Finite Group Databasebeta

← #218800

Subgroups — #218800

Order 512.

684 conjugacy classes of subgroups.

The subgroup lattice has 585 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23700C211yesmaximal, characteristic
2#21596C211yesmaximal, characteristic, complemented
3#21486C211yesmaximal, characteristic
4#9547C211yesmaximal, characteristic, complemented
5#9546C211yesmaximal, characteristic, complemented
6#9539C211yesmaximal, characteristic, complemented
7#9538C211yesmaximal, characteristic, complemented
8#9072C211yesmaximal, characteristic
9#9071C211yesmaximal, characteristic, complemented
10#8948C211yesmaximal, characteristic, complemented
11#8945C211yesmaximal, characteristic, complemented
12#8846C211yesmaximal, characteristic, complemented
13#8845C211yesmaximal, characteristic, complemented
14#8598C211yesmaximal, characteristic, complemented
15#7848C211yesmaximal, characteristic, complemented
16C2×C8×Q8C2211yescharacteristic
17C2×C4×SD16C2211yescharacteristic
18C2×#435—21no—
19C2×C8×D4C2211yescharacteristic
20C2×#431C2211yescharacteristic
21C2×C4×M16C2211yescharacteristic
22C22×C4×C8C2211yescharacteristic
23C4×C22⋊Q8C2211yescharacteristic
24C4×#121C2211yescharacteristic
25#2088C411yescharacteristic
26#2077C411yescharacteristic
27#2066C411yescharacteristic
28#2065C411yescharacteristic
29#2055C411yescharacteristic
30#2035C411yescharacteristic
31#2034C411yescharacteristic
32#2000C411yescharacteristic
33#1984C2211yescharacteristic
34#1954C2211yescharacteristic
35#1939C2211yescharacteristic
36#1939C2211yescharacteristic
37#1924—21no—
38#1917—21no—
39#1870—21no—
40#1851—21no—
41#1797C2211yescharacteristic
42#1797C2211yescharacteristic
43C4×#106C2211yescharacteristic
44#1771—21no—
45#1770C2211yescharacteristic
46#1770C2211yescharacteristic
47C4×#103C2211yescharacteristic
48C4×#102C2211yescharacteristic
49#1753—21no—
50#1752—21no—
51C2×#334C2211yescharacteristic
52C2×#334C2211yescharacteristic
53C2×Q8⋊C8C2211yescharacteristic
54C2×Q8⋊C8C2211yescharacteristic
55C2×D4⋊C8C2211yescharacteristic
56C2×D4⋊C8C2211yescharacteristic
57#1279C2211yescharacteristic
58#1279C2211yescharacteristic
59#1279C2211yescharacteristic
60#1279C2211yescharacteristic
61#1277C2211yescharacteristic
62#1277C2211yescharacteristic
63#1273C2211yescharacteristic
64#1273C2211yescharacteristic
65#1273C2211yescharacteristic
66#1273C2211yescharacteristic
67C22×SD16C2×C411yescharacteristic
68C22×M16D411yescharacteristic
69C22×M16D411yescharacteristic
70C23×C8—21no—
71C23×C8—21no—
72C2×C22⋊Q8C2×C411yescharacteristic
73C2×#121C2×C411yescharacteristic
