Finite Group Databasebeta

← #196759

Subgroups — #196759

Order 512.

1247 conjugacy classes of subgroups.

The subgroup lattice has 624 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2990C211yesmaximal, characteristic
2#35279C211yesmaximal, characteristic
3#21851C211yesmaximal, characteristic
4C2×#1556C211yesmaximal, characteristic, complemented
5C22×#332C211yesmaximal, characteristic, complemented
6#9140C212yesmaximal, complemented
7#9139C212yesmaximal, complemented
8#9132C212yesmaximal, complemented
9#9131C212yesmaximal, complemented
10#8044C211yesmaximal, characteristic, complemented
11#8044C211yesmaximal, characteristic, complemented
12C22×C4⋊Q8C2211yescharacteristic
13C2×C4×#41C2211yescharacteristic
14#2990—24no—
15#2990—24no—
16C2×#446—21no—
17C2×#435—21no—
18C2×#435—21no—
19C2×C8×D4—21no—
20C2×#432C2211yescharacteristic
21C2×#432C2211yescharacteristic
22C2×#432C2211yescharacteristic
23C2×#431C2211yescharacteristic
24C2×#423C2211yescharacteristic
25C22×C4⋊C8C2211yescharacteristic
26C22×C4⋊C8C2211yescharacteristic
27C22×C4⋊C8C2211yescharacteristic
28#2857C2212yes—
29#2857C2212yes—
30#2628C412yes—
31#2628C412yes—
32#2568—21no—
33#2567—21no—
34#2563—21no—
35#2542—21no—
36#2487—21no—
37#2487—21no—
38#2409—21no—
39#2409—21no—
40#2317C412yes—
41#2311C412yes—
42#2285C2211yescharacteristic
43#1985—21no—
44#1955—21no—
45#1869—21no—
46#1869—21no—
47#1797C2212yes—
48#1797C2212yes—
49#1556—24no—
50C2×#332—24no—
51C2×#332C2212yes—
52C2×#332C2212yes—
53C2×#332C2212yes—
54C2×#332C2212yes—
55#1296C2212yes—
56#1296C2212yes—
57#1296C2212yes—
58#1296C2212yes—
59C22×#41C2×C411yescharacteristic
60C23×Q8—21no—
61C22×M16—21no—
62C23×C8—21no—
63C23×C8—21no—
64C23×C8—21no—
65C2×C4⋊Q8—21no—
66C2×C4⋊Q8—21no—
67C2×C4⋊Q8—22no—
68C2×C4⋊Q8—22no—
69C2×C4⋊Q8—22no—
70C2×C4⋊Q8—22no—
71C2×C4⋊Q8C2×C412yes—
72C2×C4⋊Q8C2312yes—
73C2×C22⋊Q8—22no—
74C2×C22⋊Q8—22no—
75C4×#41—22no—
76C4×#41—22no—
77C4×#41—22no—
78C4×#41—22no—
79C2×C4×Q8D411yescharacteristic
80C2×C4×Q8D411yescharacteristic
81C2×C4×D4D411yescharacteristic
82C2×C4×D4D411yescharacteristic
83C2×C4×D4D411yescharacteristic
84C2×C4×D4D411yescharacteristic
85C2×C4×D4D411yescharacteristic
86C2×C4×D4D411yescharacteristic
87C2×#117C2×C412yes—
88C2×#117C2×C411yescharacteristic
89C2×#117C2×C411yescharacteristic
90C2×#117C2×C411yescharacteristic
91C22×C4⋊C4—21no—
92C22×C4⋊C4C2×C412yes—
93C22×C42C2311yescharacteristic
94C22×C42C2311yescharacteristic
95C22×C42C2311yescharacteristic
96#446—28no—
97#435—28no—
98#435—28no—
99C8×D4—28no—
100#432—22no—
101#432—22no—
102#432—22no—
103#432—22no—
104#432—22no—
105#432D414yes—
106#431—22no—
107#431—22no—
108#423—22no—
109#423—22no—
110C2×C4⋊C8—21no—
111C2×C4⋊C8—21no—
112C2×C4⋊C8—22no—
113C2×C4⋊C8—21no—
114C2×C4⋊C8—22no—
115C2×C4⋊C8—22no—
116C2×C4⋊C8—22no—
117C2×C4⋊C8—22no—
118C2×C4⋊C8—21no—
119C2×C4⋊C8D414yes—
120C2×C4⋊C8C2312yes—
121C2×C4⋊C8C2312yes—
122C2×C4⋊C8C2311yescharacteristic
123C2×C4⋊C8C2311yescharacteristic
