Finite Group Databasebeta

← #212010

Subgroups — #212010

Order 512.

574 conjugacy classes of subgroups.

The subgroup lattice has 535 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#19160C211yesmaximal, characteristic
2#17975C211yesmaximal, characteristic, complemented
3#17945C211yesmaximal, characteristic, complemented
4#17940C211yesmaximal, characteristic, complemented
5#17337C211yesmaximal, characteristic, complemented
6#9724C211yesmaximal, characteristic, complemented
7#9634C211yesmaximal, characteristic, complemented
8#9588C211yesmaximal, characteristic, complemented
9#9588C211yesmaximal, characteristic, complemented
10#9211C211yesmaximal, characteristic, complemented
11#9211C211yesmaximal, characteristic, complemented
12#8673C211yesmaximal, characteristic, complemented
13#8297C211yesmaximal, characteristic, complemented
14#8245C211yesmaximal, characteristic, complemented
15#8241C211yesmaximal, characteristic, complemented
16#3169C2211yescharacteristic
17#3106C2211yescharacteristic
18#3106C2211yescharacteristic
19#2965C2211yescharacteristic
20#2965C2211yescharacteristic
21#2931C2211yescharacteristic
22#2931C2211yescharacteristic
23C4×#131C2211yescharacteristic
24C4×#108C2211yescharacteristic
25#1698C411yescharacteristic
26#1698C411yescharacteristic
27#1697C411yescharacteristic
28#1697C411yescharacteristic
29#1696C411yescharacteristic
30#1695C411yescharacteristic
31#1688C411yescharacteristic
32#1687C411yescharacteristic
33#1594—21no—
34#1592—21no—
35#1590—21no—
36#1589—21no—
37#1586—21no—
38#1585—21no—
39#1582—21no—
40#1581—21no—
41#1581—21no—
42#1581—21no—
43#1581—21no—
44#1581—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#1455—21no—
60#1453—21no—
61#1452C2211yescharacteristic
62#1451—21no—
63#1451C2211yescharacteristic
64C8×M16—21no—
65C8×M16C2211yescharacteristic
66C8×M16C2211yescharacteristic
67C8×M16C2211yescharacteristic
68C2×#322C2211yescharacteristic
69#1299C2211yescharacteristic
70#1299C2211yescharacteristic
71#1299C2211yescharacteristic
72#1299C2211yescharacteristic
73#1278C2211yescharacteristic
74#1278C2211yescharacteristic
75C2×#131—21no—
76C2×#131—21no—
77C2×#131—21no—
78#532C2311yescharacteristic
79#532C2311yescharacteristic
80C4×#41C2311yescharacteristic
81#500C2×C411yescharacteristic
82#493C2×C411yescharacteristic
83#486C2×C411yescharacteristic
84#486C2×C411yescharacteristic
85#482—21no—
86#482—21no—
87#477C2×C411yescharacteristic
88#477C2×C411yescharacteristic
89#476C2×C411yescharacteristic
90#476C2×C411yescharacteristic
91#466—21no—
92#466—21no—
93#460C2×C411yescharacteristic
94#460C2×C411yescharacteristic
95#459C2×C411yescharacteristic
96#459C2×C411yescharacteristic
97#449—21no—
98#449—21no—
99#446C2×C411yescharacteristic
100#446C2×C411yescharacteristic
101#446D411yescharacteristic
102#446D411yescharacteristic
103C4×SD16—21no—
104C4×SD16—21no—
105C4×D8—21no—
106C4×D8—21no—
107C4×D8—21no—
108C4×D8—21no—
109#436—21no—
110#436—21no—
111#436—21no—
112#436—21no—
113#436C2×C411yescharacteristic
114#436C2×C411yescharacteristic
115#436D411yescharacteristic
116#436D411yescharacteristic
117#435—21no—
