Finite Group Databasebeta

← #217818

Subgroups — #217818

Order 512.

740 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21598C211yesmaximal, characteristic
2C4×#451C211yesmaximal, characteristic
3#21303C211yesmaximal, characteristic
4#9662C211yesmaximal, characteristic, complemented
5#9656C211yesmaximal, characteristic, complemented
6#9557C211yesmaximal, characteristic, complemented
7#9553C211yesmaximal, characteristic, complemented
8#9544C211yesmaximal, characteristic, complemented
9#9544C211yesmaximal, characteristic, complemented
10#8950C211yesmaximal, characteristic, complemented
11#8950C211yesmaximal, characteristic, complemented
12#8942C211yesmaximal, characteristic
13#8832C211yesmaximal, characteristic
14#8311C211yesmaximal, characteristic
15#8146C211yesmaximal, characteristic
16C22×C4×Q8C2211yescharacteristic
17C2×#451C411yescharacteristic
18C2×#446C2211yescharacteristic
19C2×C4×Q16C2211yescharacteristic
20C2×C4×Q16C2211yescharacteristic
21C2×#423C2211yescharacteristic
22C2×C4×M16C2211yescharacteristic
23C4×C22⋊Q8C2211yescharacteristic
24#1985—21no—
25#1985—21no—
26#1984C2211yescharacteristic
27#1984C2211yescharacteristic
28#1984C2211yescharacteristic
29#1984C2211yescharacteristic
30#1984C2211yescharacteristic
31#1939C2211yescharacteristic
32#1917—21no—
33#1878C411yescharacteristic
34#1878C411yescharacteristic
35#1872C411yescharacteristic
36#1863C411yescharacteristic
37#1863C411yescharacteristic
38#1852C411yescharacteristic
39#1851—21no—
40#1797C2211yescharacteristic
41#1787C411yescharacteristic
42#1770—21no—
43C8×C4⋊C4C2211yescharacteristic
44C4×#103C2211yescharacteristic
45C4×#103C2211yescharacteristic
46#1753—21no—
47C4×C22⋊C8C2211yescharacteristic
48C4×C22⋊C8C2211yescharacteristic
49#1728—21no—
50#1728—21no—
51C2×#335C2211yescharacteristic
52C2×#335C2211yescharacteristic
53C2×Q8⋊C8C2211yescharacteristic
54C2×Q8⋊C8C2211yescharacteristic
55C2×Q8⋊C8C2211yescharacteristic
56C2×Q8⋊C8C2211yescharacteristic
57C2×Q8⋊C8C2211yescharacteristic
58C2×Q8⋊C8C2211yescharacteristic
59C2×#322C2211yescharacteristic
60C2×C82C2211yescharacteristic
61#1279C2211yescharacteristic
62#1279C2211yescharacteristic
63#1277C2211yescharacteristic
64#1277C2211yescharacteristic
65#1273C2211yescharacteristic
66#1273C2211yescharacteristic
67C23×Q8C2×C411yescharacteristic
68C22×Q16C2×C411yescharacteristic
69C22×Q16C2×C411yescharacteristic
70C22×M16—21no—
71C22×M16D411yescharacteristic
72C22×M16D411yescharacteristic
73C2×C22⋊Q8C2×C411yescharacteristic
74C2×C4×Q8—21no—
75C2×C4×Q8—21no—
76C2×C4×Q8—41no—
77C2×C4×Q8—41no—
78C2×C4×Q8—41no—
79C2×C4×Q8—41no—
80C2×C4×Q8C2×C411yescharacteristic
81C2×C4×Q8C2×C411yescharacteristic
82C2×C4×Q8C2311yescharacteristic
83C2×C4×Q8C2311yescharacteristic
84C2×C4×Q8C2311yescharacteristic
85C22×C4⋊C4—21no—
86C22×C4⋊C4C2×C411yescharacteristic
87C22×C42—21no—
88C22×C42C2311yescharacteristic
89#451—24no—
90#446—41no—
91#446—41no—
92C4×Q16—41no—
93C4×Q16—41no—
94C4×Q16—41no—
95C4×Q16—41no—
96C2×#107C2×C411yescharacteristic
97C2×#107C2×C411yescharacteristic
98#423—41no—
99#423—41no—
100C2×C4⋊C8—21no—
101C2×C4⋊C8—21no—
102C2×C4⋊C8—21no—
103C2×C4⋊C8C2311yescharacteristic
