Finite Group Databasebeta

← #374368

Subgroups — #374368

Order 512.

2536 conjugacy classes of subgroups.

The subgroup lattice has 650 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#459C211yesmaximal, characteristic
2C2×D4×C4⋊C4C211yesmaximal, characteristic, complemented
3C2×#1957C211yesmaximal, characteristic, complemented
4C2×#1879C214yesmaximal, complemented
5C2×C4×#102C211yesmaximal, characteristic, complemented
6#14154C211yesmaximal, characteristic, complemented
7#11066C214yesmaximal, complemented
8#10849C211yesmaximal, characteristic, complemented
9#10834C211yesmaximal, characteristic, complemented
10C23×D8C2212yes—
11C22×#127C2211yescharacteristic
12C22×C4×D4C2212yes—
13C22×C4×D4C2211yescharacteristic
14C23×C4⋊C4—21no—
15C2×#459—24no—
16C2×#459—24no—
17C2×#459C418yescomplemented
18C22×C4⋊C8C2211yescharacteristic
19C22×#102C2214yes—
20C22×#102C2212yes—
21C22×#102C2212yes—
22C22×C4×C8C2211yescharacteristic
23D4×C4⋊C4—24no—
24D4×C4⋊C4—24no—
25C2×#390—21no—
26C2×#390—21no—
27C2×#390C2214yes—
28C2×#389C2212yes—
29C2×#385—22no—
30C2×#384C2211yescharacteristic
31C2×#380—21no—
32C2×C4×C4⋊C4C2211yescharacteristic
33C2×C4×C4⋊C4C2211yescharacteristic
34#1957—24no—
35#1879—216no—
36C4×#102—24no—
37C2×#340C2212yes—
38C2×#340C2214yes—
39C2×#336C2214yes—
40#1438—22no—
41C23×D4—21no—
42C23×D4—21no—
43C23×D4C2312yes—
44C23×D4C2311yescharacteristic
45C24×C4—21no—
46C24×C4—22no—
47C22×D8—28no—
48C22×D8—24no—
49C22×D8—24no—
50C22×D8—28no—
51C22×D8C2×C418yes—
52C23×C8C2312yes—
53C23×C8C2312yes—
54C2×#127—22no—
55C2×#127—22no—
56C2×#127—22no—
57C2×#127—22no—
58C2×#127—22no—
59C2×#127C2×C414yescomplemented
60C2×C4×D4—24no—
61C2×C4×D4—24no—
62C2×C4×D4—28no—
63C2×C4×D4—22no—
64C2×C4×D4—22no—
65C2×C4×D4—22no—
66C2×C4×D4—44no—
67C2×C4×D4C2×C414yes—
68C2×C4×D4D414yes—
69C2×C4×D4D414yes—
70C22×C4⋊C4—28no—
71C22×C4⋊C4—21no—
72C22×C4⋊C4—22no—
73C22×C4⋊C4—21no—
74C22×C4⋊C4—44no—
75C22×C4⋊C4—44no—
76C22×C4⋊C4D411yescharacteristic
77C22×C4⋊C4D411yescharacteristic
78C22×C4⋊C4C2312yes—
79C22×C4⋊C4C2312yes—
80C22×C4⋊C4C2311yescharacteristic
81C22×C22⋊C4—21no—
82C22×C22⋊C4—22no—
83C22×C22⋊C4—24no—
84C22×C22⋊C4—24no—
85C22×C42—21no—
86C22×C42C2312yes—
87C22×C42C2311yescharacteristic
88#459—232no—
89C2×C4⋊C8—22no—
90C2×C4⋊C8—22no—
91C2×C4⋊C8C2×C414yescomplemented
92C2×#102—24no—
93C2×#102—24no—
94C2×#102—24no—
95C2×#102—24no—
96C2×#102—28no—
97C2×#102—28no—
98C2×#102—44no—
99C2×#102C2×C418yescomplemented
100C2×#102D418yescomplemented
101C2×C4×C8—22no—
102C2×C4×C8—22no—
