Finite Group Databasebeta

← #365590

Subgroups — #365590

Order 512.

1986 conjugacy classes of subgroups.

The subgroup lattice has 642 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×D4×C4⋊C4C211yesmaximal, characteristic
2C2×#1955C211yesmaximal, characteristic
3C2×#1949C211yesmaximal, characteristic, complemented
4C2×#1853C212yesmaximal, complemented
5C2×#1852C212yesmaximal, complemented
6C2×#1847C211yesmaximal, characteristic, complemented
7#13905C211yesmaximal, characteristic
8#11096C212yesmaximal, complemented
9#11095C212yesmaximal, complemented
10#10849C211yesmaximal, characteristic, complemented
11#10703C211yesmaximal, characteristic, complemented
12C22×C4×D4—21no—
13C22×C4×D4C2211yescharacteristic
14C23×C4⋊C4C2212yes—
15C22×#107C2211yescharacteristic
16C22×#107C2212yes—
17C22×#106C2211yescharacteristic
18C22×#106C2212yes—
19C22×C4⋊C8C2211yescharacteristic
20C22×C22⋊C8C2212yes—
21C22×C22⋊C8C2212yes—
22C22×#97C2211yescharacteristic
23D4×C4⋊C4—24no—
24D4×C4⋊C4—24no—
25C2×#389—21no—
26C2×#385—22no—
27C2×#384C2211yescharacteristic
28C2×#382C2211yescharacteristic
29C2×#380C2212yes—
30C2×#380C2212yes—
31C2×#380C2212yes—
32C2×C4×C4⋊C4C2211yescharacteristic
33C2×C4×C4⋊C4C2211yescharacteristic
34#1955C418yescomplemented
35#1949—24no—
36#1853—28no—
37#1852—28no—
38#1847—24no—
39C2×#340C2211yescharacteristic
40C2×#340C2212yes—
41C2×#340C2212yes—
42C2×#340C2211yescharacteristic
43C2×#340C2212yes—
44C2×#340C2212yes—
45#1445—21no—
46#1438—21no—
47#1437—22no—
48C23×D4D411yescharacteristic
49C24×C4—22no—
50C24×C4C2312yes—
51C23×C8C2311yescharacteristic
52C23×C8C2311yescharacteristic
53C23×C8C2312yes—
54C2×C4×D4—28no—
55C2×C4×D4—22no—
56C2×C4×D4—22no—
57C2×C4×D4—22no—
58C2×C4×D4—22no—
59C2×C4×D4—48no—
60C2×C4×D4C2×C414yes—
61C2×C4×D4Q814yes—
62C22×C4⋊C4—22no—
63C22×C4⋊C4—21no—
64C22×C4⋊C4—21no—
65C22×C4⋊C4—24no—
66C22×C4⋊C4—24no—
67C22×C4⋊C4—24no—
68C22×C4⋊C4—24no—
69C22×C4⋊C4—42no—
70C22×C4⋊C4—42no—
71C22×C4⋊C4D411yescharacteristic
72C22×C4⋊C4C2311yescharacteristic
73C22×C4⋊C4C2311yescharacteristic
74C22×C4⋊C4C2312yes—
75C22×C4⋊C4C2311yescharacteristic
76C22×C4⋊C4C2312yes—
77C22×C4⋊C4C2311yescharacteristic
78C22×C22⋊C4—22no—
79C22×C22⋊C4—22no—
80C22×C22⋊C4—22no—
81C22×C22⋊C4D412yes—
82C22×C42—21no—
83C22×C42—21no—
84C22×C42C2311yescharacteristic
85C2×#107—24no—
86C2×#107—24no—
87C2×#107—24no—
88C2×#107—22no—
89C2×#107—42no—
90C2×#106—24no—
91C2×#106—24no—
92C2×#106—24no—
93C2×#106—22no—
94C2×#106—42no—
95C2×C4⋊C8—22no—
