Finite Group Databasebeta

← #203013

Subgroups — #203013

Order 512.

835 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
1C2×#1962C211yesmaximal, characteristic
2C4×#432C211yesmaximal, characteristic, complemented
3#17822C211yesmaximal, characteristic, complemented
4#17731C211yesmaximal, characteristic
5#17495C211yesmaximal, characteristic
6#9582C212yesmaximal, complemented
7#9581C212yesmaximal, complemented
8#9578C212yesmaximal, complemented
9#9577C212yesmaximal, complemented
10#8077C211yesmaximal, characteristic, complemented
11#7867C211yesmaximal, characteristic, complemented
12C2×#532C2211yescharacteristic
13#2920C2211yescharacteristic
14#2920C2211yescharacteristic
15C2×#432C2211yescharacteristic
16C2×#432C2211yescharacteristic
17C2×#423C2211yescharacteristic
18C22×#104—21no—
19C22×#104—21no—
20C2×C43C2211yescharacteristic
21#1962—21no—
22#1962—21no—
23#1962C2212yes—
24#1962C2212yes—
25#1767C2211yescharacteristic, complemented
26#1767C2211yescharacteristic, complemented
27C4×C4⋊C8C2211yescharacteristic, complemented
28C4×C4⋊C8C2211yescharacteristic, complemented
29#1749—21no—
30C4×C22⋊C8—21no—
31#1745—21no—
32C4×#97—21no—
33#1573—22no—
34#1571C2212yescomplemented
35#1570C2212yescomplemented
36#1569—22no—
37#1557C414yescomplemented
38#1556C412yescomplemented
39#1555C412yescomplemented
40#1540C2212yes—
41#1540C2212yes—
42#1496—22no—
43#1490—22no—
44#1480C2212yes—
45#1480C2212yes—
46#1301C2212yescomplemented
47#1301C2212yescomplemented
48#1276C2212yescomplemented
49#1276C2212yescomplemented
50C22×#41—21no—
51C22×M16—21no—
52#532—21no—
53#532—21no—
54#532—21no—
55#532—21no—
56#532—21no—
57#532—21no—
58#532C2×C412yes—
59#532C2312yes—
60C2×C4⋊Q8C2×C411yescharacteristic
61C2×#127C2×C411yescharacteristic
62C2×#124—21no—
63C2×#121—21no—
64C2×#121—21no—
65C2×C4×D4—21no—
66C2×C4×D4—21no—
67C22×C42—21no—
68C22×C42—21no—
69C22×C42D411yescharacteristic
70C22×C42D411yescharacteristic
71C22×C42C2311yescharacteristic
72#432—21no—
73#432—21no—
74#432—21no—
75#432—21no—
76#432—21no—
77#432—21no—
78#432—21no—
79#432C2×C412yes—
80#432C2×C412yes—
81#432C2×C412yes—
82#432D412yes—
83#432D412yes—
84#431—21no—
85#431—21no—
86#431C2312yes—
87#431C2312yes—
88#423—21no—
89#423—21no—
90#423—21no—
91#423—21no—
92#423C2×C412yes—
93#423C2312yes—
94C2×C4⋊C8—21no—
95C2×C4⋊C8—21no—
96C2×C4⋊C8C2311yescharacteristic
97C2×C4⋊C8C2311yescharacteristic
98C2×C4⋊C8C2311yescharacteristic
99C2×C4⋊C8C2311yescharacteristic
100C2×#104—22no—
101C2×#104—22no—
102C2×#104—22no—
103C2×#104—22no—
104C2×#104—44no—
105C2×#104—44no—
106#407—21no—
107#407—21no—
108C2×C22⋊C8—21no—
109C2×C22⋊C8—21no—
110C4×M16—21no—
111C4×M16—21no—
112C2×#97—21no—
113C2×#97—21no—
114C2×#97—21no—
115C2×#97—21no—
116C2×C4×C8—21no—
117C2×C4×C8—21no—
118C43—21no—
119C43—21no—
120C43D411yescharacteristic
121C43D411yescharacteristic
122C43C2311yescharacteristic
