Finite Group Databasebeta

← #211889

Subgroups — #211889

Order 512.

549 conjugacy classes of subgroups.

The subgroup lattice has 529 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22131C211yesmaximal, characteristic
2#22130C211yesmaximal, characteristic
3#19126C211yesmaximal, characteristic
4#17980C211yesmaximal, characteristic, complemented
5#17331C211yesmaximal, characteristic
6#9720C211yesmaximal, characteristic, complemented
7#9625C211yesmaximal, characteristic, complemented
8#9589C211yesmaximal, characteristic, complemented
9#9589C211yesmaximal, characteristic, complemented
10#9209C211yesmaximal, characteristic
11#9209C211yesmaximal, characteristic
12#8666C211yesmaximal, characteristic
13#8289C211yesmaximal, characteristic, complemented
14#8256C211yesmaximal, characteristic, complemented
15#8252C211yesmaximal, characteristic, complemented
16#3170C2211yescharacteristic
17#3076C2211yescharacteristic
18#3076C2211yescharacteristic
19#2965C2211yescharacteristic
20C8×#41C2211yescharacteristic
21#2960C2211yescharacteristic
22#2960C2211yescharacteristic
23#2922C2211yescharacteristic
24#1986C2211yescharacteristic
25#1986C2211yescharacteristic
26#1950C2211yescharacteristic
27#1950C2211yescharacteristic
28#1946—21no—
29#1945—21no—
30#1938—21no—
31#1937—21no—
32#1929—21no—
33#1929—21no—
34#1922—21no—
35#1922—21no—
36#1806C2211yescharacteristic
37#1806C2211yescharacteristic
38C4×#106C2211yescharacteristic
39#1770C2211yescharacteristic
40C8×C4⋊C4C2211yescharacteristic
41#1684C411yescharacteristic, complemented
42#1684C411yescharacteristic, complemented
43#1683C411yescharacteristic, complemented
44#1683C411yescharacteristic, complemented
45#1676C411yescharacteristic, complemented
46#1675C411yescharacteristic, complemented
47#1670C411yescharacteristic, complemented
48#1669C411yescharacteristic, complemented
49#1587—21no—
50#1587—21no—
51#1587—21no—
52#1587—21no—
53#1566C2211yescharacteristic
54C2×#335C2211yescharacteristic
55#1542C2211yescharacteristic
56#1542C2211yescharacteristic
57#1503C2211yescharacteristic
58#1503C2211yescharacteristic
59#1493C2211yescharacteristic
60#1493C2211yescharacteristic
61#1454—21no—
62#1453—21no—
63#1452—21no—
64#1451C2211yescharacteristic
65#1450C2211yescharacteristic
66C8×M16—21no—
67C8×M16C2211yescharacteristic
68C2×#322C2211yescharacteristic
69#1303C2211yescharacteristic
70#1303C2211yescharacteristic
71#1301C2211yescharacteristic
72#1301C2211yescharacteristic
73#1279C2211yescharacteristic
74#1279C2211yescharacteristic
75#533C2311yescharacteristic
76#533C2311yescharacteristic
77C4×#41C2311yescharacteristic
78#500C2×C411yescharacteristic
79#494C2×C411yescharacteristic
80#487C2×C411yescharacteristic
81#487C2×C411yescharacteristic
82#479—21no—
83#479—21no—
84#478—21no—
85#478—21no—
86#477C2×C411yescharacteristic
87#477C2×C411yescharacteristic
88#476C2×C411yescharacteristic
89#476C2×C411yescharacteristic
90#475C2×C411yescharacteristic
91#475C2×C411yescharacteristic
92#474C2×C411yescharacteristic
93#474C2×C411yescharacteristic
