Finite Group Databasebeta

← #229577

Subgroups — #229577

Order 512.

542 conjugacy classes of subgroups.

The subgroup lattice has 526 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#19160C211yesmaximal, characteristic
2#18084C211yesmaximal, characteristic, complemented
3#18025C211yesmaximal, characteristic, complemented
4#18022C211yesmaximal, characteristic, complemented
5#17841C211yesmaximal, characteristic, complemented
6#9849C211yesmaximal, characteristic, complemented
7#9633C211yesmaximal, characteristic, complemented
8#9588C211yesmaximal, characteristic, complemented
9#9588C211yesmaximal, characteristic, complemented
10#8970C211yesmaximal, characteristic, complemented
11#8969C211yesmaximal, characteristic, complemented
12#8935C211yesmaximal, characteristic, complemented
13#8934C211yesmaximal, characteristic, complemented
14#8269C211yesmaximal, characteristic, complemented
15#8196C211yesmaximal, characteristic, complemented
16#3169C2211yescharacteristic
17#3106C2211yescharacteristic
18#3106C2211yescharacteristic
19#2991C2211yescharacteristic
20#2990C2211yescharacteristic
21#2979C2211yescharacteristic
22#2979C2211yescharacteristic
23#2965C2211yescharacteristic
24#1962C2211yescharacteristic
25#1962C2211yescharacteristic
26C4×#104C2211yescharacteristic
27C4×#97C2211yescharacteristic
28#1698C411yescharacteristic, complemented
29#1698C411yescharacteristic, complemented
30#1697C411yescharacteristic, complemented
31#1697C411yescharacteristic, complemented
32#1696C411yescharacteristic, complemented
33#1695C411yescharacteristic, complemented
34#1688C411yescharacteristic, complemented
35#1687C411yescharacteristic, complemented
36#1594—21no—
37#1592—21no—
38#1589—21no—
39#1586—21no—
40#1586—21no—
41#1583—21no—
42#1582—21no—
43#1581—21no—
44#1581—21no—
45#1580—21no—
46SD16⋊C8—21no—
47C8×D8—21no—
48#1574C2211yescharacteristic
49#1574C2211yescharacteristic
50#1574C2211yescharacteristic
51#1574C2211yescharacteristic
52#1572—21no—
53#1570—21no—
54#1569—21no—
55#1566C2211yescharacteristic
56#1565—21no—
57#1564C2211yescharacteristic
58#1564C2211yescharacteristic
59#1543C2211yescharacteristic
60#1543C2211yescharacteristic
61#1492C2211yescharacteristic
62#1492C2211yescharacteristic
63#1484C2211yescharacteristic
64#1484C2211yescharacteristic
65#1476C2211yescharacteristic
66#1452C2211yescharacteristic
67#1452C2211yescharacteristic
68C8×M16C2211yescharacteristic
69#1299C2211yescharacteristic
70#1299C2211yescharacteristic
71#1278C2211yescharacteristic
72#1276C2211yescharacteristic
73#1276C2211yescharacteristic
74#1276C2211yescharacteristic
75#532C2311yescharacteristic
76#532C2311yescharacteristic
77C4×#41C2311yescharacteristic
78#500C2×C411yescharacteristic
79#493C2×C411yescharacteristic
80#486C2×C411yescharacteristic
81#486C2×C411yescharacteristic
82#482—21no—
83#482—21no—
84#477C2×C411yescharacteristic
85#477C2×C411yescharacteristic
86#476C2×C411yescharacteristic
87#476C2×C411yescharacteristic
