Finite Group Databasebeta

← #230405

Subgroups — #230405

Order 512.

534 conjugacy classes of subgroups.

The subgroup lattice has 524 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#19160C211yesmaximal, characteristic
2#18091C211yesmaximal, characteristic, complemented
3#18022C211yesmaximal, characteristic, complemented
4#18018C211yesmaximal, characteristic, complemented
5#17841C211yesmaximal, characteristic, complemented
6#9852C211yesmaximal, characteristic, complemented
7#9635C211yesmaximal, characteristic, complemented
8#9588C211yesmaximal, characteristic, complemented
9#9586C211yesmaximal, characteristic, complemented
10#8970C211yesmaximal, characteristic, complemented
11#8969C211yesmaximal, characteristic, complemented
12#8934C211yesmaximal, characteristic, complemented
13#8934C211yesmaximal, characteristic, complemented
14#8270C211yesmaximal, characteristic, complemented
15#8195C211yesmaximal, characteristic, complemented
16#3169C2211yescharacteristic
17#3106C2211yescharacteristic
18#3106C2211yescharacteristic
19#2991C2211yescharacteristic
20#2990C2211yescharacteristic
21#2979C2211yescharacteristic
22#2979C2211yescharacteristic
23#2965C2211yescharacteristic
24#1962C2211yescharacteristic
25#1962C2211yescharacteristic
26C4×#104C2211yescharacteristic
27#1744C2211yescharacteristic
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#1588—21no—
40#1586—21no—
41#1582—21no—
42#1581—21no—
43#1581—21no—
44#1581—21no—
45#1580—21no—
46#1580—21no—
47C8×SD16—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#1451C2211yescharacteristic
68#1450C2211yescharacteristic
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×C4×C8C2311yescharacteristic
143C2×C4×C8C2311yescharacteristic
144C43C2311yescharacteristic
145#339—21no—
146#336—21no—
147#336—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—
301C4×C8—21no—
302C4×C8—21no—
303C4×C8—21no—
304C4×C8—21no—
305C4×C8—21no—
306C4×C8—21no—
307C4×C8—21no—
308C4×C8—21no—
309C4×C8—41no—
310C4×C8C22×C411yescharacteristic
311C4×C8C22×C411yescharacteristic
312C4×C8C2×D411yescharacteristic
313C4×C8C2×D411yescharacteristic
314C4×C8C2×D411yescharacteristic
315C4×C8C2×D411yescharacteristic
316#41—41no—
317#41—41no—
318#41—81no—
319#41—81no—
320#41—81no—
321#41—81no—
322C2×Q8—21no—
323C2×Q8—21no—
324C2×Q8C4×D411yescharacteristic
325C2×D4—21no—
326C2×D4—21no—
327C2×D4—21no—
328C2×D4—21no—
329C2×D4—21no—
330C2×D4—21no—
331C2×D4—21no—
332C2×D4—41no—
333C2×D4—41no—
334C2×D4C4×D411yescharacteristic
335C22×C4—21no—
336C22×C4—21no—
337C22×C4—21no—
338C22×C4—41no—
339C22×C4—41no—
340C22×C4—41no—
341C22×C4C2×#4111yescharacteristic
342C22×C4C2×#4111yescharacteristic
343C22×C4H(5,𝔽2)11yescharacteristic
344SD16—42no—
345SD16—81no—
346D8—42no—
347D8—42no—
348D8—81no—
349D8—81no—
350M16—42no—
351M16—41no—
352M16—41no—
353M16—81no—
354M16—81no—
355M16—81no—
356M16—161no—
357M16—161no—
358M16—161no—
359C2×C8—21no—
360C2×C8—21no—
361C2×C8—21no—
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—41no—