118#435—21no—
119#435—21no—
120#435—21no—
121C8×D4—21no—
122C8×D4—21no—
123C8×D4—21no—
124C8×D4—21no—
125#431—21no—
126#431—21no—
127#431C2311yescharacteristic
128#431C2311yescharacteristic
129#431C2311yescharacteristic
130#431C2311yescharacteristic
131C2×#108—21no—
132#423C2311yescharacteristic
133#423C2311yescharacteristic
134#408—21no—
135#408—21no—
136#405—21no—
137#405—22no—
138C4×M16—21no—
139C4×M16—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—42no—
205#131—42no—
206#131—82no—
207#131—82no—
208C2×M16—21no—
209C2×M16—22no—
210C2×M16—21no—
211C2×M16—21no—
212C2×M16—21no—
213C2×M16—21no—
214C2×M16—21no—
215C2×M16—21no—
216C2×M16—21no—
217C2×M16—21no—
218C2×M16—41no—
219C2×M16—41no—
220C22×C8—22no—
221C22×C8—21no—
222C22×C8—41no—
223C22×C8—41no—
224C22×C8#4111yescharacteristic
225C22×C8#4111yescharacteristic
226C22×C8#4111yescharacteristic
227C22×C8#4111yescharacteristic
228C4⋊Q8C22×C411yescharacteristic
229C4⋊Q8C22×C411yescharacteristic
230#127C22×C411yescharacteristic
231#127C22×C411yescharacteristic
232#124—21no—
233#124—21no—
234#121—21no—
235#121—21no—
236#121—21no—
237#121—21no—
238C4×Q8C22×C411yescharacteristic
239C4×Q8C2×D411yescharacteristic
240C4×D4—21no—
241C4×D4—21no—
242C4×D4—21no—
243C4×D4—21no—
244C4×D4—21no—
245C4×D4—21no—
246C4×D4C22×C411yescharacteristic
247C4×D4C2×D411yescharacteristic
248#117—21no—
249#117#4111yescharacteristic
250C2×C42—21no—
251C2×C42C2411yescharacteristic
252#108—42no—
253#107—21no—
254#107—21no—
255#107—21no—
256#107—21no—
257#106—21no—
258#106—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⋊C8C22×C411yescharacteristic
268C4⋊C8C22×C411yescharacteristic
269C4⋊C8C22×C411yescharacteristic
270C4⋊C8C22×C411yescharacteristic
271C4⋊C8C22×C411yescharacteristic
272C4⋊C8C22×C411yescharacteristic
273C4⋊C8C2×D411yescharacteristic
274C4⋊C8C2×D411yescharacteristic
275#103—21no—
276#103—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—
285#102—21no—
286#102—21no—
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—22no—
300#97—21no—
301#97—21no—
302#97—21no—
303#97—21no—
304#97—21no—
305#97—21no—
306#97#4112yes—
307C4×C8—22no—
308C4×C8—21no—
309C4×C8—21no—
310C4×C8—21no—
311C4×C8—21no—
312C4×C8—21no—
313C4×C8—21no—
314C4×C8—21no—
315C4×C8—21no—
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×C8C2×D411yescharacteristic
325C4×C8C2×D411yescharacteristic
326#41—41no—
327#41—41no—
328#41—81no—
329#41—81no—
330#41—81no—
331#41—81no—
332C2×Q8—21no—
333C2×Q8—21no—
334C2×Q8C4×D411yescharacteristic
335C2×D4—21no—
336C2×D4—21no—
337C2×D4—21no—
338C2×D4—21no—
339C2×D4—21no—
340C2×D4—21no—
341C2×D4—21no—
342C2×D4—41no—
343C2×D4—41no—
344C2×D4C4×D411yescharacteristic
345C22×C4—21no—
346C22×C4—21no—
347C22×C4—21no—
348C22×C4—41no—
349C22×C4—41no—
350C22×C4C2×#4111yescharacteristic
351C22×C4C2×#4111yescharacteristic
352C22×C4C2×#4111yescharacteristic
353SD16—81no—
354SD16—81no—
355D8—81no—
356D8—81no—
357D8—81no—
358D8—81no—
359M16—22no—
360M16—22no—
361M16—22no—
362M16—42no—
363M16—44no—
364M16—41no—
365M16—42no—
366M16—41no—
367M16—41no—
368M16—81no—
369M16—81no—
370M16—82no—
371M16—82no—