88#466—21no—
89#466—21no—
90#460C2×C411yescharacteristic
91#460C2×C411yescharacteristic
92#459C2×C411yescharacteristic
93#459C2×C411yescharacteristic
94#449—21no—
95#449—21no—
96#446—21no—
97#446C2×C411yescharacteristic
98#446C2×C411yescharacteristic
99C8×Q8—21no—
100C4×SD16—21no—
101C4×SD16—21no—
102C4×D8—21no—
103C4×D8—21no—
104C4×D8—21no—
105C4×D8—21no—
106#436—21no—
107#436—21no—
108#436—21no—
109#436—21no—
110#436—21no—
111#436—41no—
112#436C2×C411yescharacteristic
113#436C2×C411yescharacteristic
114#435—21no—
115#435—21no—
116#435—41no—
117C8×D4—21no—
118C8×D4—21no—
119C8×D4—21no—
120C8×D4—21no—
121C8×D4—21no—
122#432—21no—
123#431—21no—
124#431—21no—
125#431C2311yescharacteristic
126#431C2311yescharacteristic
127#431C2311yescharacteristic
128#431C2311yescharacteristic
129#431C2311yescharacteristic
130#431C2311yescharacteristic
131#423—21no—
132#423C2311yescharacteristic
133#423C2311yescharacteristic
134C2×C4⋊C8—21no—
135C2×#104—21no—
136C2×#104—41no—
137C2×#104—41no—
138#408—41no—
139#408—41no—
140C4×M16—21no—
141C4×M16C2311yescharacteristic
142C2×#97—21no—
143C2×#97—21no—
144C2×C4×C8C2311yescharacteristic
145C2×C4×C8C2311yescharacteristic
146C43C2311yescharacteristic
147#339—21no—
148#335—21no—
149#335—21no—
150#335—21no—
151#335—21no—
152#335C2×C411yescharacteristic
153#335C2×C411yescharacteristic
154#335D411yescharacteristic
155#335D411yescharacteristic
156#334—21no—
157#334—21no—
158#334—21no—
159#334—21no—
160#334—21no—
161#334—21no—
162#334C2×C411yescharacteristic
163#334C2×C411yescharacteristic
164#334D411yescharacteristic
165#334D411yescharacteristic
166#333C2×C411yescharacteristic
167#333C2×C411yescharacteristic
168#333C2×C411yescharacteristic
169#333C2×C411yescharacteristic
170#331C2×C411yescharacteristic
171#331C2×C411yescharacteristic
172#331C2×C411yescharacteristic
173#331C2×C411yescharacteristic
174Q8⋊C8D411yescharacteristic
175Q8⋊C8D411yescharacteristic
176D4⋊C8—21no—
177D4⋊C8—21no—
178D4⋊C8—21no—
179D4⋊C8—21no—
180D4⋊C8—21no—
181D4⋊C8—21no—
182D4⋊C8—21no—
183D4⋊C8—21no—
184D4⋊C8D411yescharacteristic
185D4⋊C8D411yescharacteristic
186#322—21no—
187#322—21no—
188#322—21no—
189#322—21no—
190#322—21no—
191C82—21no—
192C2×#41—21no—
193C2×#41—21no—
194C2×#41#4111yescharacteristic
195C2×SD16—21no—
196C2×SD16—21no—
197C2×D8—21no—
198C2×D8—21no—
199C2×D8—21no—
200C2×D8—21no—
201C2×M16—21no—
202C2×M16—21no—
203C2×M16—21no—
204C2×M16—21no—
205C2×M16—21no—
206C2×M16—41no—
207C2×M16—41no—
208C2×M16—41no—
209C22×C8—21no—
210C22×C8—41no—
211C22×C8—41no—
212C22×C8—41no—
213C22×C8#4111yescharacteristic
214C22×C8#4111yescharacteristic
215C4⋊Q8C22×C411yescharacteristic
216C4⋊Q8C22×C411yescharacteristic
217#127C22×C411yescharacteristic
218#127C22×C411yescharacteristic
219#124—21no—
220#124—21no—
221#121—21no—
222#121—21no—
223#121—21no—
224#121—21no—
225C4×Q8C22×C411yescharacteristic
226C4×Q8C2×D411yescharacteristic
227C4×D4—21no—
228C4×D4—21no—
229C4×D4—21no—
230C4×D4—21no—
231C4×D4—21no—
