Finite Group Databasebeta

← #196720

Subgroups — #196720

Order 512.

1229 conjugacy classes of subgroups.

The subgroup lattice has 633 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2925C211yesmaximal, characteristic
2#35279C211yesmaximal, characteristic
3#21853C211yesmaximal, characteristic
4C2×#1558C211yesmaximal, characteristic, complemented
5C2×#1548C211yesmaximal, characteristic, complemented
6#9136C212yesmaximal, complemented
7#9135C212yesmaximal, complemented
8#9128C212yesmaximal, complemented
9#9127C212yesmaximal, complemented
10#8046C211yesmaximal, characteristic, complemented
11#8042C211yesmaximal, characteristic, complemented
12C22×C4⋊Q8C2211yescharacteristic
13C2×C4×#41C2211yescharacteristic
14#2925—24no—
15#2925—24no—
16C2×#433—21no—
17C2×#433—21no—
18C2×#432C2211yescharacteristic
19C2×#432C2211yescharacteristic
20C2×#432C2211yescharacteristic
21C2×#431C2211yescharacteristic
22C2×#431C2211yescharacteristic
23C2×#423C2211yescharacteristic
24C2×#423C2211yescharacteristic
25C22×C4⋊C8C2211yescharacteristic
26C2×#408—21no—
27C2×#405—21no—
28#2857C2212yes—
29#2857C2212yes—
30#2628—21no—
31#2628—21no—
32#2568C412yes—
33#2567—21no—
34#2563—21no—
35#2542C412yes—
36#2487—21no—
37#2487—21no—
38#2409C412yes—
39#2409C412yes—
40#2317—21no—
41#2311—21no—
42#2285C2211yescharacteristic
43#1985—21no—
44#1955—21no—
45#1870—21no—
46#1870—21no—
47#1797C2212yes—
48#1797C2212yes—
49#1558—24no—
50#1548—24no—
51C2×#333C2212yes—
52C2×#333C2212yes—
53C2×#333C2212yes—
54C2×#333C2212yes—
55#1296C2212yes—
56#1296C2212yes—
57#1296C2212yes—
58#1296C2212yes—
59C22×#41D411yescharacteristic
60C23×Q8C2×C412yes—
61C22×M16—21no—
62C22×M16—21no—
63C23×C8—21no—
64C2×C4⋊Q8—21no—
65C2×C4⋊Q8—21no—
66C2×C4⋊Q8—22no—
67C2×C4⋊Q8—22no—
68C2×C4⋊Q8—22no—
69C2×C4⋊Q8—22no—
70C2×C4⋊Q8C2×C412yes—
71C2×C4⋊Q8C2312yes—
72C2×C22⋊Q8C2×C414yes—
73C2×C22⋊Q8C2×C414yes—
74C4×#41—22no—
75C4×#41—22no—
76C4×#41—22no—
77C4×#41—22no—
78C2×C4×Q8C2×C411yescharacteristic
79C2×C4×Q8D411yescharacteristic
80C2×C4×D4C2×C411yescharacteristic
81C2×C4×D4C2×C411yescharacteristic
82C2×C4×D4C2×C411yescharacteristic
83C2×C4×D4D411yescharacteristic
84C2×C4×D4D411yescharacteristic
85C2×C4×D4D411yescharacteristic
86C2×#117—21no—
87C2×#117D411yescharacteristic
88C2×#117D411yescharacteristic
89C2×#117D411yescharacteristic
90C22×C4⋊C4—21no—
91C22×C4⋊C4—21no—
92C22×C42C2311yescharacteristic
93C22×C42C2311yescharacteristic
94C22×C42C2311yescharacteristic
95#433—28no—
96#433—28no—
97#432—22no—
98#432—22no—
99#432—22no—
100#432—22no—
101#432—22no—
102#432D414yes—
103#431—22no—
104#431—22no—
105#431—22no—
106#431D414yes—
107#423—22no—
108#423—22no—
109#423—22no—
