Finite Group Databasebeta

← #368381

Subgroups — #368381

Order 512.

1826 conjugacy classes of subgroups.

The subgroup lattice has 639 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#442C211yesmaximal, characteristic
2C2×#1972C211yesmaximal, characteristic
3C2×#1970C211yesmaximal, characteristic
4C2×#1943C211yesmaximal, characteristic, complemented
5C2×#1770C211yesmaximal, 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×#442—24no—
15C2×#442—24no—
16C2×#442—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#1970C418yescomplemented
34#1943—24no—
35#1770—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⋊C4D412yes—
67C22×C4⋊C4C2312yes—
68C22×C4⋊C4C2312yes—
69C22×C4⋊C4C2312yes—
70C22×C22⋊C4—22no—
71C22×C22⋊C4—22no—
72C22×C22⋊C4—22no—
73C22×C22⋊C4—22no—
74C22×C22⋊C4—22no—
75C22×C42C2311yescharacteristic
76C22×C42C2311yescharacteristic
77C22×C42C2311yescharacteristic
78#442—232no—
79C2×#107—22no—
80C2×#107—22no—
81C2×#107C2×C414yescomplemented
82C2×#106—22no—
83C2×#106—22no—
84C2×#106—22no—
85C2×#106—22no—
86C2×#106—22no—
87C2×#106D414yes—
88C2×C4⋊C8—22no—
89C2×C4⋊C8—22no—
90C2×C4⋊C8—22no—
91C2×#102—24no—
92C2×#102—24no—
93C2×#102—24no—
94C2×#102—24no—
95C2×#102—24no—
96C2×#102—42no—
97C2×#102—42no—
98C2×#102—42no—
99C2×#102C2×C418yescomplemented
100C2×#97—22no—
101C2×#97—22no—
102C2×#97—22no—
103C2×#97—22no—
104C2×#97—22no—
105C2×#97D414yes—
106C2×C4×C8—22no—
107C2×C4×C8—22no—
108C2×C4×C8C2×C414yescomplemented
109#390—24no—
110#390—24no—
111#390—42no—
112#390C2×C418yescomplemented
113#389—24no—
114#389—42no—
115#385—48no—
116C4×C4⋊C4—22no—
117C4×C4⋊C4—22no—
118C2×#95—22no—
119C2×#95—22no—
120C2×#95—22no—
121C2×#95—22no—
122C2×#95—22no—
123C2×#95D412yes—
124#340—24no—
125#340—42no—
126#340—42no—
127#336—22no—
128#336—22no—
129C25—22no—
130C22×D4—22no—
131C22×D4—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×D4C22×C414yes—
142C22×D4C2×D414yes—
143C22×D4C2×Q814yes—
144C23×C4—22no—
145C23×C4—22no—
146C23×C4—22no—
147C23×C4—44no—
148C23×C4—44no—
149C23×C4—44no—
150C23×C4—44no—
151C23×C4—44no—
152C23×C4—82no—
153C23×C4—82no—
154C23×C4C2×D412yes—
155C23×C4C2×D411yescharacteristic
156C23×C4C2×D412yes—
157C23×C4C2×D411yescharacteristic
158C23×C4C2411yescharacteristic
159C2×D8—28no—
160C2×D8—28no—
161C2×D8—28no—
162C2×D8—28no—
163C2×D8—28no—
164C2×D8—28no—
165C2×D8C4⋊C4116yescomplemented
166C22×C8—21no—
167C22×C8—21no—
168C22×C8—21no—
169C22×C8—21no—
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×C8C22×C412yes—
185C22×C8C22×C412yes—
186C22×C8C2×D412yes—
187C22×C8C2×D412yes—
188C22×C8C2×D412yes—
189C22×C8C2×D412yes—
190C22×C8C2×Q812yes—
191C22×C8#4112yes—
192C22×C8#4112yes—
193C22×C8#4112yes—
194C4×D4—216no—
195C4×D4—48no—
196C4×D4—48no—
197C4×D4—48no—
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—24no—
207C2×C4⋊C4—24no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—42no—
217C2×C4⋊C4—44no—
218C2×C4⋊C4—44no—
219C2×C4⋊C4—82no—
220C2×C4⋊C4C22×C414yes—
221C2×C4⋊C4C2×D414yes—
222C2×C22⋊C4—28no—
223C2×C22⋊C4—28no—
224C2×C22⋊C4—44no—
225C2×C22⋊C4—44no—
226C2×C22⋊C4—44no—
227C2×C22⋊C4—44no—
228C2×C22⋊C4—44no—
229C2×C22⋊C4—44no—
230C2×C22⋊C4—44no—
231C2×C22⋊C4—44no—
232C2×C22⋊C4—44no—
233C2×C22⋊C4—82no—
234C2×C22⋊C4—82no—
235C2×C22⋊C4—82no—
236C2×C22⋊C4—82no—
237C2×C42—21no—
238C2×C42—21no—
239C2×C42—21no—
240C2×C42—21no—
241C2×C42—21no—
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×C42C22×C412yes—
252C2×C42C2×D412yes—
253C2×C42#4112yes—
254C2×C42#4112yes—
255#107—28no—
256#106—28no—
257#106—44no—
258C4⋊C8—28no—
259#102—216no—
260#102—48no—
261#102—48no—
262#97—28no—
263#97—44no—