372M16—81no—
373M16—81no—
374C2×C8—22no—
375C2×C8—22no—
376C2×C8—21no—
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—41no—
387C2×C8—41no—
388C2×C8—41no—
389C2×C8—44no—
390C2×C8—41no—
391C2×C8—41no—
392C2×C8—41no—
393C2×C8—41no—
394C2×C8—41no—
395C2×C8—41no—
396C2×C8—82no—
397C2×C8—82no—
398C2×C8C4×D411yescharacteristic
399C2×C8C4×D411yescharacteristic
400C2×C8C4×D411yescharacteristic
401C2×C8C4×D411yescharacteristic
402C2×C8#12111yescharacteristic
403C2×C8#12111yescharacteristic
404C2×C8#12111yescharacteristic
405C2×C8#12111yescharacteristic
406C4⋊C4—21no—
407C4⋊C4—21no—
408C4⋊C4—21no—
409C4⋊C4—21no—
410C4⋊C4—21no—
411C4⋊C4—21no—
412C4⋊C4C4×D411yescharacteristic
413C4⋊C4C4×D411yescharacteristic
414C22⋊C4—21no—
415C22⋊C4—21no—
416C22⋊C4—21no—
417C22⋊C4—41no—
418C22⋊C4—41no—
419C22⋊C4—41no—
420C22⋊C4—41no—
421C42—21no—
422C42—21no—
423C42—21no—
424C42C23×C411yescharacteristic
425C42C22×D411yescharacteristic
426C42C22×D411yescharacteristic
427C42C2×#4111yescharacteristic
428C23—21no—
429C23—41no—
430C23—41no—
431C23#52911yescharacteristic
432Q8—21no—
433Q8—21no—
434Q8—81no—
435Q8—81no—
436D4—21no—
437D4—21no—
438D4—42no—
439D4—41no—
440D4—41no—
441D4—41no—
442D4—41no—
443D4—41no—
444D4—41no—
445D4—41no—
446D4—41no—
447D4—81no—
448D4—81no—
449D4—82no—
450D4—82no—
451C2×C4—21no—
452C2×C4—21no—
453C2×C4—21no—
454C2×C4—21no—
455C2×C4—21no—
456C2×C4—21no—
457C2×C4—21no—
458C2×C4—21no—
459C2×C4—21no—
460C2×C4—42no—
461C2×C4—41no—
462C2×C4—41no—
463C2×C4—41no—
464C2×C4—41no—
465C2×C4—41no—
466C2×C4—41no—
467C2×C4—81no—
468C2×C4—81no—
469C2×C4—81no—
470C2×C4—81no—
471C2×C4C2×C4×D411yescharacteristic
472C2×C4C2×C4×D411yescharacteristic
473C2×C4C4×#4111yescharacteristic
474C2×C4C2×#12111yescharacteristic
475C2×C4#52511yescharacteristic
476C2×C4#53211yescharacteristic
477C8—24no—
478C8—21no—
479C8—21no—
480C8—21no—
481C8—21no—
482C8—21no—
483C8—21no—
484C8—21no—
485C8—41no—
486C8—41no—
487C8—42no—
488C8—41no—
489C8—41no—
490C8—41no—
491C8—41no—
492C8—41no—
493C8—41no—
494C8—41no—
495C8—41no—
496C8—81no—
497C8—81no—
498C8—81no—
499C8#47212yes—
500C22—21no—
501C22—21no—
502C22—21no—
503C22—41no—
504C22—41no—
505C22—41no—
506C22—81no—
507C22—81no—
508C22—81no—
509C22—81no—
510C22—81no—
511C22—81no—
512C22C4×#12111yescharacteristic
513C4—21no—
514C4—21no—
515C4—21no—
516C4—41no—
517C4—41no—
518C4—41no—
519C4—81no—
520C4—81no—
521C4#295611yescharacteristic
522C4#295611yescharacteristic
523C4#295611yescharacteristic
524C4#295711yescharacteristic
525C4C2×#47211yescharacteristic
526C4#306711yescharacteristic
527C2—41no—
528C2—81no—
529C2—161no—
530C2—161no—
531C2#2151911yescharacteristic
532C2#2155111yescharacteristic
533C2C4×#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. #17945C2semidirect
4. #17940C2semidirect
5. #17337C2semidirect
6. #9724C2semidirect
7. #9634C2semidirect
8. #9588C2semidirect
9. #9588C2semidirect
10. #9211C2semidirect
11. #9211C2semidirect
12. #8673C2semidirect
13. #8297C2semidirect
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
#179752
#179452
#179402
#173372
#97242
#96342
#95882
#92112
#86732
#82972
#82452
#82412