103C2×C4×C8—22no—
104#390—28no—
105#390—44no—
106#390—44no—
107#390—44no—
108#389—24no—
109#389—24no—
110#385—216no—
111#384—22no—
112#384D414yes—
113#380—28no—
114C4×C4⋊C4—22no—
115C4×C4⋊C4—22no—
116C4×C4⋊C4—22no—
117C4×C4⋊C4—22no—
118C2×#95—22no—
119C2×#95—21no—
120C2×#95D412yes—
121#340—28no—
122#340—24no—
123#336—28no—
124C25—21no—
125C25—22no—
126C22×D4—24no—
127C22×D4—24no—
128C22×D4—21no—
129C22×D4—22no—
130C22×D4—22no—
131C22×D4—22no—
132C22×D4—24no—
133C22×D4—21no—
134C22×D4—21no—
135C22×D4—22no—
136C22×D4—24no—
137C22×D4—22no—
138C22×D4—22no—
139C22×D4—24no—
140C22×D4—22no—
141C22×D4—24no—
142C22×D4—42no—
143C22×D4—42no—
144C22×D4—42no—
145C22×D4—42no—
146C22×D4—42no—
147C22×D4—42no—
148C22×D4—42no—
149C22×D4—42no—
150C22×D4—82no—
151C22×D4—82no—
152C22×D4C22×C414yes—
153C22×D4C22×C412yes—
154C22×D4C2×D412yes—
155C22×D4C2×D412yes—
156C22×D4C2×D414yes—
157C22×D4#4114yes—
158C23×C4—21no—
159C23×C4—24no—
160C23×C4—24no—
161C23×C4—24no—
162C23×C4—22no—
163C23×C4—22no—
164C23×C4—28no—
165C23×C4—48no—
166C23×C4—44no—
167C23×C4—42no—
168C23×C4—42no—
169C23×C4—42no—
170C23×C4—42no—
171C23×C4—44no—
172C23×C4—84no—
173C23×C4C2×D412yes—
174C23×C4C2×D412yes—
175C23×C4C2×D411yescharacteristic
176C23×C4C2×D411yescharacteristic
177C23×C4C2411yescharacteristic
178C2×D8—216no—
179C2×D8—232no—
180C2×D8—416no—
181C2×D8—48no—
182C2×D8—48no—
183C22×C8—24no—
184C22×C8—22no—
185C22×C8—22no—
186C22×C8—22no—
187C22×C8—22no—
188C22×C8—24no—
189C22×C8—22no—
190C22×C8—22no—
191C22×C8—22no—
192C22×C8—22no—
193C22×C8C22×C414yes—
194C22×C8C2×D414yes—
195#127—28no—
196#127—28no—
197#127—44no—
198#127—44no—
199C4×D4—28no—
200C4×D4—28no—
201C4×D4—28no—
202C4×D4—416no—
203C4×D4—48no—
204C4×D4—48no—
205C4×D4C22⋊C4116yescomplemented
206C2×C4⋊C4—22no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—24no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—24no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—28no—
218C2×C4⋊C4—24no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—432no—
226C2×C4⋊C4—44no—
227C2×C4⋊C4—44no—
228C2×C4⋊C4—42no—
229C2×C4⋊C4—42no—
230C2×C4⋊C4—82no—
231C2×C4⋊C4D814yescomplemented
232C2×C4⋊C4C22×C412yes—
233C2×C4⋊C4C2×D414yes—
234C2×C4⋊C4C2×D414yes—
235C2×C4⋊C4C2×D412yes—
236C2×C4⋊C4C2×D412yes—
237C2×C22⋊C4—24no—
238C2×C22⋊C4—24no—
239C2×C22⋊C4—24no—
240C2×C22⋊C4—28no—
241C2×C22⋊C4—48no—
242C2×C22⋊C4—48no—
243C2×C22⋊C4—48no—
244C2×C22⋊C4—48no—
245C2×C22⋊C4—48no—
246C2×C22⋊C4—84no—
247C2×C22⋊C4—84no—
248C2×C42—24no—
249C2×C42—22no—
