Finite Group Databasebeta

← #368379

Subgroups — #368379

Order 512.

1826 conjugacy classes of subgroups.

The subgroup lattice has 660 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4×D8C211yesmaximal, characteristic
2C2×#1972C211yesmaximal, characteristic
3C2×#1972C211yesmaximal, characteristic
4C2×#1944C211yesmaximal, characteristic, complemented
5C2×#1771C211yesmaximal, characteristic, complemented
6#14018C212yesmaximal, complemented
7#11076C212yesmaximal, complemented
8#11066C212yesmaximal, complemented
9#10863C211yesmaximal, characteristic, complemented
10#10861C211yesmaximal, characteristic, complemented
11#10788C212yesmaximal, complemented
12C23×D8C2211yescharacteristic
13C22×C4×D4C2212yes—
14C2×C4×D8—24no—
15C2×C4×D8—24no—
16C2×C4×D8—24no—
17C22×#107C2211yescharacteristic
18C22×#106C2211yescharacteristic
19C22×#106C2211yescharacteristic
20C22×C4⋊C8C2211yescharacteristic
21C22×#102C2212yes—
22C22×#102C2212yes—
23C22×#102C2212yes—
24C22×#97C2211yescharacteristic
25C22×#97C2211yescharacteristic
26C22×C4×C8C2211yescharacteristic
27C2×#390C2212yes—
28C2×#390C2212yes—
29C2×#389C2212yes—
30C2×#385—22no—
31C2×C4×C4⋊C4C2211yescharacteristic
32#1972—24no—
33#1972C418yescomplemented
34#1944—24no—
35#1771—24no—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#336C2211yescharacteristic
40C2×#336C2211yescharacteristic
41#1439—22no—
42#1439—22no—
43C4⋊#95C2212yes—
44C4⋊#95C2212yes—
45C22⋊#95—22no—
46C23×D4C2312yes—
47C24×C4—22no—
48C22×D8—22no—
49C22×D8—22no—
50C22×D8—22no—
51C22×D8—22no—
52C22×D8C2×C414yes—
53C22×D8D414yes—
54C22×D8Q814yes—
55C23×C8C2311yescharacteristic
56C23×C8C2311yescharacteristic
57C23×C8C2311yescharacteristic
58C23×C8C2311yescharacteristic
59C2×C4×D4—24no—
60C2×C4×D4—24no—
61C2×C4×D4—24no—
62C2×C4×D4—24no—
63C2×C4×D4—42no—
64C2×C4×D4—42no—
65C22×C4⋊C4—22no—
66C22×C4⋊C4D411yescharacteristic
67C22×C4⋊C4D411yescharacteristic
68C22×C4⋊C4C2312yes—
69C22×C4⋊C4C2312yes—
70C22×C4⋊C4C2312yes—
71C22×C22⋊C4—22no—
72C22×C22⋊C4—22no—
73C22×C22⋊C4—22no—
74C22×C22⋊C4—22no—
75C22×C22⋊C4—22no—
76C22×C42C2311yescharacteristic
77C22×C42C2311yescharacteristic
78C22×C42C2311yescharacteristic
79C4×D8—232no—
80C2×#107—22no—
81C2×#107—22no—
82C2×#107D414yes—
83C2×#106—22no—
84C2×#106—22no—
85C2×#106—22no—
86C2×#106—22no—
87C2×#106—22no—
88C2×#106C2×C414yescomplemented
89C2×C4⋊C8—22no—
90C2×C4⋊C8—22no—
91C2×C4⋊C8—22no—
92C2×#102—24no—
93C2×#102—24no—
94C2×#102—24no—
95C2×#102—24no—
96C2×#102—24no—
97C2×#102—42no—
98C2×#102—42no—
99C2×#102—42no—
100C2×#102C2×C418yescomplemented
101C2×#97—22no—
102C2×#97—22no—
103C2×#97—22no—
104C2×#97—22no—
105C2×#97—22no—
106C2×#97C2×C414yescomplemented
107C2×C4×C8—22no—
108C2×C4×C8—22no—
109C2×C4×C8D414yes—
110#390—24no—
111#390—24no—
112#390—42no—
113#390C2×C418yescomplemented
114#389—24no—
115#389—42no—
116#385—48no—
117C4×C4⋊C4—22no—
118C4×C4⋊C4—22no—
119C2×#95—22no—
120C2×#95—22no—
121C2×#95—22no—
122C2×#95—22no—
123C2×#95—22no—
124C2×#95D411yescharacteristic
125C2×#95D411yescharacteristic
126#340—24no—
127#340—42no—
128#340—42no—
129#336—22no—
130#336—22no—
131C25—22no—
132C22×D4—22no—
133C22×D4—22no—
134C22×D4—22no—
135C22×D4—22no—
136C22×D4—22no—
137C22×D4—22no—
138C22×D4—22no—
139C22×D4—22no—
