Finite Group Databasebeta

← #209193

Subgroups — #209193

Order 512.

1181 conjugacy classes of subgroups.

The subgroup lattice has 698 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2972C211yesmaximal, characteristic
2#22098C211yesmaximal, characteristic, complemented
3C4×#450C211yesmaximal, characteristic
4C2×SD16⋊C8C211yesmaximal, characteristic, complemented
5C2×C8×SD16C211yesmaximal, characteristic, complemented
6#8945C211yesmaximal, characteristic, complemented
7#8945C211yesmaximal, characteristic, complemented
8#8938C211yesmaximal, characteristic, complemented
9#8845C211yesmaximal, characteristic, complemented
10#8843C211yesmaximal, characteristic, complemented
11#8831C211yesmaximal, characteristic, complemented
12#8310C211yesmaximal, characteristic
13#8200C211yesmaximal, characteristic, complemented
14C8×#102C211yesmaximal, characteristic, complemented
15#8148C211yesmaximal, characteristic, complemented
16C22×C4×D4C2211yescharacteristic
17C2×#450C411yescharacteristic
18#2972—24no—
19#2972—24no—
20C2×C8×Q8C2211yescharacteristic, complemented
21C2×C4×SD16C2211yescharacteristic
22C2×C4×SD16C2211yescharacteristic
23C2×C8×D4—21no—
24C2×C8×D4—21no—
25C2×C8×D4C2211yescharacteristic
26C2×C8×D4C2211yescharacteristic
27C2×C8×D4C2211yescharacteristic
28C2×C8×D4C2211yescharacteristic
29C2×#431C2211yescharacteristic
30C2×#431C2211yescharacteristic
31C22×C22⋊C8—21no—
32C22×C22⋊C8—21no—
33C4×C22⋊Q8C2211yescharacteristic
34#1984C2211yescharacteristic, complemented
35#1939—21no—
36#1939C2211yescharacteristic, complemented
37#1923—21no—
38#1923—21no—
39#1916—21no—
40#1877C411yescharacteristic
41#1877C411yescharacteristic
42#1872C411yescharacteristic
43#1864C411yescharacteristic
44#1864C411yescharacteristic
45#1853C411yescharacteristic
46#1797C2211yescharacteristic
47#1786C411yescharacteristic
48C8×C4⋊C4C2211yescharacteristic, complemented
49C4×#102C2211yescharacteristic
50C4×#102C2211yescharacteristic
51C4×C22⋊C8C2211yescharacteristic
52C4×C22⋊C8C2211yescharacteristic
53SD16⋊C8—24no—
54C8×SD16—24no—
55C2×#335C2211yescharacteristic, complemented
56C2×#334C2211yescharacteristic, complemented
57C2×Q8⋊C8C2211yescharacteristic, complemented
58C2×Q8⋊C8C2211yescharacteristic, complemented
59C2×D4⋊C8C2211yescharacteristic
60C2×D4⋊C8C2211yescharacteristic
61C2×D4⋊C8C2211yescharacteristic
62C2×D4⋊C8C2211yescharacteristic
63C2×#322C2211yescharacteristic, complemented
64C2×C82C2211yescharacteristic, complemented
65#1279C2211yescharacteristic, complemented
66#1279C2211yescharacteristic, complemented
67#1277C2211yescharacteristic, complemented
68#1277C2211yescharacteristic, complemented
69#1273C2211yescharacteristic, complemented
70#1273C2211yescharacteristic, complemented
71C23×D4C2×C411yescharacteristic
72C24×C4—21no—
73C22×SD16C2×C411yescharacteristic
74C22×SD16C2×C411yescharacteristic
75C23×C8—21no—
76C23×C8—21no—
77C23×C8—21no—
78C23×C8—21no—
79C2×C22⋊Q8C2×C411yescharacteristic
80C2×C4×Q8C2311yescharacteristic
81C2×C4×D4—22no—
82C2×C4×D4—22no—
83C2×C4×D4—22no—
84C2×C4×D4—22no—
85C2×C4×D4—21no—
86C2×C4×D4—21no—