124C2×C4⋊C8C2311yescharacteristic
125C2×C4⋊C8C2311yescharacteristic
126C2×C22⋊C8—21no—
127C2×C22⋊C8—21no—
128C2×C22⋊C8—21no—
129C2×C22⋊C8—21no—
130C2×C22⋊C8—22no—
131C2×C22⋊C8—22no—
132C2×#97—21no—
133C2×#97—21no—
134C2×#97—21no—
135C2×C4×C8—21no—
136#400—22no—
137#400—22no—
138#399—22no—
139#399—22no—
140#395—22no—
141#395—22no—
142#393—22no—
143#393—22no—
144#391—22no—
145#391—22no—
146#391—22no—
147#390—22no—
148#389—22no—
149#389—22no—
150#389C2×C414yes—
151#389C2×C414yes—
152#386—22no—
153#386—22no—
154#385—22no—
155#385—22no—
156#384—22no—
157#384C2312yes—
158#382—22no—
159#382C2×C412yes—
160#380—22no—
161#380—22no—
162#380—21no—
163#380—21no—
164#380C2×C414yes—
165#380C2×C414yes—
166C4×C4⋊C4—21no—
167C4×C4⋊C4—21no—
168#336—22no—
169#336—22no—
170#332—24no—
171#332—24no—
172#332—24no—
173#332—24no—
174C2×#41—24no—
175C2×#41—22no—
176C2×#41—22no—
177C2×#41—42no—
178C22×Q8—22no—
179C22×Q8—22no—
180C22×Q8—42no—
181C22×Q8—42no—
182C22×Q8—82no—
183C22×Q8—82no—
184C22×Q8C22⋊C411yescharacteristic
185C22×D4C22⋊C411yescharacteristic
186C22×D4C22⋊C411yescharacteristic
187C22×D4C22⋊C411yescharacteristic
188C23×C4—22no—
189C23×C4C22×C411yescharacteristic
190C23×C4C22×C411yescharacteristic
191C23×C4C22×C411yescharacteristic
192C23×C4C2×D411yescharacteristic
193C23×C4C2×D411yescharacteristic
194C23×C4C2×D411yescharacteristic
195C23×C4C2×D411yescharacteristic
196C23×C4C2×D411yescharacteristic
197C23×C4C2×D411yescharacteristic
198C2×M16—24no—
199C2×M16—82no—
200C22×C8—21no—
201C22×C8—21no—
202C22×C8—21no—
203C22×C8—21no—
204C22×C8—24no—
205C22×C8—24no—
206C22×C8—22no—
207C22×C8—22no—
208C22×C8—42no—
209C22×C8—82no—
210C22×C8—82no—
211C22×C8—82no—
212C4⋊Q8—24no—
213C4⋊Q8—24no—
214C4⋊Q8—22no—
215C4⋊Q8—22no—
216C4⋊Q8—22no—
217C4⋊Q8—22no—
218C4⋊Q8—42no—
219C4⋊Q8—42no—
220C4⋊Q8—42no—
221C4⋊Q8—42no—
222C22⋊Q8—82no—
223C22⋊Q8—82no—
224C22⋊Q8—82no—
225C22⋊Q8—82no—
226C4×Q8—22no—
227C4×Q8—22no—
228C4×Q8—22no—
229C4×Q8—22no—
230C4×D4—22no—
231C4×D4—22no—
232C4×D4—22no—
233C4×D4—22no—
234C4×D4—22no—
235C4×D4—22no—
236C4×D4—22no—
237C4×D4—22no—
238C4×D4—22no—
239C4×D4—22no—
240C4×D4—22no—
241C4×D4—22no—
242#117—22no—
243#117—22no—
244#117—22no—
245#117—22no—
246#117—22no—
247#117—22no—
248#117—44no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—42no—
259C2×C4⋊C4—42no—
260C2×C4⋊C4C22⋊C411yescharacteristic
261C2×C4⋊C4C22⋊C411yescharacteristic
262C2×C4⋊C4C22⋊C411yescharacteristic
263C2×C4⋊C4C22⋊C411yescharacteristic
264C2×C4⋊C4C22⋊C411yescharacteristic
265C2×C4⋊C4C22⋊C411yescharacteristic
266C2×C4⋊C4C22×C414yes—
267C2×C4⋊C4C22×C414yes—
268C2×C22⋊C4—22no—
269C2×C22⋊C4—24no—
270C2×C22⋊C4—24no—
271C2×C22⋊C4C22⋊C411yescharacteristic
272C2×C22⋊C4C22⋊C411yescharacteristic
273C2×C22⋊C4C22⋊C411yescharacteristic
274C2×C22⋊C4C22⋊C411yescharacteristic
275C2×C22⋊C4C22⋊C411yescharacteristic
276C2×C22⋊C4C22⋊C411yescharacteristic
277C2×C42—21no—
278C2×C42—21no—
279C2×C42—21no—
280C2×C42—21no—
281C2×C42—21no—
282C2×C42—21no—
