Finite Group Databasebeta

← #211841

Subgroups — #211841

Order 512.

589 conjugacy classes of subgroups.

The subgroup lattice has 467 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22256C211yesmaximal, characteristic, complemented
2#20562C211yesmaximal, characteristic
3#19137C211yesmaximal, characteristic
4#17905C211yesmaximal, characteristic
5#17902C211yesmaximal, characteristic
6#9749C211yesmaximal, characteristic, complemented
7#9629C211yesmaximal, characteristic, complemented
8#9580C211yesmaximal, characteristic, complemented
9#9579C211yesmaximal, characteristic, complemented
10#9235C211yesmaximal, characteristic
11#9235C211yesmaximal, characteristic
12#8664C211yesmaximal, characteristic
13#8270C211yesmaximal, characteristic, complemented
14#8256C211yesmaximal, characteristic, complemented
15#8254C211yesmaximal, characteristic, complemented
16#3160C2211yescharacteristic
17#3101C2211yescharacteristic
18#3101C2211yescharacteristic
19#2960C2211yescharacteristic
20#2960C2211yescharacteristic
21C2×#436C2211yescharacteristic
22C2×C8×D4C2211yescharacteristic
23#2293C2211yescharacteristic
24#1994C2211yescharacteristic
25#1994C2211yescharacteristic
26#1922—21no—
27#1922—21no—
28#1922—21no—
29#1922—21no—
30#1837C2211yescharacteristic
31#1770C2211yescharacteristic
32C8×C4⋊C4C2211yescharacteristic
33C4×C4⋊C8C2211yescharacteristic
34#1753—21no—
35C8×C22⋊C4—21no—
36#1750—21no—
37#1744C2211yescharacteristic
38#1729—21no—
39#1695C412yescomplemented
40#1693C412yescomplemented
41#1691C412yescomplemented
42#1691C412yescomplemented
43#1585—22no—
44#1582—22no—
45SD16⋊C8—22no—
46C8×SD16—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—
68C23×C8—21no—
69#533C2311yescharacteristic
70#533C2311yescharacteristic
71C2×C4×D4C2311yescharacteristic
72C2×#117—21no—
73#500C2×C411yescharacteristic
74#500C2×C411yescharacteristic
75#499—21no—
76#499—21no—
77#498C2×C411yescharacteristic
78#498C2×C411yescharacteristic
79#483—21no—
80#483—21no—
81#476C2×C412yes—
82#476C2×C412yes—
83#461C2×C412yes—
84#461C2×C412yes—
85#450—21no—
86#450—21no—
87#446—21no—
88#446—21no—
89#446—21no—
90#446—21no—
91C8×Q8—21no—
92C8×Q8—21no—
93C8×Q8—21no—
94C8×Q8—21no—
95C4×SD16—22no—
96C4×SD16—22no—
97#436—21no—
98#436—21no—
99#436C2×C412yes—
100#436C2×C412yes—
101C8×D4—21no—
102C8×D4—21no—
103C8×D4D412yes—
104C8×D4D412yes—
105#431C2311yescharacteristic
106#431C2311yescharacteristic
107#431C2311yescharacteristic
108#431C2311yescharacteristic
109#427—21no—
110#427—21no—
111#424—21no—
112#424—21no—
113#423C2311yescharacteristic
114#423C2311yescharacteristic
115C2×C4⋊C8C2311yescharacteristic
116C2×C4⋊C8C2311yescharacteristic
117C2×C22⋊C8—21no—
118C2×C22⋊C8—21no—
119#405—21no—
120#405—21no—
121C2×#97—21no—
122C2×C4×C8—21no—
123C2×C4×C8—21no—
124C2×C4×C8C2311yescharacteristic
125C2×C4×C8C2311yescharacteristic
126#390—21no—
127#389—21no—
128#388—22no—
129C4×C4⋊C4C2311yescharacteristic
130C4×C22⋊C4—21no—
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×C412yes—
148#334—22no—
