Finite Group Databasebeta

← #229133

Subgroups — #229133

Order 512.

750 conjugacy classes of subgroups.

The subgroup lattice has 594 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22986C211yesmaximal, characteristic
2#22953C211yesmaximal, characteristic, complemented
3C4×#465C211yesmaximal, characteristic
4C2×SD16⋊C8C211yesmaximal, characteristic, complemented
5C2×#1567C211yesmaximal, characteristic, complemented
6#9660C211yesmaximal, characteristic, complemented
7#9596C211yesmaximal, characteristic, complemented
8#9548C211yesmaximal, characteristic, complemented
9#9545C211yesmaximal, characteristic, complemented
10#8944C211yesmaximal, characteristic
11#8942C211yesmaximal, characteristic, complemented
12#8844C211yesmaximal, characteristic, complemented
13#8843C211yesmaximal, characteristic, complemented
14#8286C211yesmaximal, characteristic, complemented
15#8144C211yesmaximal, characteristic, complemented
16C2×#465C411yescharacteristic
17C2×C8×Q8C2211yescharacteristic
18C2×C4×SD16C2211yescharacteristic
19C2×#435—21no—
20C2×C8×D4C2211yescharacteristic
21C2×#423C2211yescharacteristic
22C2×#423C2211yescharacteristic
23C22×C4×C8C2211yescharacteristic
24C4×C22⋊Q8C2211yescharacteristic
25C4×#121C2211yescharacteristic
26#1984C2211yescharacteristic
27#1968C411yescharacteristic
28#1954C2211yescharacteristic
29#1941C411yescharacteristic
30#1940—21no—
31#1939C2211yescharacteristic
32#1939C2211yescharacteristic
33#1934C411yescharacteristic
34#1933C411yescharacteristic
35#1924—21no—
36#1916—21no—
37#1893C411yescharacteristic
38#1892C411yescharacteristic
39#1870—21no—
40#1851—21no—
41#1850—21no—
42#1807C411yescharacteristic
43C4×#106C2211yescharacteristic
44#1770C2211yescharacteristic
45#1770C2211yescharacteristic
46C4×#103C2211yescharacteristic
47C4×#102C2211yescharacteristic
48#1753—21no—
49C4×C22⋊C8C2211yescharacteristic
50C4×C22⋊C8C2211yescharacteristic
51SD16⋊C8—24no—
52#1567—24no—
53C2×#335C2211yescharacteristic
54C2×#335C2211yescharacteristic
55C2×#335C2211yescharacteristic
56C2×#335C2211yescharacteristic
57C2×Q8⋊C8C2211yescharacteristic
58C2×Q8⋊C8C2211yescharacteristic
59C2×D4⋊C8C2211yescharacteristic
60C2×D4⋊C8C2211yescharacteristic
61C2×#322C2211yescharacteristic
62C2×#322C2211yescharacteristic
63#1279C2211yescharacteristic
64#1279C2211yescharacteristic
65#1279C2211yescharacteristic
66#1279C2211yescharacteristic
67#1273C2211yescharacteristic
68#1273C2211yescharacteristic
69C22×SD16C2×C411yescharacteristic
70C22×M16—21no—
71C22×M16—21no—
72C23×C8—21no—
73C23×C8C2×C411yescharacteristic
74C23×C8C2×C411yescharacteristic
75C2×C22⋊Q8C2×C411yescharacteristic
76C2×#121C2×C411yescharacteristic
77C2×C4×Q8C2311yescharacteristic
78C2×C4×D4—21no—
79C2×C4×D4C2311yescharacteristic
80C22×C42C2311yescharacteristic
81#465—24no—
82C8×Q8—22no—
83C8×Q8—41no—
84C4×SD16—22no—
85C4×SD16—22no—
86#435—44no—
87C8×D4—22no—
88C8×D4—41no—
89C2×#106C2×C411yescharacteristic
90C2×#106C2×C411yescharacteristic
91C2×#106C2×C411yescharacteristic
92#423—22no—
93#423—22no—
94#423—41no—
95#423—41no—
96C2×C4⋊C8—21no—
97C2×C4⋊C8—21no—
98C2×C4⋊C8—21no—
99C2×C4⋊C8—21no—
100C2×C4⋊C8—21no—
101C2×C4⋊C8C2311yescharacteristic
102C2×C4⋊C8C2311yescharacteristic
