Finite Group Databasebeta

← #203862

Subgroups — #203862

Order 512.

740 conjugacy classes of subgroups.

The subgroup lattice has 519 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21365C211yesmaximal, characteristic
2#21170C211yesmaximal, characteristic
3#20824C211yesmaximal, characteristic
4C4⋊C4×M16C211yesmaximal, characteristic
5C2×#1569C211yesmaximal, characteristic
6#9622C211yesmaximal, characteristic, complemented
7#9621C211yesmaximal, characteristic, complemented
8#9619C211yesmaximal, characteristic, complemented
9#9617C211yesmaximal, characteristic, complemented
10#9616C211yesmaximal, characteristic, complemented
11#9615C211yesmaximal, characteristic, complemented
12#9613C211yesmaximal, characteristic, complemented
13#9611C211yesmaximal, characteristic, complemented
14#8042C211yesmaximal, characteristic
15#8041C211yesmaximal, characteristic
16C2×#432C2211yescharacteristic
17C2×#428—21no—
18C2×#423C2211yescharacteristic
19C2×#423C2211yescharacteristic
20C2×C4×M16—21no—
21C2×C4×M16C2211yescharacteristic
22#2285C2211yescharacteristic
23#2285C2211yescharacteristic
24C2×C4×C4⋊C4C2211yescharacteristic
25#1949—21no—
26#1949—21no—
27#1948C2211yescharacteristic
28#1948C2211yescharacteristic
29#1948C2211yescharacteristic
30#1948C2211yescharacteristic
31#1947C2211yescharacteristic
32#1947C2211yescharacteristic
33#1947C2211yescharacteristic
34#1947C2211yescharacteristic
35#1853—21no—
36#1852—21no—
37#1845C2211yescharacteristic
38#1844C2211yescharacteristic
39#1834C2211yescharacteristic
40#1834C2211yescharacteristic
41#1809—21no—
42#1809—21no—
43#1776—21no—
44#1776—21no—
45#1776—21no—
46#1776—21no—
47#1771—21no—
48#1771—21no—
49#1770C2211yescharacteristic
50#1770C2211yescharacteristic
51C8×C4⋊C4C2211yescharacteristic
52C8×C4⋊C4C2211yescharacteristic
53#1736—21no—
54#1728—21no—
55#1569C418yescomplemented
56C2×#335C2211yescharacteristic
57C2×#335C2211yescharacteristic
58C2×#334C2211yescharacteristic
59C2×#334C2211yescharacteristic
60#1296C2211yescharacteristic
61#1296C2211yescharacteristic
62#1296C2211yescharacteristic
63#1296C2211yescharacteristic
64#1296C2211yescharacteristic
65#1296C2211yescharacteristic
66#1296C2211yescharacteristic
67#1296C2211yescharacteristic
68C22×M16—21no—
69C22×M16—21no—
70C22×M16D411yescharacteristic
71C22×M16D411yescharacteristic
72C2×#117—21no—
73C2×#117—21no—
74C22×C4⋊C4D411yescharacteristic
75C22×C4⋊C4D411yescharacteristic
76C22×C42—21no—
77C22×C42C2311yescharacteristic
78#432—22no—
79#432C2×C414yes—
80#428—44no—
81C2×#107—21no—
82C2×#107—21no—
83C2×#107—21no—
84C2×#107—21no—
85C2×#107—21no—
86C2×#107—21no—
87C2×#106—21no—
88C2×#106—21no—
89C2×#106—21no—
90C2×#106—21no—
91C2×#106—21no—
92C2×#106—21no—
93#423—22no—
94#423—22no—
95#423C2×C414yes—
96#423Q814yes—
97C2×C4⋊C8—21no—
98C2×C4⋊C8—21no—
99C2×C4⋊C8C2311yescharacteristic
