Finite Group Databasebeta

← #368253

Subgroups — #368253

Order 512.

1352 conjugacy classes of subgroups.

The subgroup lattice has 596 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2369C211yesmaximal, characteristic
2C2×#1990C211yesmaximal, characteristic
3C2×#1987C211yesmaximal, characteristic
4C2×C4×#106C211yesmaximal, characteristic
5#14151C211yesmaximal, characteristic
6#11327C212yesmaximal
7#11292C212yesmaximal
8#10908C212yesmaximal
9#10884C212yesmaximal
10#10849C211yesmaximal, characteristic
11#10834C211yesmaximal, characteristic
12C22×C4⋊Q8C2211yescharacteristic
13C22×#106C2211yescharacteristic
14C22×#106C2211yescharacteristic
15C22×C4⋊C8C2211yescharacteristic
16C22×C4×C8C2211yescharacteristic
17#2369—24no—
18#2369—24no—
19C2×#391—22no—
20C2×#389—21no—
21C2×#389—21no—
22C2×#389C2212yes—
23C2×#389C2212yes—
24C2×#384C2211yescharacteristic
25C2×C4×C4⋊C4—21no—
26C2×C4×C4⋊C4C2211yescharacteristic
27C2×C4×C4⋊C4C2212yes—
28C2×C4×C4⋊C4C2211yescharacteristic
29C2×C4×C4⋊C4C2211yescharacteristic
30#1990C418yescomplemented
31#1987—24no—
32C4×#106—24no—
33C2×#340C2211yescharacteristic
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#340C2211yescharacteristic
38C2×#340C2212yes—
39C2×#340C2212yes—
40C2×#340C2212yes—
41C2×#336C2212yes—
42C2×#336C2212yes—
43#1441—22no—
44C4⋊#95—21no—
45C4⋊#95C2212yes—
46C23×Q8—21no—
47C23×C8C2312yes—
48C23×C8C2311yescharacteristic
49C23×C8C2311yescharacteristic
50C2×C4⋊Q8—22no—
51C2×C4⋊Q8—22no—
52C2×C4⋊Q8—22no—
53C2×C4⋊Q8C2×C414yes—
54C2×C4⋊Q8D414yes—
55C2×C4⋊Q8Q814yes—
56C22×C4⋊C4—21no—
57C22×C4⋊C4—21no—
58C22×C4⋊C4—21no—
59C22×C4⋊C4—22no—
60C22×C4⋊C4—22no—
61C22×C4⋊C4D411yescharacteristic
62C22×C4⋊C4D411yescharacteristic
63C22×C4⋊C4C2312yes—
64C22×C4⋊C4C2311yescharacteristic
65C22×C4⋊C4C2311yescharacteristic
66C22×C4⋊C4C2311yescharacteristic
67C22×C4⋊C4C2312yes—
68C22×C4⋊C4C2311yescharacteristic
69C22×C42—21no—
70C22×C42—22no—
71C22×C42—21no—
72C22×C42C2312yes—
73C22×C42C2311yescharacteristic
74C2×#106—22no—
75C2×#106—22no—
76C2×#106—22no—
77C2×#106—22no—
78C2×#106—22no—
79C2×#106Q814yescomplemented
80C2×C4⋊C8—22no—
81C2×C4⋊C8—22no—
82C2×C4⋊C8—22no—
83C2×C4×C8—22no—
84C2×C4×C8—22no—
85C2×C4×C8C2×C414yes—
86#391—216no—
87#389—28no—
88#389—24no—
89#389—24no—
90#389—24no—
91#389—42no—
92#389—44no—
93#384—22no—
94#384C2×C414yes—
95C4×C4⋊C4—28no—
96C4×C4⋊C4—24no—
97C4×C4⋊C4—24no—
98C4×C4⋊C4—22no—
99C4×C4⋊C4—22no—
100C4×C4⋊C4—22no—
101C4×C4⋊C4—22no—
102C4×C4⋊C4—22no—
103C4×C4⋊C4D414yes—
104C2×#95—22no—
105C2×#95—22no—
106C2×#95—21no—
107C2×#95—22no—
108C2×#95—22no—
109C2×#95—21no—
110C2×#95—22no—
111C2×#95D412yes—
112#340—24no—
113#340—24no—
114#340—22no—
115#340—22no—
116#340—24no—
117#340—42no—
118#340C2×C418yes—
119#340C2×C418yes—
120#336—24no—
121#336—42no—
122C22×Q8—24no—
123C22×Q8—24no—
124C22×Q8—24no—
125C22×Q8—81no—
126C22×Q8—81no—
127C22×Q8—81no—
128C22×Q8—81no—
129C23×C4—22no—
130C23×C4—21no—
131C23×C4—22no—
132C23×C4—21no—
133C23×C4—21no—
134C23×C4—21no—
135C23×C4C2×D412yes—
136C23×C4C2×D412yes—
137C23×C4C2×D411yescharacteristic