87C2×C4×D4C2×C411yescharacteristic
88C2×C4×D4C2×C411yescharacteristic
89C2×C4×D4C2311yescharacteristic
90C2×C4×D4C2311yescharacteristic
91C22×C4⋊C4C2×C411yescharacteristic
92C22×C22⋊C4—21no—
93C22×C42C2311yescharacteristic
94#450C818yescomplemented
95C8×Q8—22no—
96C8×Q8—22no—
97C4×SD16—22no—
98C4×SD16—22no—
99C4×SD16—22no—
100C4×SD16—22no—
101C8×D4—28no—
102C8×D4—28no—
103C8×D4—22no—
104C8×D4—22no—
105C8×D4—22no—
106C8×D4—22no—
107C8×D4—22no—
108C8×D4—22no—
109C8×D4D414yes—
110C8×D4D414yes—
111#431—22no—
112#431—22no—
113#431—22no—
114#431—22no—
115C2×#106C2×C411yescharacteristic
116C2×#106C2×C411yescharacteristic
117C2×C4⋊C8—21no—
118C2×C4⋊C8—21no—
119C2×C4⋊C8D411yescharacteristic
120C2×C4⋊C8D411yescharacteristic
121C2×C4⋊C8C2311yescharacteristic
122C2×C4⋊C8C2311yescharacteristic
123C2×C4⋊C8C2311yescharacteristic
124C2×C4⋊C8C2311yescharacteristic
125C2×#103C2×C411yescharacteristic
126C2×#103C2×C411yescharacteristic
127C2×#102C2×C411yescharacteristic
128C2×#102C2×C411yescharacteristic
129C2×#102C2×C411yescharacteristic
130C2×#102C2×C411yescharacteristic
131C2×#102C2×C411yescharacteristic
132C2×#102C2×C411yescharacteristic
133C2×C22⋊C8—28no—
134C2×C22⋊C8—28no—
135C2×C22⋊C8—21no—
136C2×C22⋊C8—21no—
137C2×C22⋊C8—21no—
138C2×C22⋊C8—21no—
139C2×C22⋊C8—21no—
140C2×C22⋊C8—21no—
141C2×C22⋊C8—21no—
142C2×C22⋊C8C2×C411yescharacteristic
143C2×C22⋊C8C2×C411yescharacteristic
144C2×C22⋊C8D411yescharacteristic
145C2×C22⋊C8D411yescharacteristic
146C2×C4×C8—21no—
147C2×C4×C8—21no—
148C2×C4×C8—21no—
149C2×C4×C8C2311yescharacteristic
150C2×C4×C8C2311yescharacteristic
151C2×C4×C8C2311yescharacteristic
152C2×C4×C8C2311yescharacteristic
153C2×C4×C8C2311yescharacteristic
154C2×C4×C8C2311yescharacteristic
155#391C2×C411yescharacteristic
156#389C2×C411yescharacteristic
157#387—21no—
158#385—21no—
159#380C2×C411yescharacteristic
160C4×C4⋊C4—21no—
161C4×C4⋊C4C2311yescharacteristic
162C4×C22⋊C4—21no—
163#340C2×C411yescharacteristic
164#340C2×C411yescharacteristic
165#340C2×C411yescharacteristic
166#340C2×C411yescharacteristic
167#336—21no—
168#336—21no—
169#336—21no—
170#336—21no—
171#336—21no—
172#336—21no—
173#336C2×C411yescharacteristic
174#336C2×C411yescharacteristic
175#335—22no—
176#334—22no—
177Q8⋊C8—22no—
178Q8⋊C8—22no—
179D4⋊C8—22no—
180D4⋊C8—22no—
181D4⋊C8—22no—
182D4⋊C8—22no—
183#322—22no—
184C82—22no—
185C25—21no—
186C22×Q8C22×C411yescharacteristic
187C22×D4—22no—
188C22×D4—22no—
189C22×D4—22no—
190C22×D4—22no—
191C22×D4—21no—
192C22×D4—22no—
193C22×D4C2×C814yes—
194C22×D4C22×C411yescharacteristic
195C22×D4C22×C411yescharacteristic
196C23×C4—24no—
197C23×C4—24no—
198C23×C4—21no—
199C23×C4—21no—
200C23×C4—21no—
201C23×C4—21no—
202C23×C4—21no—
203C23×C4—81no—
204C23×C4—81no—
205C23×C4—81no—
206C23×C4—81no—
207C23×C4C22×C411yescharacteristic
208C23×C4C2×D411yescharacteristic
209C23×C4#4111yescharacteristic
210C2×SD16—22no—
211C2×SD16—22no—
212C2×SD16—22no—
213C2×SD16—22no—
214C2×SD16C2×C814yescomplemented
215C2×SD16C2×C814yescomplemented
216C22×C8—21no—
217C22×C8—21no—
218C22×C8—21no—
219C22×C8—24no—
220C22×C8—24no—
221C22×C8—24no—
