Finite Group Databasebeta

← #211849

Subgroups — #211849

Order 512.

611 conjugacy classes of subgroups.

The subgroup lattice has 471 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22256C211yesmaximal, characteristic, complemented
2#20567C211yesmaximal, characteristic
3#19137C211yesmaximal, characteristic
4#17933C211yesmaximal, characteristic
5#17930C211yesmaximal, characteristic
6#9749C211yesmaximal, characteristic, complemented
7#9629C211yesmaximal, characteristic, complemented
8#9584C211yesmaximal, characteristic, complemented
9#9583C211yesmaximal, characteristic, complemented
10#9231C211yesmaximal, characteristic
11#9231C211yesmaximal, characteristic
12#8664C211yesmaximal, characteristic
13#8270C211yesmaximal, characteristic, complemented
14#8258C211yesmaximal, characteristic, complemented
15#8252C211yesmaximal, characteristic, complemented
16#3160C2211yescharacteristic
17#3101C2211yescharacteristic
18#3101C2211yescharacteristic
19#2960C2211yescharacteristic
20#2960C2211yescharacteristic
21C2×#436C2211yescharacteristic
22C2×#436C2211yescharacteristic
23C42×D4C2211yescharacteristic
24#1994C2211yescharacteristic
25#1994C2211yescharacteristic
26#1922—21no—
27#1922—21no—
28#1922—21no—
29#1922—21no—
30#1837C2211yescharacteristic
31C8×C4⋊C4C2211yescharacteristic
32C8×C4⋊C4C2211yescharacteristic
33#1767C2211yescharacteristic
34#1753—21no—
35#1753—21no—
36C4×C22⋊C8—21no—
37#1744C2211yescharacteristic
38#1728—21no—
39#1695C412yescomplemented
40#1693C412yescomplemented
41#1691C412yescomplemented
42#1691C412yescomplemented
43#1592—22no—
44#1590—22no—
45#1588—22no—
46#1583—22no—
47#1571C2211yescharacteristic
48#1570C2211yescharacteristic
49#1564C2211yescharacteristic
50#1564C2211yescharacteristic
51#1542C2211yescharacteristic
52#1542C2211yescharacteristic
53#1503C2211yescharacteristic
54#1503C2211yescharacteristic
55#1486C2211yescharacteristic
56#1486C2211yescharacteristic
57#1481C2211yescharacteristic
58#1481C2211yescharacteristic
59#1452C2211yescharacteristic
60#1450C2211yescharacteristic
61#1303C2211yescharacteristic
62#1303C2211yescharacteristic
63#1301C2211yescharacteristic
64#1301C2211yescharacteristic
65#1276C2211yescharacteristic
66#1276C2211yescharacteristic
67C22×M16—21no—
68C22×M16—21no—
69#533C2311yescharacteristic
70#533C2311yescharacteristic
71C2×C4×D4—21no—
72C2×C4×D4C2311yescharacteristic
73C22×C42—21no—
74#500C2×C411yescharacteristic, complemented
75#500C2×C411yescharacteristic, complemented
76#499—21no—
77#499—21no—
78#498C2×C411yescharacteristic, complemented
79#498C2×C411yescharacteristic, complemented
80#483—21no—
81#483—21no—
82#476C2×C412yes—
83#476C2×C412yes—
84#461C2×C412yes—