264C4×C8—28no—
265#95—24no—
266#95—44no—
267#95—44no—
268#95—44no—
269#95—82no—
270#95—82no—
271C24—24no—
272C24—24no—
273C24—24no—
274C24—42no—
275C24—42no—
276C24—42no—
277C24—42no—
278C24—42no—
279C24—42no—
280C24—42no—
281C24—42no—
282C24—42no—
283C24—42no—
284C24—42no—
285C24—42no—
286C24C22×D411yescharacteristic
287C2×D4—24no—
288C2×D4—24no—
289C2×D4—24no—
290C2×D4—24no—
291C2×D4—24no—
292C2×D4—24no—
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—42no—
303C2×D4—42no—
304C2×D4—42no—
305C2×D4—42no—
306C2×D4—42no—
307C2×D4—42no—
308C2×D4—42no—
309C2×D4—42no—
310C2×D4—42no—
311C2×D4—42no—
312C2×D4—42no—
313C2×D4—42no—
314C2×D4C2×C4⋊C418yes—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—22no—
318C22×C4—21no—
319C22×C4—21no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—24no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—24no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—416no—
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—42no—
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—88no—
358C22×C4—88no—
359C22×C4—88no—
360C22×C4—88no—
361C22×C4—88no—
362C22×C4—88no—
363C22×C4C4×D412yes—
364C22×C4C4×D412yes—
365C22×C4#12114yes—
366C22×C4AGL(1,ℤ/8)14yes—
367C22×C4C23×C411yescharacteristic
368C22×C4C22×D411yescharacteristic
369C22×C4C22×D411yescharacteristic
370C22×C4C22×D411yescharacteristic
371C22×C4C22×Q811yescharacteristic
372C22×C4C2×#4111yescharacteristic
373C22×C4C2×#4111yescharacteristic
374C22×C4C2×#4111yescharacteristic
375C22×C4C2×#4111yescharacteristic
376C22×C4C2×#4111yescharacteristic
377C22×C4H(5,𝔽2)11yescharacteristic
378C22×C4H(5,𝔽2)11yescharacteristic
379C22×C4H(5,𝔽2)11yescharacteristic
380C22×C42−1+411yescharacteristic
381D8—232no—
382C2×C8—22no—
383C2×C8—22no—
384C2×C8—22no—
385C2×C8—22no—
386C2×C8—22no—
387C2×C8—22no—
388C2×C8—22no—
389C2×C8—22no—
390C2×C8—22no—
391C2×C8—41no—
392C2×C8—41no—
393C2×C8—41no—
394C2×C8—41no—
395C2×C8—41no—
396C2×C8—41no—
397C2×C8—41no—
398C2×C8—41no—
399C2×C8—41no—
400C2×C8—41no—
401C2×C8—41no—
402C2×C8—41no—
403C2×C8C2×C4⋊C414yes—
404C2×C8C4×D414yes—
405C2×C8C4×D414yes—
406C2×C8#12014yes—
407C2×C8C22⋊Q814yes—
408C2×C8C22⋊Q814yes—
409C2×C8#12314yes—
410C4⋊C4—24no—
411C4⋊C4—24no—
412C4⋊C4—24no—
413C4⋊C4—24no—
414C4⋊C4—48no—
415C4⋊C4—42no—
416C4⋊C4—42no—
417C4⋊C4—42no—
418C4⋊C4—42no—
419C4⋊C4—42no—
420C4⋊C4—42no—
421C4⋊C4—42no—
422C4⋊C4—42no—
423C4⋊C4—88no—
424C22⋊C4—416no—
425C22⋊C4—416no—
426C22⋊C4—88no—
427C22⋊C4—88no—
428C22⋊C4—88no—
429C42—22no—
430C42—22no—
431C42—22no—
432C42—22no—
433C42—22no—
434C42—22no—
435C42—22no—
436C42—22no—
437C42—41no—
438C42—41no—
439C42—41no—
440C42—41no—
441C23—28no—
442C23—44no—
443C23—44no—
444C23—44no—
445C23—44no—
446C23—44no—
447C23—44no—
448C23—44no—
449C23—44no—
450C23—44no—
451C23—44no—
452C23—44no—
453C23—44no—
454C23—44no—
455C23—44no—
456C23—44no—
457C23—44no—
458C23—44no—
459C23—44no—
460C23—82no—
461C23—82no—
462C23—82no—
463C23—82no—
464C23—82no—
465C23—82no—
466C23—82no—
467C23—82no—
468C23—82no—
469C23—82no—
470C23—82no—
471C23—82no—
472C23—82no—
473C23—82no—
474C23—82no—
475C23—82no—
476C23C2×C4×D411yescharacteristic
477C23C2×#12111yescharacteristic
478C23#53211yescharacteristic
479C23D4211yescharacteristic
480C23D4211yescharacteristic
481C23D4×Q811yescharacteristic
482C23#55011yescharacteristic
483C23C2×AGL(1,ℤ/8)12yes—
484C23#57512yes—
485C23#57612yes—
486C23#57712yes—
487D4—28no—
488D4—28no—
489D4—44no—
490D4—44no—
491D4—44no—
492D4—44no—
493D4—44no—
494D4—44no—
495C2×C4—24no—
496C2×C4—22no—
497C2×C4—24no—
498C2×C4—22no—
499C2×C4—24no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—24no—
