Finite Group Databasebeta

← #211996

Subgroups — #211996

Order 512.

558 conjugacy classes of subgroups.

The subgroup lattice has 531 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#19160C211yesmaximal, characteristic
2#17981C211yesmaximal, characteristic, complemented
3#17917C211yesmaximal, characteristic, complemented
4#17912C211yesmaximal, characteristic, complemented
5#17329C211yesmaximal, characteristic, complemented
6#9726C211yesmaximal, characteristic, complemented
7#9635C211yesmaximal, characteristic, complemented
8#9588C211yesmaximal, characteristic, complemented
9#9588C211yesmaximal, characteristic, complemented
10#9213C211yesmaximal, characteristic, complemented
11#9213C211yesmaximal, characteristic, complemented
12#8672C211yesmaximal, characteristic, complemented
13#8296C211yesmaximal, characteristic, complemented
14#8245C211yesmaximal, characteristic, complemented
15#8241C211yesmaximal, characteristic, complemented
16#3169C2211yescharacteristic
17#3106C2211yescharacteristic
18#3106C2211yescharacteristic
19#2965C2211yescharacteristic
20C8×#41C2211yescharacteristic
21#2931C2211yescharacteristic
22#2931C2211yescharacteristic
23#2922C2211yescharacteristic
24C4×#108C2211yescharacteristic
25#1698C411yescharacteristic
26#1698C411yescharacteristic
27#1697C411yescharacteristic
28#1697C411yescharacteristic
29#1696C411yescharacteristic
30#1695C411yescharacteristic
31#1688C411yescharacteristic
32#1687C411yescharacteristic
33#1588—21no—
34#1586—21no—
35#1586—21no—
36#1586—21no—
37#1586—21no—
38#1586—21no—
39#1583—21no—
40#1581—21no—
41#1580—21no—
42SD16⋊C8—21no—
43C8×SD16—21no—
44C8×D8—21no—
45#1574C2211yescharacteristic
46#1574C2211yescharacteristic
47#1566C2211yescharacteristic
48#1564C2211yescharacteristic
49#1543C2211yescharacteristic
50#1543C2211yescharacteristic
51#1525C2211yescharacteristic
52#1525C2211yescharacteristic
53#1492C2211yescharacteristic
54#1492C2211yescharacteristic
55#1489C2211yescharacteristic
56#1489C2211yescharacteristic
57#1484C2211yescharacteristic
58#1484C2211yescharacteristic
59#1454—21no—
60#1453—21no—
61#1452—21no—
62#1452C2211yescharacteristic
63#1451—21no—
64#1451C2211yescharacteristic
65#1451C2211yescharacteristic
66C8×M16C2211yescharacteristic
67C8×M16C2211yescharacteristic
68C2×#322C2211yescharacteristic
69#1299C2211yescharacteristic
70#1299C2211yescharacteristic
71#1299C2211yescharacteristic
72#1299C2211yescharacteristic
73#1278C2211yescharacteristic
74#1278C2211yescharacteristic
75C2×#131—21no—
76C2×#131—21no—
77#532C2311yescharacteristic
78#532C2311yescharacteristic
79C4×#41C2311yescharacteristic
80#500C2×C411yescharacteristic
81#493C2×C411yescharacteristic
82#486C2×C411yescharacteristic
83#486C2×C411yescharacteristic
84#482—21no—
85#482—21no—
86#477C2×C411yescharacteristic
87#477C2×C411yescharacteristic
88#476C2×C411yescharacteristic
89#476C2×C411yescharacteristic
90#466—21no—