110#423—22no—
111C2×C4⋊C8—21no—
112C2×C4⋊C8—21no—
113C2×C4⋊C8—22no—
114C2×C4⋊C8—22no—
115C2×C4⋊C8—21no—
116C2×C4⋊C8C2312yes—
117C2×C4⋊C8C2312yes—
118C2×C4⋊C8C2311yescharacteristic
119C2×C4⋊C8C2311yescharacteristic
120C2×C4⋊C8C2311yescharacteristic
121C2×C4⋊C8C2311yescharacteristic
122#408—28no—
123C2×C22⋊C8—21no—
124C2×C22⋊C8—21no—
125C2×C22⋊C8—21no—
126C2×C22⋊C8—21no—
127C2×C22⋊C8—21no—
128C2×C22⋊C8—22no—
129C2×C22⋊C8—22no—
130#405—28no—
131C2×#97—21no—
132C2×#97—21no—
133C2×#97—21no—
134C2×C4×C8—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#391C2×C414yes—
146#391C2×C414yes—
147#390—22no—
148#389—22no—
149#389—22no—
150#389—22no—
151#389—22no—
152#386—22no—
153#386—22no—
154#385—22no—
155#385—22no—
156#384—22no—
157#384C2312yes—
158#382—22no—
159#382—21no—
160#380—22no—
161#380—22no—
162#380—22no—
163#380—22no—
164#380—21no—
165#380C2×C412yes—
166C4×C4⋊C4—21no—
167C4×C4⋊C4C2×C412yes—
168#336—22no—
169#336—22no—
170#333—24no—
171#333—24no—
172#333—24no—
173#333—24no—
174C2×#41—22no—
175C2×#41—22no—
176C2×#41—22no—
177C2×#41—22no—
178C2×#41—41no—
179C2×#41—41no—
180C22×Q8—22no—
181C22×Q8—24no—
182C22×Q8—24no—
183C22×Q8—84no—
184C22×Q8C22⋊C411yescharacteristic
185C22×Q8C22×C414yes—
186C22×D4C22⋊C411yescharacteristic
187C22×D4C22⋊C411yescharacteristic
188C22×D4C22⋊C411yescharacteristic
189C23×C4—22no—
190C23×C4C22×C411yescharacteristic
191C23×C4C22×C411yescharacteristic
192C23×C4C22×C411yescharacteristic
193C23×C4C2×D411yescharacteristic
194C23×C4C2×D411yescharacteristic
195C23×C4C2×D411yescharacteristic
196C23×C4C2×D411yescharacteristic
197C23×C4C2×D411yescharacteristic
198C23×C4C2×D411yescharacteristic
199C2×M16—24no—
200C2×M16—42no—
201C2×M16—42no—
202C2×M16—42no—
203C2×M16—82no—
204C22×C8—24no—
205C22×C8—21no—
206C22×C8—21no—
207C22×C8—21no—
208C22×C8—21no—
209C22×C8—22no—
210C22×C8—22no—
211C22×C8—42no—
212C22×C8—42no—
213C4⋊Q8—24no—
214C4⋊Q8—24no—
215C4⋊Q8—22no—
216C4⋊Q8—22no—
217C4⋊Q8—22no—
218C4⋊Q8—22no—
219C4⋊Q8—42no—
220C4⋊Q8—42no—
221C4⋊Q8—42no—
222C4⋊Q8—42no—
223C22⋊Q8—84no—
224C22⋊Q8—84no—
225C4×Q8—22no—
226C4×Q8—22no—
227C4×Q8—22no—
228C4×Q8—22no—
229C4×D4—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—
241#117—22no—
242#117—22no—
243#117—22no—
244#117—22no—
245#117—22no—
246#117—22no—
247#117—42no—
248#117—42no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—42no—
258C2×C4⋊C4—42no—
259C2×C4⋊C4—42no—
260C2×C4⋊C4—42no—
261C2×C4⋊C4C22⋊C411yescharacteristic
262C2×C4⋊C4C22⋊C411yescharacteristic
263C2×C4⋊C4C22⋊C411yescharacteristic
264C2×C4⋊C4C22⋊C411yescharacteristic
265C2×C4⋊C4C22⋊C411yescharacteristic
266C2×C4⋊C4C22⋊C411yescharacteristic