123C43C2311yescharacteristic
124#336—21no—
125#336—21no—
126#336—21no—
127#336—21no—
128#335—24no—
129#334—24no—
130#332C2×C414yescomplemented
131#332C2×C414yescomplemented
132#331C2×C414yescomplemented
133#331C2×C414yescomplemented
134D4⋊C8—24no—
135D4⋊C8—24no—
136C2×#41—22no—
137C2×#41—22no—
138C2×#41—42no—
139C2×#41—42no—
140C2×#41—42no—
141C2×#41—42no—
142C2×#41—41no—
143C2×#41—41no—
144C22×Q8—21no—
145C22×D4—21no—
146C22×D4—21no—
147C22×D4—21no—
148C22×D4—41no—
149C23×C4—21no—
150C23×C4—21no—
151C23×C4—41no—
152C23×C4C2×D411yescharacteristic
153C23×C4#4111yescharacteristic
154C23×C4#4111yescharacteristic
155C2×M16—22no—
156C2×M16—22no—
157C2×M16—41no—
158C2×M16—41no—
159C2×M16—42no—
160C2×M16—42no—
161C22×C8—21no—
162C22×C8—21no—
163C22×C8—21no—
164C22×C8—21no—
165C22×C8—41no—
166C4⋊Q8—21no—
167C4⋊Q8—21no—
168C4⋊Q8—21no—
169C4⋊Q8C22×C412yes—
170#127—21no—
171#127—21no—
172#127—21no—
173#127C22×C412yes—
174#124—22no—
175#124—22no—
176#124—41no—
177#124—41no—
178#121—22no—
179#121—22no—
180#121—22no—
181#121—22no—
182#121—41no—
183#121—41no—
184#121—41no—
185#121—41no—
186C4×D4—22no—
187C4×D4—22no—
188C4×D4—22no—
189C4×D4—22no—
190C4×D4—41no—
191C4×D4—41no—
192C4×D4—41no—
193C4×D4—41no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—21no—
196C2×C22⋊C4—41no—
197C2×C22⋊C4—41no—
198C2×C42—21no—
199C2×C42—22no—
200C2×C42—22no—
201C2×C42—21no—
202C2×C42—21no—
203C2×C42—21no—
204C2×C42—21no—
205C2×C42—21no—
206C2×C42—21no—
207C2×C42—21no—
208C2×C42—21no—
209C2×C42—21no—
210C2×C42—41no—
211C2×C42—41no—
212C2×C42—44no—
213C2×C42—44no—
214C2×C42C22×C411yescharacteristic
215C2×C42C22×C411yescharacteristic
216C2×C42C2×D411yescharacteristic
217C2×C42C2×D411yescharacteristic
218C2×C42C2×D411yescharacteristic
219C2×C42C2×D411yescharacteristic
220C2×C42C2×D411yescharacteristic
221C2×C42C2×D411yescharacteristic
222C2×C42C2×D411yescharacteristic
223C2×C42#4111yescharacteristic
224C2×C42#4111yescharacteristic
225C2×C42#4111yescharacteristic
226C2×C42#4111yescharacteristic
227C2×C42#4111yescharacteristic
228C2×C42#4111yescharacteristic
229C2×C42C2411yescharacteristic
230C4⋊C8—22no—
231C4⋊C8—22no—
232C4⋊C8—21no—
233C4⋊C8—21no—
234C4⋊C8—21no—
235C4⋊C8—21no—
236C4⋊C8—22no—
237C4⋊C8—22no—
238C4⋊C8—41no—
239C4⋊C8—41no—
240C4⋊C8—41no—
241C4⋊C8C22×C412yes—
242C4⋊C8C22×C412yes—
243C4⋊C8C22×C414yes—
244C4⋊C8C2×D412yes—
245C4⋊C8C2×D412yes—
246#104—44no—
247#104—44no—
248#104—44no—
249#104—44no—
250C22⋊C8—22no—
251C22⋊C8—22no—
252C22⋊C8—22no—
253C22⋊C8—22no—
254C22⋊C8—22no—
255C22⋊C8—22no—
256C22⋊C8—22no—
257C22⋊C8—41no—
258#97—22no—
259#97—22no—
260#97—22no—
261#97—22no—
262#97—22no—
263#97—42no—
264#97—41no—
265#97—42no—
266C4×C8—22no—
267C4×C8—22no—
268C4×C8—42no—
269C4×C8—42no—
270C24—41no—
271C24#12311yescharacteristic
272#41—44no—
273#41—44no—
274#41—42no—
275#41—42no—
276#41—42no—
277#41—42no—
278#41—82no—