232C4×D4—21no—
233C4×D4C22×C411yescharacteristic
234C4×D4C2×D411yescharacteristic
235#117—21no—
236#117#4111yescharacteristic
237C2×C42—21no—
238C2×C42—21no—
239C2×C42—21no—
240C2×C42—21no—
241C2×C42C2411yescharacteristic
242#107—21no—
243#107—21no—
244#107—21no—
245#107—21no—
246#106—21no—
247#106—21no—
248C4⋊C8—21no—
249C4⋊C8—21no—
250C4⋊C8—21no—
251C4⋊C8—21no—
252C4⋊C8—21no—
253C4⋊C8—21no—
254C4⋊C8—41no—
255C4⋊C8—41no—
256C4⋊C8C22×C411yescharacteristic
257C4⋊C8C22×C411yescharacteristic
258C4⋊C8C22×C411yescharacteristic
259C4⋊C8C22×C411yescharacteristic
260C4⋊C8C22×C411yescharacteristic
261C4⋊C8C22×C411yescharacteristic
262C4⋊C8C2×D411yescharacteristic
263C4⋊C8C2×D411yescharacteristic
264C4⋊C8C2×D411yescharacteristic
265C4⋊C8C2×D411yescharacteristic
266C4⋊C8C2×D411yescharacteristic
267C4⋊C8C2×D411yescharacteristic
268#104—42no—
269#104—82no—
270#104—82no—
271#103—21no—
272#103—21no—
273#102—21no—
274#102—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—
283C22⋊C8—21no—
284C22⋊C8—21no—
285C22⋊C8—21no—
286C22⋊C8—21no—
287C22⋊C8—21no—
288C22⋊C8—21no—
289C22⋊C8—21no—
290C22⋊C8—21no—
291C22⋊C8—41no—
292C22⋊C8—41no—
293C22⋊C8—41no—
294C22⋊C8—41no—
295C22⋊C8—41no—
296C22⋊C8—41no—
297#97—21no—
298#97—21no—
299#97—21no—
300#97—41no—
301#97—42no—
302#97—42no—
303C4×C8—21no—
304C4×C8—21no—
305C4×C8—21no—
306C4×C8—21no—
307C4×C8—21no—
308C4×C8—21no—
309C4×C8—21no—
310C4×C8—21no—
311C4×C8—41no—
312C4×C8C22×C411yescharacteristic
313C4×C8C22×C411yescharacteristic
314C4×C8C2×D411yescharacteristic
315C4×C8C2×D411yescharacteristic
316C4×C8C2×D411yescharacteristic
317C4×C8C2×D411yescharacteristic
318#41—41no—
319#41—41no—
320#41—81no—
321#41—81no—
322#41—81no—
323#41—81no—
324C2×Q8—21no—
325C2×Q8—21no—
326C2×Q8C4×D411yescharacteristic
327C2×D4—21no—
328C2×D4—21no—
329C2×D4—21no—
330C2×D4—21no—
331C2×D4—21no—
332C2×D4—21no—
333C2×D4—21no—
334C2×D4—41no—
335C2×D4—41no—
336C2×D4C4×D411yescharacteristic
337C22×C4—21no—
338C22×C4—21no—
339C22×C4—21no—
340C22×C4—41no—
341C22×C4—41no—
342C22×C4—41no—
343C22×C4C2×#4111yescharacteristic
344C22×C4C2×#4111yescharacteristic
345C22×C4H(5,𝔽2)11yescharacteristic
346SD16—42no—
347SD16—81no—
348D8—42no—
349D8—42no—
350D8—81no—
351D8—81no—
352M16—22no—
353M16—41no—
354M16—42no—
355M16—81no—
356M16—81no—
357M16—81no—
358M16—161no—
359M16—161no—
360M16—161no—
361C2×C8—22no—
362C2×C8—21no—
363C2×C8—21no—
364C2×C8—21no—
365C2×C8—21no—
366C2×C8—21no—
367C2×C8—21no—
368C2×C8—21no—
369C2×C8—21no—
370C2×C8—21no—
371C2×C8—21no—
372C2×C8—21no—
373C2×C8—21no—
374C2×C8—21no—
375C2×C8—21no—
376C2×C8—41no—
377C2×C8—41no—
378C2×C8—41no—
379C2×C8—41no—
380C2×C8—41no—
381C2×C8—41no—
382C2×C8—161no—
383C2×C8—161no—
384C2×C8—161no—
385C2×C8C4×D411yescharacteristic
386C2×C8C4×D411yescharacteristic
387C2×C8C4×D411yescharacteristic
388C2×C8C4×D411yescharacteristic
389C4⋊C4—21no—
390C4⋊C4—21no—
391C4⋊C4—21no—
392C4⋊C4—21no—
393C4⋊C4—21no—
394C4⋊C4—21no—
395C4⋊C4C4×D411yescharacteristic