283C2×C42—21no—
284C2×C42—21no—
285C2×C42—22no—
286C2×C42C22×C411yescharacteristic
287C2×C42C22×C411yescharacteristic
288C2×C42C22×C411yescharacteristic
289C2×C42C2×D411yescharacteristic
290C2×C42C2×D411yescharacteristic
291C2×C42C2×D411yescharacteristic
292C2×C42C2×D411yescharacteristic
293C2×C42C2×D412yes—
294C2×C42C2×D412yes—
295C2×C42C2×D411yescharacteristic
296C2×C42C2×D411yescharacteristic
297C2×C42#4112yes—
298C2×C42#4112yes—
299C2×C42C2411yescharacteristic
300C4⋊C8—24no—
301C4⋊C8—24no—
302C4⋊C8—24no—
303C4⋊C8—21no—
304C4⋊C8—21no—
305C4⋊C8—21no—
306C4⋊C8—21no—
307C4⋊C8—22no—
308C4⋊C8—22no—
309C4⋊C8—22no—
310C4⋊C8—22no—
311C4⋊C8—42no—
312C4⋊C8C2×D412yes—
313C4⋊C8C2×D412yes—
314C4⋊C8C2×D412yes—
315C4⋊C8C2×D412yes—
316C22⋊C8—24no—
317C22⋊C8—24no—
318C22⋊C8—24no—
319C22⋊C8—42no—
320C22⋊C8—82no—
321C22⋊C8—82no—
322#97—24no—
323#97—24no—
324#97—42no—
325C4×C8—24no—
326#95—22no—
327#95—22no—
328#95—24no—
329#95—24no—
330#95—24no—
331#95—24no—
332C24C2×C22⋊C411yescharacteristic
333C24C2×C22⋊C411yescharacteristic
334C24C2×C22⋊C411yescharacteristic
335#41—48no—
336#41—48no—
337C2×Q8—22no—
338C2×Q8—22no—
339C2×Q8—41no—
340C2×Q8—42no—
341C2×Q8—42no—
342C2×Q8—42no—
343C2×Q8—42no—
344C2×Q8—42no—
345C2×Q8—42no—
346C2×Q8—44no—
347C2×Q8—84no—
348C2×Q8—88no—
349C2×Q8—84no—
350C2×Q8—84no—
351C2×D4—22no—
352C2×D4—22no—
353C2×D4—22no—
354C2×D4—22no—
355C2×D4—22no—
356C2×D4—22no—
357C2×D4—22no—
358C2×D4—41no—
359C2×D4#10114yes—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—21no—
368C22×C4—21no—
369C22×C4—21no—
370C22×C4—21no—
371C22×C4—21no—
372C22×C4—21no—
373C22×C4—21no—
374C22×C4—21no—
375C22×C4—21no—
376C22×C4—21no—
377C22×C4—21no—
378C22×C4—21no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—41no—
391C22×C4—41no—
392C22×C4—44no—
393C22×C4—84no—
394C22×C4C2×C22⋊C411yescharacteristic
395C22×C4C2×C22⋊C411yescharacteristic
396C22×C4C2×C22⋊C411yescharacteristic
397C22×C4C2×C22⋊C411yescharacteristic
398C22×C4C2×C22⋊C411yescharacteristic
399C22×C4C2×C22⋊C411yescharacteristic
400C22×C4C2×C22⋊C411yescharacteristic
401C22×C4C2×C22⋊C411yescharacteristic
402C22×C4C2×C22⋊C411yescharacteristic
403C22×C4C4×D412yes—
404C22×C4C4×D412yes—
405C22×C4#12012yes—
406C22×C4#12112yes—
407C22×C4#12112yes—
408C22×C4#12312yes—
409C22×C4C23×C411yescharacteristic
410C22×C4C22×D411yescharacteristic
411C22×C4C22×D411yescharacteristic
412M16—84no—
413C2×C8—22no—
414C2×C8—22no—
415C2×C8—22no—
416C2×C8—22no—
417C2×C8—42no—
418C2×C8—42no—
419C2×C8—42no—
420C2×C8—42no—
421C2×C8—42no—
422C2×C8—42no—
423C2×C8—42no—
424C2×C8—42no—
425C2×C8—42no—
426C2×C8—42no—
427C2×C8—84no—
428C2×C8—84no—
429C2×C8—84no—
430C4⋊C4—22no—
431C4⋊C4—22no—
432C4⋊C4—22no—
433C4⋊C4—22no—
434C4⋊C4—22no—
435C4⋊C4—22no—
436C4⋊C4—24no—
437C4⋊C4—24no—
438C4⋊C4—44no—
439C4⋊C4—42no—
440C4⋊C4—42no—
441C4⋊C4—42no—
442C4⋊C4—42no—
443C4⋊C4—42no—
444C4⋊C4—42no—
445C4⋊C4—44no—
446C4⋊C4—44no—
447C4⋊C4—84no—
448C4⋊C4—84no—
449C22⋊C4—22no—
450C22⋊C4—22no—
