Finite Group Databasebeta

← #203009

Subgroups — #203009

Order 512.

773 conjugacy classes of subgroups.

The subgroup lattice has 523 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21209C211yesmaximal, characteristic
2#20470C211yesmaximal, characteristic, complemented
3#17822C211yesmaximal, characteristic, complemented
4#17717C211yesmaximal, characteristic
5#17495C211yesmaximal, characteristic
6#9582C212yesmaximal, complemented
7#9581C212yesmaximal, complemented
8#9578C212yesmaximal, complemented
9#9577C212yesmaximal, complemented
10#8081C211yesmaximal, characteristic, complemented
11#7868C211yesmaximal, characteristic, complemented
12C2×#532C2211yescharacteristic
13#2920C2211yescharacteristic
14#2920C2211yescharacteristic
15C2×#432C2211yescharacteristic
16C2×#431C2211yescharacteristic
17C2×#423C2211yescharacteristic
18C2×#421—21no—
19C2×#421—21no—
20C4×#117C2211yescharacteristic
21#1963—21no—
22#1963—21no—
23#1962C2212yes—
24#1962C2212yes—
25#1767C2211yescharacteristic, complemented
26#1767C2211yescharacteristic, complemented
27C4×C4⋊C8C2211yescharacteristic, complemented
28C4×C4⋊C8C2211yescharacteristic, complemented
29#1750—21no—
30#1750—21no—
31#1746—21no—
32#1726—21no—
33#1573—22no—
34#1571C2212yescomplemented
35#1570C2212yescomplemented
36#1569—22no—
37#1553C414yescomplemented
38#1549C412yescomplemented
39#1545C412yescomplemented
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—
67C2×#117D411yescharacteristic
68C2×#117D411yescharacteristic
69C22×C42C2311yescharacteristic
70#432—21no—
71#432—21no—
72#432—21no—
73#432—21no—
74#432—21no—
75#432C2×C412yes—
76#432D412yes—
77#432D412yes—
78#431—21no—
79#431—21no—
80#431—21no—
81#431—21no—
82#431C2×C412yes—
83#431C2×C412yes—
84#431C2312yes—
85#431C2312yes—
86#423—21no—
87#423—21no—
88#423—21no—
89#423—21no—
90#423C2×C412yes—
91#423C2312yes—
92C2×C4⋊C8—21no—
93C2×C4⋊C8—21no—
94C2×C4⋊C8C2311yescharacteristic
95C2×C4⋊C8C2311yescharacteristic
96C2×C4⋊C8C2311yescharacteristic
97C2×C4⋊C8C2311yescharacteristic
98#421—44no—
99#421—44no—
100C2×#104—22no—
101C2×#104—22no—
102C2×#104—22no—
103C2×#104—22no—
104#407—21no—
105#407—21no—
106C2×C22⋊C8—21no—
107C2×C22⋊C8—21no—
108C4×M16—21no—
109C4×M16—21no—
110C2×#97—21no—
111C2×#97—21no—
112C2×#97—21no—
113C2×C4×C8—21no—
114C2×C4×C8—21no—
115C2×C4×C8—21no—
116C4×C4⋊C4—21no—
117C4×C4⋊C4—21no—
118C4×C22⋊C4—21no—
119C4×C22⋊C4—21no—
120#376D411yescharacteristic
121#376D411yescharacteristic
122C43C2311yescharacteristic
123C43C2311yescharacteristic
124#336—21no—
125#336—21no—
126#336—21no—
127#336—21no—
128#335—24no—
129#334—24no—
130#332C2×C414yes—
131#332C2×C414yes—
132#331C2×C414yes—
133#331C2×C414yes—
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—41no—
150C23×C4C2×D411yescharacteristic
151C23×C4#4111yescharacteristic
152C23×C4#4111yescharacteristic
153C2×M16—22no—
154C2×M16—22no—
155C2×M16—41no—
156C2×M16—41no—
157C2×M16—42no—
158C2×M16—42no—
159C22×C8—21no—
160C22×C8—21no—
161C22×C8—21no—
162C22×C8—21no—
163C22×C8—41no—
164C4⋊Q8—21no—
165C4⋊Q8—21no—
166C4⋊Q8—21no—
167C4⋊Q8C22×C412yes—
168#127—21no—
169#127—21no—
170#127—21no—
171#127C22×C412yes—
172#124—22no—
173#124—22no—
174#124—41no—
175#124—41no—
176#121—22no—
177#121—22no—
178#121—22no—
179#121—22no—
180#121—41no—
181#121—41no—
182#121—41no—
183#121—41no—
184C4×D4—22no—
185C4×D4—22no—
186C4×D4—22no—
187C4×D4—22no—
188C4×D4—41no—
189C4×D4—41no—
190C4×D4—41no—
191C4×D4—41no—
192#117—22no—
193#117—22no—
194#117—41no—