100C2×C4⋊C8C2311yescharacteristic
101C2×C4⋊C8C2311yescharacteristic
102C2×C4⋊C8C2311yescharacteristic
103C2×C4⋊C8C2311yescharacteristic
104C2×C4⋊C8C2311yescharacteristic
105C2×C22⋊C8—21no—
106C4×M16—44no—
107C4×M16C2×C414yes—
108C4×M16Q814yes—
109C2×#97—21no—
110C2×#97—21no—
111C2×#97—21no—
112C2×#97D411yescharacteristic
113C2×#97D411yescharacteristic
114C2×C4×C8—21no—
115C2×C4×C8—21no—
116C2×C4×C8C2311yescharacteristic
117C2×C4×C8C2311yescharacteristic
118#384C2311yescharacteristic
119#384C2311yescharacteristic
120#384C2311yescharacteristic
121#384C2311yescharacteristic
122#382—21no—
123#382—21no—
124#380—21no—
125#380—21no—
126#380—21no—
127#380—21no—
128C4×C4⋊C4—21no—
129C4×C4⋊C4—21no—
130C4×C4⋊C4—21no—
131C4×C4⋊C4—21no—
132C4×C4⋊C4—21no—
133C4×C4⋊C4—21no—
134C4×C4⋊C4D411yescharacteristic
135C4×C4⋊C4D411yescharacteristic
136C4×C4⋊C4C2311yescharacteristic
137C4×C4⋊C4C2311yescharacteristic
138C2×#95—21no—
139#340—21no—
140#340—21no—
141#340—21no—
142#340—21no—
143#340—21no—
144#340—21no—
145#340—21no—
146#340—21no—
147#340—21no—
148#340—21no—
149#340—21no—
150#340—21no—
151#336—21no—
152#336—21no—
153#336—21no—
154#336—21no—
155#335C2×C414yescomplemented
156#335C2×C414yescomplemented
157#334C2×C414yescomplemented
158#334C2×C414yescomplemented
159C23×C4—21no—
160C23×C4—21no—
161C23×C4C2×D411yescharacteristic
162C23×C4C2×D411yescharacteristic
163C23×C4#4111yescharacteristic
164C2×M16—24no—
165C2×M16—24no—
166C2×M16—22no—
167C2×M16—22no—
168C2×M16—41no—
169C2×M16—41no—
170C2×M16—42no—
171C2×M16—42no—
172C2×M16—42no—
173C2×M16—42no—
174C2×M16C4⋊C414yes—
175C2×M16C4⋊C414yes—
176C22×C8—21no—
177C22×C8—21no—
178C22×C8—21no—
179C22×C8—21no—
180C22×C8—21no—
181C22×C8—21no—
182C22×C8C2×D411yescharacteristic
183C22×C8C2×D411yescharacteristic
184C22×C8C2×D411yescharacteristic
185C22×C8C2×D411yescharacteristic
186#117—42no—
187#117—42no—
188#117—82no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—41no—
206C2×C4⋊C4—41no—
207C2×C4⋊C4C2×D411yescharacteristic
208C2×C4⋊C4C2×D411yescharacteristic
209C2×C4⋊C4C2×D411yescharacteristic
210C2×C4⋊C4C2×D411yescharacteristic
211C2×C22⋊C4—41no—
212C2×C22⋊C4—41no—
213C2×C42—21no—
214C2×C42—21no—
215C2×C42—21no—
216C2×C42—21no—
217C2×C42—42no—
218C2×C42—42no—
219C2×C42—41no—
220C2×C42—41no—
221C2×C42C22×C412yes—
222C2×C42C2×D411yescharacteristic
223C2×C42C2×D411yescharacteristic
224C2×C42C2×Q812yes—
225C2×C42C2×Q812yes—
226C2×C42#4112yes—
227C2×C42#4111yescharacteristic
228C2×C42C2411yescharacteristic
229#107—42no—
230#107—42no—
231#107—42no—
232#107—42no—
233#107—42no—
234#107—42no—
235#106—42no—
236#106—42no—
237#106—42no—
238#106—42no—
239#106—42no—
240#106—42no—
241C4⋊C8—24no—
242C4⋊C8—24no—
243C4⋊C8—21no—
244C4⋊C8—21no—
245C4⋊C8—21no—
246C4⋊C8—21no—
247C4⋊C8—21no—
248C4⋊C8—21no—
249C4⋊C8C22×C412yes—
250C4⋊C8C22×C412yes—
