Finite Group Databasebeta

← #211891

Subgroups — #211891

Order 512.

578 conjugacy classes of subgroups.

The subgroup lattice has 556 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22166C211yesmaximal, characteristic
2#22130C211yesmaximal, characteristic, complemented
3#19124C211yesmaximal, characteristic
4#17979C211yesmaximal, characteristic, complemented
5#17331C211yesmaximal, characteristic, complemented
6#9722C211yesmaximal, characteristic, complemented
7#9626C211yesmaximal, characteristic, complemented
8#9590C211yesmaximal, characteristic, complemented
9#9589C211yesmaximal, characteristic, complemented
10#9210C211yesmaximal, characteristic
11#9209C211yesmaximal, characteristic, complemented
12#8667C211yesmaximal, characteristic, complemented
13#8288C211yesmaximal, characteristic, complemented
14#8256C211yesmaximal, characteristic, complemented
15#8251C211yesmaximal, characteristic, complemented
16#3167C2211yescharacteristic
17#3076C2211yescharacteristic
18#3035C2211yescharacteristic
19#2965C2211yescharacteristic
20C8×#41C2211yescharacteristic
21#2960C2211yescharacteristic
22#2931C2211yescharacteristic
23#2922C2211yescharacteristic
24#1986C2211yescharacteristic
25#1980C2211yescharacteristic
26#1975—21no—
27#1950C2211yescharacteristic
28#1950C2211yescharacteristic
29#1945—21no—
30#1938—21no—
31#1937—21no—
32#1929—21no—
33#1922—21no—
34#1876—21no—
35#1866—21no—
36#1806C2211yescharacteristic
37#1806C2211yescharacteristic
38C4×#107C2211yescharacteristic
39#1770C2211yescharacteristic
40C8×C4⋊C4C2211yescharacteristic
41#1680C411yescharacteristic, complemented
42#1679C411yescharacteristic, complemented
43#1678C411yescharacteristic, complemented
44#1677C411yescharacteristic, complemented
45#1672C411yescharacteristic, complemented
46#1671C411yescharacteristic, complemented
47#1666C411yescharacteristic, complemented
48#1665C411yescharacteristic, complemented
49#1588—21no—
50#1588—21no—
51#1585—21no—
52#1585—21no—
53#1567C2211yescharacteristic
54C2×#334C2211yescharacteristic
55#1542C2211yescharacteristic
56#1540C2211yescharacteristic
57#1503C2211yescharacteristic
58#1502C2211yescharacteristic
59#1493C2211yescharacteristic
60#1492C2211yescharacteristic
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
75C2×#135—21no—
76C2×#131—21no—
77#533C2311yescharacteristic
78#532C2311yescharacteristic
79C4×#41C2311yescharacteristic
80#498C2×C411yescharacteristic
81#492C2×C411yescharacteristic
82#487C2×C411yescharacteristic
83#486C2×C411yescharacteristic
84#479—21no—
85#478—21no—
86#477C2×C411yescharacteristic
87#476C2×C411yescharacteristic
88#475C2×C411yescharacteristic
89#474C2×C411yescharacteristic
90#465—21no—
91#465—21no—
92#464—21no—
93#463—21no—
94#462C2×C411yescharacteristic
95#461C2×C411yescharacteristic
96#460C2×C411yescharacteristic
97#459C2×C411yescharacteristic
98#446—21no—
99#446—21no—
100#446—21no—
101C8×Q8—21no—
102C8×Q8—21no—
103C8×Q8C2×C411yescharacteristic
104C8×Q8C2×C411yescharacteristic
105C4×SD16—21no—