138C23×C4C2×D411yescharacteristic
139C23×C4C2411yescharacteristic
140C22×C8—21no—
141C22×C8—21no—
142C22×C8—21no—
143C22×C8—21no—
144C22×C8—21no—
145C22×C8—21no—
146C22×C8—21no—
147C22×C8—21no—
148C22×C8—21no—
149C22×C8—21no—
150C22×C8—22no—
151C22×C8—22no—
152C22×C8—22no—
153C22×C8—22no—
154C22×C8—22no—
155C22×C8—22no—
156C22×C8—22no—
157C22×C8C22×C412yes—
158C22×C8C22×C412yes—
159C22×C8C2×Q812yes—
160C22×C8#4112yes—
161C4⋊Q8—28no—
162C4⋊Q8—28no—
163C4⋊Q8—28no—
164C4⋊Q8C4⋊C4116yescomplemented
165C2×C4⋊C4—24no—
166C2×C4⋊C4—24no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—24no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—24no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—28no—
201C2×C4⋊C4—24no—
202C2×C4⋊C4—42no—
203C2×C4⋊C4—42no—
204C2×C4⋊C4—44no—
205C2×C4⋊C4—44no—
206C2×C4⋊C4—81no—
207C2×C4⋊C4—81no—
208C2×C4⋊C4—82no—
209C2×C4⋊C4—82no—
210C2×C4⋊C4—82no—
211C2×C4⋊C4SD1614yes—
212C2×C4⋊C4C22×C414yes—
213C2×C4⋊C4C22×C414yes—
214C2×C4⋊C4C2×D412yes—
215C2×C4⋊C4C2×D414yes—
216C2×C4⋊C4C2×D412yes—
217C2×C4⋊C4C2×Q812yes—
218C2×C4⋊C4C2×Q814yes—
219C2×C4⋊C4C2×Q812yes—
220C2×C4⋊C4#4112yes—
221C2×C4⋊C4#4112yes—
222C2×C42—24no—
223C2×C42—22no—
224C2×C42—22no—
225C2×C42—22no—
226C2×C42—22no—
227C2×C42—22no—
228C2×C42—22no—
229C2×C42—28no—
230C2×C42—42no—
231C2×C42—42no—
232C2×C42—42no—
233C2×C42—42no—
234C2×C42—42no—
235C2×C42—82no—
236C2×C42—82no—
237C2×C42C22×C412yes—
238C2×C42C2×D412yes—
239C2×C42C2×D412yes—
240C2×C42C2×Q812yes—
241C2×C42#4112yes—
242C2×C42#4112yes—
243#106—28no—
244#106—28no—
245C4⋊C8—44no—
246C4×C8—28no—
247#95—28no—
248#95—28no—
249#95—24no—
250#95—24no—
251#95—44no—
252#95—42no—
253#95—44no—
254#95—82no—
255C24C22×D411yescharacteristic
256C2×Q8—216no—
257C2×Q8—84no—
258C2×Q8—84no—
259C2×Q8—84no—
260C2×Q8—84no—
261C2×Q8—84no—
262C2×Q8—84no—
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—21no—
266C22×C4—21no—
267C22×C4—24no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—24no—
271C22×C4—24no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—24no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—41no—
291C22×C4—41no—
292C22×C4—41no—
293C22×C4—42no—
294C22×C4—42no—
295C22×C4—41no—
296C22×C4—42no—
297C22×C4—42no—
298C22×C4—41no—
299C22×C4—41no—
300C22×C4—42no—
301C22×C4—41no—
302C22×C4—41no—
303C22×C4—41no—
304C22×C4—42no—
305C22×C4—41no—
306C22×C4—41no—
307C22×C4—42no—
308C22×C4—41no—
309C22×C4—41no—
310C22×C4—42no—
311C22×C4—41no—
312C22×C4—41no—
313C22×C4—41no—
314C22×C4—41no—
315C22×C4—42no—
316C22×C4—42no—
317C22×C4—41no—
318C22×C4—41no—
319C22×C4—41no—
320C22×C4—41no—
321C22×C4—41no—
322C22×C4C4×D412yes—
323C22×C4C4×D412yes—
324C22×C4#12012yes—
325C22×C4#12012yes—
326C22×C4C22⋊Q812yes—
327C22×C4C22⋊Q812yes—
328C22×C4C2×SD1614yes—
329C22×C4C2×SD1612yes—
330C22×C4#13512yes—
331C22×C4#13712yes—
332C22×C4#13712yes—
333C22×C4C23×C411yescharacteristic
334C22×C4C22×D411yescharacteristic
335C22×C4C22×D411yescharacteristic
336C22×C4C22×Q811yescharacteristic
337C22×C4C22×Q811yescharacteristic
338C22×C4C2×#4111yescharacteristic