149#334—21no—
150#334C2×C412yes—
151#333C2×C412yes—
152#333C2×C412yes—
153#332C2×C412yes—
154#332C2×C412yes—
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—
167C2×SD16—22no—
168C2×SD16—22no—
169C2×M16—22no—
170C2×M16—22no—
171C2×M16—21no—
172C2×M16—21no—
173C2×M16—81no—
174C2×M16—81no—
175C22×C8—21no—
176C22×C8—21no—
177C22×C8—22no—
178C22×C8—22no—
179C22×C8—81no—
180C22×C8—81no—
181C22×C8C2×D411yescharacteristic
182C22×C8C2×D411yescharacteristic
183C22×C8C2×D411yescharacteristic
184C22×C8C2×D411yescharacteristic
185C4⋊Q8C22×C411yescharacteristic
186C4⋊Q8C22×C411yescharacteristic
187C4⋊Q8C22×C411yescharacteristic
188C4⋊Q8C22×C411yescharacteristic
189#125—21no—
190#125—21no—
191C22⋊Q8—21no—
192C22⋊Q8—21no—
193C22⋊Q8—21no—
194C22⋊Q8—21no—
195C4×Q8—21no—
196C4×Q8—21no—
197C4×Q8—21no—
198C4×Q8—21no—
199C4×D4—21no—
200C4×D4—21no—
201C4×D4C22×C412yes—
202C4×D4C2×D412yes—
203#117—21no—
204#117—21no—
205#117—81no—
206#117—81no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4#4111yescharacteristic
210C2×C22⋊C4—21no—
211C2×C22⋊C4—22no—
212C2×C22⋊C4—22no—
213C2×C42—21no—
214C2×C42—21no—
215C2×C42—21no—
216C2×C42C2×D411yescharacteristic
217C2×C42C2×D411yescharacteristic
218C2×C42C2411yescharacteristic
219#107—21no—
220#107—21no—
221#107—21no—
222#107—21no—
223#106—21no—
224#106—21no—
225#106—21no—
226#106—21no—
227#106—22no—
228#106—22no—
229C4⋊C8—21no—
230C4⋊C8—21no—
231C4⋊C8—21no—
232C4⋊C8—21no—
233C4⋊C8—21no—
234C4⋊C8—21no—
235C4⋊C8—21no—
236C4⋊C8—21no—
237C4⋊C8—41no—
238C4⋊C8—41no—
239C4⋊C8—41no—
240C4⋊C8—41no—
241C4⋊C8C22×C412yes—
242C4⋊C8C22×C412yes—
243C4⋊C8C22×C412yes—
244C4⋊C8C2×D412yes—
245#103—22no—
246#103—22no—
247#102—22no—
248#102—22no—
249C22⋊C8—21no—
250C22⋊C8—21no—
251C22⋊C8—22no—
252C22⋊C8—22no—
253C22⋊C8—22no—
254C22⋊C8—22no—
255C22⋊C8—21no—
256C22⋊C8—21no—
257#97—41no—
258#97—41no—
259#97—81no—
260C4×C8—21no—
261C4×C8—21no—
262C4×C8—21no—
263C4×C8—21no—
264C4×C8—41no—
265C4×C8—41no—
266C4×C8—81no—
267C4×C8—81no—
268C4×C8C22×C411yescharacteristic
269C4×C8C22×C411yescharacteristic
270C4×C8C2×D411yescharacteristic
271C4×C8C2×D411yescharacteristic
272C4×C8#4111yescharacteristic
273C4×C8#4111yescharacteristic
274C4×C8#4111yescharacteristic
275C4×C8#4111yescharacteristic
276#95—22no—
277#95#4112yes—
278C24—21no—
279C2×Q8—21no—
280C2×Q8—21no—
281C2×Q8—21no—
282C2×Q8—21no—
283C2×D4—21no—
284C2×D4—42no—
285C2×D4C4×D412yes—
286C22×C4—21no—
287C22×C4—21no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—21no—
292C22×C4—41no—
293C22×C4—41no—
294C22×C4—41no—
295C22×C4—41no—
296C22×C4C22×D411yescharacteristic
297C22×C4C2×#4111yescharacteristic
298C22×C4C2×#4111yescharacteristic
299SD16—44no—
300SD16—44no—
301M16—41no—
302M16—41no—
303M16—41no—
304M16—41no—
305M16—82no—
306M16—82no—
307C2×C8—21no—
308C2×C8—21no—
309C2×C8—21no—
310C2×C8—21no—
311C2×C8—21no—
312C2×C8—21no—
313C2×C8—21no—
314C2×C8—21no—
315C2×C8—22no—
316C2×C8—22no—
317C2×C8—22no—
318C2×C8—22no—
319C2×C8—41no—
320C2×C8—41no—
321C2×C8—41no—
322C2×C8—41no—