106C4×SD16—21no—
107#436—21no—
108#436—21no—
109#436—21no—
110#435—22no—
111C8×D4—21no—
112C8×D4—21no—
113C8×D4—21no—
114C8×D4—21no—
115C8×D4C2×C411yescharacteristic
116C8×D4C2×C411yescharacteristic
117#433—21no—
118#433—21no—
119#431—21no—
120#431—21no—
121#431—21no—
122#431—21no—
123#431C2311yescharacteristic
124#431C2311yescharacteristic
125C2×#108—21no—
126C2×#108—21no—
127#427—21no—
128C2×#107—21no—
129#424—21no—
130#424D411yescharacteristic
131#424D411yescharacteristic
132#423C2311yescharacteristic
133#423C2311yescharacteristic
134C2×C4⋊C8C2311yescharacteristic
135C2×C4⋊C8C2311yescharacteristic
136C2×#104—21no—
137C2×#104—21no—
138C2×#104—21no—
139C2×#104—21no—
140#416—21no—
141#416—21no—
142#408—21no—
143#408D411yescharacteristic
144#408D411yescharacteristic
145#405—21no—
146C4×M16—21no—
147C4×M16C2311yescharacteristic
148C2×#97—21no—
149C2×C4×C8—21no—
150C2×C4×C8—21no—
151C2×C4×C8C2311yescharacteristic
152C2×C4×C8C2311yescharacteristic
153C2×C4×C8C2311yescharacteristic
154C4×C4⋊C4C2311yescharacteristic
155C4×C4⋊C4C2311yescharacteristic
156#341—21no—
157#341—21no—
158#340—21no—
159#339—21no—
160#339—21no—
161#339—21no—
162#339—21no—
163#336—21no—
164#336—21no—
165#335—21no—
166#335—21no—
167#334C2×C411yescharacteristic
168#334C2×C411yescharacteristic
169#334C2×C411yescharacteristic
170#334C2×C411yescharacteristic
171#333C2×C411yescharacteristic
172#333C2×C411yescharacteristic
173#332C2×C411yescharacteristic
174#332C2×C411yescharacteristic
175#332C2×C411yescharacteristic
176#332C2×C411yescharacteristic
177#331C2×C411yescharacteristic
178#331C2×C411yescharacteristic
179#328—22no—
180#327—22no—
181Q8⋊C8—21no—
182Q8⋊C8—21no—
183D4⋊C8—21no—
184D4⋊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×#41—21no—
196C2×#41C2×D411yescharacteristic
197#135—81no—
198#135—81no—
199C2×Q16—21no—
200C2×SD16—21no—
201C2×SD16—21no—
202C2×SD16—21no—
203C2×SD16—21no—
204C2×D8—21no—
205#131—81no—
206#131—81no—
207C2×M16—22no—
208C2×M16—21no—
209C2×M16—21no—
210C2×M16—41no—
211C2×M16C2×D411yescharacteristic
212C2×M16C2×D411yescharacteristic
213C22×C8—21no—
214C22×C8—21no—
215C22×C8—21no—
216C22×C8—21no—
217C22×C8—21no—
218C22×C8—41no—
219C22×C8C2×D411yescharacteristic
220C22×C8C2×D411yescharacteristic
221C22×C8#4111yescharacteristic
222C22×C8#4111yescharacteristic
223C4⋊Q8C22×C411yescharacteristic
224C4⋊Q8C22×C411yescharacteristic
225C4⋊Q8C22×C411yescharacteristic
226#127C22×C411yescharacteristic
227#125—21no—
228#124—21no—
229C22⋊Q8—21no—
230C22⋊Q8—21no—
231#121—21no—
232#121—21no—
233C4×Q8—21no—
234C4×Q8—21no—
235C4×Q8C22×C411yescharacteristic
236C4×Q8#4111yescharacteristic
237C4×D4—21no—
238C4×D4—21no—
239C4×D4—21no—
240C4×D4—21no—
241C4×D4C22×C411yescharacteristic
242C4×D4#4111yescharacteristic
243#117—21no—
244#117—21no—
245#117C2×D411yescharacteristic
246C2×C4⋊C4—21no—
247C2×C4⋊C4—21no—
248C2×C42—21no—
249C2×C42—21no—
250C2×C42—21no—
251C2×C42C2411yescharacteristic
252#108—81no—
253#108—81no—
254#107—21no—
255#107—21no—
256#107—42no—