195#117—41no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—21no—
200C2×C22⋊C4—21no—
201C2×C22⋊C4—21no—
202C2×C22⋊C4—41no—
203C2×C22⋊C4—41no—
204C2×C42—21no—
205C2×C42—21no—
206C2×C42—21no—
207C2×C42—21no—
208C2×C42C22×C411yescharacteristic
209C2×C42C22×C411yescharacteristic
210C2×C42C2×D411yescharacteristic
211C2×C42C2×D411yescharacteristic
212C2×C42C2×D411yescharacteristic
213C2×C42C2×D411yescharacteristic
214C2×C42C2×D411yescharacteristic
215C2×C42C2×D411yescharacteristic
216C2×C42C2×D411yescharacteristic
217C2×C42#4111yescharacteristic
218C2×C42#4111yescharacteristic
219C2×C42#4111yescharacteristic
220C2×C42#4111yescharacteristic
221C2×C42#4111yescharacteristic
222C2×C42#4111yescharacteristic
223C2×C42C2411yescharacteristic
224C4⋊C8—22no—
225C4⋊C8—22no—
226C4⋊C8—21no—
227C4⋊C8—21no—
228C4⋊C8—21no—
229C4⋊C8—21no—
230C4⋊C8—22no—
231C4⋊C8—22no—
232C4⋊C8—41no—
233C4⋊C8—41no—
234C4⋊C8—41no—
235C4⋊C8C22×C412yes—
236C4⋊C8C22×C412yes—
237C4⋊C8C22×C414yes—
238C4⋊C8C2×D412yes—
239C4⋊C8C2×D412yes—
240#104—44no—
241#104—44no—
242#104—44no—
243#104—44no—
244C22⋊C8—22no—
245C22⋊C8—22no—
246C22⋊C8—22no—
247C22⋊C8—22no—
248C22⋊C8—22no—
249C22⋊C8—22no—
250C22⋊C8—22no—
251C22⋊C8—41no—
252#97—22no—
253#97—22no—
254#97—22no—
255#97—42no—
256#97—41no—
257#97—42no—
258C4×C8—22no—
259C4×C8—22no—
260C4×C8—22no—
261C4×C8—22no—
262C4×C8—42no—
263C4×C8—42no—
264#95—21no—
265#95—21no—
266#95—21no—
267#95—21no—
268C24—41no—
269C24#12311yescharacteristic
270#41—44no—
271#41—44no—
272#41—42no—
273#41—42no—
274#41—42no—
275#41—42no—
276#41—82no—
277#41—82no—
278#41—82no—
279#41—82no—
280C2×Q8—22no—
281C2×Q8—42no—
282C2×Q8—42no—
283C2×Q8—41no—
284C2×D4—22no—
285C2×D4—22no—
286C2×D4—22no—
287C2×D4—42no—
288C2×D4—42no—
289C2×D4—42no—
290C2×D4—42no—
291C2×D4—41no—
292C2×D4—41no—
293C2×D4—41no—
294C2×D4—82no—
295C2×D4—82no—
296C2×D4—82no—
297C2×D4—82no—
298C22×C4—21no—
299C22×C4—21no—
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—21no—
306C22×C4—42no—
307C22×C4—41no—
308C22×C4—42no—
309C22×C4—42no—
310C22×C4—82no—
311C22×C4—82no—
312C22×C4—81no—
313C22×C4#11711yescharacteristic
314C22×C4#11711yescharacteristic
315C22×C4C4×D411yescharacteristic
316C22×C4C4×D411yescharacteristic
317C22×C4C4×D411yescharacteristic
318C22×C4C4×D411yescharacteristic
319C22×C4#12111yescharacteristic
320C22×C4#12111yescharacteristic
321C22×C4#12311yescharacteristic
322C22×C4#12311yescharacteristic
323C22×C4#12311yescharacteristic
324C22×C4#12411yescharacteristic
325C22×C4#12411yescharacteristic
326C22×C4#12611yescharacteristic
327C22×C4#12611yescharacteristic
328C22×C4C22×D411yescharacteristic
329C22×C4C2×#4111yescharacteristic
330C22×C4C2×#4111yescharacteristic
331M16—42no—
332M16—42no—
333M16—82no—
334M16—82no—
335M16—82no—
336C2×C8—22no—
337C2×C8—22no—
338C2×C8—24no—
339C2×C8—22no—
340C2×C8—22no—
341C2×C8—41no—
342C2×C8—41no—
343C2×C8—41no—
344C2×C8—41no—
345C2×C8—41no—
346C2×C8—41no—
347C2×C8—41no—
348C2×C8—41no—
349C2×C8—42no—
350C2×C8—42no—
351C2×C8—82no—
352C4⋊C4—22no—
353C4⋊C4—22no—
354C4⋊C4—42no—
355C4⋊C4—42no—
356C4⋊C4—41no—
357C4⋊C4—41no—
358C22⋊C4—42no—
359C22⋊C4—42no—
360C22⋊C4—42no—
361C22⋊C4—42no—
362C22⋊C4—42no—
363C22⋊C4—42no—
364C42—21no—
365C42—21no—
366C42—21no—
367C42—21no—
368C42—21no—
369C42—21no—
370C42—42no—
371C42—42no—
372C42—81no—
373C42—81no—
374C42#11711yescharacteristic
375C42#11711yescharacteristic
376C42C4×D411yescharacteristic
377C42C4×D411yescharacteristic
378C42C4×D411yescharacteristic
