Finite Group Databasebeta

← #211149

Subgroups — #211149

Order 512.

712 conjugacy classes of subgroups.

The subgroup lattice has 533 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22146C211yesmaximal, characteristic
2#22110C211yesmaximal, characteristic
3C4×#487C211yesmaximal, characteristic, complemented
4#20485C211yesmaximal, characteristic
5C2×#1679C211yesmaximal, characteristic
6#9736C211yesmaximal, characteristic, complemented
7#9613C211yesmaximal, characteristic, complemented
8#9574C211yesmaximal, characteristic, complemented
9#9570C211yesmaximal, characteristic, complemented
10#9227C211yesmaximal, characteristic
11#9225C211yesmaximal, characteristic
12#8646C211yesmaximal, characteristic
13C4×#334C211yesmaximal, characteristic, complemented
14#8237C211yesmaximal, characteristic, complemented
15#8236C211yesmaximal, characteristic, complemented
16C2×#487C2211yescharacteristic
17C2×#474C2211yescharacteristic
18C2×#461C2211yescharacteristic
19C2×#436C2211yescharacteristic
20C2×C8×D4C2211yescharacteristic
21C2×#405—21no—
22C4×C4⋊Q8C2211yescharacteristic
23C4×C4⋊Q8C2211yescharacteristic
24#2293C2211yescharacteristic
25#1990C2211yescharacteristic
26#1988C2211yescharacteristic
27#1981C2211yescharacteristic
28#1968—21no—
29#1956C2211yescharacteristic
30#1948C2211yescharacteristic
31#1947C2211yescharacteristic
32#1942—21no—
33#1934—21no—
34#1933—21no—
35#1933—21no—
36#1933—21no—
37#1925—21no—
38#1918—21no—
39#1880—21no—
40#1865—21no—
41#1834C2211yescharacteristic
42#1802C2211yescharacteristic
43#1802C2211yescharacteristic
44#1770C2211yescharacteristic
45C8×C4⋊C4C2211yescharacteristic
46#1767C2211yescharacteristic
47C4×C4⋊C8C2211yescharacteristic
48C4×C4⋊C8C2211yescharacteristic
49C4×#103—21no—
50C4×#103—21no—
51#1753—21no—
52C8×C22⋊C4—21no—
53#1750—21no—
54C42×C8C2211yescharacteristic
55#1679C418yescomplemented
56C2×#334C2211yescharacteristic
57C2×#334C2211yescharacteristic
58C2×#332C2211yescharacteristic
59C2×#332C2211yescharacteristic
60C2×#332C2211yescharacteristic
61C2×#332C2211yescharacteristic
62#1296C2211yescharacteristic
63#1296C2211yescharacteristic
64#1296C2211yescharacteristic
65#1296C2211yescharacteristic
66#1277C2211yescharacteristic
67#1277C2211yescharacteristic
68C22×SD16—21no—
69C22×M16—21no—
70C23×C8—21no—
71C2×C4⋊Q8C2311yescharacteristic
72C2×C4⋊Q8C2311yescharacteristic
73C2×C4×Q8—21no—
74C2×C4×Q8—21no—
75C2×C4×D4C2311yescharacteristic
76C2×#117—21no—
77#487—22no—
78#487C2×C414yes—
79#474—22no—
80#474C2×C414yes—
81#461—22no—
82#461C2×C414yes—
83#436—22no—
84#436—22no—
85C8×D4—22no—
86C8×D4C2×C414yes—
87C2×#107—21no—
88C2×#107—21no—
89C2×#106—21no—
90C2×C4⋊C8C2311yescharacteristic
91C2×C4⋊C8C2311yescharacteristic
92C2×C4⋊C8C2311yescharacteristic
93C2×C4⋊C8C2311yescharacteristic