140C22×D4—22no—
141C22×D4—22no—
142C22×D4—22no—
143C22×D4C22×C414yes—
144C22×D4C2×D414yes—
145C22×D4C2×Q814yes—
146C23×C4—22no—
147C23×C4—22no—
148C23×C4—22no—
149C23×C4—44no—
150C23×C4—44no—
151C23×C4—44no—
152C23×C4—44no—
153C23×C4—44no—
154C23×C4—82no—
155C23×C4—82no—
156C23×C4C2×D411yescharacteristic
157C23×C4C2×D411yescharacteristic
158C23×C4C2×D411yescharacteristic
159C23×C4C2×D411yescharacteristic
160C23×C4C2×D411yescharacteristic
161C23×C4C2×D411yescharacteristic
162C23×C4C2411yescharacteristic
163C2×D8—28no—
164C2×D8—28no—
165C2×D8—28no—
166C2×D8—28no—
167C2×D8—28no—
168C2×D8—28no—
169C2×D8C4⋊C4116yescomplemented
170C22×C8—21no—
171C22×C8—21no—
172C22×C8—21no—
173C22×C8—21no—
174C22×C8—21no—
175C22×C8—21no—
176C22×C8—21no—
177C22×C8—21no—
178C22×C8—21no—
179C22×C8—21no—
180C22×C8—21no—
181C22×C8—21no—
182C22×C8—21no—
183C22×C8—21no—
184C22×C8—21no—
185C22×C8—21no—
186C22×C8—21no—
187C22×C8—21no—
188C22×C8C22×C412yes—
189C22×C8C22×C412yes—
190C22×C8C2×D412yes—
191C22×C8C2×D412yes—
192C22×C8C2×D412yes—
193C22×C8C2×D412yes—
194C22×C8C2×Q812yes—
195C22×C8#4112yes—
196C22×C8#4112yes—
197C22×C8#4112yes—
198C4×D4—216no—
199C4×D4—48no—
200C4×D4—48no—
201C4×D4—48no—
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—22no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—24no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—42no—
222C2×C4⋊C4—44no—
223C2×C4⋊C4—44no—
224C2×C4⋊C4—82no—
225C2×C4⋊C4C22×C414yes—
226C2×C4⋊C4C2×D414yes—
227C2×C22⋊C4—28no—
228C2×C22⋊C4—28no—
229C2×C22⋊C4—44no—
230C2×C22⋊C4—44no—
231C2×C22⋊C4—44no—
232C2×C22⋊C4—44no—
233C2×C22⋊C4—44no—
234C2×C22⋊C4—44no—
235C2×C22⋊C4—44no—
236C2×C22⋊C4—44no—
237C2×C22⋊C4—44no—
238C2×C22⋊C4—82no—
239C2×C22⋊C4—82no—
240C2×C22⋊C4—82no—
241C2×C22⋊C4—82no—
242C2×C42—21no—
243C2×C42—21no—
244C2×C42—21no—
245C2×C42—21no—
246C2×C42—21no—
247C2×C42—21no—
248C2×C42—21no—
249C2×C42—21no—
250C2×C42—21no—
251C2×C42—21no—
252C2×C42—21no—
253C2×C42—21no—
254C2×C42—21no—
255C2×C42—21no—
256C2×C42C22×C412yes—
257C2×C42C2×D412yes—
258C2×C42#4112yes—
259C2×C42#4112yes—
260#107—28no—
261#106—28no—
262#106—44no—
263C4⋊C8—28no—
264#102—216no—
265#102—48no—
266#102—48no—
267#97—28no—
268#97—44no—
269C4×C8—28no—
270#95—22no—
271#95—22no—
272#95—44no—
273#95—44no—
274#95—44no—
275#95—82no—
276#95—82no—
277C24—24no—
278C24—24no—
279C24—24no—
280C24—42no—
281C24—42no—
282C24—42no—
283C24—42no—
284C24—42no—
285C24—42no—
286C24—42no—
287C24—42no—
288C24—42no—
289C24—42no—
290C24—42no—
291C24—42no—
292C24C22×D411yescharacteristic
293C2×D4—24no—
294C2×D4—24no—
295C2×D4—24no—
296C2×D4—24no—
297C2×D4—24no—
298C2×D4—24no—
299C2×D4—24no—
300C2×D4—24no—
301C2×D4—24no—
302C2×D4—24no—
303C2×D4—24no—
304C2×D4—24no—
305C2×D4—24no—
306C2×D4—24no—
307C2×D4—24no—
308C2×D4—42no—
309C2×D4—42no—
310C2×D4—42no—
311C2×D4—42no—
312C2×D4—42no—
313C2×D4—42no—
314C2×D4—42no—
315C2×D4—42no—
316C2×D4—42no—
317C2×D4—42no—
318C2×D4—42no—
319C2×D4—42no—
320C2×D4C2×C4⋊C418yes—
321C22×C4—21no—
322C22×C4—21no—
323C22×C4—22no—
324C22×C4—21no—
325C22×C4—21no—
326C22×C4—21no—
327C22×C4—21no—
328C22×C4—21no—
329C22×C4—21no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—24no—
337C22×C4—21no—
338C22×C4—21no—
339C22×C4—21no—
340C22×C4—24no—
341C22×C4—21no—