222C22×C8—24no—
223C22×C8—21no—
224C22×C8—21no—
225C22×C8—21no—
226C22×C8—81no—
227C22×C8—81no—
228C22×C8—81no—
229C22×C8—81no—
230C22×C8—81no—
231C22×C8—81no—
232C22×C8—81no—
233C22×C8—81no—
234C22×C8C22×C411yescharacteristic
235C22×C8C22×C411yescharacteristic
236C22×C8C22×C411yescharacteristic
237C22×C8C22×C411yescharacteristic
238C22×C8C2×D411yescharacteristic
239C22×C8C2×D411yescharacteristic
240C22×C8C2×D411yescharacteristic
241C22×C8C2×D411yescharacteristic
242C22⋊Q8—22no—
243C22⋊Q8C2×C814yescomplemented
244C4×Q8—21no—
245C4×Q8—21no—
246C4×Q8—21no—
247C4×Q8—21no—
248C4×D4—24no—
249C4×D4—24no—
250C4×D4—24no—
251C4×D4—24no—
252C4×D4—22no—
253C4×D4—22no—
254C4×D4—22no—
255C4×D4—22no—
256C4×D4—21no—
257C4×D4—21no—
258C4×D4—21no—
259C4×D4—21no—
260C4×D4C2×D412yes—
261C4×D4C2×D412yes—
262C4×D4#4112yes—
263C4×D4#4112yes—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—21no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4—21no—
269C2×C4⋊C4—21no—
270C2×C4⋊C4C22×C411yescharacteristic
271C2×C4⋊C4C22×C411yescharacteristic
272C2×C4⋊C4C22×C411yescharacteristic
273C2×C4⋊C4C22×C411yescharacteristic
274C2×C4⋊C4C22×C411yescharacteristic
275C2×C22⋊C4—24no—
276C2×C22⋊C4—24no—
277C2×C22⋊C4—21no—
278C2×C22⋊C4—21no—
279C2×C22⋊C4—21no—
280C2×C22⋊C4—21no—
281C2×C22⋊C4—21no—
282C2×C22⋊C4—21no—
283C2×C42—21no—
284C2×C42—21no—
285C2×C42—21no—
286C2×C42—21no—
287C2×C42—21no—
288C2×C42—21no—
289C2×C42C22×C411yescharacteristic
290C2×C42C2×D411yescharacteristic
291C2×C42#4111yescharacteristic
292C2×C42C2411yescharacteristic
293#106—22no—
294#106—22no—
295C4⋊C8—22no—
296C4⋊C8—22no—
297C4⋊C8—21no—
298C4⋊C8—21no—
299C4⋊C8—21no—
300C4⋊C8—21no—
301C4⋊C8—21no—
302C4⋊C8—21no—
303C4⋊C8—42no—
304C4⋊C8—81no—
305C4⋊C8C2×D412yes—
306C4⋊C8C2×D412yes—
307#103—22no—
308#103—22no—
309#102—22no—
310#102—22no—
311#102—22no—
312#102—22no—
313#102C2×C814yescomplemented
314#102C2×C814yescomplemented
315C22⋊C8—24no—
316C22⋊C8—24no—
317C22⋊C8—24no—
318C22⋊C8—22no—
319C22⋊C8—22no—
320C22⋊C8—22no—
321C22⋊C8—24no—
322C22⋊C8—24no—
323C22⋊C8—42no—
324C22⋊C8—81no—
325C22⋊C8C2×C814yescomplemented
326C4×C8—24no—
327C4×C8—21no—
328C4×C8—21no—
329C4×C8—21no—
330C4×C8—21no—
331C4×C8—21no—
332C4×C8—21no—
333C4×C8—21no—
334C4×C8—21no—
335C4×C8—21no—
336C4×C8—21no—
337C4×C8—42no—
338C4×C8—81no—
339C4×C8C2×D412yes—
340C4×C8C2×D412yes—
341#95—21no—
342#95—21no—
343#95—21no—
344#95—21no—
345#95—21no—
346C24—24no—
347C24—24no—
348C24—21no—
349C24—21no—
350C24—24no—
351C24—81no—
352C24—81no—
353C24—81no—
354C24—81no—
355C24C4×D411yescharacteristic
356C2×Q8—21no—
357C2×Q8—21no—
358C2×Q8—21no—
359C2×Q8—21no—
360C2×Q8—21no—
361C2×Q8C22×C812yes—
362C2×D4—24no—
363C2×D4—24no—
364C2×D4—28no—
365C2×D4—28no—
366C2×D4—28no—
367C2×D4—28no—
368C2×D4—21no—
369C2×D4—21no—
370C2×D4—21no—
371C2×D4—21no—
372C2×D4—21no—
373C2×D4—21no—
374C2×D4—24no—
375C2×D4C4×D412yes—
376C2×D4C4×D412yes—
377C2×D4C22×C812yes—
378C2×D4C22×C812yes—
379C2×D4#13112yes—
380C2×D4#13112yes—
381C22×C4—21no—
382C22×C4—21no—
383C22×C4—21no—
384C22×C4—21no—
385C22×C4—21no—
386C22×C4—21no—
387C22×C4—21no—