323C2×C8—41no—
324C2×C8—41no—
325C2×C8—41no—
326C2×C8—41no—
327C2×C8—82no—
328C2×C8—82no—
329C2×C8C4×D411yescharacteristic
330C2×C8C4×D411yescharacteristic
331C2×C8C4×D411yescharacteristic
332C2×C8C4×D411yescharacteristic
333C2×C8#12111yescharacteristic
334C2×C8#12111yescharacteristic
335C2×C8#12111yescharacteristic
336C2×C8#12111yescharacteristic
337C4⋊C4—21no—
338C4⋊C4—21no—
339C4⋊C4—21no—
340C4⋊C4—21no—
341C4⋊C4—21no—
342C4⋊C4—21no—
343C4⋊C4—21no—
344C4⋊C4—21no—
345C4⋊C4—21no—
346C4⋊C4—42no—
347C4⋊C4—41no—
348C4⋊C4—41no—
349C4⋊C4—41no—
350C4⋊C4—41no—
351C4⋊C4—82no—
352C4⋊C4C4×D412yes—
353C22⋊C4—22no—
354C22⋊C4—22no—
355C22⋊C4—44no—
356C22⋊C4—41no—
357C22⋊C4—41no—
358C22⋊C4—82no—
359C42—21no—
360C42—21no—
361C42—21no—
362C42—21no—
363C42—42no—
364C42—42no—
365C42—41no—
366C42—41no—
367C42—41no—
368C42—41no—
369C42C23×C411yescharacteristic
370C42C22×D411yescharacteristic
371C42C2×#4111yescharacteristic
372C42C2×#4111yescharacteristic
373C23—22no—
374C23—42no—
375C23—42no—
376C23—41no—
377C23—41no—
378C23#52511yescharacteristic
379Q8—42no—
380Q8—42no—
381Q8—81no—
382Q8—81no—
383D4—22no—
384D4—22no—
385D4—42no—
386C2×C4—22no—
387C2×C4—22no—
388C2×C4—22no—
389C2×C4—41no—
390C2×C4—41no—
391C2×C4—41no—
392C2×C4—41no—
393C2×C4—41no—
394C2×C4—42no—
395C2×C4—42no—
396C2×C4—42no—
397C2×C4—42no—
398C2×C4—41no—
399C2×C4—41no—
400C2×C4—42no—
401C2×C4—42no—
402C2×C4—41no—
403C2×C4—42no—
404C2×C4—41no—
405C2×C4—41no—
406C2×C4—41no—
407C2×C4—41no—
408C2×C4C2×C4×D411yescharacteristic
409C2×C4C2×C4×D411yescharacteristic
410C2×C4C4×#4111yescharacteristic
411C2×C4C2×#12111yescharacteristic
412C2×C4#52911yescharacteristic
413C2×C4#53211yescharacteristic
414C2×C4#57711yescharacteristic
415C2×C4#57711yescharacteristic
416C2×C4#57711yescharacteristic
417C2×C4#57811yescharacteristic
418C8—21no—
419C8—21no—
420C8—21no—
421C8—21no—
422C8—21no—
423C8—21no—
424C8—21no—
425C8—21no—
426C8—42no—
427C8—42no—
428C8—42no—
429C8—42no—
430C8—81no—
431C8—81no—
432C8—81no—
433C8—81no—
434C22—21no—
435C22—21no—
436C22—42no—
437C22—42no—
438C22—42no—
439C22—82no—
440C22—82no—
441C22C4×#12111yescharacteristic
442C22#312011yescharacteristic
443C22#312311yescharacteristic
444C4—21no—
445C4—21no—
446C4—21no—
447C4—42no—
448C4—42no—
449C4—81no—
450C4—82no—
451C4—81no—
452C4—81no—
453C4—81no—
454C4—81no—
455C4#294711yescharacteristic
456C4#294911yescharacteristic
457C4#295512yes—
458C4#304211yescharacteristic
459C4#304611yescharacteristic
460C2—21no—
461C2—21no—
462C2—82no—
463C2#2151411yescharacteristic
464C2#2155011yescharacteristic
465C2#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. #9580C2semidirect
9. #9579C2semidirect
13. #8270C2semidirect
14. #8256C2semidirect
15. #8254C2semidirect
39. #1695C4semidirect
40. #1693C4semidirect
41. #1691C4semidirect
42. #1691C4semidirect

Maximal subgroups

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

subgroupindex
#222562
#205622
#191372
#179052
#179022
#97492
#96292
#95802
#95792
#92352
#86642
#82702
#82562
#82542