
89C2×C4⋊C8—22no—
90C2×C4⋊C8C2×C414yes—
91C2×#102—24no—
92C2×#102—24no—
93C2×#102—24no—
94C2×#102—28no—
95C2×#102—44no—
96C2×#102—44no—
97C2×#102—44no—
98C2×#102C2×C418yes—
99#390—24no—
100#390—24no—
101#390—44no—
102#390—44no—
103#390—44no—
104#390—48no—
105#390—44no—
106#389—24no—
107#389—24no—
108#388—216no—
109#384—22no—
110#384—22no—
111C4×C4⋊C4—22no—
112C4×C4⋊C4D414yes—
113C4×C22⋊C4—28no—
114C2×#95—22no—
115C2×#95—21no—
116C43—22no—
117C43—22no—
118#340—24no—
119#340—44no—
120#336—44no—
121C25—21no—
122C25—22no—
123C22×D4—24no—
124C22×D4—24no—
125C22×D4—21no—
126C22×D4—22no—
127C22×D4—22no—
128C22×D4—22no—
129C22×D4—21no—
130C22×D4—21no—
131C22×D4—22no—
132C22×D4—22no—
133C22×D4—22no—
134C22×D4—22no—
135C22×D4—22no—
136C22×D4—24no—
137C22×D4—24no—
138C22×D4—24no—
139C22×D4—22no—
140C22×D4—24no—
141C22×D4—44no—
142C22×D4—44no—
143C22×D4—42no—
144C22×D4—42no—
145C22×D4—42no—
146C22×D4—42no—
147C22×D4—42no—
148C22×D4—42no—
149C22×D4—42no—
150C22×D4—42no—
151C22×D4C22×C414yes—
152C22×D4C22×C412yes—
153C22×D4C2×D412yes—
154C22×D4C2×D412yes—
155C23×C4—21no—
156C23×C4—24no—
157C23×C4—24no—
158C23×C4—24no—
159C23×C4—22no—
160C23×C4—22no—
161C23×C4—42no—
162C23×C4—42no—
163C23×C4—42no—
164C23×C4—42no—
165C23×C4C2×D412yes—
166C23×C4C2×D412yes—
167C23×C4C2×D411yescharacteristic
168C23×C4C2×D411yescharacteristic
169C23×C4C2411yescharacteristic
170C2×D8—216no—
171C2×D8—416no—
172C2×D8—48no—
173C2×D8—48no—
174C2×D8—416no—
175C22×C8—22no—
176C22×C8—22no—
177C22×C8—24no—
178C22×C8—24no—
179C22×C8—22no—
180C22×C8—22no—
181C22×C8—22no—
182C22×C8—24no—
183C22×C8—24no—
184C22×C8C22×C414yes—
185#127—28no—
186#127—28no—
187#127—44no—
188#127—44no—
189C4×D4—28no—
190C4×D4—28no—
191C4×D4—28no—
192C4×D4C22⋊C4116yescomplemented
193#117—432no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—28no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—24no—
206C2×C4⋊C4—24no—
207C2×C4⋊C4—44no—
208C2×C4⋊C4—44no—
209C2×C4⋊C4—44no—
210C2×C4⋊C4—42no—
211C2×C4⋊C4—42no—
212C2×C4⋊C4—42no—
213C2×C4⋊C4—42no—
214C2×C4⋊C4C22×C412yes—
215C2×C4⋊C4C2×D414yes—
216C2×C4⋊C4C2×D412yes—
217C2×C4⋊C4C2×D412yes—
218C2×C22⋊C4—24no—
219C2×C22⋊C4—24no—
220C2×C22⋊C4—24no—
221C2×C22⋊C4—28no—
222C2×C22⋊C4—28no—
223C2×C22⋊C4—44no—
224C2×C22⋊C4—44no—
225C2×C22⋊C4—44no—
226C2×C22⋊C4—44no—
227C2×C22⋊C4—48no—
228C2×C22⋊C4—48no—
229C2×C22⋊C4—48no—
230C2×C22⋊C4—48no—
231C2×C22⋊C4—48no—
232C2×C22⋊C4—48no—
233C2×C42—22no—
234C2×C42—22no—
235C2×C42—24no—