94#467—21no—
95#467—21no—
96#446—21no—
97#446—21no—
98#446—21no—
99#446—21no—
100#446—21no—
101C8×Q8—21no—
102C8×Q8—21no—
103C8×Q8—21no—
104C8×Q8—21no—
105C8×Q8C2×C411yescharacteristic
106C8×Q8C2×C411yescharacteristic
107C4×Q16—21no—
108C4×Q16—21no—
109#436—21no—
110#435—22no—
111C8×D4—21no—
112C8×D4—21no—
113C8×D4C2×C411yescharacteristic
114C8×D4C2×C411yescharacteristic
115#433—21no—
116#433—21no—
117#431—21no—
118#431—21no—
119#431—21no—
120#431—21no—
121#431C2311yescharacteristic
122#431C2311yescharacteristic
123C2×#108—21no—
124C2×#108—21no—
125#427—21no—
126#427—21no—
127C2×#106—21no—
128#424—21no—
129#424—21no—
130#424D411yescharacteristic
131#424D411yescharacteristic
132#423C2311yescharacteristic
133#423C2311yescharacteristic
134C2×C4⋊C8C2311yescharacteristic
135C2×C4⋊C8C2311yescharacteristic
136C2×#104—21no—
137C2×#104—21no—
138#416—21no—
139#416—21no—
140#408D411yescharacteristic
141#408D411yescharacteristic
142#405—21no—
143#405—21no—
144C4×M16—21no—
145C4×M16C2311yescharacteristic
146C2×#97—21no—
147C2×C4×C8—21no—
148C2×C4×C8—21no—
149C2×C4×C8C2311yescharacteristic
150C2×C4×C8C2311yescharacteristic
151C2×C4×C8C2311yescharacteristic
152C4×C4⋊C4C2311yescharacteristic
153C4×C4⋊C4C2311yescharacteristic
154#341—21no—
155#341—21no—
156#340—21no—
157#339—21no—
158#339—21no—
159#339—21no—
160#339—21no—
161#339—21no—
162#339—21no—
163#336—21no—
164#336—21no—
165#335C2×C411yescharacteristic
166#335C2×C411yescharacteristic
167#335C2×C411yescharacteristic
168#335C2×C411yescharacteristic
169#334—21no—
170#334—21no—
171#333C2×C411yescharacteristic
172#333C2×C411yescharacteristic
173#333C2×C411yescharacteristic
174#333C2×C411yescharacteristic
175#332C2×C411yescharacteristic
176#332C2×C411yescharacteristic
177#332C2×C411yescharacteristic
178#332C2×C411yescharacteristic
179#328—22no—
180#328—22no—
181Q8⋊C8—21no—
182Q8⋊C8—21no—
183Q8⋊C8—21no—
184Q8⋊C8—21no—
185#322—21no—
186#322—21no—
187#322—21no—
188#322—21no—
189#322D411yescharacteristic
190#322D411yescharacteristic
191#322D411yescharacteristic
192#322D411yescharacteristic
193C82—21no—
194C82—21no—
195C2×#41C2×D411yescharacteristic
196C2×Q16—21no—
197C2×Q16—21no—
198C2×M16—22no—
199C2×M16—21no—
200C2×M16—21no—
201C2×M16C2×D411yescharacteristic
202C2×M16C2×D411yescharacteristic
203C22×C8—21no—
204C22×C8—21no—
205C22×C8—21no—
206C22×C8—21no—
207C22×C8—21no—
208C22×C8C2×D411yescharacteristic
209C22×C8C2×D411yescharacteristic
210C22×C8#4111yescharacteristic
211C22×C8#4111yescharacteristic
212C4⋊Q8C22×C411yescharacteristic
213C4⋊Q8C22×C411yescharacteristic
214C4⋊Q8C22×C411yescharacteristic
215C4⋊Q8C22×C411yescharacteristic
216#125—21no—
217#125—21no—
218C22⋊Q8—21no—
219C22⋊Q8—21no—
220C22⋊Q8—21no—
221C22⋊Q8—21no—
222C4×Q8—21no—
223C4×Q8—21no—
224C4×Q8—21no—
225C4×Q8—21no—
226C4×Q8C22×C411yescharacteristic
227C4×Q8#4111yescharacteristic
228C4×D4—21no—
229C4×D4—21no—
230C4×D4C22×C411yescharacteristic
231C4×D4#4111yescharacteristic
232#117—21no—
233#117—21no—
234#117—21no—