251C4⋊C8C22×C412yes—
252C4⋊C8C22×C412yes—
253C4⋊C8C2×Q812yes—
254C4⋊C8C2×Q812yes—
255C22⋊C8—24no—
256#97—24no—
257#97—42no—
258#97—42no—
259#97C4⋊C414yes—
260#97C4⋊C414yes—
261C4×C8—42no—
262C4×C8—42no—
263C4×C8C22×C412yes—
264C4×C8C22×C412yes—
265C4×C8C2×Q812yes—
266C4×C8C2×Q812yes—
267#95—21no—
268#95—21no—
269#95—21no—
270#95—21no—
271#95—41no—
272#95—41no—
273#95—41no—
274#95—41no—
275C24#12111yescharacteristic
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—21no—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—21no—
282C22×C4—21no—
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—21no—
287C22×C4—21no—
288C22×C4—41no—
289C22×C4—41no—
290C22×C4—42no—
291C22×C4—42no—
292C22×C4—42no—
293C22×C4—42no—
294C22×C4—41no—
295C22×C4—41no—
296C22×C4—41no—
297C22×C4—41no—
298C22×C4—81no—
299C22×C4—81no—
300C22×C4C2×C4⋊C412yes—
301C22×C4C4×D412yes—
302C22×C4C4×Q812yes—
303C22×C4#12111yescharacteristic
304C22×C4#12111yescharacteristic
305C22×C4#12111yescharacteristic
306C22×C4C22⋊Q812yes—
307C22×C4#12512yes—
308C22×C4C4⋊Q812yes—
309C22×C4C22×D411yescharacteristic
310C22×C4C22×D411yescharacteristic
311C22×C4C2×#4111yescharacteristic
312M16—28no—
313M16—44no—
314M16—48no—
315M16—48no—
316C2×C8—21no—
317C2×C8—21no—
318C2×C8—21no—
319C2×C8—21no—
320C2×C8—21no—
321C2×C8—21no—
322C2×C8—22no—
323C2×C8—22no—
324C2×C8—22no—
325C2×C8—22no—
326C2×C8—22no—
327C2×C8—22no—
328C2×C8—41no—
329C2×C8—41no—
330C2×C8—41no—
331C2×C8—41no—
332C2×C8—41no—
333C2×C8—41no—
334C2×C8—41no—
335C2×C8—41no—
336C2×C8—41no—
337C2×C8—41no—
338C2×C8—41no—
339C2×C8—41no—
340C2×C8C2×C4⋊C412yes—
341C2×C8C2×C4⋊C412yes—
342C2×C8C2×C4⋊C412yes—
343C2×C8C2×C4⋊C412yes—
344C2×C8AGL(1,ℤ/8)12yes—
345C2×C8#13712yes—
346C4⋊C4—21no—
347C4⋊C4—21no—
348C4⋊C4—21no—
349C4⋊C4—21no—
350C4⋊C4—21no—
351C4⋊C4—21no—
352C4⋊C4—42no—
353C4⋊C4—42no—
354C4⋊C4—41no—
355C4⋊C4—41no—
356C4⋊C4—41no—
357C4⋊C4—41no—
358C4⋊C4—41no—
359C4⋊C4—41no—
360C4⋊C4—41no—
361C4⋊C4—41no—
362C4⋊C4—41no—
363C4⋊C4—41no—
364C4⋊C4—41no—
365C4⋊C4—41no—
366C4⋊C4—41no—
367C4⋊C4—41no—
368C4⋊C4—41no—
369C4⋊C4—41no—
370C4⋊C4—81no—
371C4⋊C4—81no—
372C4⋊C4—81no—
373C4⋊C4—81no—
374C4⋊C4AGL(1,ℤ/8)12yes—
375C4⋊C4#13712yes—
376C22⋊C4—82no—
377C22⋊C4—82no—
378C42—42no—
379C42—42no—
380C42—41no—
381C42—41no—
382C42—41no—
383C42—41no—
384C42—82no—
385C42—82no—
386C42C2×C4⋊C412yes—
387C42C4×D412yes—
388C42C4×Q812yes—
389C42C22⋊Q812yes—
390C42#12512yes—
391C42C4⋊Q812yes—
392C42C23×C411yescharacteristic
393C42C22×Q811yescharacteristic
394C42C22×Q811yescharacteristic
395C42C2×#4111yescharacteristic
396C23—21no—
397C23—21no—
398C23—21no—
399C23—21no—
400C23—21no—
401C23—21no—
402C23#38912yes—
403C23C2×#12111yescharacteristic
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—21no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—22no—
413C2×C4—22no—