104C2×C4⋊C8C2311yescharacteristic
105C2×C4⋊C8C2311yescharacteristic
106C2×C4⋊C8C2311yescharacteristic
107C2×#103C2×C411yescharacteristic
108C2×#103C2×C411yescharacteristic
109C2×#103C2×C411yescharacteristic
110C2×#103C2×C411yescharacteristic
111C2×#103C2×C411yescharacteristic
112C2×#103C2×C411yescharacteristic
113C2×#103C2×C411yescharacteristic
114C2×#103C2×C411yescharacteristic
115C2×C22⋊C8—21no—
116C2×C22⋊C8—21no—
117C2×C22⋊C8C2×C411yescharacteristic
118C2×C22⋊C8C2×C411yescharacteristic
119C4×M16—41no—
120C4×M16—41no—
121C2×#97—21no—
122C2×#97D411yescharacteristic
123C2×#97D411yescharacteristic
124C2×C4×C8—21no—
125C2×C4×C8C2311yescharacteristic
126C2×C4×C8C2311yescharacteristic
127C2×C4×C8C2311yescharacteristic
128C2×C4×C8C2311yescharacteristic
129C2×C4×C8C2311yescharacteristic
130C2×C4×C8C2311yescharacteristic
131#391C2×C411yescharacteristic
132#389C2×C411yescharacteristic
133#387—21no—
134#385—21no—
135#380C2×C411yescharacteristic
136C4×C4⋊C4—21no—
137C4×C4⋊C4C2311yescharacteristic
138C4×C22⋊C4—21no—
139#340C2×C411yescharacteristic
140#340C2×C411yescharacteristic
141#340C2×C411yescharacteristic
142#340C2×C411yescharacteristic
143#336—21no—
144#336—21no—
145#336—21no—
146#336—21no—
147#336—21no—
148#336—21no—
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#336C2×C411yescharacteristic
158#336C2×C411yescharacteristic
159#335—41no—
160#335—41no—
161Q8⋊C8—41no—
162Q8⋊C8—41no—
163Q8⋊C8—41no—
164Q8⋊C8—41no—
165Q8⋊C8—41no—
166Q8⋊C8—41no—
167#322—41no—
168C82—41no—
169C22×Q8—21no—
170C22×Q8—22no—
171C22×Q8—22no—
172C22×Q8—41no—
173C22×Q8—41no—
174C22×Q8—41no—
175C22×Q8—41no—
176C22×Q8C22×C411yescharacteristic
177C22×Q8C22×C411yescharacteristic
178C22×Q8C22×C411yescharacteristic
179C23×C4—21no—
180C23×C4—21no—
181C23×C4C22×C411yescharacteristic
182C23×C4C2×D411yescharacteristic
183C23×C4#4111yescharacteristic
184C2×Q16—22no—
185C2×Q16—22no—
186C2×Q16—22no—
187C2×Q16—41no—
188C2×Q16—41no—
189C2×Q16—41no—
190C2×M16—22no—
191C2×M16—22no—
192C2×M16—41no—
193C2×M16—41no—
194C2×M16—41no—
195C2×M16—42no—
196C2×M16—42no—
197C2×M16—42no—
198C2×M16Q1614yescomplemented
199C22×C8—21no—
200C22×C8—21no—
201C22×C8—21no—
202C22×C8—21no—
203C22×C8—21no—
204C22×C8—21no—
205C22×C8C22×C411yescharacteristic
206C22×C8C22×C411yescharacteristic
207C22×C8C22×C411yescharacteristic
208C22×C8C22×C411yescharacteristic
209C22×C8C2×D411yescharacteristic
210C22×C8C2×D411yescharacteristic
211C22×C8C2×D411yescharacteristic
212C22×C8C2×D411yescharacteristic
213C22⋊Q8—22no—
214C22⋊Q8—41no—
215C4×Q8—21no—
216C4×Q8—21no—
217C4×Q8—21no—
218C4×Q8—21no—
219C4×Q8—21no—
220C4×Q8—21no—
221C4×Q8—21no—
222C4×Q8—21no—
223C4×Q8—21no—
224C4×Q8—21no—
225C4×Q8—21no—
226C4×Q8—21no—
227C4×Q8—41no—
228C4×Q8—41no—
229C4×Q8—41no—
230C4×Q8—41no—
231C4×Q8—42no—
232C4×Q8—42no—
233C4×Q8—42no—
234C4×Q8—42no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4—21no—
239C2×C4⋊C4—21no—
240C2×C4⋊C4—21no—
241C2×C4⋊C4—21no—
242C2×C4⋊C4—21no—
243C2×C4⋊C4—41no—
244C2×C4⋊C4—41no—
245C2×C4⋊C4—42no—
246C2×C4⋊C4—42no—
247C2×C4⋊C4C22×C411yescharacteristic
248C2×C4⋊C4C22×C411yescharacteristic
249C2×C4⋊C4C22×C411yescharacteristic
