Finite Group Databasebeta

← #208739

Subgroups — #208739

Order 512.

930 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×#2966C211yesmaximal, characteristic
2#23572C211yesmaximal, characteristic
3#21597C211yesmaximal, characteristic, complemented
4C2×#1593C211yesmaximal, characteristic, complemented
5C2×#1592C211yesmaximal, characteristic, complemented
6#8946C211yesmaximal, characteristic, complemented
7#8945C211yesmaximal, characteristic, complemented
8#8941C211yesmaximal, characteristic, complemented
9#8845C211yesmaximal, characteristic, complemented
10#8844C211yesmaximal, characteristic, complemented
11#8834C211yesmaximal, characteristic, complemented
12#8315C211yesmaximal, characteristic
13#8209C211yesmaximal, characteristic, complemented
14#8208C211yesmaximal, characteristic, complemented
15#8145C211yesmaximal, characteristic, complemented
16C2×#529C2211yescharacteristic
17#2966—24no—
18#2966—24no—
19C2×#446C2211yescharacteristic
20C2×#446C2211yescharacteristic
21C2×C8×Q8C2211yescharacteristic
22C2×C4×Q16C2211yescharacteristic
23C2×C4×SD16C2211yescharacteristic
24C2×#436C2211yescharacteristic
25C2×#435—21no—
26C2×C8×D4—21no—
27C2×C8×D4C2211yescharacteristic
28C2×#433—21no—
29C2×#432C2211yescharacteristic
30C2×#405—21no—
31C22×C4×C8C2211yescharacteristic
32C4×C22⋊Q8C2211yescharacteristic
33#2059C411yescharacteristic
34#2042C411yescharacteristic
35#2035C411yescharacteristic
36#2033C411yescharacteristic
37#2026C411yescharacteristic
38#2015C411yescharacteristic
39#2001C411yescharacteristic
40#2000C411yescharacteristic
41#1984C2211yescharacteristic
42#1939C2211yescharacteristic
43#1916—21no—
44#1850—21no—
45#1797C2211yescharacteristic
46#1771—21no—
47#1770C2211yescharacteristic
48C4×#103C2211yescharacteristic
49C4×#102C2211yescharacteristic
50#1752—21no—
51C4×C22⋊C8C2211yescharacteristic
52C4×C22⋊C8C2211yescharacteristic
53#1593—24no—
54#1592—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×#322C2211yescharacteristic
65#1279C2211yescharacteristic
66#1279C2211yescharacteristic
67#1277C2211yescharacteristic
68#1277C2211yescharacteristic
69#1273C2211yescharacteristic
70#1273C2211yescharacteristic
71C22×Q16C2×C411yescharacteristic
72C22×SD16C2×C411yescharacteristic
73C22×M16—21no—
74C23×C8—21no—
75C23×C8D411yescharacteristic
76C23×C8D411yescharacteristic
77#529—22no—
78#529—22no—
79#529—22no—
80#529—22no—
81C2×#126—21no—
82C2×#125C2×C411yescharacteristic
83C2×#124C2×C411yescharacteristic
84C2×#123—21no—
85C2×C22⋊Q8C2×C411yescharacteristic
86C2×C22⋊Q8C2×C411yescharacteristic
87C2×#121C2×C411yescharacteristic
88C2×C4×Q8C2311yescharacteristic
89C2×C4×Q8C2311yescharacteristic
90C2×C4×D4—21no—
91C2×C4×D4C2311yescharacteristic
92C2×#117—21no—
93C22×C42C2311yescharacteristic
94#446—22no—
95#446—22no—
96#446—22no—
97#446D414yes—
98C8×Q8—22no—
99C8×Q8—22no—
100C4×Q16—22no—
101C4×Q16—22no—
102C4×SD16—22no—
103C4×SD16—22no—
104#436—22no—
105#436D414yes—
106#435—28no—
107C8×D4—28no—
108C8×D4—22no—
