Finite Group Databasebeta

← #209324

Subgroups — #209324

Order 512.

912 conjugacy classes of subgroups.

The subgroup lattice has 639 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2986C211yesmaximal, characteristic, complemented
2#23568C211yesmaximal, characteristic
3#22135C211yesmaximal, characteristic
4C2×#1585C211yesmaximal, characteristic, complemented
5C2×C8×SD16C211yesmaximal, characteristic, complemented
6#8952C211yesmaximal, characteristic, complemented
7#8948C211yesmaximal, characteristic, complemented
8#8940C211yesmaximal, characteristic
9#8848C211yesmaximal, characteristic, complemented
10#8846C211yesmaximal, characteristic, complemented
11#8831C211yesmaximal, characteristic, complemented
12#8314C211yesmaximal, characteristic, complemented
13#8207C211yesmaximal, characteristic, complemented
14#8205C211yesmaximal, characteristic, complemented
15#8148C211yesmaximal, characteristic, complemented
16C2×#533C2211yescharacteristic
17#2986—24no—
18#2986—24no—
19C2×#446—21no—
20C2×#446C2211yescharacteristic
21C2×#446C2211yescharacteristic
22C2×C8×Q8—21no—
23C2×C8×Q8C2211yescharacteristic
24C2×C8×Q8C2211yescharacteristic
25C2×C4×SD16C2211yescharacteristic
26C2×C4×SD16C2211yescharacteristic
27C2×C8×D4C2211yescharacteristic
28C2×#433—21no—
29C2×#433—21no—
30C2×#431C2211yescharacteristic
31C2×#431C2211yescharacteristic
32C4×#121C2211yescharacteristic
33#2058C411yescharacteristic
34#2043C411yescharacteristic
35#2035C411yescharacteristic
36#2035C411yescharacteristic
37#2023C411yescharacteristic
38#2013C411yescharacteristic
39#2000C411yescharacteristic
40#2000C411yescharacteristic
41#1955—21no—
42#1954C2211yescharacteristic
43#1939C2211yescharacteristic
44#1924—21no—
45#1869—21no—
46#1850—21no—
47#1797C2211yescharacteristic
48#1770C2211yescharacteristic
49C4×#103C2211yescharacteristic
50C4×#103C2211yescharacteristic
51C4×C22⋊C8C2211yescharacteristic
52C4×C22⋊C8C2211yescharacteristic
53#1585—24no—
54C8×SD16—24no—
55C2×#335C2211yescharacteristic
56C2×#334C2211yescharacteristic
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×SD16C2×C411yescharacteristic
72C22×SD16C2×C411yescharacteristic
73C23×C8—21no—
74#533—22no—
75#533—22no—
76#533—22no—
77#533—22no—
78C2×C4⋊Q8C2×C411yescharacteristic
79C2×C4⋊Q8C2×C411yescharacteristic
80C2×#125—21no—
81C2×C22⋊Q8—21no—
82C2×C22⋊Q8C2×C411yescharacteristic
83C2×C22⋊Q8C2×C411yescharacteristic
84C2×#121C2×C411yescharacteristic
85C2×C4×Q8—21no—
86C2×C4×Q8C2311yescharacteristic
87C2×C4×Q8C2311yescharacteristic
88C2×C4×D4—21no—
89C2×C4×D4C2311yescharacteristic
90C2×#117—21no—
91C22×C42C2311yescharacteristic
92#446—28no—
93#446—22no—
94#446—22no—
95#446—22no—
96#446D414yescomplemented
97C8×Q8—28no—
98C8×Q8—22no—
99C8×Q8—22no—
100C8×Q8—22no—
101C8×Q8D414yescomplemented
102C4×SD16—22no—
103C4×SD16—22no—
104C4×SD16—22no—
105C4×SD16—22no—
106C8×D4—22no—
107C8×D4—22no—
108#433—28no—
109#433—28no—
110#431—22no—
111#431—22no—
112#431—22no—
113#431—22no—
114C2×#106C2×C411yescharacteristic
115C2×#106C2×C411yescharacteristic
116C2×C4⋊C8—21no—
117C2×C4⋊C8—21no—
118C2×C4⋊C8—21no—
119C2×C4⋊C8—21no—
120C2×C4⋊C8—21no—
121C2×C4⋊C8D411yescharacteristic
122C2×C4⋊C8D411yescharacteristic
