Finite Group Databasebeta

← #203888

Subgroups — #203888

Order 512.

710 conjugacy classes of subgroups.

The subgroup lattice has 513 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21987C211yesmaximal, characteristic
2#21963C211yesmaximal, characteristic
3#20800C211yesmaximal, characteristic
4#20482C211yesmaximal, characteristic
5C2×#1569C211yesmaximal, characteristic
6#9621C211yesmaximal, characteristic, complemented
7#9620C211yesmaximal, characteristic, complemented
8#9618C211yesmaximal, characteristic, complemented
9#9617C211yesmaximal, characteristic, complemented
10#9615C211yesmaximal, characteristic, complemented
11#9614C211yesmaximal, characteristic, complemented
12#9612C211yesmaximal, characteristic, complemented
13#9611C211yesmaximal, characteristic, complemented
14#8046C211yesmaximal, characteristic
15#8043C211yesmaximal, characteristic
16C2×#432C2211yescharacteristic
17C2×#431C2211yescharacteristic
18C2×#428—21no—
19C2×#428—21no—
20C2×#423C2211yescharacteristic
21C2×C4×M16C2211yescharacteristic
22#2285C2211yescharacteristic
23C2×#384C2211yescharacteristic
24#2283C2211yescharacteristic
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#1847—21no—
38#1847—21no—
39#1847—21no—
40#1845C2211yescharacteristic
41#1845C2211yescharacteristic
42#1844C2211yescharacteristic
43#1844C2211yescharacteristic
44#1836—21no—
45#1809—21no—
46#1809—21no—
47#1770C2211yescharacteristic
48#1770C2211yescharacteristic
49C8×C4⋊C4C2211yescharacteristic
50C8×C4⋊C4C2211yescharacteristic
51#1753—21no—
52#1753—21no—
53#1746—21no—
54#1726—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×M16D411yescharacteristic
70C22×M16D411yescharacteristic
71C2×#117—21no—
72C22×C4⋊C4—21no—
73C22×C4⋊C4—21no—
74C22×C4⋊C4—21no—
75C22×C42C2311yescharacteristic
76#432—22no—
77#432C2×C414yes—
78#431—22no—
79#431Q814yes—
80#428—44no—
81#428—44no—
82C2×#107—21no—
83C2×#107—21no—
84C2×#107—21no—
85C2×#107—21no—
86C2×#107—21no—
87C2×#107—21no—
88C2×#107—21no—
89C2×#107—21no—
90C2×#106—21no—
91C2×#106—21no—
92C2×#106—21no—
93C2×#106—21no—
94C2×#106—21no—
95C2×#106—21no—
96C2×#106—21no—
97C2×#106—21no—
98#423—22no—
99#423C2×C414yes—
100C2×C4⋊C8—21no—
101C2×C4⋊C8C2311yescharacteristic
102C2×C4⋊C8C2311yescharacteristic
103C2×C4⋊C8C2311yescharacteristic
104C2×C4⋊C8C2311yescharacteristic
105C2×C4⋊C8C2311yescharacteristic
106C2×C4⋊C8C2311yescharacteristic
107C2×C22⋊C8—21no—
108C2×C22⋊C8—21no—
109C4×M16C2×C414yes—
110C4×M16Q814yes—
111C2×#97—21no—
112C2×#97—21no—
113C2×#97—21no—
114C2×#97D411yescharacteristic
115C2×#97D411yescharacteristic
116C2×C4×C8—21no—
117C2×C4×C8—21no—
118C2×C4×C8—21no—
119C2×C4×C8C2311yescharacteristic
120C2×C4×C8C2311yescharacteristic
121#384—21no—
122#384—21no—
123#384—21no—
124#384C2311yescharacteristic
125#384C2311yescharacteristic
126#384C2311yescharacteristic
127#384C2311yescharacteristic
128#383—21no—
129#382—21no—
130#382D411yescharacteristic
131#382D411yescharacteristic
132#381—21no—
133#380—21no—
134#380—21no—
135#380D411yescharacteristic
136#380D411yescharacteristic
137C4×C4⋊C4—21no—
138C4×C4⋊C4—21no—
139C4×C4⋊C4C2311yescharacteristic
140C4×C4⋊C4C2311yescharacteristic
141C4×C22⋊C4—21no—
142#376—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—41no—
160C23×C4C2×D411yescharacteristic
161C23×C4C2×D411yescharacteristic
