Finite Group Databasebeta

← #212000

Subgroups — #212000

Order 512.

570 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#17981C211yesmaximal, characteristic, complemented
3#17945C211yesmaximal, characteristic, complemented
4#17940C211yesmaximal, characteristic, complemented
5#17336C211yesmaximal, characteristic, complemented
6#9726C211yesmaximal, characteristic, complemented
7#9635C211yesmaximal, characteristic, complemented
8#9586C211yesmaximal, characteristic, complemented
9#9586C211yesmaximal, characteristic, complemented
10#9211C211yesmaximal, characteristic, complemented
11#9211C211yesmaximal, characteristic, complemented
12#8672C211yesmaximal, characteristic, complemented
13#8296C211yesmaximal, characteristic, complemented
14#8245C211yesmaximal, characteristic, complemented
15#8243C211yesmaximal, 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#1586—21no—
39#1586—21no—
40#1586—21no—
41#1586—21no—
42#1585—21no—
43#1582—21no—
44#1581—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#1455—21no—
60#1453—21no—
61#1451—21no—
62#1451—21no—
63#1451C2211yescharacteristic
64#1451C2211yescharacteristic
65#1451C2211yescharacteristic
66#1450C2211yescharacteristic
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#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—42no—
205#131—42no—
206#131—82no—
207#131—82no—
208C2×M16—22no—
209C2×M16—21no—
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—21no—
221C22×C8—22no—
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—44no—
361M16—41no—
362M16—41no—
363M16—41no—
364M16—41no—
365M16—41no—
366M16—81no—
367M16—81no—
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—22no—
387C2×C8—41no—
388C2×C8—41no—
389C2×C8—41no—
390C2×C8—41no—
391C2×C8—44no—
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—22no—
478C8—22no—
479C8—21no—
480C8—21no—
481C8—21no—
482C8—21no—
483C8—21no—
484C8—21no—
485C8—21no—
486C8—21no—
487C8—42no—
488C8—42no—
489C8—41no—
490C8—41no—
491C8—41no—
492C8—41no—
493C8—41no—
494C8—41no—
495C8—41no—
496C8—41no—
497C8—81no—
498C8—81no—
499C8—81no—
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
525C4#306811yescharacteristic
526C4#306811yescharacteristic
527C2—41no—
528C2—81no—
529C2—161no—
530C2—161no—
531C2#2151911yescharacteristic
532C2#2154911yescharacteristic
533C2#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. #17945C2semidirect
4. #17940C2semidirect
5. #17336C2semidirect
6. #9726C2semidirect
7. #9635C2semidirect
8. #9586C2semidirect
9. #9586C2semidirect
10. #9211C2semidirect
11. #9211C2semidirect
12. #8672C2semidirect
13. #8296C2semidirect
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
#179812
#179452
#179402
#173362
#97262
#96352
#95862
#92112
#86722
#82962
#82452
#82432