123C2×C4⋊C8C2311yescharacteristic
124C2×C4⋊C8C2311yescharacteristic
125C2×C4⋊C8C2311yescharacteristic
126C2×C4⋊C8C2311yescharacteristic
127C2×#103C2×C411yescharacteristic
128C2×#103C2×C411yescharacteristic
129C2×#103C2×C411yescharacteristic
130C2×#103C2×C411yescharacteristic
131C2×#103C2×C411yescharacteristic
132C2×#103C2×C411yescharacteristic
133C2×#102C2×C411yescharacteristic
134C2×#102C2×C411yescharacteristic
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⋊C8C2×C411yescharacteristic
142C2×C22⋊C8C2×C411yescharacteristic
143C2×C22⋊C8D411yescharacteristic
144C2×C22⋊C8D411yescharacteristic
145C2×#97—21no—
146C2×#97—21no—
147C2×#97—21no—
148C2×C4×C8—21no—
149C2×C4×C8—21no—
150C2×C4×C8—21no—
151C2×C4×C8C2311yescharacteristic
152C2×C4×C8C2311yescharacteristic
153C2×C4×C8C2311yescharacteristic
154C2×C4×C8C2311yescharacteristic
155C2×C4×C8C2311yescharacteristic
156C2×C4×C8C2311yescharacteristic
157#390C2×C411yescharacteristic
158#389C2×C411yescharacteristic
159#388—21no—
160#386—21no—
161#380C2×C411yescharacteristic
162C4×C4⋊C4C2311yescharacteristic
163C4×C22⋊C4—21no—
164#340C2×C411yescharacteristic
165#340C2×C411yescharacteristic
166#340C2×C411yescharacteristic
167#340C2×C411yescharacteristic
168#336—21no—
169#336—21no—
170#336—21no—
171#336—21no—
172#336C2×C411yescharacteristic
173#336C2×C411yescharacteristic
174#335—22no—
175#334—22no—
176Q8⋊C8—22no—
177Q8⋊C8—22no—
178Q8⋊C8—22no—
179Q8⋊C8—22no—
180D4⋊C8—22no—
181D4⋊C8—22no—
182#322—22no—
183C82—22no—
184C22×Q8—21no—
185C22×Q8C22×C411yescharacteristic
186C22×Q8C22×C411yescharacteristic
187C22×D4—21no—
188C22×D4C22×C411yescharacteristic
189C23×C4—21no—
190C23×C4C22×C411yescharacteristic
191C23×C4C2×D411yescharacteristic
192C23×C4#4111yescharacteristic
193C2×SD16—22no—
194C2×SD16—22no—
195C2×SD16—22no—
196C2×SD16—22no—
197C2×SD16—22no—
198C2×SD16—41no—
199C22×C8—21no—
200C22×C8—21no—
201C22×C8—21no—
202C22×C8—21no—
203C22×C8—21no—
204C22×C8—21no—
205C22×C8—42no—
206C22×C8—81no—
207C22×C8—81no—
208C22×C8C22×C411yescharacteristic
209C22×C8C22×C411yescharacteristic
210C22×C8C22×C411yescharacteristic
211C22×C8C22×C411yescharacteristic
212C22×C8C2×D411yescharacteristic
213C22×C8C2×D411yescharacteristic
214C22×C8C2×D411yescharacteristic
215C22×C8C2×D411yescharacteristic
216C4⋊Q8—22no—
217C4⋊Q8—22no—
218C4⋊Q8—22no—
219C4⋊Q8—22no—
220#125—24no—
221#125—24no—
222C22⋊Q8—24no—
223C22⋊Q8—24no—
224C22⋊Q8—22no—
225C22⋊Q8—22no—
226C22⋊Q8—22no—
227C22⋊Q8—22no—
228#121—22no—
229#121—41no—
230C4×Q8—24no—
231C4×Q8—24no—
232C4×Q8—21no—
233C4×Q8—21no—
234C4×Q8—21no—
235C4×Q8—21no—
236C4×Q8C2×D412yes—
237C4×Q8C2×D412yes—
238C4×Q8#4112yes—
239C4×Q8#4112yes—
240C4×D4—21no—
241C4×D4—21no—
242C4×D4—21no—
243C4×D4—21no—
244C4×D4—81no—
245C4×D4—81no—
246#117—24no—
247#117—24no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4—21no—
253C2×C4⋊C4—21no—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4C22×C411yescharacteristic
257C2×C4⋊C4C22×C411yescharacteristic
258C2×C4⋊C4C22×C411yescharacteristic
259C2×C4⋊C4C22×C411yescharacteristic
260C2×C4⋊C4C22×C411yescharacteristic
261C2×C22⋊C4—21no—
262C2×C22⋊C4—21no—
263C2×C22⋊C4—21no—
264C2×C22⋊C4—21no—
265C2×C22⋊C4—21no—