279#41—82no—
280#41—82no—
281#41—82no—
282C2×Q8—22no—
283C2×Q8—42no—
284C2×Q8—42no—
285C2×Q8—41no—
286C2×D4—22no—
287C2×D4—22no—
288C2×D4—22no—
289C2×D4—42no—
290C2×D4—42no—
291C2×D4—42no—
292C2×D4—42no—
293C2×D4—41no—
294C2×D4—41no—
295C2×D4—41no—
296C2×D4—82no—
297C2×D4—82no—
298C2×D4—82no—
299C2×D4—82no—
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—24no—
306C22×C4—21no—
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—42no—
310C22×C4—42no—
311C22×C4—42no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4—42no—
315C22×C4—42no—
316C22×C4—41no—
317C22×C4—42no—
318C22×C4—42no—
319C22×C4—82no—
320C22×C4—82no—
321C22×C4#11711yescharacteristic
322C22×C4#11711yescharacteristic
323C22×C4C4×D411yescharacteristic
324C22×C4C4×D411yescharacteristic
325C22×C4C4×D411yescharacteristic
326C22×C4C4×D411yescharacteristic
327C22×C4#12111yescharacteristic
328C22×C4#12111yescharacteristic
329C22×C4#12311yescharacteristic
330C22×C4#12311yescharacteristic
331C22×C4#12311yescharacteristic
332C22×C4#12411yescharacteristic
333C22×C4#12411yescharacteristic
334C22×C4#12611yescharacteristic
335C22×C4#12611yescharacteristic
336C22×C4C22×D411yescharacteristic
337C22×C4C2×#4111yescharacteristic
338C22×C4C2×#4111yescharacteristic
339M16—42no—
340M16—42no—
341M16—82no—
342M16—82no—
343M16—82no—
344C2×C8—24no—
345C2×C8—22no—
346C2×C8—22no—
347C2×C8—22no—
348C2×C8—22no—
349C2×C8—41no—
350C2×C8—41no—
351C2×C8—41no—
352C2×C8—41no—
353C2×C8—41no—
354C2×C8—41no—
355C2×C8—41no—
356C2×C8—41no—
357C2×C8—42no—
358C2×C8—42no—
359C2×C8—82no—
360C4⋊C4—22no—
361C4⋊C4—22no—
362C4⋊C4—41no—
363C4⋊C4—41no—
364C22⋊C4—42no—
365C22⋊C4—42no—
366C22⋊C4—42no—
367C22⋊C4—42no—
368C42—24no—
369C42—21no—
370C42—21no—
371C42—21no—
372C42—21no—
373C42—21no—
374C42—21no—
375C42—42no—
376C42—42no—
377C42—42no—
378C42—42no—
379C42—42no—
380C42—42no—
381C42—42no—
382C42—42no—
383C42—42no—
384C42#11711yescharacteristic
385C42#11711yescharacteristic
386C42C4×D411yescharacteristic
387C42C4×D411yescharacteristic
388C42C4×D411yescharacteristic
389C42C4×D411yescharacteristic
390C42#12111yescharacteristic
391C42#12111yescharacteristic
392C42#12411yescharacteristic
393C42#12411yescharacteristic
394C42#12611yescharacteristic
395C42#12611yescharacteristic
396C42AGL(1,ℤ/8)12yes—
397C42AGL(1,ℤ/8)12yes—
398C42C23×C411yescharacteristic
399C42C22×D411yescharacteristic
400C42C2×#4111yescharacteristic
401C42C2×#4111yescharacteristic
402C23—21no—
403C23—21no—
404C23—22no—
405C23—41no—
406C23—42no—
407C23—82no—
408C23—82no—
409C23—82no—
410C23#38811yescharacteristic
411C23#38811yescharacteristic
412C23C2×#12311yescharacteristic
413Q8—42no—
414Q8—42no—
415Q8—82no—
416D4—42no—
417D4—42no—
418D4—44no—
419D4—82no—
420D4—82no—
421D4—82no—
422D4—82no—
423D4—84no—
424D4—82no—
425D4—82no—
426C2×C4—22no—
427C2×C4—22no—
428C2×C4—22no—
429C2×C4—21no—
430C2×C4—21no—