91#466—21no—
92#460C2×C411yescharacteristic
93#460C2×C411yescharacteristic
94#459C2×C411yescharacteristic
95#459C2×C411yescharacteristic
96#449—21no—
97#449—21no—
98#446C2×C411yescharacteristic
99#446C2×C411yescharacteristic
100C8×Q8D411yescharacteristic
101C8×Q8D411yescharacteristic
102C4×SD16—21no—
103C4×SD16—21no—
104C4×D8—21no—
105C4×D8—21no—
106C4×D8—21no—
107C4×D8—21no—
108#436—21no—
109#436—21no—
110#436—21no—
111#436—21no—
112#436C2×C411yescharacteristic
113#436C2×C411yescharacteristic
114#435—21no—
115#435—21no—
116C8×D4—21no—
117C8×D4—21no—
118C8×D4—21no—
119C8×D4—21no—
120C8×D4—21no—
121C8×D4—21no—
122C8×D4D411yescharacteristic
123C8×D4D411yescharacteristic
124#433—22no—
125#431—21no—
126#431—21no—
127#431—21no—
128#431—21no—
129#431C2311yescharacteristic
130#431C2311yescharacteristic
131#431C2311yescharacteristic
132#431C2311yescharacteristic
133C2×#108—21no—
134#424—21no—
135#423C2311yescharacteristic
136#423C2311yescharacteristic
137#408—21no—
138#408—21no—
139#408—21no—
140C4×M16—21no—
141C4×M16C2311yescharacteristic
142C4×M16C2311yescharacteristic
143C4×M16C2311yescharacteristic
144C2×C4×C8—21no—
145C2×C4×C8C2311yescharacteristic
146C2×C4×C8C2311yescharacteristic
147C2×C4×C8C2311yescharacteristic
148#341—21no—
149#335—21no—
150#335—21no—
151#335C2×C411yescharacteristic
152#335C2×C411yescharacteristic
153#334—21no—
154#334—21no—
155#334—21no—
156#334—21no—
157#334C2×C411yescharacteristic
158#334C2×C411yescharacteristic
159#333C2×C411yescharacteristic
160#333C2×C411yescharacteristic
161#333C2×C411yescharacteristic
162#333C2×C411yescharacteristic
163#331C2×C411yescharacteristic
164#331C2×C411yescharacteristic
165#331C2×C411yescharacteristic
166#331C2×C411yescharacteristic
167#329—22no—
168#329—22no—
169Q8⋊C8—21no—
170Q8⋊C8—21no—
171D4⋊C8—21no—
172D4⋊C8—21no—
173D4⋊C8—21no—
174D4⋊C8—21no—
175D4⋊C8—21no—
176D4⋊C8—21no—
177D4⋊C8—21no—
178D4⋊C8—21no—
179D4⋊C8—21no—
180D4⋊C8—21no—
181#322—21no—
182#322—21no—
183#322—21no—
184#322—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×#41—21no—
197C2×#41#4111yescharacteristic
198C2×SD16—21no—
199C2×SD16—21no—
200C2×D8—21no—
201C2×D8—21no—
202C2×D8—21no—
203C2×D8—21no—
204#131—82no—
205#131—82no—
206C2×M16—21no—
207C2×M16—21no—
208C2×M16—21no—
209C2×M16—21no—
210C2×M16—21no—
211C2×M16—21no—
212C2×M16—21no—
213C2×M16—41no—
214C2×M16—41no—
215C22×C8—21no—
216C22×C8—21no—
217C22×C8—21no—
218C22×C8—41no—
219C22×C8—41no—
220C22×C8#4111yescharacteristic
221C22×C8#4111yescharacteristic
222C22×C8#4111yescharacteristic
223C22×C8#4111yescharacteristic
224C4⋊Q8C22×C411yescharacteristic
225C4⋊Q8C22×C411yescharacteristic
226#127C22×C411yescharacteristic
227#127C22×C411yescharacteristic
228#124—21no—
229#124—21no—
230#121—21no—
231#121—21no—
232#121—21no—
233#121—21no—
234C4×Q8C22×C411yescharacteristic
235C4×Q8C2×D411yescharacteristic
236C4×D4—21no—
237C4×D4—21no—
238C4×D4—21no—