396C4⋊C4C4×D411yescharacteristic
397C22⋊C4—21no—
398C22⋊C4—21no—
399C22⋊C4—21no—
400C22⋊C4—41no—
401C22⋊C4—41no—
402C22⋊C4—41no—
403C22⋊C4—41no—
404C42—21no—
405C42—21no—
406C42—21no—
407C42—22no—
408C42—41no—
409C42—41no—
410C42—41no—
411C42—41no—
412C42—41no—
413C42C23×C411yescharacteristic
414C42C22×D411yescharacteristic
415C42C22×D411yescharacteristic
416C422−1+411yescharacteristic
417C23—21no—
418C23—41no—
419C23—41no—
420C23#55311yescharacteristic
421Q8—21no—
422Q8—21no—
423Q8—81no—
424Q8—81no—
425D4—21no—
426D4—21no—
427D4—41no—
428D4—41no—
429D4—41no—
430D4—41no—
431D4—41no—
432D4—41no—
433D4—41no—
434D4—41no—
435D4—81no—
436D4—81no—
437D4—81no—
438D4—161no—
439D4—161no—
440C2×C4—21no—
441C2×C4—21no—
442C2×C4—21no—
443C2×C4—21no—
444C2×C4—21no—
445C2×C4—21no—
446C2×C4—21no—
447C2×C4—21no—
448C2×C4—21no—
449C2×C4—41no—
450C2×C4—41no—
451C2×C4—41no—
452C2×C4—41no—
453C2×C4—42no—
454C2×C4—41no—
455C2×C4—41no—
456C2×C4—81no—
457C2×C4—81no—
458C2×C4—81no—
459C2×C4—81no—
460C2×C4—81no—
461C2×C4—81no—
462C2×C4—81no—
463C2×C4C2×C4×D411yescharacteristic
464C2×C4C2×C4×D411yescharacteristic
465C2×C4#52011yescharacteristic
466C2×C4D4211yescharacteristic
467C2×C4#54711yescharacteristic
468C2×C4#54811yescharacteristic
469C8—21no—
470C8—21no—
471C8—21no—
472C8—41no—
473C8—41no—
474C8—42no—
475C8—41no—
476C8—41no—
477C8—41no—
478C8—41no—
479C8—41no—
480C8—41no—
481C8—41no—
482C8—41no—
483C8—41no—
484C8—41no—
485C8—41no—
486C8—41no—
487C8—81no—
488C8—81no—
489C8#44412yes—
490C22—21no—
491C22—21no—
492C22—21no—
493C22—41no—
494C22—41no—
495C22—41no—
496C22—81no—
497C22—81no—
498C22—81no—
499C22—81no—
500C22—81no—
501C22—81no—
502C22#236111yescharacteristic
503C4—21no—
504C4—21no—
505C4—21no—
506C4—41no—
507C4—41no—
508C4—41no—
509C4—82no—
510C4—81no—
511C4—81no—
512C4C2×#44411yescharacteristic
513C4#295511yescharacteristic
514C4#295611yescharacteristic
515C4#295611yescharacteristic
516C4#328811yescharacteristic
517C4#328811yescharacteristic
518C2—41no—
519C2—81no—
520C2—161no—
521C2—161no—
522C2#2301611yescharacteristic
523C2#2302011yescharacteristic
524C2#2304811yescharacteristic
subgroupG/H
centerC2×C4D42
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
socleC22#2361
prime radical#19160C2
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. #18084C2semidirect
3. #18025C2semidirect
4. #18022C2semidirect
5. #17841C2semidirect
6. #9849C2semidirect
7. #9633C2semidirect
8. #9588C2semidirect
9. #9588C2semidirect
10. #8970C2semidirect
11. #8969C2semidirect
12. #8935C2semidirect
13. #8934C2semidirect
14. #8269C2semidirect
15. #8196C2semidirect
28. #1698C4semidirect
29. #1698C4semidirect
30. #1697C4semidirect
31. #1697C4semidirect
32. #1696C4semidirect
33. #1695C4semidirect
34. #1688C4semidirect
35. #1687C4semidirect

Maximal subgroups

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

subgroupindex
#191602
#180842
#180252
#180222
#178412
#98492
#96332
#95882
#89702
#89692
#89352
#89342
#82692
#81962