379C42C4×D411yescharacteristic
380C42#12111yescharacteristic
381C42#12111yescharacteristic
382C42#12411yescharacteristic
383C42#12411yescharacteristic
384C42#12611yescharacteristic
385C42#12611yescharacteristic
386C42AGL(1,ℤ/8)12yes—
387C42AGL(1,ℤ/8)12yes—
388C42C23×C411yescharacteristic
389C42C22×D411yescharacteristic
390C42C2×#4111yescharacteristic
391C42C2×#4111yescharacteristic
392C23—21no—
393C23—21no—
394C23—41no—
395C23—41no—
396C23—42no—
397C23—82no—
398C23—82no—
399C23—82no—
400C23#38811yescharacteristic
401C23#38811yescharacteristic
402C23C2×#12311yescharacteristic
403Q8—42no—
404Q8—42no—
405Q8—82no—
406D4—42no—
407D4—42no—
408D4—44no—
409D4—82no—
410D4—82no—
411D4—82no—
412D4—82no—
413D4—84no—
414D4—82no—
415D4—82no—
416C2×C4—21no—
417C2×C4—21no—
418C2×C4—21no—
419C2×C4—21no—
420C2×C4—21no—
421C2×C4—21no—
422C2×C4—22no—
423C2×C4—22no—
424C2×C4—22no—
425C2×C4—21no—
426C2×C4—21no—
427C2×C4—21no—
428C2×C4—21no—
429C2×C4—22no—
430C2×C4—22no—
431C2×C4—41no—
432C2×C4—41no—
433C2×C4—41no—
434C2×C4—41no—
435C2×C4—41no—
436C2×C4—41no—
437C2×C4—41no—
438C2×C4—41no—
439C2×C4—41no—
440C2×C4—41no—
441C2×C4—41no—
442C2×C4—41no—
443C2×C4—41no—
444C2×C4—41no—
445C2×C4—41no—
446C2×C4—41no—
447C2×C4—41no—
448C2×C4—41no—
449C2×C4—41no—
450C2×C4—42no—
451C2×C4—84no—
452C2×C4—82no—
453C2×C4—82no—
454C2×C4—82no—
455C2×C4#38811yescharacteristic
456C2×C4#38811yescharacteristic
457C2×C4#38811yescharacteristic
458C2×C4#38811yescharacteristic
459C2×C4#38811yescharacteristic
460C2×C4#38811yescharacteristic
461C2×C4C2×#11711yescharacteristic
462C2×C4C2×C4×D411yescharacteristic
463C2×C4C2×C4×D411yescharacteristic
464C2×C4C2×#12111yescharacteristic
465C2×C4C2×#12411yescharacteristic
466C2×C4C2×#12611yescharacteristic
467C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
468C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
469C2×C4#57511yescharacteristic
470C2×C4#57511yescharacteristic
471C8—42no—
472C8—42no—
473C8—44no—
474C8—42no—
475C8—42no—
476C8—82no—
477C8—82no—
478C22—21no—
479C22—21no—
480C22—21no—
481C22—21no—
482C22—21no—
483C22—21no—
484C22—21no—
485C22—21no—
486C22—81no—
487C22—81no—
488C22—82no—
489C22—82no—
490C22—82no—
491C22—164no—
492C22C2×#38811yescharacteristic
493C22#309411yescharacteristic
494C22#309411yescharacteristic
495C4—21no—
496C4—21no—
497C4—21no—
498C4—21no—
499C4—21no—
500C4—21no—
501C4—21no—
502C4—21no—
503C4—21no—
504C4—42no—
505C4—42no—
506C4—81no—
507C4—81no—
508C4—82no—
509C4C4×AGL(1,ℤ/8)11yescharacteristic
510C4C4×AGL(1,ℤ/8)11yescharacteristic
511C4#295712yes—
512C4#303811yescharacteristic
513C4#304011yescharacteristic
514C2—21no—
515C2—21no—
516C2—41no—
517C2—41no—
518C2—162no—
519C2#2106011yescharacteristic
520C2#2108011yescharacteristic
521C2#2109811yescharacteristic
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. #20470C2semidirect
3. #17822C2semidirect
6. #9582C2semidirect
7. #9581C2semidirect
8. #9578C2semidirect
9. #9577C2semidirect
10. #8081C2semidirect
11. #7868C2semidirect
25. #1767C22semidirect
26. #1767C22semidirect
27. C4×C4⋊C8C22semidirect
28. C4×C4⋊C8C22semidirect
34. #1571C22semidirect
35. #1570C22semidirect
37. #1553C4semidirect
38. #1549C4semidirect
39. #1545C4semidirect
46. #1301C22semidirect
47. #1301C22semidirect
48. #1276C22semidirect
49. #1276C22semidirect

Maximal subgroups

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

subgroupindex
#212092
#204702
#178222
#177172
#174952
#95822
#95812
#95782
#95772
#80812
#78682