85#461C2×C412yes—
86#450—21no—
87#450—21no—
88#446—21no—
89#446—21no—
90#446—21no—
91#446—21no—
92C8×Q8—21no—
93C8×Q8—21no—
94C8×Q8—21no—
95C8×Q8—21no—
96C4×SD16—22no—
97C4×SD16—22no—
98#436—21no—
99#436—21no—
100#436—21no—
101#436—21no—
102#436C2×C412yes—
103#436C2×C412yes—
104#436D412yes—
105#436D412yes—
106#431C2311yescharacteristic
107#431C2311yescharacteristic
108#431C2311yescharacteristic
109#431C2311yescharacteristic
110#427—21no—
111#427—21no—
112#424—21no—
113#424—21no—
114#423C2311yescharacteristic
115#423C2311yescharacteristic
116C2×C4⋊C8C2311yescharacteristic
117C2×C4⋊C8C2311yescharacteristic
118C2×C22⋊C8—21no—
119C2×C22⋊C8—21no—
120#405—21no—
121#405—21no—
122C2×#97—21no—
123C2×C4×C8—21no—
124C2×C4×C8—21no—
125C2×C4×C8C2311yescharacteristic
126C2×C4×C8C2311yescharacteristic
127C4×C4⋊C4—21no—
128C4×C4⋊C4C2311yescharacteristic
129C4×C22⋊C4—21no—
130C4×C22⋊C4—22no—
131C43C2311yescharacteristic
132#341—21no—
133#341—21no—
134#339—21no—
135#339—21no—
136#339—21no—
137#339—21no—
138#336—21no—
139#336—21no—
140#336—21no—
141#336D411yescharacteristic
142#336D411yescharacteristic
143#336D411yescharacteristic
144#336D411yescharacteristic
145#335—22no—
146#335—21no—
147#335C2×C412yescomplemented
148#334—22no—
149#334—21no—
150#334C2×C412yescomplemented
151#333C2×C412yescomplemented
152#333C2×C412yescomplemented
153#332C2×C412yescomplemented
154#332C2×C412yescomplemented
155#328—22no—
156#328—22no—
157Q8⋊C8—22no—
158Q8⋊C8—22no—
159D4⋊C8—22no—
160D4⋊C8—22no—
161#322—21no—
162#322—21no—
163#322—21no—
164C82—21no—
165C22×D4#4111yescharacteristic
166C23×C4—21no—
167C23×C4—22no—
168C2×SD16—22no—
169C2×SD16—22no—
170C2×M16—22no—
171C2×M16—22no—
172C2×M16—22no—
173C2×M16—22no—
174C2×M16—21no—
175C2×M16—21no—
176C2×M16—81no—
177C2×M16—81no—
178C2×M16—81no—
179C2×M16—81no—
180C22×C8—21no—
181C22×C8—21no—
182C22×C8C2×D411yescharacteristic
183C22×C8C2×D411yescharacteristic
184C22×C8C2×D411yescharacteristic
185C22×C8C2×D411yescharacteristic
186C4⋊Q8C22×C411yescharacteristic
187C4⋊Q8C22×C411yescharacteristic
188C4⋊Q8C22×C411yescharacteristic
189C4⋊Q8C22×C411yescharacteristic
190#125—21no—
191#125—21no—
192C22⋊Q8—21no—
193C22⋊Q8—21no—
194C22⋊Q8—21no—
195C22⋊Q8—21no—
196C4×Q8—21no—
197C4×Q8—21no—
198C4×Q8—21no—
199C4×Q8—21no—
200C4×D4—21no—
201C4×D4—21no—
202C4×D4—44no—
203C4×D4C22×C412yes—
204C4×D4C2×D412yes—
205#117—21no—
206#117—21no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4#4111yescharacteristic
209C2×C22⋊C4—21no—
210C2×C22⋊C4—22no—
211C2×C42—21no—
212C2×C42—21no—
213C2×C42—21no—
214C2×C42—22no—
215C2×C42—81no—
216C2×C42—81no—
217C2×C42C2×D411yescharacteristic
218C2×C42C2×D411yescharacteristic