339C22×C4C2×#4111yescharacteristic
340C22×C4C2×#4111yescharacteristic
341C22×C4C2×#4111yescharacteristic
342C22×C4C2×#4111yescharacteristic
343C22×C4H(5,𝔽2)11yescharacteristic
344C22×C42−1+411yescharacteristic
345C22×C42−1+411yescharacteristic
346C22×C42−1+411yescharacteristic
347C2×C8—22no—
348C2×C8—22no—
349C2×C8—22no—
350C2×C8—22no—
351C2×C8—22no—
352C2×C8—22no—
353C2×C8—22no—
354C2×C8—22no—
355C2×C8—22no—
356C2×C8—41no—
357C2×C8—41no—
358C2×C8—41no—
359C2×C8—41no—
360C2×C8—42no—
361C2×C8—42no—
362C2×C8—42no—
363C2×C8—42no—
364C2×C8—42no—
365C2×C8—42no—
366C2×C8—42no—
367C2×C8C4×Q814yes—
368C4⋊C4—24no—
369C4⋊C4—24no—
370C4⋊C4—24no—
371C4⋊C4—22no—
372C4⋊C4—22no—
373C4⋊C4—22no—
374C4⋊C4—216no—
375C4⋊C4—28no—
376C4⋊C4—24no—
377C4⋊C4—24no—
378C4⋊C4—22no—
379C4⋊C4—22no—
380C4⋊C4—22no—
381C4⋊C4—22no—
382C4⋊C4—22no—
383C4⋊C4—22no—
384C4⋊C4—41no—
385C4⋊C4—41no—
386C4⋊C4—41no—
387C4⋊C4—41no—
388C4⋊C4—41no—
389C4⋊C4—41no—
390C4⋊C4—42no—
391C4⋊C4—42no—
392C4⋊C4—48no—
393C4⋊C4—44no—
394C4⋊C4—84no—
395C4⋊C4—88no—
396C4⋊C4—88no—
397C4⋊C4C2×C4⋊C418yes—
398C4⋊C4C22⋊Q814yes—
399C42—42no—
400C42—42no—
401C42—42no—
402C42—42no—
403C42—48no—
404C42—88no—
405C42—84no—
406C42C2×C4⋊C414yes—
407C42C4×D414yes—
408C42C22⋊Q814yes—
409C42#12314yes—
410C23C2×C4×D411yescharacteristic
411C23C2×#12011yescharacteristic
412C23C2×C22⋊Q811yescharacteristic
413C23#54611yescharacteristic
414C23#54711yescharacteristic
415C23#54811yescharacteristic
416C23D4×Q811yescharacteristic
417C23C22×SD1611yescharacteristic
418C23C2×#13511yescharacteristic
419C23C2×#13711yescharacteristic
420C23C2×#13711yescharacteristic
421C23#57611yescharacteristic
422C23#57611yescharacteristic
423C23#57711yescharacteristic
424C23#57711yescharacteristic
425Q8—816no—
426C2×C4—22no—
427C2×C4—22no—
428C2×C4—22no—
429C2×C4—22no—
430C2×C4—22no—
431C2×C4—22no—
432C2×C4—24no—
433C2×C4—24no—
434C2×C4—24no—
435C2×C4—24no—
436C2×C4—28no—
437C2×C4—42no—
438C2×C4—44no—
439C2×C4—44no—
440C2×C4—42no—
441C2×C4—42no—
442C2×C4—42no—
443C2×C4—42no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—44no—
447C2×C4—44no—
448C2×C4—42no—
449C2×C4—42no—
450C2×C4—42no—
451C2×C4—44no—
452C2×C4—44no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—44no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—44no—
465C2×C4—44no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—81no—
473C2×C4—81no—
474C2×C4—81no—
475C2×C4—81no—
476C2×C4—81no—
477C2×C4—82no—
478C2×C4—81no—
479C2×C4—82no—
480C2×C4—81no—
481C2×C4—81no—
482C2×C4—81no—
483C2×C4—81no—
484C2×C4—81no—
485C2×C4—82no—
486C2×C4—81no—
487C2×C4—82no—
488C2×C4#38514yes—
489C2×C4#38514yes—
490C2×C4C4×SD1614yes—
491C2×C4#44114yes—
492C2×C4#44814yes—
493C2×C4#45214yes—
494C2×C4#47514yes—
495C2×C4#47914yes—
496C2×C4C22×C4⋊C411yescharacteristic
497C2×C4C2×C4×D411yescharacteristic
498C2×C4C2×C4×Q811yescharacteristic
499C2×C4C4×#4111yescharacteristic
500C2×C4#51911yescharacteristic
501C2×C4#52011yescharacteristic
502C2×C4C2×C22⋊Q811yescharacteristic
503C2×C4C2×C22⋊Q811yescharacteristic