96C2×C4⋊C8—22no—
97C2×C4⋊C8C2×C414yescomplemented
98C2×C22⋊C8—24no—
99C2×C22⋊C8—24no—
100C2×C22⋊C8—24no—
101C2×C22⋊C8—24no—
102C2×C22⋊C8C2×C418yescomplemented
103C2×C22⋊C8C2×C418yescomplemented
104C2×#97—22no—
105C2×#97—22no—
106C2×#97C2×C414yescomplemented
107#389—28no—
108#385—216no—
109#384—22no—
110#384D414yes—
111#382—22no—
112#382—22no—
113#380—24no—
114#380—24no—
115#380—24no—
116#380—24no—
117#380—42no—
118#380—42no—
119C4×C4⋊C4—22no—
120C4×C4⋊C4—22no—
121C4×C4⋊C4—22no—
122C4×C4⋊C4D414yes—
123C2×#95—21no—
124C2×#95—22no—
125C2×#95—21no—
126C2×#95—22no—
127C2×#95—22no—
128C2×#95—21no—
129C2×#95—21no—
130#340—24no—
131#340—24no—
132#340—24no—
133#340—22no—
134#340—22no—
135#340D418yescomplemented
136C25C2×D412yes—
137C22×D4—24no—
138C22×D4—24no—
139C22×D4—22no—
140C22×D4—22no—
141C22×D4C4⋊C414yes—
142C23×C4—21no—
143C23×C4—22no—
144C23×C4—22no—
145C23×C4—22no—
146C23×C4—22no—
147C23×C4—22no—
148C23×C4—22no—
149C23×C4—22no—
150C23×C4—22no—
151C23×C4—22no—
152C23×C4—22no—
153C23×C4—22no—
154C23×C4—22no—
155C23×C4—21no—
156C23×C4—22no—
157C23×C4—22no—
158C23×C4—28no—
159C23×C4—21no—
160C23×C4—21no—
161C23×C4—44no—
162C23×C4—44no—
163C23×C4—44no—
164C23×C4—44no—
165C23×C4—82no—
166C23×C4—82no—
167C23×C4C22×C414yes—
168C23×C4C2×D412yes—
169C23×C4C2×D411yescharacteristic
170C23×C4C2×D411yescharacteristic
171C23×C4C2×Q814yes—
172C23×C4C2411yescharacteristic
173C22×C8—22no—
174C22×C8—22no—
175C22×C8—22no—
176C22×C8—22no—
177C22×C8—22no—
178C22×C8—22no—
179C22×C8—21no—
180C22×C8—21no—
181C22×C8—22no—
182C22×C8—22no—
183C22×C8—21no—
184C22×C8—21no—
185C22×C8C22×C412yes—
186C22×C8C22×C412yes—
187C22×C8C22×C414yes—
188C22×C8C2×D414yes—
189C22×C8#4114yes—
190C4×D4—28no—
191C4×D4—28no—
192C4×D4—28no—
193C4×D4—28no—
194C4×D4—432no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—28no—
210C2×C4⋊C4—24no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—48no—
224C2×C4⋊C4—48no—
225C2×C4⋊C4—48no—
226C2×C4⋊C4—44no—
227C2×C4⋊C4—44no—
228C2×C4⋊C4—44no—
229C2×C4⋊C4—48no—
230C2×C4⋊C4—81no—
231C2×C4⋊C4—81no—
232C2×C4⋊C4C4⋊C414yes—
233C2×C4⋊C4C2×D414yes—
234C2×C4⋊C4C2×D414yes—
235C2×C4⋊C4C2×D412yes—
236C2×C4⋊C4C2×D414yes—
237C2×C4⋊C4C2×D412yes—
238C2×C22⋊C4—24no—
239C2×C22⋊C4—24no—
240C2×C22⋊C4—28no—
241C2×C22⋊C4—44no—
242C2×C22⋊C4—44no—
243C2×C22⋊C4—44no—
244C2×C22⋊C4—44no—
245C2×C22⋊C4—44no—
246C2×C22⋊C4—44no—
247C2×C22⋊C4—82no—
248C2×C22⋊C4—82no—
249C2×C22⋊C4C4⋊C418yes—
250C2×C42—24no—
251C2×C42—44no—
252C2×C42—42no—