250C2×C4⋊C4C22×C411yescharacteristic
251C2×C4⋊C4C22×C411yescharacteristic
252C2×C22⋊C4—21no—
253C2×C22⋊C4—21no—
254C2×C42—21no—
255C2×C42—21no—
256C2×C42—21no—
257C2×C42—21no—
258C2×C42—21no—
259C2×C42—21no—
260C2×C42—21no—
261C2×C42—21no—
262C2×C42—21no—
263C2×C42—21no—
264C2×C42—42no—
265C2×C42—42no—
266C2×C42C22×C411yescharacteristic
267C2×C42C2×D411yescharacteristic
268C2×C42#4111yescharacteristic
269C2×C42C2411yescharacteristic
270#107—41no—
271#107—41no—
272C4⋊C8—21no—
273C4⋊C8—21no—
274C4⋊C8—21no—
275C4⋊C8—21no—
276C4⋊C8—21no—
277C4⋊C8—21no—
278C4⋊C8—21no—
279C4⋊C8—21no—
280C4⋊C8—42no—
281C4⋊C8—42no—
282C4⋊C8—42no—
283#103—22no—
284#103—22no—
285#103—41no—
286#103—41no—
287#103—41no—
288#103—41no—
289#103—41no—
290#103M1614yescomplemented
291C22⋊C8—22no—
292C22⋊C8—41no—
293C22⋊C8—42no—
294C22⋊C8—42no—
295#97—41no—
296#97—41no—
297#97—42no—
298C4×C8—21no—
299C4×C8—21no—
300C4×C8—21no—
301C4×C8—21no—
302C4×C8—21no—
303C4×C8—21no—
304C4×C8—21no—
305C4×C8—21no—
306C4×C8—21no—
307C4×C8—21no—
308C4×C8—21no—
309C4×C8—21no—
310C4×C8—42no—
311#95—21no—
312#95—21no—
313#95—21no—
314#95—21no—
315#95—21no—
316C24C4×D411yescharacteristic
317C2×Q8—21no—
318C2×Q8—21no—
319C2×Q8—21no—
320C2×Q8—21no—
321C2×Q8—21no—
322C2×Q8—21no—
323C2×Q8—21no—
324C2×Q8—21no—
325C2×Q8—21no—
326C2×Q8—21no—
327C2×Q8—21no—
328C2×Q8—21no—
329C2×Q8—21no—
330C2×Q8—21no—
331C2×Q8—21no—
332C2×Q8—21no—
333C2×Q8—42no—
334C2×Q8—42no—
335C2×Q8—42no—
336C2×Q8—44no—
337C2×Q8—44no—
338C2×Q8—44no—
339C2×Q8—44no—
340C2×Q8C2×M1612yes—
341C2×Q8#13112yes—
342C22×C4—21no—
343C22×C4—21no—
344C22×C4—21no—
345C22×C4—21no—
346C22×C4—21no—
347C22×C4—21no—
348C22×C4—21no—
349C22×C4—21no—
350C22×C4—21no—
351C22×C4—21no—
352C22×C4—21no—
353C22×C4—21no—
354C22×C4—21no—
355C22×C4—21no—
356C22×C4—21no—
357C22×C4—21no—
358C22×C4—21no—
359C22×C4—21no—
360C22×C4—21no—
361C22×C4—21no—
362C22×C4—41no—
363C22×C4—41no—
364C22×C4—42no—
365C22×C4—42no—
366C22×C4—42no—
367C22×C4—42no—
368C22×C4—81no—
369C22×C4—81no—
370C22×C4C4×D411yescharacteristic
371C22×C4C4×D411yescharacteristic
372C22×C4C4×D411yescharacteristic
373C22×C4C2×Q1612yes—
374C22×C4#13512yes—
375C22×C4C23×C411yescharacteristic
376C22×C4C22×D411yescharacteristic
377C22×C4C2×#4111yescharacteristic
378Q16—44no—
379Q16—44no—
380M16—44no—
381M16—44no—
382M16—84no—
383C2×C8—21no—
384C2×C8—21no—
385C2×C8—21no—
386C2×C8—21no—
387C2×C8—21no—
388C2×C8—21no—
389C2×C8—21no—
390C2×C8—21no—
391C2×C8—22no—
392C2×C8—21no—
393C2×C8—21no—
394C2×C8—21no—
395C2×C8—21no—
396C2×C8—21no—
397C2×C8—21no—
398C2×C8—21no—
399C2×C8—21no—
400C2×C8—21no—
401C2×C8—21no—
402C2×C8—21no—
403C2×C8—21no—
404C2×C8—41no—
405C2×C8—41no—
406C2×C8—41no—
407C2×C8—41no—
408C2×C8—41no—
409C2×C8—41no—
410C2×C8—41no—
411C2×C8—41no—
412C2×C8—41no—
413C2×C8—41no—
414C2×C8—41no—
415C2×C8—41no—
416C2×C8—41no—
417C2×C8—41no—
418C2×C8—41no—
419C2×C8—41no—
420C2×C8—41no—
421C2×C8C2×M1612yes—
422C2×C8C2×Q1612yes—
423C2×C8C2×Q1612yes—
424C2×C8#13712yes—
425C4⋊C4—21no—
426C4⋊C4—21no—
427C4⋊C4—21no—
428C4⋊C4—21no—
429C4⋊C4—21no—
430C4⋊C4—21no—
431C4⋊C4—21no—
432C4⋊C4—21no—
433C4⋊C4—21no—
434C4⋊C4—42no—
435C4⋊C4—42no—
436C4⋊C4—42no—
437C4⋊C4—42no—