257#106—21no—
258#106—21no—
259#106—21no—
260#106—21no—
261C4⋊C8—22no—
262C4⋊C8—21no—
263C4⋊C8—21no—
264C4⋊C8—21no—
265C4⋊C8—21no—
266C4⋊C8—21no—
267C4⋊C8—21no—
268C4⋊C8—21no—
269C4⋊C8—21no—
270C4⋊C8—21no—
271C4⋊C8—21no—
272C4⋊C8—21no—
273C4⋊C8—21no—
274C4⋊C8—21no—
275C4⋊C8—41no—
276C4⋊C8—41no—
277C4⋊C8C22×C411yescharacteristic
278C4⋊C8C22×C411yescharacteristic
279C4⋊C8C22×C411yescharacteristic
280C4⋊C8C22×C411yescharacteristic
281C4⋊C8C22×C411yescharacteristic
282C4⋊C8C22×C411yescharacteristic
283#104—81no—
284#104—81no—
285#104—81no—
286#104—81no—
287#103—21no—
288#103—21no—
289#103—21no—
290#103—21no—
291#103—21no—
292#103—21no—
293#102—21no—
294#102—21no—
295#102—21no—
296#102—21no—
297#102—21no—
298#102—21no—
299C22⋊C8—22no—
300C22⋊C8—21no—
301C22⋊C8—21no—
302C22⋊C8—21no—
303C22⋊C8—21no—
304C22⋊C8—21no—
305C22⋊C8—21no—
306C22⋊C8—21no—
307C22⋊C8—41no—
308C22⋊C8—41no—
309C22⋊C8#4112yes—
310#97—22no—
311#97—21no—
312#97—41no—
313#97—81no—
314C4×C8—21no—
315C4×C8—21no—
316C4×C8—21no—
317C4×C8—21no—
318C4×C8—21no—
319C4×C8—21no—
320C4×C8—41no—
321C4×C8—81no—
322C4×C8C22×C411yescharacteristic
323C4×C8C22×C411yescharacteristic
324C4×C8C2×D411yescharacteristic
325C4×C8C2×D411yescharacteristic
326C4×C8C2×D411yescharacteristic
327C4×C8C2×D411yescharacteristic
328C4×C8C2×D411yescharacteristic
329C4×C8C2×D411yescharacteristic
330C4×C8#4111yescharacteristic
331C4×C8#4111yescharacteristic
332#95—21no—
333#95—21no—
334#41—21no—
335#41—21no—
336#41—21no—
337#41—21no—
338#41—41no—
339#41—41no—
340#41—41no—
341#41—41no—
342C2×Q8—21no—
343C2×Q8—21no—
344C2×Q8—21no—
345C2×Q8C4×D411yescharacteristic
346C2×D4—21no—
347C2×D4—21no—
348C2×D4—21no—
349C2×D4—21no—
350C2×D4—41no—
351C2×D4C4×D411yescharacteristic
352C22×C4—21no—
353C22×C4—21no—
354C22×C4—21no—
355C22×C4—41no—
356C22×C4—41no—
357C22×C4—41no—
358C22×C4C22×D411yescharacteristic
359C22×C4C2×#4111yescharacteristic
360C22×C4C2×#4111yescharacteristic
361Q16—81no—
362SD16—81no—
363SD16—81no—
364SD16—81no—
365SD16—81no—
366D8—81no—
367M16—21no—
368M16—21no—
369M16—21no—
370M16—21no—
371M16—44no—
372M16—41no—
373M16—41no—
374M16—41no—
375M16—41no—
376M16—82no—
377C2×C8—21no—
378C2×C8—21no—
379C2×C8—21no—
380C2×C8—21no—
381C2×C8—21no—
382C2×C8—21no—
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—41no—
393C2×C8—41no—
394C2×C8—42no—
395C2×C8—41no—
396C2×C8—41no—
397C2×C8—41no—
398C2×C8—41no—
399C2×C8—41no—
400C2×C8—41no—
401C2×C8—81no—
402C2×C8—81no—
403C2×C8—82no—
404C2×C8C4×D411yescharacteristic
405C2×C8C4×D411yescharacteristic
406C2×C8C4×D411yescharacteristic
407C2×C8C4×D411yescharacteristic
408C2×C8#12111yescharacteristic
409C2×C8#12111yescharacteristic
410C2×C8#12111yescharacteristic
411C2×C8#12111yescharacteristic
412C4⋊C4—21no—
413C4⋊C4—21no—
414C4⋊C4—22no—
415C4⋊C4—21no—
416C4⋊C4—21no—
417C4⋊C4—21no—
418C4⋊C4—21no—
419C4⋊C4—21no—
420C4⋊C4—21no—
421C4⋊C4—41no—
422C4⋊C4—41no—
423C4⋊C4—41no—
424C4⋊C4—41no—
425C4⋊C4—41no—
426C4⋊C4C4×D411yescharacteristic
427C4⋊C4C4×D411yescharacteristic