253C2×C42—81no—
254C2×C42—81no—
255C2×C42C22×C412yes—
256C2×C42C2×D412yes—
257C2×C42C2×D412yes—
258C2×C42C2×Q812yes—
259C2×C42#4112yes—
260C2×C42#4112yes—
261#107—48no—
262#107—44no—
263#106—48no—
264#106—44no—
265C4⋊C8—28no—
266C22⋊C8—216no—
267C22⋊C8—216no—
268#97—28no—
269#95—28no—
270#95—24no—
271#95—44no—
272#95—44no—
273#95—42no—
274#95—42no—
275#95—42no—
276C24—22no—
277C24—22no—
278C24—22no—
279C24—22no—
280C24—22no—
281C24—22no—
282C24—22no—
283C24—22no—
284C24—22no—
285C24—22no—
286C24—22no—
287C24—22no—
288C24C2×C4⋊C414yes—
289C24AGL(1,ℤ/8)14yes—
290C24#13714yes—
291C24C22×D411yescharacteristic
292C2×D4—28no—
293C2×D4—216no—
294C2×D4—28no—
295C2×D4—216no—
296C2×D4—28no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—21no—
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—24no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—24no—
312C22×C4—24no—
313C22×C4—24no—
314C22×C4—24no—
315C22×C4—24no—
316C22×C4—24no—
317C22×C4—24no—
318C22×C4—24no—
319C22×C4—24no—
320C22×C4—24no—
321C22×C4—24no—
322C22×C4—24no—
323C22×C4—22no—
324C22×C4—24no—
325C22×C4—24no—
326C22×C4—416no—
327C22×C4—42no—
328C22×C4—41no—
329C22×C4—41no—
330C22×C4—42no—
331C22×C4—42no—
332C22×C4—42no—
333C22×C4—42no—
334C22×C4—42no—
335C22×C4—42no—
336C22×C4—42no—
337C22×C4—42no—
338C22×C4—42no—
339C22×C4—42no—
340C22×C4—42no—
341C22×C4—42no—
342C22×C4—42no—
343C22×C4—42no—
344C22×C4—42no—
345C22×C4—42no—
346C22×C4—416no—
347C22×C4—42no—
348C22×C4—42no—
349C22×C4—42no—
350C22×C4—42no—
351C22×C4—42no—
352C22×C4—42no—
353C22×C4—42no—
354C22×C4—42no—
355C22×C4—42no—
356C22×C4—42no—
357C22×C4—42no—
358C22×C4—42no—
359C22×C4—42no—
360C22×C4—42no—
361C22×C4—41no—
362C22×C4—41no—
363C22×C4—41no—
364C22×C4—41no—
365C22×C4—42no—
366C22×C4—42no—
367C22×C4—42no—
368C22×C4—42no—
369C22×C4—41no—
370C22×C4—41no—
371C22×C4—41no—
372C22×C4—41no—
373C22×C4—88no—
374C22×C4—88no—
375C22×C4—88no—
376C22×C4—88no—
377C22×C4—88no—
378C22×C4C2×C4⋊C414yes—
379C22×C4C2×C4⋊C412yes—
380C22×C4C2×C4⋊C412yes—
381C22×C4#12012yes—
382C22×C4#12012yes—
383C22×C4#12312yes—
384C22×C4#12312yes—
385C22×C4C23×C411yescharacteristic
386C22×C4C22×D411yescharacteristic
387C22×C4C22×D411yescharacteristic
388C22×C4C22×D412yes—
389C22×C4C22×Q811yescharacteristic
390C22×C4C2×#4111yescharacteristic
391C22×C4C2×#4111yescharacteristic
392C22×C4C2×#4112yes—
393C22×C4H(5,𝔽2)12yes—
394C22×C42−1+412yes—
395C2×C8—24no—
396C2×C8—24no—
397C2×C8—24no—
398C2×C8—24no—
399C2×C8—24no—