162C23×C4#4111yescharacteristic
163C2×M16—24no—
164C2×M16—22no—
165C2×M16—22no—
166C2×M16—22no—
167C2×M16—22no—
168C2×M16—81no—
169C2×M16—81no—
170C2×M16C4⋊C414yes—
171C2×M16C4⋊C414yes—
172C22×C8—21no—
173C22×C8—21no—
174C22×C8—21no—
175C22×C8—21no—
176C22×C8—21no—
177C22×C8—21no—
178C22×C8C2×D411yescharacteristic
179C22×C8C2×D411yescharacteristic
180C22×C8C2×D411yescharacteristic
181C22×C8C2×D411yescharacteristic
182#117—42no—
183#117—42no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—42no—
197C2×C4⋊C4—42no—
198C2×C4⋊C4—41no—
199C2×C4⋊C4—41no—
200C2×C4⋊C4—41no—
201C2×C4⋊C4—81no—
202C2×C4⋊C4—81no—
203C2×C4⋊C4—82no—
204C2×C4⋊C4C2×D411yescharacteristic
205C2×C4⋊C4C2×D411yescharacteristic
206C2×C4⋊C4C2×D411yescharacteristic
207C2×C4⋊C4C2×D411yescharacteristic
208C2×C22⋊C4—21no—
209C2×C22⋊C4—21no—
210C2×C22⋊C4—41no—
211C2×C42—21no—
212C2×C42—21no—
213C2×C42—41no—
214C2×C42C22×C412yes—
215C2×C42C2×D411yescharacteristic
216C2×C42C2×D411yescharacteristic
217C2×C42C2×Q812yes—
218C2×C42C2×Q812yes—
219C2×C42#4112yes—
220C2×C42#4111yescharacteristic
221C2×C42C2411yescharacteristic
222#107—42no—
223#107—42no—
224#107—42no—
225#107—42no—
226#107—81no—
227#107—81no—
228#107—81no—
229#107—81no—
230#106—42no—
231#106—42no—
232#106—42no—
233#106—42no—
234#106—81no—
235#106—81no—
236#106—81no—
237#106—81no—
238C4⋊C8—24no—
239C4⋊C8—21no—
240C4⋊C8—21no—
241C4⋊C8—21no—
242C4⋊C8—21no—
243C4⋊C8—21no—
244C4⋊C8—21no—
245C4⋊C8C22×C412yes—
246C4⋊C8C22×C412yes—
247C4⋊C8C22×C412yes—
248C4⋊C8C22×C412yes—
249C4⋊C8C2×Q812yes—
250C4⋊C8C2×Q812yes—
251C22⋊C8—24no—
252C22⋊C8—24no—
253#97—24no—
254#97—42no—
255#97—42no—
256#97C4⋊C414yes—
257#97C4⋊C414yes—
258C4×C8—24no—
259C4×C8—42no—
260C4×C8—42no—
261C4×C8C22×C412yes—
262C4×C8C22×C412yes—
263C4×C8C2×Q812yes—
264C4×C8C2×Q812yes—
265#95—21no—
266#95—21no—
267#95—21no—
268#95—21no—
269#95—21no—
270#95—21no—
271#95—21no—
272#95—21no—
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—41no—
291C22×C4—41no—
292C22×C4—41no—
293C22×C4—41no—
294C22×C4—82no—
295C22×C4—82no—
296C22×C4—82no—
297C22×C4C2×C4⋊C412yes—
298C22×C4C4×D412yes—
299C22×C4C4×Q812yes—
300C22×C4#12111yescharacteristic
301C22×C4#12111yescharacteristic
302C22×C4#12111yescharacteristic
303C22×C4C22⋊Q812yes—
304C22×C4#12512yes—
305C22×C4C4⋊Q812yes—
306C22×C4C22×D411yescharacteristic
307C22×C4C22×D411yescharacteristic
308C22×C4C2×#4111yescharacteristic
309M16—28no—
310M16—28no—
311M16—84no—
312C2×C8—21no—
313C2×C8—21no—
314C2×C8—21no—
315C2×C8—21no—
316C2×C8—22no—
317C2×C8—22no—
318C2×C8—22no—
319C2×C8—22no—
320C2×C8—22no—
321C2×C8—22no—
322C2×C8—41no—
323C2×C8—41no—
324C2×C8—41no—
325C2×C8—41no—
326C2×C8—41no—
327C2×C8—41no—
328C2×C8—41no—
329C2×C8—41no—
330C2×C8—41no—
331C2×C8—41no—
332C2×C8—41no—
333C2×C8—41no—
334C2×C8C2×C4⋊C412yes—
335C2×C8C2×C4⋊C412yes—
336C2×C8C2×C4⋊C412yes—
337C2×C8C2×C4⋊C412yes—
338C2×C8AGL(1,ℤ/8)12yes—
339C2×C8AGL(1,ℤ/8)12yes—
340C2×C8#13712yes—
341C2×C8#13712yes—
342C4⋊C4—21no—
343C4⋊C4—21no—
344C4⋊C4—21no—
345C4⋊C4—21no—
346C4⋊C4—21no—
347C4⋊C4—21no—
348C4⋊C4—21no—
349C4⋊C4—21no—
350C4⋊C4—41no—
351C4⋊C4—41no—
352C4⋊C4—41no—
353C4⋊C4—41no—
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—81no—
367C4⋊C4—81no—
368C4⋊C4—81no—
369C4⋊C4—81no—
370C4⋊C4—82no—
371C4⋊C4—82no—
372C4⋊C4—82no—
373C22⋊C4—81no—
374C22⋊C4—81no—