109C8×D4—22no—
110#433—28no—
111#432—22no—
112#432—22no—
113C2×#107C2×C411yescharacteristic
114C2×#106C2×C411yescharacteristic
115C2×C4⋊C8—21no—
116C2×C4⋊C8—21no—
117C2×C4⋊C8—21no—
118C2×C4⋊C8—21no—
119C2×C4⋊C8C2311yescharacteristic
120C2×C4⋊C8C2311yescharacteristic
121C2×C4⋊C8C2311yescharacteristic
122C2×C4⋊C8C2311yescharacteristic
123C2×#103C2×C411yescharacteristic
124C2×#103C2×C411yescharacteristic
125C2×#103C2×C411yescharacteristic
126C2×#103C2×C411yescharacteristic
127C2×#103C2×C411yescharacteristic
128C2×#102C2×C411yescharacteristic
129C2×#102C2×C411yescharacteristic
130C2×#102C2×C411yescharacteristic
131C2×C22⋊C8—21no—
132C2×C22⋊C8—21no—
133C2×C22⋊C8—21no—
134C2×C22⋊C8—21no—
135C2×C22⋊C8—21no—
136C2×C22⋊C8C2×C411yescharacteristic
137C2×C22⋊C8C2×C411yescharacteristic
138#405—28no—
139C2×#97—21no—
140C2×#97—21no—
141C2×#97—21no—
142C2×#97—21no—
143C2×C4×C8—21no—
144C2×C4×C8—22no—
145C2×C4×C8—22no—
146C2×C4×C8D411yescharacteristic
147C2×C4×C8D411yescharacteristic
148C2×C4×C8C2311yescharacteristic
149C2×C4×C8C2311yescharacteristic
150C2×C4×C8C2311yescharacteristic
151C2×C4×C8C2311yescharacteristic
152C2×C4×C8C2311yescharacteristic
153C2×C4×C8C2311yescharacteristic
154#391C2×C411yescharacteristic
155#389C2×C411yescharacteristic
156#387—21no—
157#385—21no—
158#380C2×C411yescharacteristic
159C4×C4⋊C4—21no—
160C4×C4⋊C4C2311yescharacteristic
161C4×C22⋊C4—21no—
162#340C2×C411yescharacteristic
163#340C2×C411yescharacteristic
164#340C2×C411yescharacteristic
165#340C2×C411yescharacteristic
166#336—21no—
167#336—21no—
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—
183#322—22no—
184C22×Q8C22×C411yescharacteristic
185C22×Q8C22×C411yescharacteristic
186C22×D4—21no—
187C22×D4C22×C411yescharacteristic
188C23×C4—21no—
189C23×C4C22×C411yescharacteristic
190C23×C4C2×D411yescharacteristic
191C23×C4#4111yescharacteristic
192C2×Q16—22no—
193C2×Q16—22no—
194C2×Q16—41no—
195C2×SD16—22no—
196C2×SD16—22no—
197C2×SD16—41no—
198C2×M16—24no—
199C2×M16—42no—
200C2×M16—42no—
201C22×C8—24no—
202C22×C8—21no—
203C22×C8—21no—
204C22×C8—22no—
205C22×C8—22no—
206C22×C8—22no—
207C22×C8—21no—
208C22×C8—21no—
209C22×C8—21no—
210C22×C8—21no—
211C22×C8—41no—
212C22×C8—41no—
213C22×C8—41no—
214C22×C8—42no—
215C22×C8—42no—
216C22×C8C22×C411yescharacteristic
217C22×C8C22×C411yescharacteristic
218C22×C8C22×C411yescharacteristic
219C22×C8C22×C411yescharacteristic
220C22×C8C2×D411yescharacteristic
221C22×C8C2×D411yescharacteristic
222C22×C8C2×D411yescharacteristic
223C22×C8C2×D411yescharacteristic
224#126—24no—
225#126—24no—
226#125—22no—
227#125—22no—
228#124—22no—
229#124—22no—
230#123—24no—
231#123—24no—
232C22⋊Q8—22no—
233C22⋊Q8—22no—
234C22⋊Q8—41no—
235C22⋊Q8—41no—
236#121—22no—
237#121—22no—
238C4×Q8—21no—
239C4×Q8—21no—
240C4×Q8—21no—
241C4×Q8—21no—
242C4×Q8—21no—
243C4×Q8—21no—
244C4×Q8C2×D412yes—
245C4×Q8#4112yes—
246C4×D4—24no—
247C4×D4—24no—
248C4×D4—21no—
249C4×D4—21no—
250C4×D4C2×D412yes—
251C4×D4#4112yes—