94C2×C4⋊C8C2311yescharacteristic
95C2×C4⋊C8C2311yescharacteristic
96C2×#103—21no—
97C2×#103—21no—
98C2×#103—21no—
99C2×#103—21no—
100C2×#103—21no—
101C2×#103—21no—
102C2×#103—21no—
103C2×#102—21no—
104C2×#102—21no—
105C2×#102—21no—
106C2×C22⋊C8—21no—
107C2×C22⋊C8—21no—
108#405—44no—
109C2×#97—21no—
110C2×#97—21no—
111C2×C4×C8—21no—
112C2×C4×C8—21no—
113C2×C4×C8—21no—
114C2×C4×C8D411yescharacteristic
115C2×C4×C8D411yescharacteristic
116C2×C4×C8D411yescharacteristic
117C2×C4×C8D411yescharacteristic
118C2×C4×C8C2311yescharacteristic
119C2×C4×C8C2311yescharacteristic
120#391—21no—
121#391—21no—
122#390D411yescharacteristic
123#390D411yescharacteristic
124#389—21no—
125#389—21no—
126#389—21no—
127#389—21no—
128#389D411yescharacteristic
129#389D411yescharacteristic
130#388—21no—
131#388—21no—
132#384C2311yescharacteristic
133#384C2311yescharacteristic
134C4×C4⋊C4—21no—
135C4×C4⋊C4—21no—
136C4×C4⋊C4—21no—
137C4×C4⋊C4—21no—
138C4×C4⋊C4C2311yescharacteristic
139C4×C22⋊C4—21no—
140C43C2311yescharacteristic
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#336—21no—
156#334—22no—
157#334C2×C414yes—
158#332—22no—
159#332—22no—
160#332C2×C414yes—
161#332C2×C414yes—
162C22×Q8—21no—
163C22×Q8—21no—
164C22×D4C2×D411yescharacteristic
165C23×C4—21no—
166C2×SD16—24no—
167C2×SD16—81no—
168C2×SD16—81no—
169C2×M16—24no—
170C2×M16—42no—
171C2×M16—42no—
172C22×C8—21no—
173C22×C8—21no—
174C22×C8—21no—
175C22×C8—21no—
176C22×C8—24no—
177C22×C8—21no—
178C22×C8—21no—
179C22×C8—42no—
180C22×C8—42no—
181C22×C8C2×D411yescharacteristic
182C22×C8C2×D411yescharacteristic
183C22×C8C2×D411yescharacteristic
184C22×C8C2×D411yescharacteristic
185C4⋊Q8—21no—
186C4⋊Q8—21no—
187C4⋊Q8—21no—
188C4⋊Q8—21no—
189C4⋊Q8—21no—
190C4⋊Q8—21no—
191C4⋊Q8C22×C412yes—
192C4⋊Q8C22×C412yes—
193C4×Q8—42no—
194C4×Q8—42no—
195C4×Q8—81no—
196C4×Q8—81no—
197C4×D4—21no—
198C4×D4—21no—
199C4×D4C22×C412yes—
200C4×D4#4112yes—
201#117—42no—
202#117—42no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—41no—
216C2×C4⋊C4—41no—
217C2×C4⋊C4C2×D411yescharacteristic
218C2×C4⋊C4C2×D411yescharacteristic
219C2×C4⋊C4C2×D411yescharacteristic
220C2×C22⋊C4—21no—
221C2×C22⋊C4—21no—
222C2×C22⋊C4—21no—
223C2×C22⋊C4—21no—
224C2×C22⋊C4—21no—
225C2×C42—21no—
226C2×C42—41no—
227C2×C42—41no—
228C2×C42C2×D411yescharacteristic
229C2×C42C2×D411yescharacteristic
230C2×C42C2×D411yescharacteristic
231C2×C42C2×D411yescharacteristic
232C2×C42#4111yescharacteristic
233C2×C42#4111yescharacteristic
234C2×C42C2411yescharacteristic
235#107—24no—
236#107—81no—
237#106—81no—
238C4⋊C8—21no—
239C4⋊C8—21no—
240C4⋊C8—21no—
241C4⋊C8—21no—
242C4⋊C8—21no—
243C4⋊C8—21no—
244C4⋊C8—21no—
245C4⋊C8—21no—
246C4⋊C8C22×C412yes—
247C4⋊C8C22×C412yes—
248C4⋊C8C22×C412yes—
249C4⋊C8#4112yes—