219C2×C42C2411yescharacteristic
220#107—21no—
221#107—21no—
222#107—21no—
223#107—21no—
224#106—21no—
225#106—21no—
226#106—21no—
227#106—21no—
228#106—22no—
229#106—22no—
230C4⋊C8—21no—
231C4⋊C8—21no—
232C4⋊C8—21no—
233C4⋊C8—21no—
234C4⋊C8—21no—
235C4⋊C8—21no—
236C4⋊C8—21no—
237C4⋊C8—21no—
238C4⋊C8—41no—
239C4⋊C8—41no—
240C4⋊C8—41no—
241C4⋊C8—41no—
242C4⋊C8C22×C412yes—
243C4⋊C8C22×C412yes—
244C4⋊C8C22×C412yes—
245C4⋊C8C2×D412yes—
246#103—22no—
247#103—22no—
248#102—22no—
249#102—22no—
250C22⋊C8—21no—
251C22⋊C8—21no—
252C22⋊C8—22no—
253C22⋊C8—22no—
254C22⋊C8—22no—
255C22⋊C8—22no—
256C22⋊C8—21no—
257C22⋊C8—21no—
258#97—41no—
259#97—41no—
260#97—81no—
261C4×C8—21no—
262C4×C8—21no—
263C4×C8—21no—
264C4×C8—21no—
265C4×C8—41no—
266C4×C8—41no—
267C4×C8—81no—
268C4×C8—81no—
269C4×C8C22×C411yescharacteristic
270C4×C8C22×C411yescharacteristic
271C4×C8C2×D411yescharacteristic
272C4×C8C2×D411yescharacteristic
273C4×C8#4111yescharacteristic
274C4×C8#4111yescharacteristic
275C4×C8#4111yescharacteristic
276C4×C8#4111yescharacteristic
277#95—22no—
278#95#4112yes—
279C24—21no—
280C2×Q8—21no—
281C2×Q8—21no—
282C2×Q8—21no—
283C2×Q8—21no—
284C2×D4—21no—
285C2×D4—42no—
286C2×D4C4×D412yes—
287C22×C4—21no—
288C22×C4—22no—
289C22×C4—21no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—21no—
293C22×C4—44no—
294C22×C4—44no—
295C22×C4—41no—
296C22×C4—41no—
297C22×C4—41no—
298C22×C4—41no—
299C22×C4—82no—
300C22×C4C22×D411yescharacteristic
301C22×C4C2×#4111yescharacteristic
302C22×C4C2×#4111yescharacteristic
303SD16—82no—
304SD16—82no—
305M16—41no—
306M16—41no—
307M16—41no—
308M16—41no—
309M16—82no—
310M16—82no—
311M16—82no—
312M16—82no—
313C2×C8—21no—
314C2×C8—21no—
315C2×C8—21no—
316C2×C8—21no—
317C2×C8—21no—
318C2×C8—21no—
319C2×C8—21no—
320C2×C8—21no—
321C2×C8—22no—
322C2×C8—22no—
323C2×C8—22no—
324C2×C8—22no—
325C2×C8—41no—
326C2×C8—41no—
327C2×C8—41no—
328C2×C8—41no—
329C2×C8—41no—
330C2×C8—41no—
331C2×C8—41no—
332C2×C8—41no—
333C2×C8C4×D411yescharacteristic
334C2×C8C4×D411yescharacteristic
335C2×C8C4×D411yescharacteristic
336C2×C8C4×D411yescharacteristic
337C2×C8#12111yescharacteristic
338C2×C8#12111yescharacteristic
339C2×C8#12111yescharacteristic
340C2×C8#12111yescharacteristic
341C4⋊C4—24no—
342C4⋊C4—21no—
343C4⋊C4—21no—
344C4⋊C4—21no—
345C4⋊C4—21no—
346C4⋊C4—21no—
347C4⋊C4—21no—
348C4⋊C4—21no—
349C4⋊C4—21no—
350C4⋊C4—21no—
351C4⋊C4—41no—
352C4⋊C4—41no—
353C4⋊C4—41no—
354C4⋊C4—41no—
355C4⋊C4C4×D412yes—
356C22⋊C4—22no—
357C22⋊C4—22no—
358C22⋊C4—44no—
359C22⋊C4—41no—
360C22⋊C4—41no—
361C42—21no—
362C42—21no—
363C42—21no—
364C42—21no—
365C42—42no—
366C42—42no—
367C42—41no—
368C42—41no—
369C42—41no—
370C42—41no—