74C2×C4×Q8C2311yescharacteristic
75C2×C4×D4—21no—
76C2×C4×D4C2311yescharacteristic
77C22×C42C2311yescharacteristic
78C8×Q8—41no—
79C8×Q8—41no—
80C4×SD16—41no—
81C4×SD16—41no—
82#435—44no—
83C8×D4—41no—
84C8×D4—41no—
85#431—41no—
86#431—41no—
87C2×#106C2×C411yescharacteristic
88C2×#106C2×C411yescharacteristic
89C2×#106C2×C411yescharacteristic
90C2×C4⋊C8—21no—
91C2×C4⋊C8—21no—
92C2×C4⋊C8—21no—
93C2×C4⋊C8C2×C411yescharacteristic
94C2×C4⋊C8C2×C411yescharacteristic
95C2×C4⋊C8C2311yescharacteristic
96C2×C4⋊C8C2311yescharacteristic
97C2×C4⋊C8C2311yescharacteristic
98C2×C4⋊C8C2311yescharacteristic
99C2×#103C2×C411yescharacteristic
100C2×#103C2×C411yescharacteristic
101C2×#103C2×C411yescharacteristic
102C2×#102C2×C411yescharacteristic
103C2×#102C2×C411yescharacteristic
104C2×#102C2×C411yescharacteristic
105C2×C22⋊C8—21no—
106C2×C22⋊C8—21no—
107C2×C22⋊C8—21no—
108C2×C22⋊C8—21no—
109C2×C22⋊C8—21no—
110C2×C22⋊C8C2×C411yescharacteristic
111C2×C22⋊C8C2×C411yescharacteristic
112C4×M16—41no—
113C4×M16—41no—
114C2×#97—21no—
115C2×#97—21no—
116C2×#97D411yescharacteristic
117C2×#97D411yescharacteristic
118C2×C4×C8—21no—
119C2×C4×C8—21no—
120C2×C4×C8—41no—
121C2×C4×C8—41no—
122C2×C4×C8C2311yescharacteristic
123C2×C4×C8C2311yescharacteristic
124C2×C4×C8C2311yescharacteristic
125C2×C4×C8C2311yescharacteristic
126C2×C4×C8C2311yescharacteristic
127C2×C4×C8C2311yescharacteristic
128#391C2×C411yescharacteristic
129#390C2×C411yescharacteristic
130#389C2×C411yescharacteristic
131#389C2×C411yescharacteristic
132#388—21no—
133#387—21no—
134#386—21no—
135#385—21no—
136#380C2×C411yescharacteristic
137#380C2×C411yescharacteristic
138C4×C4⋊C4—21no—
139C4×C4⋊C4C2311yescharacteristic
140C4×C4⋊C4C2311yescharacteristic
141C4×C22⋊C4—21no—
142C4×C22⋊C4—21no—
143#340C2×C411yescharacteristic
144#340C2×C411yescharacteristic
145#340C2×C411yescharacteristic
146#340C2×C411yescharacteristic
147#340C2×C411yescharacteristic
148#340C2×C411yescharacteristic
149#336—21no—
150#336—21no—
151#336—21no—
152#336—21no—
153#336—21no—
154#336—21no—
155#336—21no—
156#336—21no—
157#334—41no—
158#334—41no—
159Q8⋊C8—41no—
160Q8⋊C8—41no—
161D4⋊C8—41no—
162D4⋊C8—41no—
163C22×Q8C22×C411yescharacteristic
164C22×D4—21no—
165C22×D4C22×C411yescharacteristic
166C23×C4—21no—
167C23×C4C22×C411yescharacteristic
168C23×C4C2×D411yescharacteristic
169C23×C4#4111yescharacteristic
170C2×SD16—22no—
171C2×SD16—22no—
172C2×SD16—41no—
173C2×M16—22no—
174C2×M16—41no—
175C2×M16—41no—
176C2×M16—41no—
177C2×M16—41no—
178C2×M16SD1614yescomplemented
179C22×C8—21no—
180C22×C8—21no—
181C22×C8—21no—