250#103—24no—
251#103—24no—
252#103—24no—
253#103—42no—
254#103—42no—
255#103—81no—
256#103—81no—
257#102—24no—
258#102—81no—
259#102—81no—
260C22⋊C8—24no—
261C22⋊C8—24no—
262#97—42no—
263#97—42no—
264C4×C8—22no—
265C4×C8—22no—
266C4×C8—22no—
267C4×C8—21no—
268C4×C8—21no—
269C4×C8—41no—
270C4×C8—42no—
271C4×C8—81no—
272C4×C8—81no—
273C4×C8C22×C412yes—
274C4×C8#4112yes—
275#95—21no—
276#95—21no—
277#95—21no—
278#95—41no—
279#95—41no—
280#95—41no—
281#95—41no—
282C24—21no—
283C2×Q8—22no—
284C2×Q8—22no—
285C2×Q8—41no—
286C2×Q8—41no—
287C2×Q8—41no—
288C2×Q8—41no—
289C2×Q8—41no—
290C2×Q8—41no—
291C2×Q8—41no—
292C2×Q8—41no—
293C2×Q8—41no—
294C2×Q8—41no—
295C2×D4—21no—
296C2×D4—21no—
297C2×D4—21no—
298C2×D4—21no—
299C2×D4—21no—
300C2×D4C4×D412yes—
301C22×C4—21no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—21no—
305C22×C4—21no—
306C22×C4—41no—
307C22×C4—41no—
308C22×C4—41no—
309C22×C4—41no—
310C22×C4—41no—
311C22×C4—41no—
312C22×C4—41no—
313C22×C4—41no—
314C22×C4#12111yescharacteristic
315C22×C4#12111yescharacteristic
316C22×C4#12111yescharacteristic
317C22×C4#12111yescharacteristic
318C22×C4C22×D411yescharacteristic
319C22×C4C22×D411yescharacteristic
320C22×C4C2×#4111yescharacteristic
321SD16—84no—
322M16—48no—
323C2×C8—21no—
324C2×C8—21no—
325C2×C8—21no—
326C2×C8—21no—
327C2×C8—21no—
328C2×C8—21no—
329C2×C8—22no—
330C2×C8—22no—
331C2×C8—22no—
332C2×C8—21no—
333C2×C8—22no—
334C2×C8—21no—
335C2×C8—21no—
336C2×C8—48no—
337C2×C8—41no—
338C2×C8—41no—
339C2×C8—41no—
340C2×C8—41no—
341C2×C8—41no—
342C2×C8—41no—
343C2×C8—41no—
344C2×C8—41no—
345C2×C8—41no—
346C2×C8—41no—
347C2×C8—41no—
348C2×C8—41no—
349C2×C8—41no—
350C2×C8—41no—
351C2×C8C4×D412yes—
352C2×C8C4×D412yes—
353C2×C8#13512yes—
354C4⋊C4—21no—
355C4⋊C4—22no—
356C4⋊C4—22no—
357C4⋊C4—22no—
358C4⋊C4—22no—
359C4⋊C4—21no—
360C4⋊C4—21no—
361C4⋊C4—21no—
362C4⋊C4—21no—
363C4⋊C4—41no—
364C4⋊C4—41no—
365C4⋊C4—41no—
366C4⋊C4—41no—
367C4⋊C4—42no—
368C4⋊C4—42no—
369C4⋊C4—41no—
370C4⋊C4—41no—
371C4⋊C4—41no—
372C4⋊C4—41no—
373C4⋊C4—41no—
374C4⋊C4—41no—
375C4⋊C4—41no—
376C4⋊C4—41no—
377C4⋊C4—81no—
378C4⋊C4—81no—
379C4⋊C4—82no—
380C4⋊C4—82no—
381C4⋊C4C4×D412yes—
382C22⋊C4—22no—
383C22⋊C4—22no—
384C22⋊C4—42no—
385C22⋊C4—42no—
386C22⋊C4—42no—
387C22⋊C4—42no—
388C42—21no—
389C42—21no—
390C42—21no—
391C42—21no—
392C42—21no—
393C42—21no—
394C42—22no—
395C42—41no—
396C42—41no—
397C42—41no—
398C42—82no—
399C42—82no—
400C42#13512yes—
401C42AGL(1,ℤ/8)12yes—
402C42C23×C411yescharacteristic
403C42C2×#4111yescharacteristic
404C42C2×#4111yescharacteristic
405C42C2×#4111yescharacteristic
406C23—22no—
407C23—41no—
408C23—41no—
409C23—41no—
410C23—41no—
411C23—41no—
412C23—41no—
413C23C2×#12111yescharacteristic
414Q8—42no—
415Q8—42no—
416Q8—42no—
417Q8—42no—
418Q8—81no—