374C2×C8—41no—
375C2×C8—41no—
376C2×C8—41no—
377C2×C8—41no—
378C2×C8—41no—
379C2×C8—41no—
380C2×C8—161no—
381C2×C8—161no—
382C2×C8—161no—
383C2×C8C4×D411yescharacteristic
384C2×C8C4×D411yescharacteristic
385C2×C8C4×D411yescharacteristic
386C2×C8C4×D411yescharacteristic
387C4⋊C4—21no—
388C4⋊C4—21no—
389C4⋊C4—21no—
390C4⋊C4—21no—
391C4⋊C4—21no—
392C4⋊C4—21no—
393C4⋊C4C4×D411yescharacteristic
394C4⋊C4C4×D411yescharacteristic
395C22⋊C4—21no—
396C22⋊C4—21no—
397C22⋊C4—21no—
398C22⋊C4—41no—
399C22⋊C4—41no—
400C22⋊C4—41no—
401C22⋊C4—41no—
402C42—21no—
403C42—21no—
404C42—21no—
405C42—22no—
406C42—41no—
407C42—41no—
408C42—41no—
409C42—41no—
410C42—41no—
411C42C23×C411yescharacteristic
412C42C22×D411yescharacteristic
413C42C22×D411yescharacteristic
414C422−1+411yescharacteristic
415C23—21no—
416C23—41no—
417C23—41no—
418C23#55311yescharacteristic
419Q8—21no—
420Q8—21no—
421Q8—81no—
422Q8—81no—
423D4—21no—
424D4—21no—
425D4—41no—
426D4—41no—
427D4—41no—
428D4—41no—
429D4—41no—
430D4—41no—
431D4—41no—
432D4—41no—
433D4—81no—
434D4—81no—
435D4—81no—
436D4—161no—
437D4—161no—
438C2×C4—21no—
439C2×C4—21no—
440C2×C4—21no—
441C2×C4—21no—
442C2×C4—21no—
443C2×C4—21no—
444C2×C4—21no—
445C2×C4—21no—
446C2×C4—21no—
447C2×C4—41no—
448C2×C4—41no—
449C2×C4—41no—
450C2×C4—41no—
451C2×C4—42no—
452C2×C4—41no—
453C2×C4—41no—
454C2×C4—81no—
455C2×C4—81no—
456C2×C4—81no—
457C2×C4—81no—
458C2×C4—81no—
459C2×C4—81no—
460C2×C4—81no—
461C2×C4C2×C4×D411yescharacteristic
462C2×C4C2×C4×D411yescharacteristic
463C2×C4#52011yescharacteristic
464C2×C4D4211yescharacteristic
465C2×C4#54711yescharacteristic
466C2×C4#54811yescharacteristic
467C8—21no—
468C8—21no—
469C8—21no—
470C8—21no—
471C8—41no—
472C8—41no—
473C8—41no—
474C8—41no—
475C8—41no—
476C8—41no—
477C8—41no—
478C8—41no—
479C8—41no—
480C8—41no—
481C8—41no—
482C8—41no—
483C8—41no—
484C8—41no—
485C8—81no—
486C8—81no—
487C8—81no—
488C22—21no—
489C22—21no—
490C22—21no—
491C22—41no—
492C22—41no—
493C22—41no—
494C22—81no—
495C22—81no—
496C22—81no—
497C22—81no—
498C22—81no—
499C22—81no—
500C22#236111yescharacteristic
501C4—21no—
502C4—21no—
503C4—21no—
504C4—41no—
505C4—41no—
506C4—41no—
507C4—82no—
508C4—81no—
509C4—81no—
510C4#295511yescharacteristic
511C4#295511yescharacteristic
512C4#295611yescharacteristic
513C4#295711yescharacteristic
514C4#328811yescharacteristic
515C4#328811yescharacteristic
516C2—41no—
517C2—81no—
518C2—161no—
519C2—161no—
520C2#2301711yescharacteristic
521C2#2302111yescharacteristic
522C2#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. #18091C2semidirect
3. #18022C2semidirect
4. #18018C2semidirect
5. #17841C2semidirect
6. #9852C2semidirect
7. #9635C2semidirect
8. #9588C2semidirect
9. #9586C2semidirect
10. #8970C2semidirect
11. #8969C2semidirect
12. #8934C2semidirect
13. #8934C2semidirect
14. #8270C2semidirect
15. #8195C2semidirect
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
#180912
#180222
#180182
#178412
#98522
#96352
#95882
#95862
#89702
#89692
#89342
#82702
#81952