Finite Group Databasebeta

← #209231

Subgroups — #209231

Order 512.

980 conjugacy classes of subgroups.

The subgroup lattice has 651 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2991C211yesmaximal, characteristic
2#22100C211yesmaximal, characteristic, complemented
3C4×#452C211yesmaximal, characteristic
4C2×#1588C211yesmaximal, characteristic, complemented
5C2×C8×Q16C211yesmaximal, characteristic, complemented
6#8949C211yesmaximal, characteristic, complemented
7#8946C211yesmaximal, characteristic, complemented
8#8942C211yesmaximal, characteristic, complemented
9#8847C211yesmaximal, characteristic, complemented
10#8844C211yesmaximal, characteristic, complemented
11#8832C211yesmaximal, characteristic, complemented
12#8312C211yesmaximal, characteristic
13#8206C211yesmaximal, characteristic, complemented
14#8205C211yesmaximal, characteristic, complemented
15#8148C211yesmaximal, characteristic, complemented
16C2×C4×#41C2211yescharacteristic
17C2×#452C411yescharacteristic
18#2991—24no—
19#2991—24no—
20C2×C8×Q8C2211yescharacteristic
21C2×C8×Q8C2211yescharacteristic
22C2×C8×Q8C2211yescharacteristic
23C2×C4×Q16C2211yescharacteristic
24C2×C4×SD16C2211yescharacteristic
25C2×#436—21no—
26C2×#436—21no—
27C2×#436C2211yescharacteristic
28C2×#436C2211yescharacteristic
29C2×#431—21no—
30C2×#431C2211yescharacteristic
31C2×#423—21no—
32C2×#423C2211yescharacteristic
33C4×C22⋊Q8C2211yescharacteristic
34#1984C2211yescharacteristic
35#1940—21no—
36#1939C2211yescharacteristic
37#1924—21no—
38#1923—21no—
39#1916—21no—
40#1878C411yescharacteristic
41#1877C411yescharacteristic
42#1873C411yescharacteristic
43#1865C411yescharacteristic
44#1862C411yescharacteristic
45#1854C411yescharacteristic
46#1797C2211yescharacteristic
47#1788C411yescharacteristic
48#1770C2211yescharacteristic
49C4×#103C2211yescharacteristic
50C4×#102C2211yescharacteristic
51C4×C22⋊C8C2211yescharacteristic
52C4×C22⋊C8C2211yescharacteristic
53#1588—24no—
54C8×Q16—24no—
55C2×#335C2211yescharacteristic
56C2×#335C2211yescharacteristic
57C2×Q8⋊C8C2211yescharacteristic
58C2×Q8⋊C8C2211yescharacteristic
59C2×Q8⋊C8C2211yescharacteristic
60C2×Q8⋊C8C2211yescharacteristic
61C2×D4⋊C8C2211yescharacteristic
62C2×D4⋊C8C2211yescharacteristic
63C2×#322C2211yescharacteristic
64C2×C82C2211yescharacteristic
65#1279C2211yescharacteristic
66#1279C2211yescharacteristic
67#1277C2211yescharacteristic
68#1277C2211yescharacteristic
69#1273C2211yescharacteristic
70#1273C2211yescharacteristic
71C22×#41C2×C411yescharacteristic
72C22×Q16C2×C411yescharacteristic
73C22×SD16C2×C411yescharacteristic
74C22×M16—21no—