438C4⋊C4—41no—
439C4⋊C4—41no—
440C4⋊C4—41no—
441C4⋊C4—41no—
442C4⋊C4—41no—
443C4⋊C4—41no—
444C4⋊C4—81no—
445C4⋊C4C2×M1612yes—
446C22⋊C4—42no—
447C22⋊C4—81no—
448C42—21no—
449C42—21no—
450C42—21no—
451C42—21no—
452C42—21no—
453C42—21no—
454C42—42no—
455C42—42no—
456C42—41no—
457C42—41no—
458C42—41no—
459C42—41no—
460C42—41no—
461C42—41no—
462C42—41no—
463C42—41no—
464C42H(5,𝔽2)11yescharacteristic
465C42H(5,𝔽2)11yescharacteristic
466C42H(5,𝔽2)11yescharacteristic
467C42H(5,𝔽2)11yescharacteristic
468C23—21no—
469C23—21no—
470C23—21no—
471C23—21no—
472C23—21no—
473C23—21no—
474C23C4×Q1612yes—
475C23C2×C4×D411yescharacteristic
476Q8—22no—
477Q8—22no—
478Q8—22no—
479Q8—22no—
480Q8—48no—
481Q8—41no—
482Q8—41no—
483Q8—41no—
484Q8—41no—
485Q8—41no—
486Q8—41no—
487Q8—41no—
488Q8—41no—
489C2×C4—21no—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—21no—
503C2×C4—22no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—48no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—81no—
533C2×C4—81no—
534C2×C4—84no—
535C2×C4C4×Q1612yes—
536C2×C4#51811yescharacteristic
537C2×C4#51811yescharacteristic
538C2×C4#53411yescharacteristic
539C2×C4#53511yescharacteristic
540C2×C4#53811yescharacteristic
541C2×C4#53811yescharacteristic
542C2×C4C22×M1611yescharacteristic
543C2×C4C2×#13111yescharacteristic
544C2×C4#56811yescharacteristic
545C2×C4#56811yescharacteristic
546C2×C4C22×Q1611yescharacteristic
547C2×C4C2×#13511yescharacteristic
548C2×C4C2×#13711yescharacteristic
549C2×C4#57511yescharacteristic
550C8—22no—
551C8—22no—
552C8—22no—
553C8—22no—
554C8—42no—
555C8—41no—
556C8—41no—
557C8—41no—
558C8—41no—
559C8—81no—
560C8—81no—
561C8—81no—
562C8—81no—
563C8—81no—
564C22—22no—
565C22—22no—
566C22—22no—
567C22—41no—
568C22—41no—
569C22—41no—
570C22—41no—
571C22#232111yescharacteristic
572C22#293311yescharacteristic
573C22D4×M1611yescharacteristic
574C22C2×C4×Q1611yescharacteristic
575C22C4×#13711yescharacteristic
576C22#295611yescharacteristic
577C22#295711yescharacteristic
578C4—22no—
579C4—21no—
580C4—21no—
581C4—41no—
582C4—42no—
583C4—42no—
584C4—41no—
585C4—42no—
586C4—81no—
587C4—81no—
588C4—81no—
589C4—81no—
590C4—81no—
591C4#297411yescharacteristic
592C4#297511yescharacteristic
593C4#297611yescharacteristic
594C4#297811yescharacteristic
595C4#318911yescharacteristic
596C4#319011yescharacteristic
597C4#320911yescharacteristic
598C4#321111yescharacteristic
599C2—42no—
600C2#1799811yescharacteristic
601C2#1801211yescharacteristic
602C2M16×Q1611yescharacteristic
603C2#1804711yescharacteristic
604C2#2203811yescharacteristic
605C2#2204411yescharacteristic
606C2#2206211yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C2×C4×D4
prime radicalC24C4×D4
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
4. #9662C2semidirect
5. #9656C2semidirect
6. #9557C2semidirect
7. #9553C2semidirect
8. #9544C2semidirect
9. #9544C2semidirect
10. #8950C2semidirect
11. #8950C2semidirect
198. C2×M16Q16semidirect
290. #103M16semidirect

Maximal subgroups

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

subgroupindex
#215982
C4×#4512
#213032
#96622
#96562
#95572
#95532
#95442
#89502
#89422
#88322
#83112
#81462