250C2×C42—22no—
251C2×C42—24no—
252C2×C42—42no—
253C2×C42—42no—
254C2×C42—42no—
255C2×C42C22×C412yes—
256C2×C42C2×D412yes—
257C2×C42C2×D412yes—
258C2×C42C2×D412yes—
259C2×C42#4112yes—
260C2×C42#4112yes—
261C4⋊C8—28no—
262#102—216no—
263#102—216no—
264#102—416no—
265C4×C8—44no—
266#95—28no—
267#95—24no—
268#95—42no—
269C24—22no—
270C24—22no—
271C24—22no—
272C24—24no—
273C24—24no—
274C24—24no—
275C24—41no—
276C24—41no—
277C24—41no—
278C24—41no—
279C24—42no—
280C24—42no—
281C24—42no—
282C24—42no—
283C24—44no—
284C24—44no—
285C24—42no—
286C24—42no—
287C24—42no—
288C24—42no—
289C24—44no—
290C24—44no—
291C24C22×D411yescharacteristic
292C2×D4—24no—
293C2×D4—22no—
294C2×D4—24no—
295C2×D4—28no—
296C2×D4—24no—
297C2×D4—28no—
298C2×D4—22no—
299C2×D4—22no—
300C2×D4—22no—
301C2×D4—22no—
302C2×D4—24no—
303C2×D4—24no—
304C2×D4—24no—
305C2×D4—24no—
306C2×D4—24no—
307C2×D4—24no—
308C2×D4—24no—
309C2×D4—44no—
310C2×D4—42no—
311C2×D4—42no—
312C2×D4—42no—
313C2×D4—42no—
314C2×D4—42no—
315C2×D4—42no—
316C2×D4—44no—
317C2×D4—44no—
318C2×D4—44no—
319C2×D4—44no—
320C2×D4—44no—
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—88no—
331C2×D4—84no—
332C2×D4—84no—
333C2×D4—84no—
334C2×D4—84no—
335C2×D4C2×C22⋊C414yes—
336C2×D4C4×D418yes—
337C2×D4#12118yes—
338C2×D4#12414yes—
339C2×D4#12714yes—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—24no—
347C22×C4—22no—
348C22×C4—24no—
349C22×C4—216no—
350C22×C4—22no—
351C22×C4—24no—
352C22×C4—24no—
353C22×C4—24no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—24no—
357C22×C4—416no—
358C22×C4—41no—
359C22×C4—41no—
360C22×C4—41no—
361C22×C4—41no—
362C22×C4—48no—
363C22×C4—48no—
364C22×C4—48no—
365C22×C4—48no—
366C22×C4—48no—
367C22×C4—48no—
368C22×C4—416no—
369C22×C4—42no—
370C22×C4—44no—
371C22×C4—42no—
372C22×C4—42no—
373C22×C4—42no—
374C22×C4—42no—
375C22×C4—42no—
376C22×C4—42no—
377C22×C4—44no—
378C22×C4—42no—
379C22×C4—816no—
380C22×C4—88no—
381C22×C4—816no—
382C22×C4C4×D412yes—
383C22×C4C4×D412yes—
384C22×C4#12012yes—
385C22×C4#12012yes—
386C22×C4#12112yes—
387C22×C4#12112yes—
388C22×C4C2×D814yes—
389C22×C4C2×D812yes—
390C22×C4#13512yes—
391C22×C4AGL(1,ℤ/8)12yes—
392C22×C4AGL(1,ℤ/8)12yes—
393C22×C4C23×C411yescharacteristic
394C22×C4C22×D411yescharacteristic
395C22×C4C22×D411yescharacteristic
396C22×C4C22×D412yes—
397C22×C4C22×D411yescharacteristic
398C22×C4C2×#4112yes—
399C22×C4C2×#4111yescharacteristic
400C22×C4C2×#4111yescharacteristic
401C22×C4H(5,𝔽2)12yes—
402C22×C42−1+412yes—
403D8—432no—
404C2×C8—24no—