182C22×C8—21no—
183C22×C8—21no—
184C22×C8—21no—
185C22×C8—42no—
186C22×C8—42no—
187C22×C8—42no—
188C22×C8—42no—
189C22×C8—42no—
190C22×C8—42no—
191C22×C8C22×C411yescharacteristic
192C22×C8C22×C411yescharacteristic
193C22×C8C22×C411yescharacteristic
194C22×C8C22×C411yescharacteristic
195C22×C8C2×D411yescharacteristic
196C22×C8C2×D411yescharacteristic
197C22×C8C2×D411yescharacteristic
198C22×C8C2×D411yescharacteristic
199C22⋊Q8—22no—
200C22⋊Q8—41no—
201#121—22no—
202#121—41no—
203C4×Q8—21no—
204C4×Q8—21no—
205C4×Q8—21no—
206C4×Q8—21no—
207C4×D4—21no—
208C4×D4—21no—
209C4×D4—21no—
210C4×D4—21no—
211C4×D4—42no—
212C4×D4—42no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4C22×C411yescharacteristic
218C2×C4⋊C4C22×C411yescharacteristic
219C2×C4⋊C4C22×C411yescharacteristic
220C2×C4⋊C4C22×C411yescharacteristic
221C2×C4⋊C4C22×C411yescharacteristic
222C2×C4⋊C4C22×C411yescharacteristic
223C2×C22⋊C4—21no—
224C2×C22⋊C4—21no—
225C2×C22⋊C4—21no—
226C2×C22⋊C4—21no—
227C2×C22⋊C4—21no—
228C2×C22⋊C4—21no—
229C2×C22⋊C4—21no—
230C2×C42—21no—
231C2×C42—21no—
232C2×C42—21no—
233C2×C42—21no—
234C2×C42—21no—
235C2×C42—21no—
236C2×C42—21no—
237C2×C42C22×C411yescharacteristic
238C2×C42C2×D411yescharacteristic
239C2×C42#4111yescharacteristic
240C2×C42C2411yescharacteristic
241#106—41no—
242#106—41no—
243#106—41no—
244C4⋊C8—21no—
245C4⋊C8—21no—
246C4⋊C8—21no—
247C4⋊C8—21no—
248C4⋊C8—21no—
249C4⋊C8—21no—
250C4⋊C8—21no—
251C4⋊C8—21no—
252C4⋊C8—41no—
253C4⋊C8—41no—
254C4⋊C8—42no—
255C4⋊C8—42no—
256C4⋊C8—42no—
257#103—22no—
258#103—41no—
259#103—41no—
260#102—22no—
261#102—41no—
262#102—41no—
263C22⋊C8—41no—
264C22⋊C8—41no—
265C22⋊C8—42no—
266C22⋊C8—42no—
267C22⋊C8—42no—
268C22⋊C8—42no—
269C22⋊C8—42no—
270#97—41no—
271#97—41no—
272#97—42no—
273#97—42no—
274C4×C8—21no—
275C4×C8—21no—
276C4×C8—21no—
277C4×C8—21no—
278C4×C8—21no—
279C4×C8—21no—
280C4×C8—21no—
281C4×C8—21no—
282C4×C8—21no—
283C4×C8—21no—
284C4×C8—21no—
285C4×C8—21no—
286C4×C8—42no—
287C4×C8—42no—
288#95—21no—
289#95—21no—
290#95—21no—
291#95—21no—
292#95—21no—
293#95—21no—
294#95—21no—
295#95—21no—
296C24—21no—
297C24C4×D411yescharacteristic
298C2×Q8—21no—
299C2×Q8—21no—
300C2×Q8—21no—
301C2×Q8—21no—
302C2×Q8—21no—
303C2×Q8#13112yes—
304C2×D4—21no—
305C2×D4—21no—
306C2×D4—21no—
307C2×D4—21no—
308C2×D4—21no—
309C2×D4—42no—
310C2×D4—42no—
311C2×D4—81no—
312C2×D4#13112yes—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—21no—
318C22×C4—21no—
319C22×C4—21no—
320C22×C4—21no—
321C22×C4—21no—
322C22×C4—21no—
323C22×C4—21no—
324C22×C4—21no—
325C22×C4—21no—
326C22×C4—21no—
327C22×C4—21no—