103C2×C4⋊C8C2311yescharacteristic
104C2×C4⋊C8C2311yescharacteristic
105C2×#103C2×C411yescharacteristic
106C2×#103C2×C411yescharacteristic
107C2×#103C2×C411yescharacteristic
108C2×#102C2×C411yescharacteristic
109C2×#102C2×C411yescharacteristic
110C2×#102C2×C411yescharacteristic
111C2×C22⋊C8—21no—
112C2×C22⋊C8—21no—
113C2×C22⋊C8—21no—
114C2×C22⋊C8—21no—
115C2×C22⋊C8D411yescharacteristic
116C2×C22⋊C8D411yescharacteristic
117C2×#97—21no—
118C2×#97—21no—
119C2×C4×C8—22no—
120C2×C4×C8—22no—
121C2×C4×C8—21no—
122C2×C4×C8C2×C411yescharacteristic
123C2×C4×C8C2×C411yescharacteristic
124C2×C4×C8C2311yescharacteristic
125C2×C4×C8C2311yescharacteristic
126C2×C4×C8C2311yescharacteristic
127C2×C4×C8C2311yescharacteristic
128C2×C4×C8C2311yescharacteristic
129C2×C4×C8C2311yescharacteristic
130#391C2×C411yescharacteristic
131#390C2×C411yescharacteristic
132#389C2×C411yescharacteristic
133#389C2×C411yescharacteristic
134#388—21no—
135#387—21no—
136#386—21no—
137#385—21no—
138#380C2×C411yescharacteristic
139#380C2×C411yescharacteristic
140C4×C4⋊C4—21no—
141C4×C4⋊C4C2311yescharacteristic
142C4×C4⋊C4C2311yescharacteristic
143C4×C22⋊C4—21no—
144C4×C22⋊C4—21no—
145#340C2×C411yescharacteristic
146#340C2×C411yescharacteristic
147#340C2×C411yescharacteristic
148#340C2×C411yescharacteristic
149#340C2×C411yescharacteristic
150#340C2×C411yescharacteristic
151#336—21no—
152#336—21no—
153#336—21no—
154#336—21no—
155#336—21no—
156#336—21no—
157#336—21no—
158#336D411yescharacteristic
159#336D411yescharacteristic
160#335—22no—
161#335—22no—
162#335—22no—
163#335D414yescomplemented
164Q8⋊C8—22no—
165Q8⋊C8—41no—
166D4⋊C8—22no—
167D4⋊C8—41no—
168#322—22no—
169#322—22no—
170C22×Q8C22×C411yescharacteristic
171C22×D4—21no—
172C22×D4C22×C411yescharacteristic
173C23×C4—21no—
174C23×C4C22×C411yescharacteristic
175C23×C4C2×D411yescharacteristic
176C23×C4#4111yescharacteristic
177C2×SD16—22no—
178C2×SD16—22no—
179C2×SD16M1614yescomplemented
180C2×M16—42no—
181C2×M16—42no—
182C2×M16—42no—
183C2×M16—42no—
184C2×M16—81no—
185C2×M16—81no—
186C22×C8—21no—
187C22×C8—21no—
188C22×C8—22no—
189C22×C8—22no—
190C22×C8—22no—
191C22×C8—22no—
192C22×C8—22no—
193C22×C8—21no—
194C22×C8—21no—
195C22×C8—21no—
196C22×C8—21no—
197C22×C8—41no—
198C22×C8—42no—
199C22×C8—42no—
200C22×C8—81no—
201C22×C8C22×C411yescharacteristic
202C22×C8C22×C411yescharacteristic
203C22×C8C22×C411yescharacteristic
204C22×C8C22×C411yescharacteristic
205C22×C8C2×D411yescharacteristic
206C22×C8C2×D411yescharacteristic
207C22×C8C2×D411yescharacteristic
208C22×C8C2×D411yescharacteristic
209C22⋊Q8—22no—
210C22⋊Q8—41no—
211#121—22no—
212#121—41no—
213C4×Q8—21no—
214C4×Q8—21no—
215C4×Q8—21no—
216C4×Q8—21no—
217C4×D4—21no—
218C4×D4—21no—
219C4×D4—21no—
220C4×D4—21no—
221C4×D4—42no—
222C4×D4—42no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4C22×C411yescharacteristic
228C2×C4⋊C4C22×C411yescharacteristic
229C2×C4⋊C4C22×C411yescharacteristic
230C2×C4⋊C4C22×C411yescharacteristic
231C2×C4⋊C4C22×C411yescharacteristic
232C2×C4⋊C4C22×C411yescharacteristic
233C2×C22⋊C4—21no—
234C2×C22⋊C4—21no—
235C2×C22⋊C4—21no—
236C2×C22⋊C4—21no—
237C2×C22⋊C4—21no—
238C2×C22⋊C4—21no—
239C2×C22⋊C4—21no—
240C2×C42—21no—
241C2×C42—21no—
242C2×C42—21no—