400C2×C8—24no—
401C2×C8—24no—
402C2×C8—24no—
403C2×C8—22no—
404C2×C8—22no—
405C2×C8—41no—
406C2×C8—41no—
407C2×C8—41no—
408C2×C8—41no—
409C2×C8—42no—
410C2×C8—42no—
411C2×C8C4×D418yescomplemented
412C4⋊C4—22no—
413C4⋊C4—24no—
414C4⋊C4—28no—
415C4⋊C4—24no—
416C4⋊C4—22no—
417C4⋊C4—22no—
418C4⋊C4—24no—
419C4⋊C4—24no—
420C4⋊C4—24no—
421C4⋊C4—416no—
422C4⋊C4—48no—
423C4⋊C4—42no—
424C4⋊C4—42no—
425C4⋊C4—42no—
426C4⋊C4—42no—
427C4⋊C4—42no—
428C4⋊C4—41no—
429C4⋊C4—41no—
430C4⋊C4—41no—
431C4⋊C4—41no—
432C4⋊C4—42no—
433C4⋊C4—42no—
434C4⋊C4—84no—
435C4⋊C4#12714yes—
436C22⋊C4—216no—
437C22⋊C4—416no—
438C22⋊C4—88no—
439C22⋊C4—88no—
440C42—48no—
441C42—84no—
442C42C4×D414yes—
443C42C4×D414yes—
444C42C22⋊Q814yes—
445C42C22⋊Q814yes—
446C23—24no—
447C23—24no—
448C23—24no—
449C23—24no—
450C23—24no—
451C23—24no—
452C23—24no—
453C23—24no—
454C23—24no—
455C23—24no—
456C23—24no—
457C23—24no—
458C23—24no—
459C23—24no—
460C23—24no—
461C23—24no—
462C23—24no—
463C23—24no—
464C23—42no—
465C23—42no—
466C23—42no—
467C23—42no—
468C23—42no—
469C23—42no—
470C23—42no—
471C23—42no—
472C23—42no—
473C23—42no—
474C23—42no—
475C23—42no—
476C23—42no—
477C23—42no—
478C23—42no—
479C23—42no—
480C23#42818yes—
481C23C22×C4⋊C411yescharacteristic
482C23C2×#12011yescharacteristic
483C23C2×#12311yescharacteristic
484C23D4212yes—
485C23#54712yes—
486C23C2×AGL(1,ℤ/8)12yes—
487C23C2×#13712yes—
488C23#57611yescharacteristic
489C23#57711yescharacteristic
490C23#57711yescharacteristic
491C23#57811yescharacteristic
492D4—232no—
493C2×C4—24no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—24no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—24no—
504C2×C4—24no—
505C2×C4—24no—
506C2×C4—24no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—44no—
513C2×C4—44no—
514C2×C4—42no—
515C2×C4—44no—
516C2×C4—44no—
517C2×C4—44no—
518C2×C4—44no—
519C2×C4—44no—
520C2×C4—44no—
521C2×C4—44no—
522C2×C4—44no—
523C2×C4—44no—
524C2×C4—42no—
525C2×C4—44no—
526C2×C4—42no—
527C2×C4—44no—
528C2×C4—44no—
529C2×C4—44no—
530C2×C4—44no—
531C2×C4—42no—
532C2×C4—832no—
533C2×C4—82no—
534C2×C4—82no—
535C2×C4—82no—
536C2×C4—82no—
537C2×C4—82no—
538C2×C4—82no—
539C2×C4—82no—
540C2×C4—82no—
541C2×C4—82no—
542C2×C4—82no—
543C2×C4—81no—
544C2×C4—81no—
545C2×C4—81no—
546C2×C4—81no—
547C2×C4—82no—
548C2×C4—82no—
549C2×C4—82no—
550C2×C4—81no—
551C2×C4—81no—
552C2×C4—82no—
553C2×C4#38514yes—
554C2×C4#38514yes—
555C2×C4C2×C4×D411yescharacteristic
556C2×C4C2×C4×D412yes—
557C2×C4C2×C4×D411yescharacteristic