266C2×C22⋊C4—21no—
267C2×C22⋊C4—21no—
268C2×C42—21no—
269C2×C42—21no—
270C2×C42—21no—
271C2×C42—21no—
272C2×C42—21no—
273C2×C42—21no—
274C2×C42—21no—
275C2×C42C22×C411yescharacteristic
276C2×C42C2×D411yescharacteristic
277C2×C42#4111yescharacteristic
278C2×C42C2411yescharacteristic
279#106—22no—
280#106—22no—
281C4⋊C8—24no—
282C4⋊C8—24no—
283C4⋊C8—24no—
284C4⋊C8—24no—
285C4⋊C8—22no—
286C4⋊C8—22no—
287C4⋊C8—21no—
288C4⋊C8—21no—
289C4⋊C8—21no—
290C4⋊C8—21no—
291C4⋊C8—21no—
292C4⋊C8—21no—
293C4⋊C8—81no—
294C4⋊C8C2×D412yes—
295C4⋊C8C2×D412yes—
296#103—22no—
297#103—22no—
298#103—22no—
299#103—41no—
300#103—41no—
301#103—41no—
302#102—22no—
303#102—22no—
304C22⋊C8—24no—
305C22⋊C8—24no—
306C22⋊C8—22no—
307C22⋊C8—22no—
308C22⋊C8—22no—
309C22⋊C8—41no—
310C22⋊C8—42no—
311C22⋊C8—81no—
312C22⋊C8—81no—
313C22⋊C8—81no—
314#97—24no—
315#97—24no—
316#97—81no—
317C4×C8—24no—
318C4×C8—24no—
319C4×C8—24no—
320C4×C8—21no—
321C4×C8—21no—
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×C8C2×D412yes—
331C4×C8C2×D412yes—
332#95—21no—
333#95—21no—
334#95—21no—
335C24—21no—
336C24C4×D411yescharacteristic
337C2×Q8—24no—
338C2×Q8—21no—
339C2×Q8—21no—
340C2×Q8—21no—
341C2×Q8—21no—
342C2×Q8—21no—
343C2×Q8—21no—
344C2×Q8—21no—
345C2×Q8—21no—
346C2×Q8—21no—
347C2×Q8—21no—
348C2×Q8—42no—
349C2×Q8—42no—
350C2×Q8C4×D412yes—
351C2×Q8C4×D412yes—
352C2×D4—21no—
353C2×D4—21no—
354C2×D4—21no—
355C2×D4—21no—
356C2×D4—21no—
357C2×D4—42no—
358C2×D4—42no—
359C2×D4—81no—
360C2×D4#13112yes—
361C22×C4—21no—
362C22×C4—21no—
363C22×C4—21no—
364C22×C4—21no—
365C22×C4—21no—
366C22×C4—21no—
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—21no—
381C22×C4—21no—
382C22×C4—21no—
383C22×C4—41no—
384C22×C4—41no—
385C22×C4—41no—
386C22×C4—41no—
387C22×C4—41no—
388C22×C4—41no—
389C22×C4—41no—
390C22×C4C4×D411yescharacteristic
391C22×C4C4×D411yescharacteristic
392C22×C4C4×D411yescharacteristic
393C22×C4C23×C411yescharacteristic
394C22×C4C22×D411yescharacteristic
395C22×C4C2×#4111yescharacteristic
396SD16—28no—
397SD16—44no—
398C2×C8—21no—
399C2×C8—21no—
400C2×C8—21no—
401C2×C8—21no—
402C2×C8—21no—
403C2×C8—21no—
404C2×C8—22no—
405C2×C8—22no—
406C2×C8—21no—
407C2×C8—21no—
408C2×C8—21no—
409C2×C8—21no—
410C2×C8—22no—
411C2×C8—21no—
412C2×C8—22no—
413C2×C8—21no—
414C2×C8—21no—
415C2×C8—21no—
416C2×C8—21no—
417C2×C8—21no—
418C2×C8—21no—
419C2×C8—21no—
420C2×C8—21no—
421C2×C8—21no—
422C2×C8—21no—
423C2×C8—41no—
424C2×C8—41no—
425C2×C8—41no—
426C2×C8—41no—
427C2×C8—41no—
428C2×C8—41no—
429C2×C8—41no—
430C2×C8—41no—
431C2×C8—41no—
432C2×C8—41no—
433C2×C8—41no—
434C2×C8—41no—
435C2×C8—41no—
436C2×C8—41no—
437C2×C8—84no—
438C2×C8#12012yes—
439C2×C8#12012yes—
440C2×C8#13112yes—
441C4⋊C4—24no—
442C4⋊C4—24no—
443C4⋊C4—24no—
444C4⋊C4—24no—
445C4⋊C4—24no—
446C4⋊C4—21no—
447C4⋊C4—21no—
448C4⋊C4—22no—
449C4⋊C4—22no—
450C4⋊C4—22no—
451C4⋊C4—22no—
452C4⋊C4—21no—
453C4⋊C4—21no—
454C4⋊C4—21no—
455C4⋊C4—21no—
456C4⋊C4—21no—
457C4⋊C4—21no—
458C4⋊C4—81no—
459C4⋊C4C4×D412yes—
460C4⋊C4C4×D412yes—
461C22⋊C4—24no—
462C22⋊C4—24no—
463C22⋊C4—42no—
464C22⋊C4—41no—
465C22⋊C4—41no—
466C22⋊C4—42no—
467C22⋊C4—81no—
468C22⋊C4—81no—
469C42—21no—
470C42—21no—
471C42—21no—