236C2×C42—22no—
237C2×C42—24no—
238C2×C42—22no—
239C2×C42—24no—
240C2×C42—22no—
241C2×C42—22no—
242C2×C42—22no—
243C2×C42—22no—
244C2×C42—22no—
245C2×C42—22no—
246C2×C42—22no—
247C2×C42—42no—
248C2×C42—42no—
249C2×C42—42no—
250C2×C42D814yescomplemented
251C2×C42C22×C412yes—
252C2×C42C2×D412yes—
253C2×C42C2×D412yes—
254C2×C42C2×D412yes—
255C2×C42#4112yes—
256C2×C42#4112yes—
257C4⋊C8—28no—
258C4⋊C8—44no—
259#102—216no—
260#102—416no—
261#102—48no—
262#95—28no—
263#95—24no—
264C24—22no—
265C24—22no—
266C24—22no—
267C24—24no—
268C24—42no—
269C24—42no—
270C24—41no—
271C24—41no—
272C24—41no—
273C24—42no—
274C24—41no—
275C24—42no—
276C24—42no—
277C24—42no—
278C24—44no—
279C24—44no—
280C24—42no—
281C24—42no—
282C24—42no—
283C24—44no—
284C24—42no—
285C24—44no—
286C24C22×D411yescharacteristic
287C2×D4—24no—
288C2×D4—22no—
289C2×D4—24no—
290C2×D4—24no—
291C2×D4—28no—
292C2×D4—24no—
293C2×D4—28no—
294C2×D4—22no—
295C2×D4—22no—
296C2×D4—22no—
297C2×D4—22no—
298C2×D4—22no—
299C2×D4—24no—
300C2×D4—22no—
301C2×D4—24no—
302C2×D4—416no—
303C2×D4—42no—
304C2×D4—42no—
305C2×D4—42no—
306C2×D4—44no—
307C2×D4—42no—
308C2×D4—42no—
309C2×D4—44no—
310C2×D4—42no—
311C2×D4—42no—
312C2×D4—42no—
313C2×D4—44no—
314C2×D4—44no—
315C2×D4—42no—
316C2×D4—42no—
317C2×D4—44no—
318C2×D4—44no—
319C2×D4—42no—
320C2×D4—42no—
321C2×D4—44no—
322C2×D4—44no—
323C2×D4—48no—
324C2×D4—48no—
325C2×D4—48no—
326C2×D4—48no—
327C2×D4—88no—
328C2×D4—84no—
329C2×D4—84no—
330C2×D4—84no—
331C2×D4—84no—
332C2×D4—84no—
333C2×D4—84no—
334C2×D4C2×C22⋊C414yes—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—216no—
353C22×C4—24no—
354C22×C4—41no—
355C22×C4—41no—
356C22×C4—41no—
357C22×C4—41no—
358C22×C4—48no—
359C22×C4—48no—
360C22×C4—48no—
361C22×C4—48no—
362C22×C4—48no—
363C22×C4—48no—
364C22×C4—42no—
365C22×C4—42no—
366C22×C4—42no—
367C22×C4—42no—
368C22×C4—42no—
369C22×C4—42no—
370C22×C4—42no—
371C22×C4—44no—
372C22×C4—42no—
373C22×C4—44no—
374C22×C4—42no—
375C22×C4C4×D412yes—
376C22×C4C4×D412yes—
377C22×C4#12012yes—
378C22×C4#12012yes—
379C22×C4#12112yes—
380C22×C4#12112yes—
381C22×C4C2×D814yes—
382C22×C4C2×D812yes—
383C22×C4#13512yes—
384C22×C4AGL(1,ℤ/8)14yes—
385C22×C4AGL(1,ℤ/8)12yes—
386C22×C4AGL(1,ℤ/8)12yes—
387C22×C4C23×C411yescharacteristic
388C22×C4C22×D411yescharacteristic
389C22×C4C22×D411yescharacteristic
390C22×C4C22×D411yescharacteristic
391C22×C4C2×#4111yescharacteristic
392C22×C4C2×#4111yescharacteristic
393C22×C4H(5,𝔽2)12yes—