342C22×C4—21no—
343C22×C4—21no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—416no—
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—42no—
362C22×C4—42no—
363C22×C4—42no—
364C22×C4—42no—
365C22×C4—42no—
366C22×C4—42no—
367C22×C4—42no—
368C22×C4—88no—
369C22×C4—88no—
370C22×C4—88no—
371C22×C4—88no—
372C22×C4—88no—
373C22×C4—88no—
374C22×C4C4×D412yes—
375C22×C4C4×D412yes—
376C22×C4#12112yes—
377C22×C4#12112yes—
378C22×C4AGL(1,ℤ/8)14yes—
379C22×C4C23×C411yescharacteristic
380C22×C4C22×D411yescharacteristic
381C22×C4C22×D411yescharacteristic
382C22×C4C22×D411yescharacteristic
383C22×C4C22×Q811yescharacteristic
384C22×C4C2×#4111yescharacteristic
385C22×C4C2×#4111yescharacteristic
386C22×C4C2×#4111yescharacteristic
387C22×C4C2×#4111yescharacteristic
388C22×C4C2×#4111yescharacteristic
389C22×C4H(5,𝔽2)11yescharacteristic
390C22×C4H(5,𝔽2)11yescharacteristic
391C22×C4H(5,𝔽2)11yescharacteristic
392C22×C42−1+411yescharacteristic
393D8—232no—
394C2×C8—22no—
395C2×C8—22no—
396C2×C8—22no—
397C2×C8—22no—
398C2×C8—22no—
399C2×C8—22no—
400C2×C8—22no—
401C2×C8—22no—
402C2×C8—22no—
403C2×C8—41no—
404C2×C8—41no—
405C2×C8—41no—
406C2×C8—41no—
407C2×C8—41no—
408C2×C8—41no—
409C2×C8—41no—
410C2×C8—41no—
411C2×C8—41no—
412C2×C8—41no—
413C2×C8—41no—
414C2×C8—41no—
415C2×C8C2×C4⋊C414yes—
416C2×C8C4×D414yes—
417C2×C8C4×D414yes—
418C2×C8#12014yes—
419C2×C8C22⋊Q814yes—
420C2×C8C22⋊Q814yes—
421C2×C8#12314yes—
422C4⋊C4—24no—
423C4⋊C4—24no—
424C4⋊C4—24no—
425C4⋊C4—24no—
426C4⋊C4—44no—
427C4⋊C4—42no—
428C4⋊C4—42no—
429C4⋊C4—42no—
430C4⋊C4—42no—
431C4⋊C4—44no—
432C4⋊C4—42no—
433C4⋊C4—42no—
434C4⋊C4—42no—
435C4⋊C4—42no—
436C4⋊C4—88no—
437C22⋊C4—416no—
438C22⋊C4—416no—
439C22⋊C4—88no—
440C22⋊C4—88no—
441C22⋊C4—88no—
442C42—22no—
443C42—22no—
444C42—22no—
445C42—22no—
446C42—22no—
447C42—22no—
448C42—22no—
449C42—22no—
450C42—41no—
451C42—41no—
452C42—41no—
453C42—41no—
454C23—28no—
455C23—44no—
456C23—44no—
457C23—44no—
458C23—44no—
459C23—44no—
460C23—44no—
461C23—44no—
462C23—44no—
463C23—44no—
464C23—44no—
465C23—44no—
466C23—44no—
467C23—44no—
468C23—44no—
469C23—44no—
470C23—44no—
471C23—44no—
472C23—44no—
473C23—82no—
474C23—82no—
475C23—82no—
476C23—82no—
477C23—82no—
478C23—82no—
479C23—82no—
480C23—82no—
481C23—82no—
482C23—82no—
483C23—82no—
484C23—82no—
485C23—82no—
486C23—82no—
487C23—82no—
488C23—82no—
489C23C2×C4×D411yescharacteristic
490C23C2×#12111yescharacteristic
491C23#53211yescharacteristic
492C23D4211yescharacteristic
493C23D4211yescharacteristic
494C23D4×Q811yescharacteristic
495C23#55011yescharacteristic
496C23C2×AGL(1,ℤ/8)12yes—
497C23#57512yes—
498C23#57612yes—
499C23#57712yes—
500D4—28no—
501D4—28no—
502D4—44no—
503D4—44no—
504D4—44no—
505D4—44no—
506D4—44no—
507D4—44no—
508C2×C4—22no—
509C2×C4—24no—
510C2×C4—24no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—24no—
514C2×C4—24no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—44no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—44no—
531C2×C4—44no—
532C2×C4—44no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—44no—
536C2×C4—44no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—832no—
544C2×C4—82no—
545C2×C4—82no—
546C2×C4—82no—
547C2×C4—82no—
548C2×C4—82no—
