Finite Group Databasebeta

← #209379

Subgroups — #209379

Order 512.

913 conjugacy classes of subgroups.

The subgroup lattice has 636 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2994C211yesmaximal, characteristic
2#23572C211yesmaximal, characteristic
3#22102C211yesmaximal, characteristic, complemented
4C2×#1591C211yesmaximal, characteristic, complemented
5C2×#1585C211yesmaximal, characteristic, complemented
6#8949C211yesmaximal, characteristic, complemented
7#8946C211yesmaximal, characteristic, complemented
8#8937C211yesmaximal, characteristic, complemented
9#8847C211yesmaximal, characteristic, complemented
10#8844C211yesmaximal, characteristic, complemented
11#8829C211yesmaximal, characteristic, complemented
12#8315C211yesmaximal, characteristic
13#8213C211yesmaximal, characteristic, complemented
14C8×#102C211yesmaximal, characteristic, complemented
15#8148C211yesmaximal, characteristic, complemented
16C2×#529C2211yescharacteristic
17#2994—24no—
18#2994—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×#431C2211yescharacteristic
30C2×#424—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#1939—21no—
43#1939C2211yescharacteristic
44#1924—21no—
45#1923—21no—
46#1917—21no—
47#1797C2211yescharacteristic
48C8×C4⋊C4C2211yescharacteristic
49C4×#103C2211yescharacteristic
50C4×#102C2211yescharacteristic
51C4×C22⋊C8C2211yescharacteristic
52C4×C22⋊C8C2211yescharacteristic
53#1591—24no—
54#1585—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×Q16C2×C411yescharacteristic
72C22×SD16C2×C411yescharacteristic
73C22×M16—21no—
74C23×C8—21no—
75C23×C8—21no—
76#529—22no—
77#529—22no—
78#529—22no—
79#529—22no—
80C2×#126—21no—
81C2×#125C2×C411yescharacteristic
82C2×#124C2×C411yescharacteristic
83C2×#123—21no—
84C2×C22⋊Q8C2×C411yescharacteristic
85C2×C22⋊Q8C2×C411yescharacteristic
86C2×#121C2×C411yescharacteristic
87C2×C4×Q8C2311yescharacteristic
88C2×C4×Q8C2311yescharacteristic
89C2×C4×D4—21no—
90C2×C4×D4C2311yescharacteristic
91C2×#117—21no—
92C22×C42C2311yescharacteristic
93#446—22no—
94#446—22no—
95#446—22no—
96#446D414yes—
97C8×Q8—22no—
98C8×Q8—22no—
99C4×Q16—22no—
100C4×Q16—22no—
101C4×SD16—22no—
102C4×SD16—22no—
103#436—22no—
104#436—22no—
105#435—28no—
106C8×D4—28no—
107C8×D4—22no—
108C8×D4D414yes—
109#433—28no—
110#431—22no—
111#431—22no—
112C2×#107C2×C411yescharacteristic
113C2×#106C2×C411yescharacteristic
114#424—28no—
115C2×C4⋊C8—21no—
116C2×C4⋊C8—21no—
117C2×C4⋊C8—21no—
118C2×C4⋊C8—22no—
119C2×C4⋊C8—22no—
120C2×C4⋊C8—21no—
121C2×C4⋊C8—21no—
122C2×C4⋊C8D411yescharacteristic
123C2×C4⋊C8D411yescharacteristic
124C2×C4⋊C8C2311yescharacteristic
125C2×C4⋊C8C2311yescharacteristic
126C2×C4⋊C8C2311yescharacteristic
127C2×C4⋊C8C2311yescharacteristic
128C2×#103C2×C411yescharacteristic
129C2×#103C2×C411yescharacteristic
130C2×#103C2×C411yescharacteristic
131C2×#103C2×C411yescharacteristic
132C2×#103C2×C411yescharacteristic
133C2×#102C2×C411yescharacteristic
134C2×#102C2×C411yescharacteristic
135C2×#102C2×C411yescharacteristic
136C2×C22⋊C8—21no—
137C2×C22⋊C8—21no—
138C2×C22⋊C8—21no—
139C2×C22⋊C8—21no—
140C2×C22⋊C8C2×C411yescharacteristic
141C2×C22⋊C8C2×C411yescharacteristic
142C2×C22⋊C8D411yescharacteristic
143C2×C22⋊C8D411yescharacteristic
144C2×#97—21no—
145C2×#97—21no—
146C2×C4×C8—21no—
147C2×C4×C8—21no—
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#335—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×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—41no—
197C2×SD16—41no—
198C2×M16—24no—
199C2×M16—81no—
200C2×M16—81no—
201C22×C8—21no—
202C22×C8—21no—
203C22×C8—21no—