405C2×C8—28no—
406C2×C8—24no—
407C2×C8—24no—
408C2×C8—24no—
409C2×C8—42no—
410C2×C8—44no—
411C2×C8—42no—
412C2×C8—42no—
413C2×C8—42no—
414C2×C8—42no—
415C2×C8—42no—
416C4⋊C4—22no—
417C4⋊C4—22no—
418C4⋊C4—22no—
419C4⋊C4—28no—
420C4⋊C4—24no—
421C4⋊C4—28no—
422C4⋊C4—24no—
423C4⋊C4—416no—
424C4⋊C4—42no—
425C4⋊C4—42no—
426C4⋊C4—42no—
427C4⋊C4—42no—
428C4⋊C4—48no—
429C4⋊C4—44no—
430C4⋊C4—84no—
431C4⋊C4C2×C22⋊C414yes—
432C22⋊C4—216no—
433C22⋊C4—416no—
434C22⋊C4—816no—
435C22⋊C4—816no—
436C42—42no—
437C42—42no—
438C42—44no—
439C42—84no—
440C42C2×C22⋊C414yes—
441C42C4×D414yes—
442C42#12114yes—
443C42#12314yes—
444C23—28no—
445C23—28no—
446C23—24no—
447C23—24no—
448C23—24no—
449C23—44no—
450C23—44no—
451C23—44no—
452C23—44no—
453C23—42no—
454C23—42no—
455C23—44no—
456C23—42no—
457C23—42no—
458C23—42no—
459C23—42no—
460C23—44no—
461C23—44no—
462C23—44no—
463C23—44no—
464C23—42no—
465C23—42no—
466C23—44no—
467C23—48no—
468C23—44no—
469C23—44no—
470C23—48no—
471C23—44no—
472C23—84no—
473C23—84no—
474C23—84no—
475C23—84no—
476C23—84no—
477C23—82no—
478C23—82no—
479C23—82no—
480C23—82no—
481C23—82no—
482C23—82no—
483C23—84no—
484C23—84no—
485C23—84no—
486C23—84no—
487C23—82no—
488C23—82no—
489C23—84no—
490C23C2×C4×D411yescharacteristic
491C23C2×#12011yescharacteristic
492C23C2×#12111yescharacteristic
493C23D4212yes—
494C23#54812yes—
495C23C22×D811yescharacteristic
496C23C2×#13511yescharacteristic
497C23C2×AGL(1,ℤ/8)11yescharacteristic
498C23C2×AGL(1,ℤ/8)11yescharacteristic
499C23#57612yes—
500C23#57712yes—
501D4—24no—
502D4—28no—
503D4—28no—
504D4—24no—
505D4—24no—
506D4—44no—
507D4—48no—
508D4—44no—
509D4—816no—
510D4—816no—
511D4—84no—
512D4—82no—
513D4—82no—
514D4#39018yescomplemented
515C2×C4—24no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—24no—
519C2×C4—24no—
520C2×C4—28no—
521C2×C4—24no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—44no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—432no—
532C2×C4—44no—
533C2×C4—42no—
534C2×C4—44no—
535C2×C4—44no—
536C2×C4—42no—
537C2×C4—44no—
538C2×C4—42no—
539C2×C4—44no—
540C2×C4—44no—
541C2×C4—44no—
542C2×C4—44no—
543C2×C4—832no—
544C2×C4—82no—
545C2×C4—82no—
546C2×C4—84no—
547C2×C4—84no—
548C2×C4—82no—
549C2×C4—82no—
550C2×C4#38614yes—
551C2×C4#38614yes—
552C2×C4C4×D814yes—
553C2×C4#44214yes—
554C2×C4#44714yes—
555C2×C4#44914yes—
556C2×C4#45914yes—
557C2×C4#46314yes—
558C2×C4C22×C22⋊C411yescharacteristic
559C2×C4C2×C4×D411yescharacteristic
560C2×C4C2×C4×D412yes—
561C2×C4#52012yes—