235#117C2×D411yescharacteristic
236C2×C4⋊C4—21no—
237C2×C4⋊C4—21no—
238C2×C42—21no—
239C2×C42—21no—
240C2×C42—21no—
241C2×C42C2411yescharacteristic
242#108—81no—
243#108—81no—
244#107—21no—
245#107—21no—
246#107—21no—
247#107—21no—
248#107—21no—
249#107—21no—
250#106—21no—
251#106—21no—
252#106—21no—
253#106—21no—
254#106—42no—
255C4⋊C8—22no—
256C4⋊C8—21no—
257C4⋊C8—21no—
258C4⋊C8—21no—
259C4⋊C8—21no—
260C4⋊C8—21no—
261C4⋊C8—21no—
262C4⋊C8—21no—
263C4⋊C8—21no—
264C4⋊C8—21no—
265C4⋊C8—21no—
266C4⋊C8—21no—
267C4⋊C8—21no—
268C4⋊C8—21no—
269C4⋊C8—41no—
270C4⋊C8—41no—
271C4⋊C8—41no—
272C4⋊C8—41no—
273C4⋊C8C22×C411yescharacteristic
274C4⋊C8C22×C411yescharacteristic
275C4⋊C8C22×C411yescharacteristic
276C4⋊C8C22×C411yescharacteristic
277C4⋊C8C22×C411yescharacteristic
278C4⋊C8C22×C411yescharacteristic
279#104—81no—
280#104—81no—
281#103—21no—
282#103—21no—
283#103—21no—
284#103—21no—
285#103—21no—
286#103—21no—
287#103—21no—
288#103—21no—
289#102—21no—
290#102—21no—
291#102—21no—
292#102—21no—
293C22⋊C8—22no—
294C22⋊C8—21no—
295C22⋊C8—21no—
296C22⋊C8—21no—
297C22⋊C8—21no—
298C22⋊C8—21no—
299C22⋊C8—21no—
300C22⋊C8—21no—
301C22⋊C8#4112yes—
302#97—22no—
303#97—21no—
304#97—41no—
305#97—41no—
306#97—81no—
307C4×C8—21no—
308C4×C8—21no—
309C4×C8—21no—
310C4×C8—21no—
311C4×C8—21no—
312C4×C8—21no—
313C4×C8—41no—
314C4×C8—41no—
315C4×C8—81no—
316C4×C8C22×C411yescharacteristic
317C4×C8C22×C411yescharacteristic
318C4×C8C2×D411yescharacteristic
319C4×C8C2×D411yescharacteristic
320C4×C8C2×D411yescharacteristic
321C4×C8C2×D411yescharacteristic
322C4×C8C2×D411yescharacteristic
323C4×C8C2×D411yescharacteristic
324C4×C8#4111yescharacteristic
325C4×C8#4111yescharacteristic
326#95—21no—
327#95—21no—
328#41—21no—
329#41—21no—
330#41—21no—
331#41—21no—
332C2×Q8—21no—
333C2×Q8—21no—
334C2×Q8—21no—
335C2×Q8—21no—
336C2×Q8C4×D411yescharacteristic
337C2×D4—21no—
338C2×D4C4×D411yescharacteristic
339C22×C4—21no—
340C22×C4—21no—
341C22×C4—21no—
342C22×C4—41no—
343C22×C4—41no—
344C22×C4C22×D411yescharacteristic
345C22×C4C2×#4111yescharacteristic
346C22×C4C2×#4111yescharacteristic
347Q16—81no—
348Q16—81no—
349M16—21no—
350M16—21no—
351M16—21no—
352M16—21no—
353M16—44no—
354M16—41no—
355M16—41no—
356M16—41no—
357M16—41no—
358C2×C8—21no—
359C2×C8—21no—
360C2×C8—21no—
361C2×C8—21no—
362C2×C8—21no—
363C2×C8—21no—
364C2×C8—21no—
365C2×C8—21no—
366C2×C8—21no—
367C2×C8—21no—
368C2×C8—21no—
369C2×C8—21no—
370C2×C8—21no—
371C2×C8—21no—
372C2×C8—22no—
373C2×C8—41no—
374C2×C8—41no—
375C2×C8—42no—
376C2×C8—41no—
377C2×C8—41no—
378C2×C8—41no—
379C2×C8—41no—
380C2×C8—41no—
381C2×C8—41no—
382C2×C8—81no—
383C2×C8—81no—
384C2×C8C4×D411yescharacteristic
385C2×C8C4×D411yescharacteristic
386C2×C8C4×D411yescharacteristic
387C2×C8C4×D411yescharacteristic
388C2×C8#12111yescharacteristic
389C2×C8#12111yescharacteristic
390C2×C8#12111yescharacteristic
391C2×C8#12111yescharacteristic
392C4⋊C4—21no—
393C4⋊C4—21no—
394C4⋊C4—22no—
395C4⋊C4—21no—
396C4⋊C4—21no—
397C4⋊C4—21no—
398C4⋊C4—21no—
399C4⋊C4—21no—
400C4⋊C4—21no—