394C22×C4H(5,𝔽2)12yes—
395C22×C4H(5,𝔽2)12yes—
396C22×C42−1+412yes—
397D8—432no—
398C2×C8—24no—
399C2×C8—28no—
400C2×C8—42no—
401C2×C8—42no—
402C2×C8—44no—
403C2×C8—42no—
404C2×C8—42no—
405C2×C8—44no—
406C2×C8—42no—
407C2×C8—44no—
408C4⋊C4—22no—
409C4⋊C4—22no—
410C4⋊C4—22no—
411C4⋊C4—24no—
412C4⋊C4—24no—
413C4⋊C4—416no—
414C4⋊C4—42no—
415C4⋊C4—42no—
416C4⋊C4—42no—
417C4⋊C4—42no—
418C4⋊C4—44no—
419C4⋊C4—84no—
420C4⋊C4—84no—
421C4⋊C4C2×C22⋊C414yes—
422C22⋊C4—216no—
423C22⋊C4—416no—
424C22⋊C4—416no—
425C22⋊C4—816no—
426C22⋊C4—816no—
427C42—28no—
428C42—48no—
429C42—42no—
430C42—42no—
431C42—42no—
432C42—42no—
433C42—48no—
434C42—44no—
435C42C2×C22⋊C414yes—
436C42C4×D414yes—
437C42#12114yes—
438C42#12314yes—
439C23—24no—
440C23—44no—
441C23—44no—
442C23—42no—
443C23—42no—
444C23—42no—
445C23—42no—
446C23—44no—
447C23—42no—
448C23—42no—
449C23—44no—
450C23—42no—
451C23—42no—
452C23—42no—
453C23—42no—
454C23—48no—
455C23—44no—
456C23—44no—
457C23—48no—
458C23—44no—
459C23—82no—
460C23—82no—
461C23—82no—
462C23—82no—
463C23—82no—
464C23—82no—
465C23—82no—
466C23—82no—
467C23—84no—
468C23—82no—
469C23—82no—
470C23—84no—
471C23—84no—
472C23—84no—
473C23—84no—
474C23—84no—
475C23—84no—
476C23—84no—
477C23—82no—
478C23—84no—
479C23—82no—
480C23—84no—
481C23—82no—
482C23—82no—
483C23—82no—
484C23—82no—
485C23C2×C4×D411yescharacteristic
486C23C2×#12011yescharacteristic
487C23C2×#12111yescharacteristic
488C23#53512yes—
489C23#53712yes—
490C23C22×D811yescharacteristic
491C23C2×#13511yescharacteristic
492C23C2×AGL(1,ℤ/8)11yescharacteristic
493C23C2×AGL(1,ℤ/8)11yescharacteristic
494C23C2×AGL(1,ℤ/8)12yes—
495C23#57512yes—
496D4—24no—
497D4—24no—
498D4—44no—
499D4—44no—
500D4—44no—
501D4—48no—
502D4—42no—
503D4—42no—
504D4—44no—
505D4—816no—
506D4—82no—
507D4—82no—
508D4—84no—
509D4—816no—
510C2×C4—24no—
511C2×C4—24no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—24no—
515C2×C4—24no—
516C2×C4—24no—
517C2×C4—24no—
518C2×C4—24no—
519C2×C4—24no—
520C2×C4—24no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—432no—
536C2×C4—44no—
537C2×C4—44no—
538C2×C4—44no—
539C2×C4—82no—
540C2×C4—82no—
541C2×C4—82no—
542C2×C4—82no—
543C2×C4—82no—
544C2×C4—84no—
545C2×C4—84no—
546C2×C4—82no—
547C2×C4—82no—
548C2×C4#38614yes—
549C2×C4#38614yes—
550C2×C4C4×D814yes—
551C2×C4#44214yes—
552C2×C4#44714yes—
553C2×C4#44914yes—
554C2×C4#45914yes—
555C2×C4#46314yes—
556C2×C4C22×C22⋊C411yescharacteristic
557C2×C4C2×C4×D411yescharacteristic