451C22⋊C4—22no—
452C22⋊C4—22no—
453C22⋊C4—22no—
454C22⋊C4—22no—
455C22⋊C4—44no—
456C22⋊C4—84no—
457C22⋊C4—84no—
458C42—21no—
459C42—21no—
460C42—21no—
461C42—21no—
462C42—21no—
463C42—21no—
464C42—21no—
465C42—21no—
466C42—21no—
467C42—21no—
468C42—21no—
469C42—21no—
470C42—44no—
471C42C4×D412yes—
472C42C4×D412yes—
473C42#12012yes—
474C42#12112yes—
475C42#12112yes—
476C42#12312yes—
477C42C22×D411yescharacteristic
478C42C22×D411yescharacteristic
479C42C2×#4111yescharacteristic
480C42C2×#4111yescharacteristic
481C23—22no—
482C23—22no—
483C23—21no—
484C23—22no—
485C23—22no—
486C23—21no—
487C23—21no—
488C23—21no—
489C23—22no—
490C23—22no—
491C23—21no—
492C23#38612yes—
493C23C2×#10112yes—
494C23C2×#10112yes—
495C23#41312yes—
496C23C22×C22⋊C411yescharacteristic
497Q8—44no—
498Q8—88no—
499Q8—82no—
500Q8—82no—
501Q8—82no—
502Q8—82no—
503D4—28no—
504D4—44no—
505D4—44no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—24no—
514C2×C4—24no—
515C2×C4—21no—
516C2×C4—24no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—24no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—44no—
550C2×C4—44no—
551C2×C4—44no—
552C2×C4—44no—
553C2×C4—44no—
554C2×C4—44no—
555C2×C4—88no—
556C2×C4#38612yes—
557C2×C4#38612yes—
558C2×C4#38612yes—
559C2×C4C2×#10112yes—
560C2×C4C2×C4×D411yescharacteristic
561C2×C4C2×C4×D411yescharacteristic
562C2×C4C2×#12011yescharacteristic
563C2×C4C2×#12111yescharacteristic
564C2×C4C2×#12111yescharacteristic
565C2×C4C2×#12311yescharacteristic
566C2×C4#57612yes—
567C2×C4#57712yes—
568C2×C4#57712yes—
569C2×C4#57812yes—
570C8—42no—
571C8—42no—
572C8—42no—
573C8—42no—
574C8—82no—
575C8—82no—
576C22—24no—
577C22—24no—
578C22—22no—
579C22—22no—
580C22—24no—
581C22—22no—
582C22—42no—
583C22—41no—
584C22—41no—
585C22—42no—
586C22—42no—
587C22—41no—
588C22—41no—
589C22—41no—
590C22#190114yes—
591C22C2×#38611yescharacteristic
592C22C22×#10111yescharacteristic
593C22C2×#41311yescharacteristic
594C22#289712yes—
595C22#289812yes—
596C4—22no—
597C4—22no—
598C4—21no—
599C4—21no—
600C4—22no—
601C4—22no—
602C4—41no—
603C4—41no—
604C4—41no—
605C4—41no—
606C4—41no—
607C4—84no—
608C4—84no—
609C4#300811yescharacteristic
610C4#301011yescharacteristic
611C4#305711yescharacteristic
612C4#305911yescharacteristic
613C4#306312yes—
614C4#309612yes—
615C2—24no—
616C2—42no—
617C2—81no—
618C2#1777412yes—
619C2#1777512yes—
620C2#2091411yescharacteristic
621C2#2097411yescharacteristic
622C2C2×#190111yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC22×#41C2×C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×#41 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. C2×#1556C2semidirect
5. C22×#332C2semidirect
6. #9140C2semidirect
7. #9139C2semidirect
8. #9132C2semidirect
9. #9131C2semidirect
10. #8044C2semidirect
11. #8044C2semidirect

Maximal subgroups

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

subgroupindex
C2×#29902
#352792
#218512
C2×#15562
C22×#3322
#91402
#91392
#91322
#91312
#80442