75C22×M16—21no—
76C22×M16—21no—
77C2×C22⋊Q8C2×C411yescharacteristic
78C4×#41—22no—
79C4×#41—22no—
80C4×#41—22no—
81C4×#41—22no—
82C2×C4×Q8C2×C411yescharacteristic
83C2×C4×Q8C2311yescharacteristic
84C2×C4×Q8C2311yescharacteristic
85C2×C4×D4—21no—
86C2×C4×D4—21no—
87C2×C4×D4C2×C411yescharacteristic
88C2×C4×D4C2311yescharacteristic
89C2×#117—21no—
90C2×#117C2×C411yescharacteristic
91C22×C42—21no—
92C22×C42C2311yescharacteristic
93#452—24no—
94C8×Q8—22no—
95C8×Q8—22no—
96C8×Q8—22no—
97C8×Q8—22no—
98C8×Q8—22no—
99C8×Q8D414yes—
100C4×Q16—22no—
101C4×Q16—22no—
102C4×SD16—22no—
103C4×SD16—22no—
104#436—28no—
105#436—28no—
106#436—22no—
107#436—22no—
108#436—22no—
109#436D414yes—
110#431—28no—
111#431—22no—
112#431—22no—
113C2×#107C2×C411yescharacteristic
114C2×#106C2×C411yescharacteristic
115#423—28no—
116#423—22no—
117#423—22no—
118C2×C4⋊C8—21no—
119C2×C4⋊C8—21no—
120C2×C4⋊C8—21no—
121C2×C4⋊C8—21no—
122C2×C4⋊C8—21no—
123C2×C4⋊C8D411yescharacteristic
124C2×C4⋊C8D411yescharacteristic
125C2×C4⋊C8C2311yescharacteristic
126C2×C4⋊C8C2311yescharacteristic
127C2×C4⋊C8C2311yescharacteristic
128C2×C4⋊C8C2311yescharacteristic
129C2×#103C2×C411yescharacteristic
130C2×#103C2×C411yescharacteristic
131C2×#103C2×C411yescharacteristic
132C2×#103C2×C411yescharacteristic
133C2×#103C2×C411yescharacteristic
134C2×#102C2×C411yescharacteristic
135C2×#102C2×C411yescharacteristic
136C2×#102C2×C411yescharacteristic
137C2×C22⋊C8—21no—
138C2×C22⋊C8—21no—
139C2×C22⋊C8—21no—
140C2×C22⋊C8—21no—
141C2×C22⋊C8C2×C411yescharacteristic
142C2×C22⋊C8C2×C411yescharacteristic
143C2×C22⋊C8D411yescharacteristic
144C2×C22⋊C8D411yescharacteristic
145C2×#97—21no—
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#335—22no—
177Q8⋊C8—22no—
178Q8⋊C8—22no—
179Q8⋊C8—22no—
180Q8⋊C8—22no—
181D4⋊C8—22no—
182D4⋊C8—22no—
183#322—22no—
184C82—22no—
185C2×#41—22no—
186C2×#41—22no—
187C2×#41—22no—
188C2×#41—22no—
189C2×#41—41no—
190C2×#41—41no—
191C22×Q8C22×C411yescharacteristic
192C22×Q8C22×C411yescharacteristic
193C22×D4—21no—
194C22×D4C22×C411yescharacteristic
195C23×C4—21no—
196C23×C4—21no—
197C23×C4—21no—
198C23×C4C22×C411yescharacteristic
199C23×C4C2×D411yescharacteristic
200C23×C4#4111yescharacteristic
201C2×Q16—22no—
202C2×Q16—22no—
203C2×Q16M1614yescomplemented
204C2×SD16—22no—
205C2×SD16—22no—
206C2×SD16—41no—
207C2×M16—24no—
208C2×M16—24no—
209C2×M16—24no—
210C2×M16—81no—
211C2×M16—81no—
212C2×M16—81no—
213C2×M16—81no—
214C2×M16—81no—
215C2×M16—81no—
216C22×C8—21no—
217C22×C8—21no—
218C22×C8—21no—
219C22×C8—21no—
220C22×C8—21no—
221C22×C8—21no—
222C22×C8C22×C411yescharacteristic