243C2×C42—21no—
244C2×C42—21no—
245C2×C42—21no—
246C2×C42—21no—
247C2×C42C22×C411yescharacteristic
248C2×C42C2×D411yescharacteristic
249C2×C42#4111yescharacteristic
250C2×C42C2411yescharacteristic
251#106—22no—
252#106—22no—
253#106—22no—
254C4⋊C8—21no—
255C4⋊C8—21no—
256C4⋊C8—21no—
257C4⋊C8—21no—
258C4⋊C8—21no—
259C4⋊C8—21no—
260C4⋊C8—42no—
261C4⋊C8—42no—
262C4⋊C8—42no—
263C4⋊C8—42no—
264C4⋊C8—81no—
265C4⋊C8C2×D412yes—
266C4⋊C8C2×D412yes—
267#103—22no—
268#103—22no—
269#103—22no—
270#102—22no—
271#102—22no—
272#102—22no—
273C22⋊C8—22no—
274C22⋊C8—22no—
275C22⋊C8—42no—
276C22⋊C8—42no—
277C22⋊C8—42no—
278C22⋊C8—81no—
279#97—42no—
280#97—81no—
281C4×C8—22no—
282C4×C8—22no—
283C4×C8—21no—
284C4×C8—21no—
285C4×C8—21no—
286C4×C8—21no—
287C4×C8—21no—
288C4×C8—21no—
289C4×C8—21no—
290C4×C8—21no—
291C4×C8—21no—
292C4×C8—21no—
293C4×C8—42no—
294C4×C8C2×D412yes—
295C4×C8#4112yes—
296#95—21no—
297#95—21no—
298#95—21no—
299#95—21no—
300#95—21no—
301#95—21no—
302#95—21no—
303#95—21no—
304C24—21no—
305C24C4×D411yescharacteristic
306C2×Q8—21no—
307C2×Q8—21no—
308C2×Q8—21no—
309C2×Q8—21no—
310C2×Q8—21no—
311C2×Q8C2×M1612yes—
312C2×D4—21no—
313C2×D4—21no—
314C2×D4—21no—
315C2×D4—21no—
316C2×D4—21no—
317C2×D4—42no—
318C2×D4—42no—
319C2×D4—81no—
320C2×D4C2×M1612yes—
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—21no—
333C22×C4—21no—
334C22×C4—22no—
335C22×C4—21no—
336C22×C4—21no—
337C22×C4—21no—
338C22×C4—21no—
339C22×C4—21no—
340C22×C4—21no—
341C22×C4—41no—
342C22×C4—41no—
343C22×C4—41no—
344C22×C4—41no—
345C22×C4—41no—
346C22×C4—41no—
347C22×C4—41no—
348C22×C4C4×D411yescharacteristic
349C22×C4C4×D411yescharacteristic
350C22×C4C4×D411yescharacteristic
351C22×C4AGL(1,ℤ/8)12yes—
352C22×C4#13712yes—
353C22×C4C23×C411yescharacteristic
354C22×C4C22×D411yescharacteristic
355C22×C4C2×#4111yescharacteristic
356SD16—28no—
357M16—84no—
358M16—84no—
359C2×C8—21no—
360C2×C8—21no—
361C2×C8—21no—
362C2×C8—21no—
363C2×C8—21no—
364C2×C8—21no—
365C2×C8—21no—
366C2×C8—22no—
367C2×C8—22no—
368C2×C8—21no—
369C2×C8—21no—
370C2×C8—21no—
371C2×C8—21no—
372C2×C8—21no—
373C2×C8—21no—
374C2×C8—21no—
375C2×C8—21no—
376C2×C8—21no—
377C2×C8—21no—
378C2×C8—21no—
379C2×C8—21no—
380C2×C8—44no—
381C2×C8—41no—
382C2×C8—41no—
383C2×C8—41no—
384C2×C8—41no—
385C2×C8—44no—
386C2×C8—41no—
387C2×C8—41no—
388C2×C8—41no—
389C2×C8—41no—
390C2×C8—41no—
391C2×C8—41no—
392C2×C8—41no—
393C2×C8—41no—
394C2×C8—41no—
395C2×C8—41no—
396C2×C8—41no—
397C2×C8—41no—
398C2×C8—84no—
399C2×C8C4×D412yes—
400C2×C8C4×D412yes—
401C2×C8C2×M1612yes—
402C2×C8#13112yes—
403C2×C8#13112yes—
404C4⋊C4—21no—
405C4⋊C4—21no—
406C4⋊C4—21no—
407C4⋊C4—21no—
408C4⋊C4—21no—
409C4⋊C4—21no—
410C4⋊C4—21no—
411C4⋊C4—21no—
412C4⋊C4—21no—
413C4⋊C4—21no—
414C4⋊C4—21no—
415C4⋊C4—21no—
416C4⋊C4—42no—
417C4⋊C4—41no—
418C4⋊C4—41no—
419C4⋊C4—42no—
420C4⋊C4—42no—
421C22⋊C4—42no—
422C22⋊C4—42no—
423C22⋊C4—42no—
424C22⋊C4—41no—
425C22⋊C4—41no—
426C22⋊C4—42no—
427C22⋊C4—42no—
428C22⋊C4—81no—
429C42—21no—
430C42—21no—
431C42—21no—
432C42—21no—
433C42—21no—
434C42—21no—
435C42—41no—
436C42—41no—
437C42—41no—
438C42—41no—
439C42—41no—
440C42—41no—
441C42C22×D411yescharacteristic
442C42C2×#4111yescharacteristic
443C42H(5,𝔽2)11yescharacteristic
