Finite Group Databasebeta

← #212006

Subgroups — #212006

Order 512.

562 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#17975C211yesmaximal, characteristic, complemented
3#17917C211yesmaximal, characteristic, complemented
4#17912C211yesmaximal, characteristic, complemented
5#17330C211yesmaximal, characteristic, complemented
6#9724C211yesmaximal, characteristic, complemented
7#9634C211yesmaximal, characteristic, complemented
8#9586C211yesmaximal, characteristic, complemented
9#9586C211yesmaximal, characteristic, complemented
10#9213C211yesmaximal, characteristic, complemented
11#9213C211yesmaximal, characteristic, complemented
12#8673C211yesmaximal, characteristic, complemented
13#8297C211yesmaximal, characteristic, complemented
14#8245C211yesmaximal, characteristic, complemented
15#8243C211yesmaximal, 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#1583—21no—
36#1581—21no—
37#1581—21no—
38#1581—21no—
39#1581—21no—
40#1581—21no—
41#1580—21no—
42SD16⋊C8—21no—
43C8×SD16—21no—
44C8×D8—21no—
45#1574C2211yescharacteristic
46#1574C2211yescharacteristic
47#1567C2211yescharacteristic
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#1451—21no—
62#1451C2211yescharacteristic
63#1451C2211yescharacteristic
64#1450—21no—
65#1450C2211yescharacteristic
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#335—21no—
152#335—21no—
153#335C2×C411yescharacteristic
154#335C2×C411yescharacteristic
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#322D411yescharacteristic
189#322D411yescharacteristic
190#322D411yescharacteristic
191#322D411yescharacteristic
192C82—21no—
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—22no—
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—21no—
380C2×C8—22no—
381C2×C8—41no—
382C2×C8—41no—
383C2×C8—41no—
384C2×C8—42no—
385C2×C8—41no—
386C2×C8—41no—
387C2×C8—41no—
388C2×C8—42no—
389C2×C8—41no—
390C2×C8—41no—
391C2×C8—41no—
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—41no—
481C8—41no—
482C8—42no—
483C8—42no—
484C8—41no—
485C8—41no—
486C8—41no—
487C8—41no—
488C8—41no—
489C8—41no—
490C8—41no—
491C8—41no—
492C8—81no—
493C8—81no—
494C8—81no—
495C8#47212yes—
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
521C4C2×#47211yescharacteristic
522C4#306711yescharacteristic
523C2—41no—
524C2—81no—
525C2—161no—
526C2—161no—
527C2#2151911yescharacteristic
528C2#2155011yescharacteristic
529C2C4×#47211yescharacteristic
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. #17975C2semidirect
3. #17917C2semidirect
4. #17912C2semidirect
5. #17330C2semidirect
6. #9724C2semidirect
7. #9634C2semidirect
8. #9586C2semidirect
9. #9586C2semidirect
10. #9213C2semidirect
11. #9213C2semidirect
12. #8673C2semidirect
13. #8297C2semidirect
14. #8245C2semidirect
15. #8243C2semidirect

Maximal subgroups

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

subgroupindex
#191602
#179752
#179172
#179122
#173302
#97242
#96342
#95862
#92132
#86732
#82972
#82452
#82432