223C22×C8C22×C411yescharacteristic
224C22×C8C22×C411yescharacteristic
225C22×C8C22×C411yescharacteristic
226C22×C8C2×D411yescharacteristic
227C22×C8C2×D411yescharacteristic
228C22×C8C2×D411yescharacteristic
229C22×C8C2×D411yescharacteristic
230C22⋊Q8—22no—
231C22⋊Q8—41no—
232C4×Q8—22no—
233C4×Q8—22no—
234C4×Q8—21no—
235C4×Q8—21no—
236C4×Q8—21no—
237C4×Q8—21no—
238C4×Q8—21no—
239C4×Q8—21no—
240C4×Q8C2×D412yes—
241C4×Q8#4112yes—
242C4×D4—24no—
243C4×D4—24no—
244C4×D4—24no—
245C4×D4—24no—
246C4×D4—22no—
247C4×D4—22no—
248C4×D4—21no—
249C4×D4—21no—
250C4×D4C2×D412yes—
251C4×D4#4112yes—
252#117—24no—
253#117—24no—
254#117—22no—
255#117—22no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4—21no—
258C2×C4⋊C4—21no—
259C2×C4⋊C4—21no—
260C2×C4⋊C4—21no—
261C2×C4⋊C4C22×C411yescharacteristic
262C2×C4⋊C4C22×C411yescharacteristic
263C2×C4⋊C4C22×C411yescharacteristic
264C2×C4⋊C4C22×C411yescharacteristic
265C2×C4⋊C4C22×C411yescharacteristic
266C2×C22⋊C4—21no—
267C2×C22⋊C4—21no—
268C2×C22⋊C4—21no—
269C2×C22⋊C4—21no—
270C2×C22⋊C4—21no—
271C2×C42—24no—
272C2×C42—24no—
273C2×C42—21no—
274C2×C42—21no—
275C2×C42—21no—
276C2×C42—21no—
277C2×C42—21no—
278C2×C42—21no—
279C2×C42—21no—
280C2×C42—21no—
281C2×C42—21no—
282C2×C42C22×C411yescharacteristic
283C2×C42C2×D411yescharacteristic
284C2×C42#4111yescharacteristic
285C2×C42C2411yescharacteristic
286#107—22no—
287#106—22no—
288C4⋊C8—24no—
289C4⋊C8—24no—
290C4⋊C8—24no—
291C4⋊C8—22no—
292C4⋊C8—22no—
293C4⋊C8—21no—
294C4⋊C8—21no—
295C4⋊C8—21no—
296C4⋊C8—21no—
297C4⋊C8—21no—
298C4⋊C8—21no—
299C4⋊C8—42no—
300C4⋊C8—81no—
301C4⋊C8C2×D412yes—
302C4⋊C8C2×D412yes—
303#103—22no—
304#103—22no—
305#103—22no—
306#103—22no—
307#103—22no—
308#102—22no—
309#102—22no—
310#102—41no—
311C22⋊C8—24no—
312C22⋊C8—24no—
313C22⋊C8—24no—
314C22⋊C8—22no—
315C22⋊C8—22no—
316C22⋊C8—22no—
317C22⋊C8—41no—
318C22⋊C8—81no—
319#97—81no—
320C4×C8—24no—
321C4×C8—24no—
322C4×C8—21no—
323C4×C8—21no—
324C4×C8—21no—
325C4×C8—21no—
326C4×C8—21no—
327C4×C8—21no—
328C4×C8—21no—
329C4×C8—21no—
330C4×C8—21no—
331C4×C8—21no—
332C4×C8—42no—
333C4×C8C2×D412yes—
334C4×C8C2×D412yes—
335#95—21no—
336#95—21no—
337#95—21no—
338#95—21no—
339#95—21no—
340C24—21no—
341C24C4×D411yescharacteristic
342#41—44no—
343#41—44no—
344#41—44no—
345#41—44no—
346C2×Q8—21no—
347C2×Q8—21no—
348C2×Q8—21no—
349C2×Q8—21no—
350C2×Q8—21no—
351C2×Q8—21no—
352C2×Q8—21no—
353C2×Q8—21no—
354C2×Q8C4×D412yes—
355C2×Q8C2×M1612yes—
356C2×Q8C2×M1612yes—
357C2×Q8#13112yes—
358C2×D4—24no—
359C2×D4—21no—
360C2×D4—21no—
361C2×D4—21no—
362C2×D4—21no—
363C2×D4—21no—
364C2×D4—42no—