558C2×C4#52012yes—
559C2×C4C2×C22⋊Q811yescharacteristic
560C2×C4C2×C22⋊Q811yescharacteristic
561C2×C4C2×#12711yescharacteristic
562C2×C4#53211yescharacteristic
563C2×C4#53212yes—
564C2×C4#53511yescharacteristic
565C2×C4#53611yescharacteristic
566C2×C4#54212yes—
567C2×C4D4212yes—
568C2×C4#54712yes—
569C2×C4#54812yes—
570C2×C4D4×Q812yes—
571C2×C4#55312yes—
572C2×C4#55412yes—
573C8—28no—
574C8—42no—
575C8—42no—
576C22—28no—
577C22—28no—
578C22—28no—
579C22—44no—
580C22—44no—
581C22—44no—
582C22—44no—
583C22—44no—
584C22—44no—
585C22—44no—
586C22—44no—
587C22—44no—
588C22—44no—
589C22—44no—
590C22—44no—
591C22C2×#38511yescharacteristic
592C22D4×C4⋊C412yes—
593C22#242212yes—
594C22#243211yescharacteristic
595C22#243311yescharacteristic
596C22C2×#42812yes—
597C22#291211yescharacteristic
598C22#291311yescharacteristic
599C22#300311yescharacteristic
600C22#300411yescharacteristic
601C22#301211yescharacteristic
602C22#301311yescharacteristic
603C22#309212yes—
604C22#309611yescharacteristic
605C22#309711yescharacteristic
606C22#326412yes—
607C22#326512yes—
608C22#326612yes—
609C22#327012yes—
610C22#334312yes—
611C22#334412yes—
612C22#334512yes—
613C22#334612yes—
614C4—24no—
615C4—24no—
616C4—44no—
617C4—44no—
618C4—82no—
619C4—84no—
620C4—84no—
621C4—161no—
622C4—161no—
623C4C4×#12711yescharacteristic
624C4#232611yescharacteristic
625C4#236112yes—
626C4#245212yes—
627C4#254211yescharacteristic
628C4#256611yescharacteristic
629C2—48no—
630C2#1397711yescharacteristic
631C2#2084411yescharacteristic
632C2#2084911yescharacteristic
633C2#2265912yes—
634C2#2266112yes—
635C2#2502212yes—
636C2#2502412yes—
637C2#2570911yescharacteristic
638C2#2571211yescharacteristic
639C2#2576311yescharacteristic
640C2#2576611yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×D4D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×C4×D4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C2×#1949C2semidirect
4. C2×#1853C2semidirect
5. C2×#1852C2semidirect
6. C2×#1847C2semidirect
8. #11096C2semidirect
9. #11095C2semidirect
10. #10849C2semidirect
11. #10703C2semidirect
34. #1955C4semidirect
97. C2×C4⋊C8C2×C4semidirect
102. C2×C22⋊C8C2×C4semidirect
103. C2×C22⋊C8C2×C4semidirect
106. C2×#97C2×C4semidirect
135. #340D4semidirect
411. C2×C8C4×D4semidirect

Maximal subgroups

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

subgroupindex
C2×D4×C4⋊C42
C2×#19552
C2×#19492
C2×#18532
C2×#18522
C2×#18472
#139052
#110962
#110952
#108492
#107032