503C2×C4—24no—
504C2×C4—22no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—44no—
512C2×C4—42no—
513C2×C4—44no—
514C2×C4—44no—
515C2×C4—44no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—44no—
520C2×C4—42no—
521C2×C4—44no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—832no—
525C2×C4—82no—
526C2×C4—82no—
527C2×C4—82no—
528C2×C4—82no—
529C2×C4—82no—
530C2×C4—82no—
531C2×C4—82no—
532C2×C4—82no—
533C2×C4—82no—
534C2×C4—82no—
535C2×C4—82no—
536C2×C4—82no—
537C2×C4—82no—
538C2×C4—82no—
539C2×C4—82no—
540C2×C4#44214yescomplemented
541C2×C4C22×C4⋊C411yescharacteristic
542C2×C4C2×C4×D411yescharacteristic
543C2×C4C2×C4×D411yescharacteristic
544C2×C4C4×#4111yescharacteristic
545C2×C4#51811yescharacteristic
546C2×C4#52011yescharacteristic
547C2×C4C2×#12011yescharacteristic
548C2×C4C2×C22⋊Q811yescharacteristic
549C2×C4C2×C22⋊Q811yescharacteristic
550C2×C4C2×#12311yescharacteristic
551C2×C4#52511yescharacteristic
552C2×C4#52911yescharacteristic
553C2×C4#53511yescharacteristic
554C2×C4#53711yescharacteristic
555C2×C4#54011yescharacteristic
556C2×C4#54211yescharacteristic
557C2×C4#54611yescharacteristic
558C2×C4#54611yescharacteristic
559C2×C4#54611yescharacteristic
560C2×C4#54711yescharacteristic
561C2×C4#54811yescharacteristic
562C2×C4#55011yescharacteristic
563C2×C4#55411yescharacteristic
564C2×C4#55411yescharacteristic
565C2×C4#55411yescharacteristic
566C2×C4#55611yescharacteristic
567C2×C4#55911yescharacteristic
568C2×C4#55911yescharacteristic
569C8—42no—
570C8—42no—
571C8—42no—
572C8#38518yescomplemented
573C22—48no—
574C22—48no—
575C22—48no—
576C22—84no—
577C22—84no—
578C22—84no—
579C22—84no—
580C22—84no—
581C22—84no—
582C22—84no—
583C22—84no—
584C22—84no—
585C22—84no—
586C22—84no—
587C22—84no—
588C22C4×#12111yescharacteristic
589C22D4×C4⋊C411yescharacteristic
590C22#236611yescharacteristic
591C22#240811yescharacteristic
592C22#244511yescharacteristic
593C22#245211yescharacteristic
594C22#253811yescharacteristic
595C22C2×#44211yescharacteristic
596C22#294311yescharacteristic
597C22#294811yescharacteristic
598C22#295211yescharacteristic
599C22#305512yes—
600C22#306212yes—
601C22#317912yes—
602C22#318012yes—
603C22#327212yes—
604C22#327312yes—
605C22#327412yes—
606C22#327512yes—
607C22#338412yes—
608C22#338912yes—
609C22#339912yes—
610C22#340112yes—
611C4—24no—
612C4—42no—
613C4—44no—
614C4—84no—
615C4—82no—
616C4—84no—
617C4—162no—
618C4C2×#38511yescharacteristic
619C4#232411yescharacteristic
620C4#234911yescharacteristic
621C4#236711yescharacteristic
622C4#240711yescharacteristic
623C4#244711yescharacteristic
624C4#245411yescharacteristic
625C4#258011yescharacteristic
626C2—88no—
627C2#1401811yescharacteristic
628C2#2152611yescharacteristic
629C2#2152911yescharacteristic
630C2#2267011yescharacteristic
631C2#2267211yescharacteristic
632C2#2318811yescharacteristic
633C2#2318911yescharacteristic
634C2#2427712yes—
635C2#2626812yes—
636C2#2649612yes—
637C2#2827612yes—
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×#1943C2semidirect
5. C2×#1770C2semidirect
6. #14018C2semidirect
7. #11076C2semidirect
8. #11066C2semidirect
9. #10863C2semidirect
10. #10861C2semidirect
11. #10788C2semidirect
33. #1970C4semidirect
81. C2×#107C2×C4semidirect
99. C2×#102C2×C4semidirect
108. C2×C4×C8C2×C4semidirect
112. #390C2×C4semidirect
165. C2×D8C4⋊C4semidirect
540. C2×C4#442semidirect
572. C8#385semidirect

Maximal subgroups

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

subgroupindex
C22×#4422
C2×#19722
C2×#19702
C2×#19432
C2×#17702
#140182
#110762
#110662
#108632
#108612
#107882