388C22×C4—21no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—24no—
392C22×C4—21no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—21no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—21no—
399C22×C4—21no—
400C22×C4—21no—
401C22×C4—21no—
402C22×C4—21no—
403C22×C4—21no—
404C22×C4—21no—
405C22×C4—21no—
406C22×C4—41no—
407C22×C4—41no—
408C22×C4—41no—
409C22×C4—41no—
410C22×C4—41no—
411C22×C4—41no—
412C22×C4—41no—
413C22×C4—41no—
414C22×C4—41no—
415C22×C4—41no—
416C22×C4—81no—
417C22×C4—81no—
418C22×C4—81no—
419C22×C4—81no—
420C22×C4—84no—
421C22×C4—84no—
422C22×C4—84no—
423C22×C4—84no—
424C22×C4—81no—
425C22×C4—81no—
426C22×C4C4×D411yescharacteristic
427C22×C4C4×D411yescharacteristic
428C22×C4C4×D411yescharacteristic
429C22×C4C22×C812yes—
430C22×C4#13112yes—
431C22×C4C23×C411yescharacteristic
432C22×C4C22×D411yescharacteristic
433C22×C4C2×#4111yescharacteristic
434SD16—28no—
435SD16—28no—
436C2×C8—21no—
437C2×C8—21no—
438C2×C8—21no—
439C2×C8—21no—
440C2×C8—21no—
441C2×C8—21no—
442C2×C8—21no—
443C2×C8—21no—
444C2×C8—22no—
445C2×C8—21no—
446C2×C8—21no—
447C2×C8—21no—
448C2×C8—21no—
449C2×C8—22no—
450C2×C8—22no—
451C2×C8—21no—
452C2×C8—21no—
453C2×C8—21no—
454C2×C8—21no—
455C2×C8—21no—
456C2×C8—21no—
457C2×C8—21no—
458C2×C8—21no—
459C2×C8—41no—
460C2×C8—41no—
461C2×C8—41no—
462C2×C8—41no—
463C2×C8—41no—
464C2×C8—41no—
465C2×C8—41no—
466C2×C8—41no—
467C2×C8—41no—
468C2×C8—41no—
469C2×C8—41no—
470C2×C8—41no—
471C2×C8—41no—
472C2×C8—41no—
473C2×C8—41no—
474C2×C8—84no—
475C2×C8—84no—
476C2×C8—84no—
477C2×C8—84no—
478C2×C8#12012yes—
479C2×C8#12012yes—
480C2×C8C22×C812yes—
481C2×C8C22×C812yes—
482C4⋊C4—24no—
483C4⋊C4—21no—
484C4⋊C4—21no—
485C4⋊C4—21no—
486C4⋊C4—24no—
487C4⋊C4—21no—
488C4⋊C4—21no—
489C4⋊C4—21no—
490C4⋊C4—21no—
491C4⋊C4—41no—
492C4⋊C4—41no—
493C4⋊C4—81no—
494C4⋊C4C4×D412yes—
495C4⋊C4C4×D412yes—
496C4⋊C4C22×C812yes—
497C22⋊C4—24no—
498C22⋊C4—24no—
499C22⋊C4—24no—
500C22⋊C4—22no—
501C22⋊C4—22no—
502C22⋊C4—22no—
503C22⋊C4—22no—
504C22⋊C4—81no—
505C42—21no—
506C42—21no—
507C42—21no—
508C42—21no—
509C42—21no—
510C42—21no—
511C42—41no—
512C42—41no—
513C42—41no—
514C42—41no—
515C42C22×D411yescharacteristic
516C42C22×D411yescharacteristic
517C42C2×#4111yescharacteristic
518C42C2×#4111yescharacteristic
519C23—216no—
520C23—22no—
521C23—22no—
522C23—21no—
523C23—22no—
524C23—22no—
525C23—21no—
526C23—21no—
527C23—21no—
528C23—21no—
529C23—22no—
530C23—22no—
531C23—21no—
532C23—41no—
533C23—41no—
534C23—41no—
535C23—41no—
536C23—41no—
537C23—41no—
538C23—41no—
539C23—41no—
540C23—84no—
541C23—84no—
542C23—84no—
543C23—84no—
544C23—84no—
545C23—84no—
546C23C8×D412yes—
547C23C2×C4×D411yescharacteristic
548Q8—22no—
549Q8—22no—
550Q8—41no—
551Q8—41no—
552D4—216no—
553D4—22no—
554D4—22no—
555D4—22no—
556D4—22no—
557D4—22no—
558D4—22no—
559D4C8×D414yes—
560D4C8×D414yes—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—41no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—41no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—81no—
606C2×C4—81no—
607C2×C4—84no—
608C2×C4—84no—
609C2×C4—84no—