562C2×C4C2×#12111yescharacteristic
563C2×C4C2×#12112yes—
564C2×C4C2×#12311yescharacteristic
565C2×C4C2×#12411yescharacteristic
566C2×C4C2×#12711yescharacteristic
567C2×C4#53511yescharacteristic
568C2×C4#53611yescharacteristic
569C2×C4#53712yes—
570C2×C4D4212yes—
571C2×C4#54712yes—
572C2×C4#54712yes—
573C2×C4#54812yes—
574C2×C4#55312yes—
575C2×C4#55312yes—
576C8—44no—
577C8—44no—
578C22—28no—
579C22—44no—
580C22—44no—
581C22—48no—
582C22—44no—
583C22—48no—
584C22—44no—
585C22—44no—
586C22—44no—
587C22—48no—
588C22—48no—
589C22—84no—
590C22—84no—
591C22—84no—
592C22—88no—
593C22—88no—
594C22—162no—
595C22—162no—
596C22—164no—
597C22—162no—
598C22—164no—
599C22—162no—
600C22C2×#38611yescharacteristic
601C22#236112yes—
602C22#241612yes—
603C22#243311yescharacteristic
604C22#243611yescharacteristic
605C22C2×C4×D811yescharacteristic
606C22C2×#44211yescharacteristic
607C22#294612yes—
608C22C2×#44711yescharacteristic
609C22C2×#44911yescharacteristic
610C22#300811yescharacteristic
611C22#301011yescharacteristic
612C22C2×#45911yescharacteristic
613C22C2×#46311yescharacteristic
614C22#304512yes—
615C22#327212yes—
616C22#327412yes—
617C22D4×D812yes—
618C22#328412yes—
619C22#335812yes—
620C22#336112yes—
621C22#336612yes—
622C22#337312yes—
623C4—24no—
624C4—44no—
625C4—44no—
626C4—84no—
627C4—82no—
628C4—84no—
629C4#187918yescomplemented
630C4C2×#39011yescharacteristic
631C4#231111yescharacteristic
632C4#236112yes—
633C4#241612yes—
634C4#250111yescharacteristic
635C4#250411yescharacteristic
636C2—48no—
637C2—164no—
638C2#1413711yescharacteristic
639C2C2×#187911yescharacteristic
640C2#2093211yescharacteristic
641C2#2299512yes—
642C2#2300112yes—
643C2#2472612yes—
644C2#2473112yes—
645C2#2575411yescharacteristic
646C2#2575711yescharacteristic
647C2#2593011yescharacteristic
648C2#2593311yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC22×#459C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×D4×C4⋊C4C2semidirect
3. C2×#1957C2semidirect
4. C2×#1879C2semidirect
5. C2×C4×#102C2semidirect
6. #14154C2semidirect
7. #11066C2semidirect
8. #10849C2semidirect
9. #10834C2semidirect
17. C2×#459C4semidirect
59. C2×#127C2×C4semidirect
91. C2×C4⋊C8C2×C4semidirect
99. C2×#102C2×C4semidirect
100. C2×#102D4semidirect
205. C4×D4C22⋊C4semidirect
231. C2×C4⋊C4D8semidirect
514. D4#390semidirect
629. C4#1879semidirect

Maximal subgroups

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

subgroupindex
C22×#4592
C2×D4×C4⋊C42
C2×#19572
C2×#18792
C2×C4×#1022
#141542
#110662
#108492
#108342