431C2×C4—22no—
432C2×C4—22no—
433C2×C4—21no—
434C2×C4—21no—
435C2×C4—21no—
436C2×C4—21no—
437C2×C4—22no—
438C2×C4—22no—
439C2×C4—22no—
440C2×C4—21no—
441C2×C4—21no—
442C2×C4—21no—
443C2×C4—21no—
444C2×C4—22no—
445C2×C4—22no—
446C2×C4—22no—
447C2×C4—41no—
448C2×C4—41no—
449C2×C4—41no—
450C2×C4—41no—
451C2×C4—41no—
452C2×C4—41no—
453C2×C4—41no—
454C2×C4—48no—
455C2×C4—41no—
456C2×C4—41no—
457C2×C4—41no—
458C2×C4—41no—
459C2×C4—48no—
460C2×C4—41no—
461C2×C4—41no—
462C2×C4—42no—
463C2×C4—84no—
464C2×C4—82no—
465C2×C4—82no—
466C2×C4—82no—
467C2×C4#38811yescharacteristic
468C2×C4#38811yescharacteristic
469C2×C4#38811yescharacteristic
470C2×C4#38811yescharacteristic
471C2×C4#38811yescharacteristic
472C2×C4#38811yescharacteristic
473C2×C4C2×#11711yescharacteristic
474C2×C4C2×C4×D411yescharacteristic
475C2×C4C2×C4×D411yescharacteristic
476C2×C4C2×#12111yescharacteristic
477C2×C4C2×#12411yescharacteristic
478C2×C4C2×#12611yescharacteristic
479C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
480C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
481C2×C4#57511yescharacteristic
482C2×C4#57511yescharacteristic
483C8—42no—
484C8—42no—
485C8—44no—
486C8—42no—
487C8—42no—
488C8—82no—
489C8—82no—
490C22—21no—
491C22—21no—
492C22—21no—
493C22—21no—
494C22—21no—
495C22—21no—
496C22—21no—
497C22—42no—
498C22—42no—
499C22—82no—
500C22—82no—
501C22—82no—
502C22—164no—
503C22#193212yes—
504C22C2×#38811yescharacteristic
505C22#309411yescharacteristic
506C22#309411yescharacteristic
507C4—24no—
508C4—21no—
509C4—21no—
510C4—21no—
511C4—21no—
512C4—21no—
513C4—21no—
514C4—21no—
515C4—21no—
516C4—21no—
517C4—42no—
518C4—42no—
519C4—42no—
520C4—82no—
521C4C4×AGL(1,ℤ/8)11yescharacteristic
522C4C4×AGL(1,ℤ/8)11yescharacteristic
523C4#295612yes—
524C4#303811yescharacteristic
525C4#304011yescharacteristic
526C2—21no—
527C2—21no—
528C2—22no—
529C2—41no—
530C2—162no—
531C2#2106011yescharacteristic
532C2C2×#193211yescharacteristic
533C2#2109911yescharacteristic
subgroupG/H
centerC2×C4C2×#121
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
socleC22C2×#388
prime radicalC2×#532C22
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C4×#432C2semidirect
3. #17822C2semidirect
6. #9582C2semidirect
7. #9581C2semidirect
8. #9578C2semidirect
9. #9577C2semidirect
10. #8077C2semidirect
11. #7867C2semidirect
25. #1767C22semidirect
26. #1767C22semidirect
27. C4×C4⋊C8C22semidirect
28. C4×C4⋊C8C22semidirect
34. #1571C22semidirect
35. #1570C22semidirect
37. #1557C4semidirect
38. #1556C4semidirect
39. #1555C4semidirect
46. #1301C22semidirect
47. #1301C22semidirect
48. #1276C22semidirect
49. #1276C22semidirect
130. #332C2×C4semidirect
131. #332C2×C4semidirect
132. #331C2×C4semidirect
133. #331C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#19622
C4×#4322
#178222
#177312
#174952
#95822
#95812
#95782
#95772
#80772
#78672