401C4⋊C4—21no—
402C4⋊C4—21no—
403C4⋊C4—41no—
404C4⋊C4—41no—
405C4⋊C4—41no—
406C4⋊C4—41no—
407C4⋊C4—41no—
408C4⋊C4—41no—
409C4⋊C4—41no—
410C4⋊C4C4×D411yescharacteristic
411C4⋊C4C4×D411yescharacteristic
412C22⋊C4—21no—
413C22⋊C4—21no—
414C22⋊C4—21no—
415C22⋊C4—41no—
416C22⋊C4—41no—
417C42—21no—
418C42—21no—
419C42—21no—
420C42—41no—
421C42—41no—
422C42—41no—
423C42—41no—
424C42—41no—
425C42—41no—
426C42C23×C411yescharacteristic
427C42C22×D411yescharacteristic
428C42C2×#4111yescharacteristic
429C42C2×#4111yescharacteristic
430C23—21no—
431C23#53211yescharacteristic
432Q8—21no—
433Q8—21no—
434Q8—41no—
435Q8—41no—
436Q8—41no—
437Q8—41no—
438Q8—81no—
439Q8—81no—
440D4—21no—
441D4—21no—
442D4—41no—
443D4—41no—
444C2×C4—21no—
445C2×C4—21no—
446C2×C4—21no—
447C2×C4—21no—
448C2×C4—21no—
449C2×C4—41no—
450C2×C4—42no—
451C2×C4—41no—
452C2×C4—41no—
453C2×C4—41no—
454C2×C4—41no—
455C2×C4—42no—
456C2×C4—41no—
457C2×C4—41no—
458C2×C4—41no—
459C2×C4—41no—
460C2×C4—81no—
461C2×C4—81no—
462C2×C4—81no—
463C2×C4—81no—
464C2×C4—81no—
465C2×C4C2×C4×D411yescharacteristic
466C2×C4C2×C4×D411yescharacteristic
467C2×C4C4×#4111yescharacteristic
468C2×C4C2×#12111yescharacteristic
469C2×C4#52511yescharacteristic
470C2×C4#52911yescharacteristic
471C2×C4#57711yescharacteristic
472C2×C4#57711yescharacteristic
473C2×C4#57811yescharacteristic
474C2×C4#57811yescharacteristic
475C8—22no—
476C8—21no—
477C8—21no—
478C8—21no—
479C8—21no—
480C8—21no—
481C8—21no—
482C8—21no—
483C8—21no—
484C8—41no—
485C8—42no—
486C8—41no—
487C8—41no—
488C8—41no—
489C8—41no—
490C8—41no—
491C8—41no—
492C8—41no—
493C8—41no—
494C8—81no—
495C8—81no—
496C22—21no—
497C22—21no—
498C22—41no—
499C22—41no—
500C22—41no—
501C22C4×#12111yescharacteristic
502C22#317811yescharacteristic
503C22#317811yescharacteristic
504C4—21no—
505C4—21no—
506C4—21no—
507C4—41no—
508C4—41no—
509C4—41no—
510C4—82no—
511C4—81no—
512C4—81no—
513C4—81no—
514C4—81no—
515C4—161no—
516C4#295011yescharacteristic
517C4#295211yescharacteristic
518C4#295611yescharacteristic
519C4#295711yescharacteristic
520C4#306311yescharacteristic
521C4#306311yescharacteristic
522C2—21no—
523C2—21no—
524C2—81no—
525C2#2151911yescharacteristic
526C2#2153011yescharacteristic
527C2#2155811yescharacteristic
subgroupG/H
centerC2×C4C2×#121
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
socleC22C4×#121
prime radicalC2×#41C2×D4
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C2×C4×C8 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #17980C2semidirect
6. #9720C2semidirect
7. #9625C2semidirect
8. #9589C2semidirect
9. #9589C2semidirect
13. #8289C2semidirect
14. #8256C2semidirect
15. #8252C2semidirect
41. #1684C4semidirect
42. #1684C4semidirect
43. #1683C4semidirect
44. #1683C4semidirect
45. #1676C4semidirect
46. #1675C4semidirect
47. #1670C4semidirect
48. #1669C4semidirect

Maximal subgroups

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

subgroupindex
#221312
#221302
#191262
#179802
#173312
#97202
#96252
#95892
#92092
#86662
#82892
#82562
#82522