414C2×C4—22no—
415C2×C4—22no—
416C2×C4—22no—
417C2×C4—22no—
418C2×C4—22no—
419C2×C4—22no—
420C2×C4—22no—
421C2×C4—22no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—41no—
425C2×C4—41no—
426C2×C4—41no—
427C2×C4—41no—
428C2×C4—41no—
429C2×C4—41no—
430C2×C4—41no—
431C2×C4—41no—
432C2×C4—41no—
433C2×C4—41no—
434C2×C4—84no—
435C2×C4—84no—
436C2×C4—81no—
437C2×C4—81no—
438C2×C4—81no—
439C2×C4—81no—
440C2×C4—81no—
441C2×C4—81no—
442C2×C4—81no—
443C2×C4—81no—
444C2×C4—81no—
445C2×C4—81no—
446C2×C4—81no—
447C2×C4—81no—
448C2×C4#38912yes—
449C2×C4#38912yes—
450C2×C4#38912yes—
451C2×C4C22×C4⋊C411yescharacteristic
452C2×C4C2×C4×D411yescharacteristic
453C2×C4C2×C4×Q811yescharacteristic
454C2×C4C2×C22⋊Q811yescharacteristic
455C2×C4C2×#12511yescharacteristic
456C2×C4C2×C4⋊Q811yescharacteristic
457C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
458C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
459C2×C4C2×#13711yescharacteristic
460C2×C4C2×#13711yescharacteristic
461C2×C4#57511yescharacteristic
462C2×C4#57511yescharacteristic
463C2×C4#57511yescharacteristic
464C2×C4#57511yescharacteristic
465C8—22no—
466C8—22no—
467C8—22no—
468C8—42no—
469C8—42no—
470C8—42no—
471C8—42no—
472C8—42no—
473C8—42no—
474C8#42814yescomplemented
475C22—22no—
476C22—22no—
477C22—22no—
478C22—41no—
479C22—41no—
480C22—41no—
481C22—41no—
482C22C2×#38911yescharacteristic
483C22#303611yescharacteristic
484C22#303711yescharacteristic
485C22#305511yescharacteristic
486C22#305611yescharacteristic
487C22#306811yescharacteristic
488C22#306911yescharacteristic
489C4—22no—
490C4—22no—
491C4—21no—
492C4—21no—
493C4—22no—
494C4—42no—
495C4—42no—
496C4—81no—
497C4—81no—
498C4—161no—
499C4—161no—
500C4—161no—
501C4—161no—
502C4C2×#42811yescharacteristic
503C4#291111yescharacteristic
504C4C4×AGL(1,ℤ/8)11yescharacteristic
505C4C4×#13711yescharacteristic
506C4#307711yescharacteristic
507C4#307811yescharacteristic
508C4#316511yescharacteristic
509C4#316611yescharacteristic
510C2—42no—
511C2#1930411yescharacteristic
512C2#1930511yescharacteristic
513C2#1931011yescharacteristic
514C2#1931111yescharacteristic
515C2#2115911yescharacteristic
516C2#2117011yescharacteristic
517C2#2118211yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC24#121
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
6. #9622C2semidirect
7. #9621C2semidirect
8. #9619C2semidirect
9. #9617C2semidirect
10. #9616C2semidirect
11. #9615C2semidirect
12. #9613C2semidirect
13. #9611C2semidirect
55. #1569C4semidirect
155. #335C2×C4semidirect
156. #335C2×C4semidirect
157. #334C2×C4semidirect
158. #334C2×C4semidirect
474. C8#428semidirect

Maximal subgroups

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

subgroupindex
#213652
#211702
#208242
C4⋊C4×M162
C2×#15692
#96222
#96212
#96192
#96172
#96162
#96152
#96132
#96112
#80422
#80412