558C2×C4#51812yes—
559C2×C4#52012yes—
560C2×C4C2×#12111yescharacteristic
561C2×C4C2×#12311yescharacteristic
562C2×C4#53511yescharacteristic
563C2×C4#53511yescharacteristic
564C2×C4#53512yes—
565C2×C4#53511yescharacteristic
566C2×C4#53512yes—
567C2×C4#53611yescharacteristic
568C2×C4#53712yes—
569C2×C4#53712yes—
570C2×C4#54012yes—
571C2×C4#54012yes—
572C2×C4#54212yes—
573C2×C4#54212yes—
574C8—44no—
575C8—82no—
576C22—44no—
577C22—44no—
578C22—44no—
579C22—82no—
580C22—82no—
581C22—82no—
582C22—84no—
583C22—82no—
584C22—84no—
585C22—84no—
586C22—84no—
587C22—88no—
588C22—88no—
589C22—84no—
590C22—84no—
591C22—84no—
592C22—162no—
593C22—162no—
594C22—162no—
595C22—162no—
596C22—164no—
597C22—164no—
598C22C2×#38611yescharacteristic
599C22#232412yes—
600C22#261512yes—
601C22#261911yescharacteristic
602C22#262011yescharacteristic
603C22C2×C4×D811yescharacteristic
604C22C2×#44211yescharacteristic
605C22C4×AGL(1,ℤ/8)12yes—
606C22C2×#44711yescharacteristic
607C22C2×#44911yescharacteristic
608C22#300612yes—
609C22C2×#45911yescharacteristic
610C22C2×#46311yescharacteristic
611C22#303811yescharacteristic
612C22#304011yescharacteristic
613C22#320512yes—
614C22#321212yes—
615C22#321412yes—
616C22#321712yes—
617C22#324512yes—
618C22#324712yes—
619C22#325112yes—
620C22#325312yes—
621C4—24no—
622C4—44no—
623C4—44no—
624C4—44no—
625C4—84no—
626C4—162no—
627C4#187918yes—
628C4#230911yescharacteristic
629C4#231111yescharacteristic
630C4#232412yes—
631C4#261911yescharacteristic
632C4#262011yescharacteristic
633C4#262412yes—
634C2—84no—
635C2—164no—
636C2#1425711yescharacteristic
637C2C2×#187911yescharacteristic
638C2#2092811yescharacteristic
639C2#2214512yes—
640C2#2215112yes—
641C2#2966012yes—
642C2#2966212yes—
643C2#2979411yescharacteristic
644C2#2979511yescharacteristic
645C2#2982411yescharacteristic
646C2#2982511yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC22×#459C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C43 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2293C2semidirect
3. C2×#1982C2semidirect
4. C2×#1802C2semidirect
5. #14713C2semidirect
6. #11066C2semidirect
7. #11022C2semidirect
8. #10849C2semidirect
9. #10832C2semidirect
16. C2×#459C4semidirect
192. C4×D4C22⋊C4semidirect
250. C2×C42D8semidirect

Maximal subgroups

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

subgroupindex
C22×#4592
C2×#22932
C2×#19822
C2×#18022
#147132
#110662
#110222
#108492
#108322