371C42—82no—
372C42C23×C411yescharacteristic
373C42C22×D411yescharacteristic
374C42C2×#4111yescharacteristic
375C42C2×#4111yescharacteristic
376C23—22no—
377C23—42no—
378C23—42no—
379C23—41no—
380C23—41no—
381C23#52511yescharacteristic
382Q8—42no—
383Q8—42no—
384Q8—81no—
385Q8—81no—
386D4—22no—
387D4—22no—
388D4—42no—
389C2×C4—22no—
390C2×C4—22no—
391C2×C4—24no—
392C2×C4—22no—
393C2×C4—41no—
394C2×C4—41no—
395C2×C4—42no—
396C2×C4—41no—
397C2×C4—41no—
398C2×C4—42no—
399C2×C4—41no—
400C2×C4—42no—
401C2×C4—42no—
402C2×C4—42no—
403C2×C4—42no—
404C2×C4—41no—
405C2×C4—41no—
406C2×C4—41no—
407C2×C4—41no—
408C2×C4—41no—
409C2×C4—41no—
410C2×C4—41no—
411C2×C4—88no—
412C2×C4C2×C4×D411yescharacteristic
413C2×C4C2×C4×D411yescharacteristic
414C2×C4C4×#4111yescharacteristic
415C2×C4C2×#12111yescharacteristic
416C2×C4#52911yescharacteristic
417C2×C4#53211yescharacteristic
418C2×C4#57711yescharacteristic
419C2×C4#57711yescharacteristic
420C2×C4#57711yescharacteristic
421C2×C4#57811yescharacteristic
422C8—21no—
423C8—21no—
424C8—21no—
425C8—21no—
426C8—21no—
427C8—21no—
428C8—21no—
429C8—21no—
430C8—42no—
431C8—42no—
432C8—42no—
433C8—42no—
434C8—81no—
435C8—81no—
436C8—81no—
437C8—81no—
438C22—21no—
439C22—21no—
440C22—42no—
441C22—42no—
442C22—42no—
443C22—82no—
444C22—82no—
445C22C4×#12111yescharacteristic
446C22#312011yescharacteristic
447C22#312311yescharacteristic
448C4—21no—
449C4—21no—
450C4—21no—
451C4—44no—
452C4—42no—
453C4—42no—
454C4—81no—
455C4—81no—
456C4—81no—
457C4—81no—
458C4—81no—
459C4#294711yescharacteristic
460C4#294911yescharacteristic
461C4#295612yes—
462C4#304211yescharacteristic
463C4#304611yescharacteristic
464C2—21no—
465C2—21no—
466C2—82no—
467C2#2151411yescharacteristic
468C2#2155111yescharacteristic
469C2#2155811yescharacteristic
subgroupG/H
centerC2×C4C2×#121
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
socleC22C4×#121
prime radicalC22×D4#41
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
1. #22256C2semidirect
6. #9749C2semidirect
7. #9629C2semidirect
8. #9584C2semidirect
9. #9583C2semidirect
13. #8270C2semidirect
14. #8258C2semidirect
15. #8252C2semidirect
39. #1695C4semidirect
40. #1693C4semidirect
41. #1691C4semidirect
42. #1691C4semidirect
74. #500C2×C4semidirect
75. #500C2×C4semidirect
78. #498C2×C4semidirect
79. #498C2×C4semidirect
147. #335C2×C4semidirect
150. #334C2×C4semidirect
151. #333C2×C4semidirect
152. #333C2×C4semidirect
153. #332C2×C4semidirect
154. #332C2×C4semidirect

Maximal subgroups

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

subgroupindex
#222562
#205672
#191372
#179332
#179302
#97492
#96292
#95842
#95832
#92312
#86642
#82702
#82582
#82522