472C42—22no—
473C42—22no—
474C42—21no—
475C42—21no—
476C42—21no—
477C42—22no—
478C42—22no—
479C42—41no—
480C42—41no—
481C42C22×D411yescharacteristic
482C42C22×D411yescharacteristic
483C42C2×#4111yescharacteristic
484C42C2×#4111yescharacteristic
485C23—22no—
486C23—21no—
487C23—21no—
488C23—21no—
489C23—21no—
490C23—21no—
491C23—21no—
492C23—21no—
493C23—41no—
494C23—41no—
495C23—41no—
496C23—41no—
497C23—41no—
498C23—41no—
499C23C2×C4×D411yescharacteristic
500Q8—22no—
501Q8—22no—
502Q8—22no—
503Q8—22no—
504Q8—48no—
505Q8—41no—
506Q8—41no—
507Q8—41no—
508Q8—41no—
509D4—22no—
510D4—22no—
511D4—41no—
512D4—41no—
513D4—84no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
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—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—81no—
556C2×C4—81no—
557C2×C4—81no—
558C2×C4—81no—
559C2×C4#44312yes—
560C2×C4C2×C4×D411yescharacteristic
561C2×C4C2×C4×D411yescharacteristic
562C2×C4C2×#12011yescharacteristic
563C2×C4#52511yescharacteristic
564C2×C4#52511yescharacteristic
565C2×C4#52511yescharacteristic
566C2×C4C2×#13111yescharacteristic
567C2×C4C2×#13111yescharacteristic
568C2×C4#56811yescharacteristic
569C2×C4#56811yescharacteristic
570C2×C4#57711yescharacteristic
571C2×C4#57711yescharacteristic
572C2×C4#57711yescharacteristic
573C2×C4#57811yescharacteristic
574C8—22no—
575C8—22no—
576C8—22no—
577C8—42no—
578C8—42no—
579C8—41no—
580C8—41no—
581C8—41no—
582C8—41no—
583C8—42no—
584C8—41no—
585C8—42no—
586C8—81no—
587C8—81no—
588C22—41no—
589C22—42no—
590C22—41no—
591C22—42no—
592C22—41no—
593C22—41no—
594C22—41no—
595C22—42no—
596C22—41no—
597C22—41no—
598C22—81no—
599C22—81no—
600C22—81no—
601C22—81no—
602C22C4×#12011yescharacteristic
603C22#293311yescharacteristic
604C22#293511yescharacteristic
605C22#294711yescharacteristic
606C22#294911yescharacteristic
607C22C2×#44311yescharacteristic
608C22#295511yescharacteristic
609C4—21no—
610C4—21no—
611C4—22no—
612C4—41no—
613C4—42no—
614C4—42no—
615C4—41no—
616C4—42no—
617C4—42no—
618C4—81no—
619C4—81no—
620C4—81no—
621C4#292911yescharacteristic
622C4#293211yescharacteristic
623C4#293311yescharacteristic
624C4#293511yescharacteristic
625C4#301011yescharacteristic
626C4#301311yescharacteristic
627C4#312111yescharacteristic
628C4#312611yescharacteristic
629C2—81no—
630C2—82no—
631C2#1801611yescharacteristic
632C2#1802811yescharacteristic
633C2#1808111yescharacteristic
634C2#1808511yescharacteristic
635C2#2144311yescharacteristic
636C2#2146211yescharacteristic
637C2#2146611yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C2×C4×D4
prime radicalC2×#121C2×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
1. C2×#2986C2semidirect
4. C2×#1585C2semidirect
5. C2×C8×SD16C2semidirect
6. #8952C2semidirect
7. #8948C2semidirect
9. #8848C2semidirect
10. #8846C2semidirect
11. #8831C2semidirect
12. #8314C2semidirect
13. #8207C2semidirect
14. #8205C2semidirect
15. #8148C2semidirect
96. #446D4semidirect
101. C8×Q8D4semidirect

Maximal subgroups

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

subgroupindex
C2×#29862
#235682
#221352
C2×#15852
C2×C8×SD162
#89522
#89482
#89402
#88482
#88462
#88312
#83142
#82072
#82052
#81482