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—21no—
303C4⋊C8—21no—
304C4⋊C8—21no—
305C4⋊C8—21no—
306C4⋊C8—22no—
307C4⋊C8—22no—
308C4⋊C8—22no—
309C4⋊C8—22no—
310C4⋊C8—42no—
311C4⋊C8C2×D412yes—
312C4⋊C8C2×D412yes—
313C4⋊C8C2×D412yes—
314C4⋊C8C2×D412yes—
315C22⋊C8—24no—
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—
326C4×C8—24no—
327#95—22no—
328#95—22no—
329#95—24no—
330#95—24no—
331#95—24no—
332#95—24no—
333C24C2×C22⋊C411yescharacteristic
334C24C2×C22⋊C411yescharacteristic
335C24C2×C22⋊C411yescharacteristic
336#41—44no—
337#41—44no—
338#41—44no—
339#41—44no—
340C2×Q8—22no—
341C2×Q8—22no—
342C2×Q8—24no—
343C2×Q8—24no—
344C2×Q8—41no—
345C2×Q8—44no—
346C2×Q8—44no—
347C2×Q8—44no—
348C2×Q8—84no—
349C2×Q8—88no—
350C2×Q8—88no—
351C2×D4—22no—
352C2×D4—22no—
353C2×D4—22no—
354C2×D4—22no—
355C2×D4—22no—
356C2×D4—22no—
357C2×D4—41no—
358C2×D4—41no—
359C2×D4—41no—
360C22×C4—21no—
361C22×C4—21no—
362C22×C4—21no—
363C22×C4—21no—
364C22×C4—21no—
365C22×C4—21no—
366C22×C4—21no—
367C22×C4—21no—
368C22×C4—22no—
369C22×C4—22no—
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—21no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—21no—
385C22×C4—21no—
386C22×C4—21no—
387C22×C4—21no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—42no—
391C22×C4—42no—
392C22×C4—42no—
393C22×C4—44no—
394C22×C4—84no—
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×C4C2×C22⋊C411yescharacteristic
404C22×C4C4×D412yes—
405C22×C4C4×D412yes—
406C22×C4#12012yes—
407C22×C4#12112yes—
408C22×C4#12112yes—
409C22×C4#12312yes—
410C22×C4#13712yes—
411C22×C4#13712yes—
412C22×C4C23×C411yescharacteristic
413C22×C4C22×D411yescharacteristic
414C22×C4C22×D411yescharacteristic
415M16—48no—
416M16—84no—
417C2×C8—22no—
418C2×C8—22no—
419C2×C8—22no—
420C2×C8—22no—
421C2×C8—48no—
422C2×C8—41no—
423C2×C8—41no—
424C2×C8—41no—
425C2×C8—41no—
426C2×C8—41no—
427C2×C8—41no—
428C2×C8—41no—
429C2×C8—41no—
430C2×C8—42no—
431C2×C8—42no—
432C2×C8—42no—
433C2×C8—42no—
434C2×C8—42no—
435C2×C8—42no—
436C4⋊C4—22no—
437C4⋊C4—22no—
438C4⋊C4—22no—
439C4⋊C4—22no—
440C4⋊C4—22no—
441C4⋊C4—22no—
442C4⋊C4—44no—
443C4⋊C4—42no—
444C4⋊C4—42no—
445C4⋊C4—42no—
446C4⋊C4—42no—
447C4⋊C4—42no—
448C4⋊C4—42no—
449C4⋊C4—42no—
450C4⋊C4—42no—
451C4⋊C4—44no—
452C4⋊C4—44no—
453C4⋊C4—84no—
454C4⋊C4—84no—
455C22⋊C4—22no—
456C22⋊C4—22no—
457C22⋊C4—22no—
458C22⋊C4—22no—
459C22⋊C4—22no—
460C22⋊C4—22no—
461C22⋊C4—44no—
462C22⋊C4—84no—
463C22⋊C4—84no—
464C42—21no—
465C42—21no—
466C42—21no—
467C42—21no—
468C42—21no—
469C42—21no—
470C42—21no—
471C42—21no—
472C42—21no—
473C42—21no—
474C42—21no—
475C42—21no—
476C42—44no—
477C42C4×D412yes—
478C42C4×D412yes—
479C42#12012yes—
480C42#12112yes—