252#117—24no—
253#117—24no—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—21no—
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×C22⋊C4—21no—
272C2×C22⋊C4—21no—
273C2×C42—21no—
274C2×C42—21no—
275C2×C42—21no—
276C2×C42—21no—
277C2×C42—21no—
278C2×C42—21no—
279C2×C42—21no—
280C2×C42C22×C411yescharacteristic
281C2×C42C2×D411yescharacteristic
282C2×C42#4111yescharacteristic
283C2×C42C2411yescharacteristic
284#107—22no—
285#106—22no—
286C4⋊C8—24no—
287C4⋊C8—24no—
288C4⋊C8—21no—
289C4⋊C8—21no—
290C4⋊C8—21no—
291C4⋊C8—21no—
292C4⋊C8—21no—
293C4⋊C8—21no—
294C4⋊C8—42no—
295C4⋊C8—42no—
296C4⋊C8C2×D412yes—
297C4⋊C8C2×D412yes—
298#103—22no—
299#103—22no—
300#103—22no—
301#103—22no—
302#103—41no—
303#102—22no—
304#102—22no—
305#102—41no—
306C22⋊C8—24no—
307C22⋊C8—24no—
308C22⋊C8—24no—
309C22⋊C8—22no—
310C22⋊C8—22no—
311C22⋊C8—42no—
312C22⋊C8—42no—
313#97—24no—
314#97—24no—
315#97—42no—
316#97—42no—
317C4×C8—24no—
318C4×C8—22no—
319C4×C8—22no—
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—
335#95—21no—
336#95—21no—
337C24—21no—
338C24C4×D411yescharacteristic
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—21no—
349C2×Q8—21no—
350C2×Q8C4×D412yes—
351C2×D4—24no—
352C2×D4—21no—
353C2×D4—21no—
354C2×D4—21no—
355C2×D4—21no—
356C2×D4—21no—
357C2×D4—42no—
358C2×D4—42no—
359C2×D4C4×D412yes—
360C22×C4—21no—
361C22×C4—21no—
362C22×C4—21no—
363C22×C4—22no—
364C22×C4—21no—
365C22×C4—21no—
366C22×C4—21no—
367C22×C4—22no—
368C22×C4—21no—
369C22×C4—21no—
370C22×C4—21no—
371C22×C4—22no—
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—21no—
381C22×C4—21no—
382C22×C4—21no—
383C22×C4—21no—
384C22×C4—41no—
385C22×C4—41no—
386C22×C4—41no—
387C22×C4—41no—
388C22×C4—41no—
389C22×C4C4×D411yescharacteristic
390C22×C4C4×D411yescharacteristic
391C22×C4C4×D411yescharacteristic
392C22×C4C23×C411yescharacteristic
393C22×C4C22×D411yescharacteristic
394C22×C4C2×#4111yescharacteristic
395Q16—44no—
396SD16—44no—
397M16—48no—
398C2×C8—21no—
399C2×C8—21no—
400C2×C8—21no—
401C2×C8—21no—
402C2×C8—21no—
403C2×C8—21no—
404C2×C8—21no—
405C2×C8—21no—
406C2×C8—21no—
407C2×C8—21no—
408C2×C8—21no—
409C2×C8—21no—
410C2×C8—21no—
411C2×C8—22no—
412C2×C8—22no—
413C2×C8—21no—
414C2×C8—21no—
415C2×C8—21no—
416C2×C8—21no—
417C2×C8—21no—
418C2×C8—21no—
419C2×C8—44no—
420C2×C8—44no—
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—41no—
431C2×C8—41no—
432C2×C8—41no—
433C2×C8—41no—
434C2×C8—41no—
435C2×C8—41no—
436C2×C8—41no—
437C2×C8—41no—
438C2×C8#12012yes—
439C2×C8#12012yes—
440C2×C8#13112yes—
441C2×C8#13112yes—
442C2×C8#13712yes—
443C4⋊C4—24no—
444C4⋊C4—24no—
445C4⋊C4—24no—
446C4⋊C4—21no—
447C4⋊C4—21no—
448C4⋊C4—22no—
449C4⋊C4—22no—
450C4⋊C4—21no—
451C4⋊C4—21no—
452C4⋊C4—21no—
453C4⋊C4—21no—
454C4⋊C4—21no—
455C4⋊C4—21no—
456C4⋊C4—42no—
457C4⋊C4—42no—
458C4⋊C4—41no—
459C4⋊C4—41no—
460C4⋊C4C4×D412yes—
461C4⋊C4C4×D412yes—
462C22⋊C4—24no—
463C22⋊C4—24no—
464C22⋊C4—24no—
465C22⋊C4—24no—
466C22⋊C4—22no—
467C22⋊C4—22no—