504C2×C4C2×#12311yescharacteristic
505C2×C4#52511yescharacteristic
506C2×C4#52911yescharacteristic
507C2×C4#53311yescharacteristic
508C2×C4#53611yescharacteristic
509C2×C4#53711yescharacteristic
510C2×C4#54111yescharacteristic
511C2×C4#54211yescharacteristic
512C2×C4#54811yescharacteristic
513C2×C4#54811yescharacteristic
514C2×C4D4×Q811yescharacteristic
515C2×C4#55011yescharacteristic
516C2×C4#55611yescharacteristic
517C2×C4#55611yescharacteristic
518C2×C4#55711yescharacteristic
519C2×C4#55711yescharacteristic
520C2×C4#55711yescharacteristic
521C2×C4#55711yescharacteristic
522C2×C4Q8211yescharacteristic
523C2×C4#55911yescharacteristic
524C8—24no—
525C8—42no—
526C8—82no—
527C22C2×#38511yescharacteristic
528C22#236511yescharacteristic
529C22#236711yescharacteristic
530C22#240711yescharacteristic
531C22#240911yescharacteristic
532C22#245111yescharacteristic
533C22#245311yescharacteristic
534C22C2×C4×SD1611yescharacteristic
535C22C2×#44111yescharacteristic
536C22#294811yescharacteristic
537C22#294911yescharacteristic
538C22C2×#44811yescharacteristic
539C22C2×#45211yescharacteristic
540C22#301211yescharacteristic
541C22#301211yescharacteristic
542C22C2×#47511yescharacteristic
543C22C2×#47911yescharacteristic
544C22#308111yescharacteristic
545C22#308111yescharacteristic
546C22#329511yescharacteristic
547C22#330511yescharacteristic
548C22#332911yescharacteristic
549C22#333111yescharacteristic
550C22#333511yescharacteristic
551C22#334011yescharacteristic
552C22#335711yescharacteristic
553C22#335711yescharacteristic
554C22#336111yescharacteristic
555C22#336211yescharacteristic
556C22#336911yescharacteristic
557C22#337211yescharacteristic
558C22Q8×SD1611yescharacteristic
559C22#338111yescharacteristic
560C22#338511yescharacteristic
561C22#338511yescharacteristic
562C4—24no—
563C4—44no—
564C4—44no—
565C4—48no—
566C4—82no—
567C4—82no—
568C4—82no—
569C4—82no—
570C4—84no—
571C4#186518yes—
572C4C4×C22⋊Q811yescharacteristic
573C4#232311yescharacteristic
574C4Q8×C4⋊C411yescharacteristic
575C4#235111yescharacteristic
576C4#244611yescharacteristic
577C4#244811yescharacteristic
578C4#258111yescharacteristic
579C4#258211yescharacteristic
580C2#1401711yescharacteristic
581C2C2×#186511yescharacteristic
582C2#2087911yescharacteristic
583C2#2315611yescharacteristic
584C2#2316211yescharacteristic
585C2#2321511yescharacteristic
586C2#2322111yescharacteristic
587C2#2411811yescharacteristic
588C2#2412111yescharacteristic
589C2#2432911yescharacteristic
590C2#2433111yescharacteristic
591C2#2644211yescharacteristic
592C2#2644511yescharacteristic
593C2#2650811yescharacteristic
594C2#2651111yescharacteristic
subgroupG/H
center, socle, prime radicalC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
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
30. #1990C4semidirect
79. C2×#106Q8semidirect
164. C4⋊Q8C4⋊C4semidirect

Maximal subgroups

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

subgroupindex
C2×#23692
C2×#19902
C2×#19872
C2×C4×#1062
#141512
#113272
#112922
#109082
#108842
#108492
#108342