481C42#12112yes—
482C42#12312yes—
483C42C22×D411yescharacteristic
484C42C22×D411yescharacteristic
485C42C2×#4111yescharacteristic
486C42C2×#4111yescharacteristic
487C23—22no—
488C23—22no—
489C23—21no—
490C23—21no—
491C23—21no—
492C23—22no—
493C23—22no—
494C23—21no—
495C23—22no—
496C23—21no—
497C23—21no—
498C23—22no—
499C23—21no—
500C23#38612yes—
501C23#41912yes—
502C23C22×C22⋊C411yescharacteristic
503Q8—44no—
504Q8—84no—
505Q8—84no—
506Q8—88no—
507D4—44no—
508D4—44no—
509D4—44no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—22no—
516C2×C4—21no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—24no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—44no—
552C2×C4—44no—
553C2×C4—44no—
554C2×C4—44no—
555C2×C4—44no—
556C2×C4—44no—
557C2×C4—81no—
558C2×C4—81no—
559C2×C4—81no—
560C2×C4—81no—
561C2×C4—88no—
562C2×C4#38612yes—
563C2×C4#38612yes—
564C2×C4#38612yes—
565C2×C4#41912yes—
566C2×C4C2×C4×D411yescharacteristic
567C2×C4C2×C4×D411yescharacteristic
568C2×C4C2×#12011yescharacteristic
569C2×C4C2×#12111yescharacteristic
570C2×C4C2×#12111yescharacteristic
571C2×C4C2×#12311yescharacteristic
572C2×C4C2×#13711yescharacteristic
573C2×C4C2×#13711yescharacteristic
574C2×C4#57511yescharacteristic
575C2×C4#57511yescharacteristic
576C2×C4#57712yes—
577C2×C4#57812yes—
578C8—42no—
579C8—42no—
580C8—42no—
581C8—42no—
582C8—82no—
583C8—82no—
584C22—24no—
585C22—22no—
586C22—22no—
587C22—22no—
588C22—22no—
589C22—42no—
590C22—42no—
591C22—42no—
592C22—41no—
593C22—41no—
594C22—42no—
595C22—42no—
596C22—41no—
597C22—41no—
598C22—41no—
599C22C2×#38611yescharacteristic
600C22#288812yes—
601C22C2×#41911yescharacteristic
602C22#289611yescharacteristic
603C22#289812yes—
604C4—22no—
605C4—22no—
606C4—22no—
607C4—21no—
608C4—21no—
609C4—22no—
610C4—41no—
611C4—41no—
612C4—41no—
613C4—41no—
614C4—41no—
615C4—84no—
616C4—84no—
617C4#300711yescharacteristic
618C4#300711yescharacteristic
619C4#305711yescharacteristic
620C4#305911yescharacteristic
621C4#306312yes—
622C4#309511yescharacteristic
623C4#309511yescharacteristic
624C2—42no—
625C2—42no—
626C2—81no—
627C2#1776012yes—
628C2#1776812yes—
629C2#2091111yescharacteristic
630C2#2097411yescharacteristic
631C2#2099011yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC22×#41D4
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×#1558C2semidirect
5. C2×#1548C2semidirect
6. #9136C2semidirect
7. #9135C2semidirect
8. #9128C2semidirect
9. #9127C2semidirect
10. #8046C2semidirect
11. #8042C2semidirect

Maximal subgroups

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

subgroupindex
C2×#29252
#352792
#218532
C2×#15582
C2×#15482
#91362
#91352
#91282
#91272
#80462
#80422