204C22×C8—21no—
205C22×C8—24no—
206C22×C8—24no—
207C22×C8—21no—
208C22×C8—21no—
209C22×C8—81no—
210C22×C8—81no—
211C22×C8—81no—
212C22×C8—81no—
213C22×C8C22×C411yescharacteristic
214C22×C8C22×C411yescharacteristic
215C22×C8C22×C411yescharacteristic
216C22×C8C22×C411yescharacteristic
217C22×C8C2×D411yescharacteristic
218C22×C8C2×D411yescharacteristic
219C22×C8C2×D411yescharacteristic
220C22×C8C2×D411yescharacteristic
221#126—24no—
222#126—24no—
223#125—22no—
224#125—22no—
225#124—22no—
226#124—22no—
227#123—24no—
228#123—24no—
229C22⋊Q8—22no—
230C22⋊Q8—22no—
231C22⋊Q8—22no—
232C22⋊Q8—41no—
233#121—22no—
234#121—22no—
235C4×Q8—21no—
236C4×Q8—21no—
237C4×Q8—21no—
238C4×Q8—21no—
239C4×Q8—21no—
240C4×Q8—21no—
241C4×Q8C2×D412yes—
242C4×Q8#4112yes—
243C4×D4—24no—
244C4×D4—24no—
245C4×D4—21no—
246C4×D4—21no—
247C4×D4C2×D412yes—
248C4×D4#4112yes—
249#117—24no—
250#117—24no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4—21no—
253C2×C4⋊C4—21no—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4—21no—
258C2×C4⋊C4C22×C411yescharacteristic
259C2×C4⋊C4C22×C411yescharacteristic
260C2×C4⋊C4C22×C411yescharacteristic
261C2×C4⋊C4C22×C411yescharacteristic
262C2×C4⋊C4C22×C411yescharacteristic
263C2×C22⋊C4—21no—
264C2×C22⋊C4—21no—
265C2×C22⋊C4—21no—
266C2×C22⋊C4—21no—
267C2×C22⋊C4—21no—
268C2×C22⋊C4—21no—
269C2×C22⋊C4—21no—
270C2×C42—21no—
271C2×C42—21no—
272C2×C42—21no—
273C2×C42—21no—
274C2×C42—21no—
275C2×C42—21no—
276C2×C42—21no—
277C2×C42C22×C411yescharacteristic
278C2×C42C2×D411yescharacteristic
279C2×C42#4111yescharacteristic
280C2×C42C2411yescharacteristic
281#107—22no—
282#106—22no—
283C4⋊C8—24no—
284C4⋊C8—24no—
285C4⋊C8—24no—
286C4⋊C8—22no—
287C4⋊C8—22no—
288C4⋊C8—21no—
289C4⋊C8—21no—
290C4⋊C8—21no—
291C4⋊C8—21no—
292C4⋊C8—21no—
293C4⋊C8—21no—
294C4⋊C8—42no—
295C4⋊C8—81no—
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—22no—
306C22⋊C8—24no—
307C22⋊C8—24no—
308C22⋊C8—24no—
309C22⋊C8—22no—
310C22⋊C8—22no—
311C22⋊C8—22no—
312C22⋊C8—41no—
313C22⋊C8—81no—
314#97—24no—
315#97—42no—
316C4×C8—24no—
317C4×C8—21no—
318C4×C8—21no—
319C4×C8—21no—
320C4×C8—21no—
321C4×C8—21no—
322C4×C8—21no—
323C4×C8—21no—
324C4×C8—21no—
325C4×C8—21no—
326C4×C8—21no—
327C4×C8—81no—
328C4×C8C2×D412yes—
329C4×C8C2×D412yes—
330#95—21no—
331#95—21no—
332#95—21no—
333#95—21no—
334#95—21no—
335C24—21no—
336C24C4×D411yescharacteristic
337C2×Q8—21no—
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×Q8C4×D412yes—
349C2×D4—24no—
350C2×D4—21no—
351C2×D4—21no—
352C2×D4—21no—
353C2×D4—21no—
354C2×D4—21no—
355C2×D4—42no—
356C2×D4—42no—
357C2×D4C4×D412yes—
358C22×C4—21no—
359C22×C4—21no—
360C22×C4—21no—
361C22×C4—21no—
362C22×C4—22no—
363C22×C4—21no—
364C22×C4—21no—
365C22×C4—21no—
366C22×C4—22no—
367C22×C4—21no—
368C22×C4—21no—
369C22×C4—21no—
370C22×C4—22no—
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—21no—
381C22×C4—21no—
382C22×C4—41no—
383C22×C4—41no—
384C22×C4—41no—
385C22×C4—41no—
386C22×C4—41no—
387C22×C4C4×D411yescharacteristic
388C22×C4C4×D411yescharacteristic
389C22×C4C4×D411yescharacteristic
390C22×C4C23×C411yescharacteristic
391C22×C4C22×D411yescharacteristic
392C22×C4C2×#4111yescharacteristic
393Q16—44no—
394SD16—44no—
395M16—84no—
396C2×C8—21no—
397C2×C8—21no—
398C2×C8—21no—
399C2×C8—21no—
400C2×C8—21no—
401C2×C8—21no—
402C2×C8—22no—
403C2×C8—22no—
404C2×C8—21no—
405C2×C8—21no—
406C2×C8—21no—
407C2×C8—21no—
408C2×C8—21no—
409C2×C8—22no—
410C2×C8—22no—
411C2×C8—21no—
412C2×C8—21no—
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—41no—
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—84no—