419Q8—81no—
420Q8—81no—
421Q8—81no—
422D4—22no—
423D4—22no—
424D4—41no—
425D4—41no—
426C2×C4—21no—
427C2×C4—21no—
428C2×C4—21no—
429C2×C4—21no—
430C2×C4—21no—
431C2×C4—21no—
432C2×C4—21no—
433C2×C4—21no—
434C2×C4—21no—
435C2×C4—21no—
436C2×C4—21no—
437C2×C4—22no—
438C2×C4—41no—
439C2×C4—41no—
440C2×C4—42no—
441C2×C4—42no—
442C2×C4—42no—
443C2×C4—42no—
444C2×C4—41no—
445C2×C4—41no—
446C2×C4—41no—
447C2×C4—41no—
448C2×C4—41no—
449C2×C4—41no—
450C2×C4—42no—
451C2×C4—42no—
452C2×C4—81no—
453C2×C4—81no—
454C2×C4—81no—
455C2×C4—81no—
456C2×C4—81no—
457C2×C4—81no—
458C2×C4—81no—
459C2×C4—81no—
460C2×C4—81no—
461C2×C4—81no—
462C2×C4#46312yes—
463C2×C4#46312yes—
464C2×C4#47312yes—
465C2×C4C2×C4×D411yescharacteristic
466C2×C4C2×C4×D411yescharacteristic
467C2×C4C4×#4111yescharacteristic
468C2×C4#52511yescharacteristic
469C2×C4#52911yescharacteristic
470C2×C4#53211yescharacteristic
471C2×C4C2×#13511yescharacteristic
472C2×C4C2×#13511yescharacteristic
473C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
474C2×C4#57511yescharacteristic
475C2×C4#57711yescharacteristic
476C2×C4#57711yescharacteristic
477C2×C4#57711yescharacteristic
478C2×C4#57711yescharacteristic
479C8—22no—
480C8—22no—
481C8—22no—
482C8—41no—
483C8—42no—
484C8—42no—
485C8—42no—
486C8—42no—
487C8—81no—
488C8—81no—
489C22—42no—
490C22—42no—
491C22—42no—
492C22—81no—
493C22—81no—
494C22—81no—
495C22—81no—
496C22C4×#12111yescharacteristic
497C22C2×#46311yescharacteristic
498C22#303511yescharacteristic
499C22#305711yescharacteristic
500C22#306011yescharacteristic
501C22C2×#47311yescharacteristic
502C22#306711yescharacteristic
503C4—22no—
504C4—22no—
505C4—21no—
506C4—21no—
507C4—22no—
508C4—41no—
509C4—42no—
510C4—81no—
511C4—82no—
512C4—81no—
513C4—82no—
514C4—161no—
515C4—161no—
516C4C4×#13511yescharacteristic
517C4C4×AGL(1,ℤ/8)11yescharacteristic
518C4#294911yescharacteristic
519C4#295211yescharacteristic
520C4#310211yescharacteristic
521C4#310511yescharacteristic
522C4#317311yescharacteristic
523C4#317411yescharacteristic
524C2—82no—
525C2#1919711yescharacteristic
526C2#1920011yescharacteristic
527C2#1922411yescharacteristic
528C2#1922811yescharacteristic
529C2C4×#46311yescharacteristic
530C2#2152311yescharacteristic
531C2C4×#47311yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC22×D4C2×D4
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C4×#487C2semidirect
6. #9736C2semidirect
7. #9613C2semidirect
8. #9574C2semidirect
9. #9570C2semidirect
13. C4×#334C2semidirect
14. #8237C2semidirect
15. #8236C2semidirect
55. #1679C4semidirect

Maximal subgroups

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

subgroupindex
#221462
#221102
C4×#4872
#204852
C2×#16792
#97362
#96132
#95742
#95702
#92272
#92252
#86462
C4×#3342
#82372
#82362