365C2×D4—42no—
366C2×D4C4×D412yes—
367C22×C4—21no—
368C22×C4—24no—
369C22×C4—24no—
370C22×C4—21no—
371C22×C4—21no—
372C22×C4—21no—
373C22×C4—21no—
374C22×C4—21no—
375C22×C4—21no—
376C22×C4—22no—
377C22×C4—21no—
378C22×C4—21no—
379C22×C4—21no—
380C22×C4—22no—
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—22no—
390C22×C4—21no—
391C22×C4—21no—
392C22×C4—21no—
393C22×C4—41no—
394C22×C4—42no—
395C22×C4—42no—
396C22×C4—42no—
397C22×C4—41no—
398C22×C4—41no—
399C22×C4—41no—
400C22×C4—41no—
401C22×C4—42no—
402C22×C4C4×D411yescharacteristic
403C22×C4C4×D411yescharacteristic
404C22×C4C4×D411yescharacteristic
405C22×C4C23×C411yescharacteristic
406C22×C4C22×D411yescharacteristic
407C22×C4C2×#4111yescharacteristic
408Q16—28no—
409SD16—44no—
410M16—84no—
411M16—84no—
412M16—84no—
413C2×C8—21no—
414C2×C8—21no—
415C2×C8—21no—
416C2×C8—21no—
417C2×C8—21no—
418C2×C8—21no—
419C2×C8—22no—
420C2×C8—22no—
421C2×C8—21no—
422C2×C8—21no—
423C2×C8—21no—
424C2×C8—21no—
425C2×C8—21no—
426C2×C8—22no—
427C2×C8—22no—
428C2×C8—21no—
429C2×C8—21no—
430C2×C8—21no—
431C2×C8—21no—
432C2×C8—21no—
433C2×C8—21no—
434C2×C8—21no—
435C2×C8—21no—
436C2×C8—21no—
437C2×C8—21no—
438C2×C8—41no—
439C2×C8—41no—
440C2×C8—41no—
441C2×C8—41no—
442C2×C8—41no—
443C2×C8—41no—
444C2×C8—41no—
445C2×C8—41no—
446C2×C8—41no—
447C2×C8—41no—
448C2×C8—41no—
449C2×C8—41no—
450C2×C8—41no—
451C2×C8—41no—
452C2×C8#12012yes—
453C2×C8#12012yes—
454C2×C8C2×M1612yes—
455C4⋊C4—24no—
456C4⋊C4—21no—
457C4⋊C4—21no—
458C4⋊C4—22no—
459C4⋊C4—22no—
460C4⋊C4—21no—
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—42no—
467C4⋊C4—41no—
468C4⋊C4—41no—
469C4⋊C4—42no—
470C4⋊C4C4×D412yes—
471C4⋊C4C4×D412yes—
472C22⋊C4—24no—
473C22⋊C4—24no—
474C22⋊C4—22no—
475C22⋊C4—22no—
476C22⋊C4—42no—
477C22⋊C4—42no—
478C42—24no—
479C42—24no—
480C42—21no—
481C42—21no—
482C42—22no—
483C42—21no—
484C42—21no—
485C42—22no—
486C42—21no—
487C42—21no—
488C42—41no—
489C42—41no—
490C42—41no—
491C42—41no—
492C42C22×D411yescharacteristic
493C42C22×D411yescharacteristic
494C42C2×#4111yescharacteristic
495C42C2×#4111yescharacteristic
496C23—21no—
497C23—21no—
498C23—21no—
499C23—21no—
500C23—21no—
501C23—21no—
502C23—21no—
503C23—22no—
504C23—41no—
505C23—41no—
506C23—41no—
507C23—41no—
508C23—41no—
509C23—41no—
510C23C2×C4×D411yescharacteristic
511Q8—22no—
512Q8—22no—
513Q8—22no—
514Q8—22no—
515Q8—22no—
516Q8—41no—
517Q8—41no—
518Q8#43514yes—
519D4—22no—
520D4—22no—
521D4—48no—
522D4—41no—
523D4—41no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—22no—
531C2×C4—21no—
532C2×C4—22no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—48no—