239C4×D4—21no—
240C4×D4—21no—
241C4×D4—21no—
242C4×D4C22×C411yescharacteristic
243C4×D4C2×D411yescharacteristic
244#117—21no—
245#117#4111yescharacteristic
246C2×C42—21no—
247C2×C42C2411yescharacteristic
248#108—42no—
249#107—21no—
250#107—21no—
251#107—21no—
252#107—21no—
253#106—21no—
254#106—21no—
255C4⋊C8—22no—
256C4⋊C8—22no—
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⋊C8C22×C411yescharacteristic
266C4⋊C8C22×C411yescharacteristic
267C4⋊C8C22×C411yescharacteristic
268C4⋊C8C22×C411yescharacteristic
269C4⋊C8C22×C411yescharacteristic
270C4⋊C8C22×C411yescharacteristic
271C4⋊C8C2×D411yescharacteristic
272C4⋊C8C2×D411yescharacteristic
273#103—21no—
274#103—21no—
275#102—21no—
276#102—21no—
277#102—21no—
278#102—21no—
279#102—21no—
280#102—21no—
281#102—21no—
282#102—21no—
283#102—21no—
284#102—21no—
285C22⋊C8—22no—
286C22⋊C8—22no—
287C22⋊C8—21no—
288C22⋊C8—21no—
289C22⋊C8—21no—
290C22⋊C8—21no—
291C22⋊C8—21no—
292C22⋊C8—21no—
293C22⋊C8—21no—
294C22⋊C8—21no—
295C22⋊C8—41no—
296C22⋊C8—41no—
297C22⋊C8—41no—
298C22⋊C8—41no—
299#97—21no—
300#97—21no—
301#97—21no—
302#97—21no—
303#97#4112yes—
304C4×C8—21no—
305C4×C8—21no—
306C4×C8—21no—
307C4×C8—21no—
308C4×C8—21no—
309C4×C8—21no—
310C4×C8—21no—
311C4×C8—21no—
312C4×C8—21no—
313C4×C8—21no—
314C4×C8C22×C411yescharacteristic
315C4×C8C22×C411yescharacteristic
316C4×C8C2×D411yescharacteristic
317C4×C8C2×D411yescharacteristic
318C4×C8C2×D411yescharacteristic
319C4×C8C2×D411yescharacteristic
320C4×C8C2×D411yescharacteristic
321C4×C8C2×D411yescharacteristic
322C4×C8C2×D411yescharacteristic
323C4×C8C2×D411yescharacteristic
324#41—41no—
325#41—41no—
326#41—81no—
327#41—81no—
328#41—81no—
329#41—81no—
330C2×Q8—21no—
331C2×Q8—21no—
332C2×Q8C4×D411yescharacteristic
333C2×D4—21no—
334C2×D4—21no—
335C2×D4—21no—
336C2×D4—21no—
337C2×D4—21no—
338C2×D4—21no—
339C2×D4—21no—
340C2×D4—41no—
341C2×D4—41no—
342C2×D4C4×D411yescharacteristic
343C22×C4—21no—
344C22×C4—21no—
345C22×C4—21no—
346C22×C4—41no—
347C22×C4—41no—
348C22×C4C2×#4111yescharacteristic
349C22×C4C2×#4111yescharacteristic
350C22×C4C2×#4111yescharacteristic
351SD16—42no—
352SD16—42no—
353D8—42no—
354D8—42no—
355D8—81no—
356D8—81no—
357M16—22no—
358M16—41no—
359M16—41no—
360M16—41no—
361M16—41no—
362M16—41no—
363M16—81no—
364M16—81no—
365M16—82no—
366M16—82no—
367M16—81no—
368M16—81no—
369C2×C8—21no—
370C2×C8—21no—
371C2×C8—21no—
372C2×C8—21no—
373C2×C8—21no—
374C2×C8—21no—
375C2×C8—21no—
376C2×C8—21no—
377C2×C8—21no—
378C2×C8—21no—
379C2×C8—22no—
380C2×C8—41no—
381C2×C8—41no—
382C2×C8—41no—
383C2×C8—41no—
384C2×C8—41no—
385C2×C8—41no—
386C2×C8—41no—
387C2×C8—41no—
388C2×C8—41no—
389C2×C8—41no—
390C2×C8—81no—
391C2×C8—81no—
392C2×C8—82no—
393C2×C8—82no—
394C2×C8C4×D411yescharacteristic
395C2×C8C4×D411yescharacteristic
396C2×C8C4×D411yescharacteristic
397C2×C8C4×D411yescharacteristic
398C2×C8#12111yescharacteristic