610C2×C4C8×D412yes—
611C2×C4C2×C4×D411yescharacteristic
612C2×C4C2×C4×D411yescharacteristic
613C2×C4C2×#12011yescharacteristic
614C2×C4#52511yescharacteristic
615C2×C4#52511yescharacteristic
616C2×C4#52511yescharacteristic
617C2×C4C23×C811yescharacteristic
618C2×C4C2×#13111yescharacteristic
619C2×C4C2×#13111yescharacteristic
620C2×C4C2×#13111yescharacteristic
621C2×C4#57611yescharacteristic
622C2×C4#57711yescharacteristic
623C2×C4#57711yescharacteristic
624C2×C4#57711yescharacteristic
625C8—22no—
626C8—22no—
627C8—22no—
628C8—22no—
629C8—42no—
630C8—41no—
631C8—41no—
632C8—41no—
633C8—42no—
634C8—41no—
635C8—42no—
636C8—81no—
637C8—81no—
638C8—81no—
639C22—22no—
640C22—22no—
641C22—24no—
642C22—24no—
643C22—22no—
644C22—41no—
645C22—42no—
646C22—42no—
647C22—42no—
648C22—42no—
649C22—42no—
650C22—42no—
651C22—41no—
652C22—42no—
653C22—42no—
654C22—41no—
655C22—42no—
656C22—42no—
657C22—41no—
658C22—42no—
659C22—42no—
660C22—816no—
661C22C4×#12011yescharacteristic
662C22C2×C8×D411yescharacteristic
663C22#292911yescharacteristic
664C22#294611yescharacteristic
665C22#294911yescharacteristic
666C22#295611yescharacteristic
667C22#295711yescharacteristic
668C4—22no—
669C4—21no—
670C4—21no—
671C4—42no—
672C4—42no—
673C4—41no—
674C4—42no—
675C4—42no—
676C4—41no—
677C4—81no—
678C4—81no—
679C4C2×C8×D411yescharacteristic
680C4C2×C8×D411yescharacteristic
681C4#292911yescharacteristic
682C4#292911yescharacteristic
683C4#300911yescharacteristic
684C4#301211yescharacteristic
685C4#311811yescharacteristic
686C4#312311yescharacteristic
687C2—42no—
688C2—44no—
689C2—44no—
690C2#1785211yescharacteristic
691C2#1785511yescharacteristic
692C2#1791811yescharacteristic
693C2#1792411yescharacteristic
694C2C8×#12011yescharacteristic
695C2#2145911yescharacteristic
696C2#2146811yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC2×C4C23×C8
Frattini, cosocleC2×C42C24
socleC23C2×C4×D4
prime radicalC23×D4C2×C4
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C2×#431 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #22098C2semidirect
4. C2×SD16⋊C8C2semidirect
5. C2×C8×SD16C2semidirect
6. #8945C2semidirect
7. #8945C2semidirect
8. #8938C2semidirect
9. #8845C2semidirect
10. #8843C2semidirect
11. #8831C2semidirect
13. #8200C2semidirect
14. C8×#102C2semidirect
15. #8148C2semidirect
20. C2×C8×Q8C22semidirect
34. #1984C22semidirect
36. #1939C22semidirect
48. C8×C4⋊C4C22semidirect
55. C2×#335C22semidirect
56. C2×#334C22semidirect
57. C2×Q8⋊C8C22semidirect
58. C2×Q8⋊C8C22semidirect
63. C2×#322C22semidirect
64. C2×C82C22semidirect
65. #1279C22semidirect
66. #1279C22semidirect
67. #1277C22semidirect
68. #1277C22semidirect
69. #1273C22semidirect
70. #1273C22semidirect
94. #450C8semidirect
214. C2×SD16C2×C8semidirect
215. C2×SD16C2×C8semidirect
243. C22⋊Q8C2×C8semidirect
313. #102C2×C8semidirect
314. #102C2×C8semidirect
325. C22⋊C8C2×C8semidirect

Maximal subgroups

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

subgroupindex
C2×#29722
#220982
C4×#4502
C2×SD16⋊C82
C2×C8×SD162
#89452
#89382
#88452
#88432
#88312
#83102
#82002
C8×#1022
#81482