375C22⋊C4—82no—
376C42—41no—
377C42—41no—
378C42—41no—
379C42—41no—
380C42—82no—
381C42C2×C4⋊C412yes—
382C42C4×D412yes—
383C42C4×Q812yes—
384C42C22⋊Q812yes—
385C42#12512yes—
386C42C4⋊Q812yes—
387C42C23×C411yescharacteristic
388C42C22×Q811yescharacteristic
389C42C22×Q811yescharacteristic
390C42C2×#4111yescharacteristic
391C23—21no—
392C23—21no—
393C23—21no—
394C23—21no—
395C23—21no—
396C23—21no—
397C23#38912yes—
398C23C2×#12111yescharacteristic
399C2×C4—21no—
400C2×C4—21no—
401C2×C4—21no—
402C2×C4—21no—
403C2×C4—21no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—22no—
408C2×C4—22no—
409C2×C4—22no—
410C2×C4—22no—
411C2×C4—22no—
412C2×C4—22no—
413C2×C4—22no—
414C2×C4—22no—
415C2×C4—42no—
416C2×C4—42no—
417C2×C4—41no—
418C2×C4—41no—
419C2×C4—41no—
420C2×C4—41no—
421C2×C4—41no—
422C2×C4—41no—
423C2×C4—41no—
424C2×C4—41no—
425C2×C4—41no—
426C2×C4—41no—
427C2×C4—41no—
428C2×C4—41no—
429C2×C4—81no—
430C2×C4—81no—
431C2×C4—81no—
432C2×C4—81no—
433C2×C4—81no—
434C2×C4—81no—
435C2×C4—81no—
436C2×C4—81no—
437C2×C4—81no—
438C2×C4—81no—
439C2×C4—81no—
440C2×C4—81no—
441C2×C4—164no—
442C2×C4#38912yes—
443C2×C4#38912yes—
444C2×C4#38912yes—
445C2×C4C22×C4⋊C411yescharacteristic
446C2×C4C2×C4×D411yescharacteristic
447C2×C4C2×C4×Q811yescharacteristic
448C2×C4C2×C22⋊Q811yescharacteristic
449C2×C4C2×#12511yescharacteristic
450C2×C4C2×C4⋊Q811yescharacteristic
451C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
452C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
453C2×C4C2×#13711yescharacteristic
454C2×C4C2×#13711yescharacteristic
455C2×C4#57611yescharacteristic
456C2×C4#57711yescharacteristic
457C2×C4#57711yescharacteristic
458C2×C4#57811yescharacteristic
459C8—22no—
460C8—22no—
461C8—42no—
462C8—42no—
463C8—42no—
464C8—42no—
465C8—42no—
466C8—42no—
467C8#42814yes—
468C8#42814yes—
469C22—22no—
470C22—22no—
471C22—22no—
472C22—41no—
473C22—41no—
474C22—41no—
475C22—41no—
476C22C2×#38911yescharacteristic
477C22#303811yescharacteristic
478C22#303911yescharacteristic
479C22#306211yescharacteristic
480C22#306311yescharacteristic
481C22#306811yescharacteristic
482C22#306911yescharacteristic
483C4—22no—
484C4—22no—
485C4—21no—
486C4—21no—
487C4—22no—
488C4—81no—
489C4—81no—
490C4—81no—
491C4—81no—
492C4—161no—
493C4—161no—
494C4—161no—
495C4—161no—
496C4C2×#42811yescharacteristic
497C4C2×#42811yescharacteristic
498C4#294611yescharacteristic
499C4#294711yescharacteristic
500C4#307911yescharacteristic
501C4#307911yescharacteristic
502C4#316311yescharacteristic
503C4#316411yescharacteristic
504C2—42no—
505C2#1937611yescharacteristic
506C2#1937711yescharacteristic
507C2#1937811yescharacteristic
508C2#1937911yescharacteristic
509C2#2116011yescharacteristic
510C2#2117111yescharacteristic
511C2#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. #9621C2semidirect
7. #9620C2semidirect
8. #9618C2semidirect
9. #9617C2semidirect
10. #9615C2semidirect
11. #9614C2semidirect
12. #9612C2semidirect
13. #9611C2semidirect
55. #1569C4semidirect
155. #335C2×C4semidirect
156. #335C2×C4semidirect
157. #334C2×C4semidirect
158. #334C2×C4semidirect

Maximal subgroups

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

subgroupindex
#219872
#219632
#208002
#204822
C2×#15692
#96212
#96202
#96182
#96172
#96152
#96142
#96122
#96112
#80462
#80432