468C22⋊C4—42no—
469C22⋊C4—42no—
470C42—21no—
471C42—21no—
472C42—21no—
473C42—22no—
474C42—21no—
475C42—21no—
476C42—21no—
477C42—22no—
478C42—41no—
479C42—41no—
480C42—41no—
481C42—41no—
482C42C22×D411yescharacteristic
483C42C22×D411yescharacteristic
484C42C2×#4111yescharacteristic
485C42C2×#4111yescharacteristic
486C23—21no—
487C23—21no—
488C23—21no—
489C23—21no—
490C23—21no—
491C23—21no—
492C23—22no—
493C23—41no—
494C23—41no—
495C23—41no—
496C23—41no—
497C23—41no—
498C23—41no—
499C23#44412yes—
500C23C2×C4×D411yescharacteristic
501Q8—22no—
502Q8—22no—
503Q8—41no—
504Q8—41no—
505Q8—41no—
506Q8—41no—
507Q8—41no—
508Q8—41no—
509D4—22no—
510D4—22no—
511D4—48no—
512D4—41no—
513D4—41no—
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—21no—
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—42no—
538C2×C4—41no—
539C2×C4—42no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—42no—
546C2×C4—41no—
547C2×C4—42no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—41no—
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×C4C2×#13711yescharacteristic
571C2×C4#57511yescharacteristic
572C2×C4#57511yescharacteristic
573C2×C4#57511yescharacteristic
574C8—22no—
575C8—22no—
576C8—22no—
577C8—22no—
578C8—22no—
579C8—41no—
580C8—41no—
581C8—41no—
582C8—42no—
583C8—42no—
584C8—81no—
585C8—81no—
586C8—81no—
587C8—81no—
588C22—22no—
589C22—22no—
590C22—22no—
591C22—41no—
592C22—41no—
593C22—41no—
594C22—41no—
595C22—42no—
596C22—42no—
597C22—42no—
598C22—81no—
599C22—81no—
600C22—81no—
601C22—81no—
602C22C4×#12011yescharacteristic
603C22#293011yescharacteristic
604C22#293111yescharacteristic
605C22#294311yescharacteristic
606C22C4×#13711yescharacteristic
607C22C2×#44411yescharacteristic
608C22#295511yescharacteristic
609C4—21no—
610C4—21no—
611C4—41no—
612C4—42no—
613C4—41no—
614C4—42no—
615C4—41no—
616C4—42no—
617C4—81no—
618C4—81no—
619C4—81no—
620C4—81no—
621C4#293311yescharacteristic
622C4#293311yescharacteristic
623C4#293511yescharacteristic
624C4#293511yescharacteristic
625C4#300511yescharacteristic
626C4#300711yescharacteristic
627C4#311111yescharacteristic
628C4#311611yescharacteristic
629C2—42no—
630C2—82no—
631C2#1793711yescharacteristic
632C2#1793911yescharacteristic
633C2#1796311yescharacteristic
634C2#1796511yescharacteristic
635C2#2144511yescharacteristic
636C2#2145711yescharacteristic
637C2#2146711yescharacteristic
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 ◁ C22×C4×C8 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #21597C2semidirect
4. C2×#1593C2semidirect
5. C2×#1592C2semidirect
6. #8946C2semidirect
7. #8945C2semidirect
8. #8941C2semidirect
9. #8845C2semidirect
10. #8844C2semidirect
11. #8834C2semidirect
13. #8209C2semidirect
14. #8208C2semidirect
15. #8145C2semidirect

Maximal subgroups

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

subgroupindex
C2×#29662
#235722
#215972
C2×#15932
C2×#15922
#89462
#89452
#89412
#88452
#88442
#88342
#83152
#82092
#82082
#81452