328C22×C4—21no—
329C22×C4—21no—
330C22×C4—21no—
331C22×C4—21no—
332C22×C4—41no—
333C22×C4—41no—
334C22×C4—41no—
335C22×C4—41no—
336C22×C4—41no—
337C22×C4—41no—
338C22×C4—41no—
339C22×C4—41no—
340C22×C4C4×D411yescharacteristic
341C22×C4C4×D411yescharacteristic
342C22×C4C4×D411yescharacteristic
343C22×C4C2×SD1612yes—
344C22×C4#13512yes—
345C22×C4C23×C411yescharacteristic
346C22×C4C22×D411yescharacteristic
347C22×C4C2×#4111yescharacteristic
348SD16—44no—
349M16—44no—
350M16—44no—
351C2×C8—21no—
352C2×C8—21no—
353C2×C8—21no—
354C2×C8—21no—
355C2×C8—22no—
356C2×C8—22no—
357C2×C8—21no—
358C2×C8—21no—
359C2×C8—21no—
360C2×C8—21no—
361C2×C8—21no—
362C2×C8—21no—
363C2×C8—21no—
364C2×C8—21no—
365C2×C8—21no—
366C2×C8—21no—
367C2×C8—21no—
368C2×C8—21no—
369C2×C8—21no—
370C2×C8—21no—
371C2×C8—21no—
372C2×C8—21no—
373C2×C8—21no—
374C2×C8—21no—
375C2×C8—41no—
376C2×C8—41no—
377C2×C8—48no—
378C2×C8—41no—
379C2×C8—41no—
380C2×C8—41no—
381C2×C8—41no—
382C2×C8—41no—
383C2×C8—41no—
384C2×C8—41no—
385C2×C8—41no—
386C2×C8—41no—
387C2×C8—41no—
388C2×C8—41no—
389C2×C8—41no—
390C2×C8—41no—
391C2×C8—41no—
392C2×C8—84no—
393C2×C8C2×SD1612yes—
394C2×C8C2×SD1612yes—
395C4⋊C4—21no—
396C4⋊C4—21no—
397C4⋊C4—21no—
398C4⋊C4—21no—
399C4⋊C4—21no—
400C4⋊C4—21no—
401C4⋊C4—21no—
402C4⋊C4—21no—
403C4⋊C4—21no—
404C4⋊C4—21no—
405C4⋊C4—21no—
406C4⋊C4—21no—
407C4⋊C4—42no—
408C4⋊C4—41no—
409C4⋊C4—41no—
410C4⋊C4—42no—
411C4⋊C4—81no—
412C22⋊C4—42no—
413C22⋊C4—42no—
414C22⋊C4—42no—
415C22⋊C4—41no—
416C22⋊C4—41no—
417C22⋊C4—42no—
418C22⋊C4—81no—
419C22⋊C4—81no—
420C42—21no—
421C42—21no—
422C42—21no—
423C42—21no—
424C42—21no—
425C42—21no—
426C42—41no—
427C42—41no—
428C42—41no—
429C42—41no—
430C42—41no—
431C42—41no—
432C42H(5,𝔽2)11yescharacteristic
433C42H(5,𝔽2)11yescharacteristic
434C42H(5,𝔽2)11yescharacteristic
435C422−1+411yescharacteristic
436C23—21no—
437C23—21no—
438C23—21no—
439C23—21no—
440C23—21no—
441C23—21no—
442C23—22no—
443C23—41no—
444C23—41no—
445C23—41no—
446C23—41no—
447C23—41no—
448C23—41no—
449C23C4×SD1612yes—
450C23C2×C4×D411yescharacteristic
451Q8—22no—
452Q8—22no—
453Q8—41no—
454Q8—41no—
455D4—22no—
456D4—22no—
457D4—41no—
458D4—41no—
459D4—84no—
460C2×C4—21no—
461C2×C4—21no—
462C2×C4—21no—
463C2×C4—21no—
464C2×C4—22no—
465C2×C4—22no—
466C2×C4—22no—
467C2×C4—22no—
468C2×C4—21no—
469C2×C4—21no—
470C2×C4—21no—
471C2×C4—22no—
472C2×C4—21no—
473C2×C4—21no—
474C2×C4—21no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—41no—
478C2×C4—41no—
479C2×C4—41no—
480C2×C4—41no—
481C2×C4—41no—
482C2×C4—41no—
483C2×C4—41no—
484C2×C4—41no—
485C2×C4—41no—