543C2×C4—48no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—42no—
552C2×C4—41no—
553C2×C4—42no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—42no—
560C2×C4—41no—
561C2×C4—42no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4#44312yes—
572C2×C4C2×C4×D411yescharacteristic
573C2×C4C2×C4×D411yescharacteristic
574C2×C4C2×#12011yescharacteristic
575C2×C4#52511yescharacteristic
576C2×C4#52511yescharacteristic
577C2×C4#52511yescharacteristic
578C2×C4C22×M1611yescharacteristic
579C2×C4C2×#13111yescharacteristic
580C2×C4#56811yescharacteristic
581C2×C4#56811yescharacteristic
582C2×C4#57611yescharacteristic
583C2×C4#57711yescharacteristic
584C2×C4#57811yescharacteristic
585C2×C4#57811yescharacteristic
586C8—22no—
587C8—22no—
588C8—22no—
589C8—42no—
590C8—42no—
591C8—41no—
592C8—41no—
593C8—41no—
594C8—41no—
595C8—41no—
596C8—42no—
597C8—42no—
598C8—81no—
599C8—81no—
600C22—41no—
601C22—41no—
602C22—41no—
603C22—41no—
604C22—42no—
605C22—41no—
606C22—41no—
607C22—42no—
608C22—41no—
609C22—42no—
610C22—81no—
611C22—81no—
612C22—81no—
613C22—81no—
614C22C4×#12011yescharacteristic
615C22#293311yescharacteristic
616C22D4×M1611yescharacteristic
617C22#294711yescharacteristic
618C22#295111yescharacteristic
619C22C2×#44311yescharacteristic
620C22#295511yescharacteristic
621C4—24no—
622C4—21no—
623C4—21no—
624C4—22no—
625C4—41no—
626C4—42no—
627C4—41no—
628C4—42no—
629C4—42no—
630C4—81no—
631C4—81no—
632C4—81no—
633C4C2×#43511yescharacteristic
634C4#293011yescharacteristic
635C4#293311yescharacteristic
636C4D4×M1611yescharacteristic
637C4#301111yescharacteristic
638C4#301211yescharacteristic
639C4#312011yescharacteristic
640C4#312411yescharacteristic
641C2—81no—
642C2—82no—
643C2#1801911yescharacteristic
644C2#1803311yescharacteristic
645C2#1805311yescharacteristic
646C2#1805911yescharacteristic
647C2#2143911yescharacteristic
648C2#2146511yescharacteristic
649C2#2146611yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C2×C4×D4
prime radicalC22×#41C2×C4
LCSG ▷ C22×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. #22100C2semidirect
4. C2×#1588C2semidirect
5. C2×C8×Q16C2semidirect
6. #8949C2semidirect
7. #8946C2semidirect
8. #8942C2semidirect
9. #8847C2semidirect
10. #8844C2semidirect
11. #8832C2semidirect
13. #8206C2semidirect
14. #8205C2semidirect
15. #8148C2semidirect
203. C2×Q16M16semidirect

Maximal subgroups

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

subgroupindex
C2×#29912
#221002
C4×#4522
C2×#15882
C2×C8×Q162
#89492
#89462
#89422
#88472
#88442
#88322
#83122
#82062
#82052
#81482