549C2×C4—82no—
550C2×C4—82no—
551C2×C4—82no—
552C2×C4—82no—
553C2×C4—82no—
554C2×C4—82no—
555C2×C4—82no—
556C2×C4—82no—
557C2×C4—82no—
558C2×C4—82no—
559C2×C4#46914yes—
560C2×C4C22×C4⋊C411yescharacteristic
561C2×C4C2×C4×D411yescharacteristic
562C2×C4C2×C4×D411yescharacteristic
563C2×C4C4×#4111yescharacteristic
564C2×C4#51811yescharacteristic
565C2×C4#52011yescharacteristic
566C2×C4C2×#12011yescharacteristic
567C2×C4C2×C22⋊Q811yescharacteristic
568C2×C4C2×C22⋊Q811yescharacteristic
569C2×C4C2×#12311yescharacteristic
570C2×C4#52511yescharacteristic
571C2×C4#52911yescharacteristic
572C2×C4#53511yescharacteristic
573C2×C4#53711yescharacteristic
574C2×C4#54011yescharacteristic
575C2×C4#54211yescharacteristic
576C2×C4#54611yescharacteristic
577C2×C4#54611yescharacteristic
578C2×C4#54611yescharacteristic
579C2×C4#54711yescharacteristic
580C2×C4#54811yescharacteristic
581C2×C4#55011yescharacteristic
582C2×C4#55411yescharacteristic
583C2×C4#55411yescharacteristic
584C2×C4#55411yescharacteristic
585C2×C4#55611yescharacteristic
586C2×C4#55911yescharacteristic
587C2×C4#55911yescharacteristic
588C8—42no—
589C8—42no—
590C8—42no—
591C8#38518yescomplemented
592C22—48no—
593C22—48no—
594C22—48no—
595C22—84no—
596C22—84no—
597C22—84no—
598C22—84no—
599C22—84no—
600C22—84no—
601C22—84no—
602C22—84no—
603C22—84no—
604C22—84no—
605C22—84no—
606C22—84no—
607C22C4×#12111yescharacteristic
608C22D4×C4⋊C411yescharacteristic
609C22#236611yescharacteristic
610C22#240811yescharacteristic
611C22#244511yescharacteristic
612C22#245211yescharacteristic
613C22#253811yescharacteristic
614C22C4×AGL(1,ℤ/8)12yes—
615C22#294612yes—
616C22C2×#46911yescharacteristic
617C22#305411yescharacteristic
618C22#305811yescharacteristic
619C22#306011yescharacteristic
620C22#317112yes—
621C22#317912yes—
622C22#326412yes—
623C22#326412yes—
624C22#326712yes—
625C22#327112yes—
626C22#338412yes—
627C22#338912yes—
628C22#339712yes—
629C22#340312yes—
630C4—24no—
631C4—42no—
632C4—42no—
633C4—42no—
634C4—84no—
635C4—81no—
636C4—81no—
637C4—84no—
638C4—162no—
639C4C2×#38511yescharacteristic
640C4#232411yescharacteristic
641C4#234911yescharacteristic
642C4#236711yescharacteristic
643C4#240711yescharacteristic
644C4#244711yescharacteristic
645C4#245411yescharacteristic
646C4#258011yescharacteristic
647C2—88no—
648C2#1401811yescharacteristic
649C2C4×#46911yescharacteristic
650C2#2153611yescharacteristic
651C2#2267612yes—
652C2#2319512yes—
653C2#2427111yescharacteristic
654C2#2427211yescharacteristic
655C2#2625812yes—
656C2#2648812yes—
657C2#2826911yescharacteristic
658C2#2827311yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×D8C22
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
4. C2×#1944C2semidirect
5. C2×#1771C2semidirect
6. #14018C2semidirect
7. #11076C2semidirect
8. #11066C2semidirect
9. #10863C2semidirect
10. #10861C2semidirect
11. #10788C2semidirect
33. #1972C4semidirect
88. C2×#106C2×C4semidirect
100. C2×#102C2×C4semidirect
106. C2×#97C2×C4semidirect
113. #390C2×C4semidirect
169. C2×D8C4⋊C4semidirect
591. C8#385semidirect

Maximal subgroups

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

subgroupindex
C22×C4×D82
C2×#19722
C2×#19442
C2×#17712
#140182
#110762
#110662
#108632
#108612
#107882