486C2×C4—41no—
487C2×C4—41no—
488C2×C4—41no—
489C2×C4—41no—
490C2×C4—41no—
491C2×C4—41no—
492C2×C4—41no—
493C2×C4—41no—
494C2×C4—41no—
495C2×C4—41no—
496C2×C4—41no—
497C2×C4—41no—
498C2×C4—41no—
499C2×C4—41no—
500C2×C4—41no—
501C2×C4—81no—
502C2×C4—81no—
503C2×C4—81no—
504C2×C4—81no—
505C2×C4C4×SD1612yes—
506C2×C4#51811yescharacteristic
507C2×C4#52011yescharacteristic
508C2×C4#53411yescharacteristic
509C2×C4#53711yescharacteristic
510C2×C4#53811yescharacteristic
511C2×C4#53911yescharacteristic
512C2×C4C2×#13111yescharacteristic
513C2×C4C2×#13111yescharacteristic
514C2×C4#56811yescharacteristic
515C2×C4#56811yescharacteristic
516C2×C4C22×SD1611yescharacteristic
517C2×C4C2×#13511yescharacteristic
518C2×C4#57511yescharacteristic
519C2×C4#57511yescharacteristic
520C8—22no—
521C8—22no—
522C8—42no—
523C8—42no—
524C8—41no—
525C8—41no—
526C8—41no—
527C8—41no—
528C8—41no—
529C8—41no—
530C8—81no—
531C8—81no—
532C8—81no—
533C8—81no—
534C22—22no—
535C22—22no—
536C22—22no—
537C22—41no—
538C22—41no—
539C22—41no—
540C22—41no—
541C22—42no—
542C22—42no—
543C22—42no—
544C22—81no—
545C22—81no—
546C22—81no—
547C22—81no—
548C22#232511yescharacteristic
549C22#293311yescharacteristic
550C22#293311yescharacteristic
551C22C2×C4×SD1611yescharacteristic
552C22#294311yescharacteristic
553C22#295611yescharacteristic
554C22#295611yescharacteristic
555C4—22no—
556C4—21no—
557C4—21no—
558C4—41no—
559C4—42no—
560C4—41no—
561C4—81no—
562C4—81no—
563C4—81no—
564C4—81no—
565C4—81no—
566C4—81no—
567C4#297711yescharacteristic
568C4#298011yescharacteristic
569C4#299211yescharacteristic
570C4#299211yescharacteristic
571C4#318711yescharacteristic
572C4#319111yescharacteristic
573C4#324611yescharacteristic
574C4#325011yescharacteristic
575C2—42no—
576C2—82no—
577C2#1799511yescharacteristic
578C2#1799711yescharacteristic
579C2#1800211yescharacteristic
580C2#1800511yescharacteristic
581C2#2217111yescharacteristic
582C2#2218011yescharacteristic
583C2#2220711yescharacteristic
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×C4×M16 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21596C2semidirect
4. #9547C2semidirect
5. #9546C2semidirect
6. #9539C2semidirect
7. #9538C2semidirect
9. #9071C2semidirect
10. #8948C2semidirect
11. #8945C2semidirect
12. #8846C2semidirect
13. #8845C2semidirect
14. #8598C2semidirect
15. #7848C2semidirect
178. C2×M16SD16semidirect

Maximal subgroups

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

subgroupindex
#237002
#215962
#214862
#95472
#95462
#95392
#95382
#90722
#90712
#89482
#89452
#88462
#88452
#85982
#78482