436C2×C8—84no—
437C2×C8#12012yes—
438C2×C8#12012yes—
439C2×C8#13112yes—
440C4⋊C4—24no—
441C4⋊C4—24no—
442C4⋊C4—24no—
443C4⋊C4—21no—
444C4⋊C4—21no—
445C4⋊C4—22no—
446C4⋊C4—22no—
447C4⋊C4—21no—
448C4⋊C4—21no—
449C4⋊C4—21no—
450C4⋊C4—21no—
451C4⋊C4—21no—
452C4⋊C4—42no—
453C4⋊C4—42no—
454C4⋊C4—41no—
455C4⋊C4—41no—
456C4⋊C4C4×D412yes—
457C4⋊C4C4×D412yes—
458C4⋊C4#13112yes—
459C22⋊C4—24no—
460C22⋊C4—24no—
461C22⋊C4—24no—
462C22⋊C4—24no—
463C22⋊C4—22no—
464C22⋊C4—22no—
465C22⋊C4—42no—
466C22⋊C4—42no—
467C42—21no—
468C42—21no—
469C42—22no—
470C42—21no—
471C42—21no—
472C42—21no—
473C42—21no—
474C42—22no—
475C42—41no—
476C42—41no—
477C42—41no—
478C42—41no—
479C42C22×D411yescharacteristic
480C42C22×D411yescharacteristic
481C42C2×#4111yescharacteristic
482C42C2×#4111yescharacteristic
483C23—21no—
484C23—21no—
485C23—21no—
486C23—21no—
487C23—21no—
488C23—21no—
489C23—22no—
490C23—41no—
491C23—41no—
492C23—41no—
493C23—41no—
494C23—41no—
495C23—41no—
496C23#44312yes—
497C23C2×C4×D411yescharacteristic
498Q8—22no—
499Q8—22no—
500Q8—41no—
501Q8—41no—
502Q8—41no—
503Q8—41no—
504Q8—41no—
505Q8—41no—
506D4—22no—
507D4—22no—
508D4—48no—
509D4—41no—
510D4—41no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—22no—
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—22no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—41no—
529C2×C4—41no—
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×C4C2×C4×D411yescharacteristic
558C2×C4C2×C4×D411yescharacteristic
559C2×C4C2×#12011yescharacteristic
560C2×C4#52511yescharacteristic
561C2×C4#52511yescharacteristic
562C2×C4#52511yescharacteristic
563C2×C4C2×#13111yescharacteristic
564C2×C4C2×#13111yescharacteristic
565C2×C4#56811yescharacteristic
566C2×C4#56811yescharacteristic
567C2×C4#57611yescharacteristic
568C2×C4#57711yescharacteristic
569C2×C4#57811yescharacteristic
570C2×C4#57811yescharacteristic
571C8—22no—
572C8—22no—
573C8—22no—
574C8—42no—
575C8—42no—
576C8—41no—
577C8—41no—
578C8—41no—
579C8—41no—
580C8—41no—
581C8—42no—
582C8—42no—
583C8—81no—
584C8—81no—
585C22—22no—
586C22—22no—
587C22—22no—
588C22—41no—
589C22—41no—
590C22—41no—
591C22—41no—
592C22—42no—
593C22—42no—
594C22—42no—
595C22—81no—
596C22—81no—
597C22—81no—
598C22—81no—
599C22C4×#12011yescharacteristic
600C22#293311yescharacteristic
601C22#293511yescharacteristic
602C22#294711yescharacteristic
603C22#295111yescharacteristic
604C22C2×#44311yescharacteristic
605C22#295511yescharacteristic
606C4—21no—
607C4—21no—
608C4—42no—
609C4—41no—
610C4—42no—
611C4—41no—
612C4—42no—
613C4—41no—
614C4—42no—
615C4—81no—
616C4—81no—
617C4—81no—
618C4#293011yescharacteristic
619C4#293111yescharacteristic
620C4#293311yescharacteristic
621C4#293511yescharacteristic
622C4#301111yescharacteristic
623C4#301211yescharacteristic
624C4#312011yescharacteristic
625C4#312411yescharacteristic
626C2—42no—
627C2—82no—
628C2#1801711yescharacteristic
629C2#1802711yescharacteristic
630C2#1808111yescharacteristic
631C2#1808711yescharacteristic
632C2#2144511yescharacteristic
633C2#2146511yescharacteristic
634C2#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
3. #22102C2semidirect
4. C2×#1591C2semidirect
5. C2×#1585C2semidirect
6. #8949C2semidirect
7. #8946C2semidirect
8. #8937C2semidirect
9. #8847C2semidirect
10. #8844C2semidirect
11. #8829C2semidirect
13. #8213C2semidirect
14. C8×#102C2semidirect
15. #8148C2semidirect

Maximal subgroups

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

subgroupindex
C2×#29942
#235722
#221022
C2×#15912
C2×#15852
#89492
#89462
#89372
#88472
#88442
#88292
#83152
#82132
C8×#1022
#81482