428C22⋊C4—21no—
429C22⋊C4—21no—
430C22⋊C4—21no—
431C22⋊C4—41no—
432C22⋊C4—41no—
433C22⋊C4—41no—
434C42—21no—
435C42—21no—
436C42—21no—
437C42—41no—
438C42—41no—
439C42—41no—
440C42—41no—
441C42—41no—
442C42C23×C411yescharacteristic
443C42C22×D411yescharacteristic
444C42C2×#4111yescharacteristic
445C42C2×#4111yescharacteristic
446C23—21no—
447C23—41no—
448C23#53211yescharacteristic
449Q8—21no—
450Q8—21no—
451Q8—41no—
452Q8—41no—
453Q8—41no—
454Q8—41no—
455Q8—81no—
456D4—21no—
457D4—21no—
458D4—41no—
459D4—41no—
460D4—41no—
461D4—41no—
462D4—41no—
463D4—41no—
464D4—81no—
465D4—81no—
466D4—81no—
467C2×C4—21no—
468C2×C4—21no—
469C2×C4—21no—
470C2×C4—21no—
471C2×C4—21no—
472C2×C4—41no—
473C2×C4—42no—
474C2×C4—41no—
475C2×C4—41no—
476C2×C4—41no—
477C2×C4—41no—
478C2×C4—42no—
479C2×C4—41no—
480C2×C4—41no—
481C2×C4—41no—
482C2×C4—81no—
483C2×C4—81no—
484C2×C4—81no—
485C2×C4—81no—
486C2×C4—81no—
487C2×C4—81no—
488C2×C4—81no—
489C2×C4C2×C4×D411yescharacteristic
490C2×C4C2×C4×D411yescharacteristic
491C2×C4C4×#4111yescharacteristic
492C2×C4C2×#12111yescharacteristic
493C2×C4#52511yescharacteristic
494C2×C4#52911yescharacteristic
495C2×C4#57611yescharacteristic
496C2×C4#57711yescharacteristic
497C2×C4#57711yescharacteristic
498C2×C4#57811yescharacteristic
499C8—22no—
500C8—21no—
501C8—21no—
502C8—21no—
503C8—21no—
504C8—21no—
505C8—21no—
506C8—21no—
507C8—21no—
508C8—41no—
509C8—42no—
510C8—41no—
511C8—41no—
512C8—41no—
513C8—41no—
514C8—41no—
515C8—41no—
516C8—41no—
517C8—41no—
518C8—81no—
519C8—81no—
520C22—21no—
521C22—21no—
522C22—41no—
523C22—41no—
524C22—41no—
525C22—81no—
526C22—81no—
527C22—81no—
528C22C4×#12111yescharacteristic
529C22#317811yescharacteristic
530C22#317911yescharacteristic
531C4—21no—
532C4—21no—
533C4—21no—
534C4—41no—
535C4—41no—
536C4—41no—
537C4—82no—
538C4—81no—
539C4—81no—
540C4—81no—
541C4—161no—
542C4#294911yescharacteristic
543C4#295111yescharacteristic
544C4#295611yescharacteristic
545C4#295711yescharacteristic
546C4#306211yescharacteristic
547C4#306311yescharacteristic
548C2—21no—
549C2—21no—
550C2—81no—
551C2—161no—
552C2#2151911yescharacteristic
553C2#2153111yescharacteristic
554C2#2155811yescharacteristic
subgroupG/H
centerC2×C4C2×#121
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
socleC22C4×#121
prime radical#3035C22
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
2. #22130C2semidirect
4. #17979C2semidirect
5. #17331C2semidirect
6. #9722C2semidirect
7. #9626C2semidirect
8. #9590C2semidirect
9. #9589C2semidirect
11. #9209C2semidirect
12. #8667C2semidirect
13. #8288C2semidirect
14. #8256C2semidirect
15. #8251C2semidirect
41. #1680C4semidirect
42. #1679C4semidirect
43. #1678C4semidirect
44. #1677C4semidirect
45. #1672C4semidirect
46. #1671C4semidirect
47. #1666C4semidirect
48. #1665C4semidirect

Maximal subgroups

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

subgroupindex
#221662
#221302
#191242
#179792
#173312
#97222
#96262
#95902
#95892
#92102
#92092
#86672
#82882
#82562
#82512