444C422−1+411yescharacteristic
445C23—21no—
446C23—21no—
447C23—21no—
448C23—21no—
449C23—21no—
450C23—21no—
451C23—22no—
452C23—41no—
453C23—41no—
454C23—41no—
455C23—41no—
456C23—41no—
457C23—41no—
458C23#44012yes—
459C23C2×C4×D411yescharacteristic
460Q8—22no—
461Q8—22no—
462Q8—41no—
463Q8—41no—
464D4—22no—
465D4—22no—
466D4—41no—
467D4—41no—
468D4—84no—
469C2×C4—21no—
470C2×C4—21no—
471C2×C4—21no—
472C2×C4—21no—
473C2×C4—21no—
474C2×C4—21no—
475C2×C4—22no—
476C2×C4—22no—
477C2×C4—22no—
478C2×C4—22no—
479C2×C4—21no—
480C2×C4—21no—
481C2×C4—21no—
482C2×C4—22no—
483C2×C4—42no—
484C2×C4—42no—
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—41no—
502C2×C4—41no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—81no—
510C2×C4—81no—
511C2×C4—81no—
512C2×C4—81no—
513C2×C4#44012yes—
514C2×C4#44412yes—
515C2×C4C2×C4×D411yescharacteristic
516C2×C4#52011yescharacteristic
517C2×C4D4211yescharacteristic
518C2×C4#54711yescharacteristic
519C2×C4#54811yescharacteristic
520C2×C4#55311yescharacteristic
521C2×C4C22×M1611yescharacteristic
522C2×C4C2×#13111yescharacteristic
523C2×C4C2×#13111yescharacteristic
524C2×C4#56811yescharacteristic
525C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
526C2×C4C2×#13711yescharacteristic
527C2×C4#57611yescharacteristic
528C2×C4#57811yescharacteristic
529C8—22no—
530C8—22no—
531C8—41no—
532C8—41no—
533C8—41no—
534C8—41no—
535C8—42no—
536C8—42no—
537C8—41no—
538C8—81no—
539C8—81no—
540C8—81no—
541C8—81no—
542C8#43514yescomplemented
543C22—22no—
544C22—22no—
545C22—22no—
546C22—41no—
547C22—41no—
548C22—41no—
549C22—41no—
550C22—42no—
551C22—42no—
552C22—42no—
553C22—81no—
554C22—81no—
555C22—81no—
556C22—81no—
557C22#236111yescharacteristic
558C22#293111yescharacteristic
559C22D4×M1611yescharacteristic
560C22C2×#44011yescharacteristic
561C22#295111yescharacteristic
562C22C2×#44411yescharacteristic
563C22#295711yescharacteristic
564C4—21no—
565C4—21no—
566C4—22no—
567C4—22no—
568C4—41no—
569C4—42no—
570C4—81no—
571C4—81no—
572C4—81no—
573C4—81no—
574C4—81no—
575C4—81no—
576C4C2×#43511yescharacteristic
577C4#293311yescharacteristic
578C4#299011yescharacteristic
579C4#299311yescharacteristic
580C4#326411yescharacteristic
581C4#327011yescharacteristic
582C4#334311yescharacteristic
583C4#334611yescharacteristic
584C2—42no—
585C2—82no—
586C2#1796011yescharacteristic
587C2#1797311yescharacteristic
588C2#1804011yescharacteristic
589C2#1806111yescharacteristic
590C2#2298611yescharacteristic
591C2#2301911yescharacteristic
592C2#2303211yescharacteristic
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 ◁ C4×C22⋊C8 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #22953C2semidirect
4. C2×SD16⋊C8C2semidirect
5. C2×#1567C2semidirect
6. #9660C2semidirect
7. #9596C2semidirect
8. #9548C2semidirect
9. #9545C2semidirect
11. #8942C2semidirect
12. #8844C2semidirect
13. #8843C2semidirect
14. #8286C2semidirect
15. #8144C2semidirect
163. #335D4semidirect
179. C2×SD16M16semidirect
542. C8#435semidirect

Maximal subgroups

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

subgroupindex
#229862
#229532
C4×#4652
C2×SD16⋊C82
C2×#15672
#96602
#95962
#95482
#95452
#89442
#89422
#88442
#88432
#82862
#81442