399C2×C8#12111yescharacteristic
400C2×C8#12111yescharacteristic
401C2×C8#12111yescharacteristic
402C4⋊C4—21no—
403C4⋊C4—21no—
404C4⋊C4—21no—
405C4⋊C4—21no—
406C4⋊C4—21no—
407C4⋊C4—21no—
408C4⋊C4C4×D411yescharacteristic
409C4⋊C4C4×D411yescharacteristic
410C22⋊C4—21no—
411C22⋊C4—21no—
412C22⋊C4—21no—
413C22⋊C4—41no—
414C22⋊C4—41no—
415C22⋊C4—41no—
416C22⋊C4—41no—
417C42—21no—
418C42—21no—
419C42—21no—
420C42C23×C411yescharacteristic
421C42C22×D411yescharacteristic
422C42C22×D411yescharacteristic
423C42C2×#4111yescharacteristic
424C23—21no—
425C23—41no—
426C23—41no—
427C23#52911yescharacteristic
428Q8—21no—
429Q8—21no—
430Q8—81no—
431Q8—81no—
432D4—21no—
433D4—21no—
434D4—42no—
435D4—41no—
436D4—41no—
437D4—41no—
438D4—41no—
439D4—41no—
440D4—41no—
441D4—41no—
442D4—41no—
443D4—81no—
444D4—81no—
445D4—82no—
446D4—82no—
447C2×C4—21no—
448C2×C4—21no—
449C2×C4—21no—
450C2×C4—21no—
451C2×C4—21no—
452C2×C4—21no—
453C2×C4—21no—
454C2×C4—21no—
455C2×C4—21no—
456C2×C4—42no—
457C2×C4—41no—
458C2×C4—41no—
459C2×C4—41no—
460C2×C4—41no—
461C2×C4—41no—
462C2×C4—41no—
463C2×C4—81no—
464C2×C4—81no—
465C2×C4—81no—
466C2×C4—81no—
467C2×C4C2×C4×D411yescharacteristic
468C2×C4C2×C4×D411yescharacteristic
469C2×C4C4×#4111yescharacteristic
470C2×C4C2×#12111yescharacteristic
471C2×C4#52511yescharacteristic
472C2×C4#53211yescharacteristic
473C8—21no—
474C8—21no—
475C8—21no—
476C8—21no—
477C8—21no—
478C8—21no—
479C8—21no—
480C8—21no—
481C8—41no—
482C8—41no—
483C8—42no—
484C8—42no—
485C8—41no—
486C8—41no—
487C8—41no—
488C8—41no—
489C8—41no—
490C8—41no—
491C8—41no—
492C8—41no—
493C8—81no—
494C8—81no—
495C8—81no—
496C22—21no—
497C22—21no—
498C22—21no—
499C22—41no—
500C22—41no—
501C22—41no—
502C22—81no—
503C22—81no—
504C22—81no—
505C22—81no—
506C22—81no—
507C22—81no—
508C22C4×#12111yescharacteristic
509C4—21no—
510C4—21no—
511C4—21no—
512C4—41no—
513C4—41no—
514C4—41no—
515C4—81no—
516C4—81no—
517C4#295511yescharacteristic
518C4#295511yescharacteristic
519C4#295611yescharacteristic
520C4#295711yescharacteristic
521C4#306811yescharacteristic
522C4#306811yescharacteristic
523C2—41no—
524C2—81no—
525C2—161no—
526C2—161no—
527C2#2151911yescharacteristic
528C2#2154811yescharacteristic
529C2#2155511yescharacteristic
subgroupG/H
centerC2×C4C2×#121
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
socleC22C4×#121
prime radical#19160C2
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. #17981C2semidirect
3. #17917C2semidirect
4. #17912C2semidirect
5. #17329C2semidirect
6. #9726C2semidirect
7. #9635C2semidirect
8. #9588C2semidirect
9. #9588C2semidirect
10. #9213C2semidirect
11. #9213C2semidirect
12. #8672C2semidirect
13. #8296C2semidirect
14. #8245C2semidirect
15. #8241C2semidirect

Maximal subgroups

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

subgroupindex
#191602
#179812
#179172
#179122
#173292
#97262
#96352
#95882
#92132
#86722
#82962
#82452
#82412