Finite Group Databasebeta

← #211715

Subgroups — #211715

Order 512.

579 conjugacy classes of subgroups.

The subgroup lattice has 463 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21658C211yesmaximal, characteristic
2#20516C211yesmaximal, characteristic, complemented
3#19123C211yesmaximal, characteristic
4#17936C211yesmaximal, characteristic
5#17934C211yesmaximal, characteristic, complemented
6#9751C211yesmaximal, characteristic
7#9629C211yesmaximal, characteristic, complemented
8#9584C211yesmaximal, characteristic, complemented
9#9582C211yesmaximal, characteristic
10#9238C211yesmaximal, characteristic
11#9236C211yesmaximal, characteristic, complemented
12#8661C211yesmaximal, characteristic, complemented
13#8267C211yesmaximal, characteristic, complemented
14#8254C211yesmaximal, characteristic, complemented
15#8253C211yesmaximal, characteristic
16#3131C2211yescharacteristic
17#3104C2211yescharacteristic
18#3103C2211yescharacteristic
19#2960C2211yescharacteristic
20C2×#446C2211yescharacteristic
21C2×C8×Q8C2211yescharacteristic
22#2931C2211yescharacteristic
23#2295C2211yescharacteristic
24#1994C2211yescharacteristic
25#1962C2211yescharacteristic
26#1937—21no—
27#1937—21no—
28#1837C2211yescharacteristic
29#1771—21no—
30#1770C2211yescharacteristic
31C8×C4⋊C4—21no—
32C8×C4⋊C4C2211yescharacteristic
33#1768—21no—
34#1767C2211yescharacteristic
35#1764—21no—
36#1764—21no—
37#1726—21no—
38C42×C8C2211yescharacteristic
39#1684C412yescomplemented
40#1682C412yescomplemented
41#1680C412yescomplemented
42#1678C412yescomplemented
43#1593—22no—
44#1592—22no—
45#1587—22no—
46#1585—22no—
47#1571C2211yescharacteristic
48#1570C2211yescharacteristic
49#1564C2211yescharacteristic
50#1564C2211yescharacteristic
51#1542C2211yescharacteristic
52#1540C2211yescharacteristic
53#1503C2211yescharacteristic
54#1502C2211yescharacteristic
55#1487C2211yescharacteristic
56#1487C2211yescharacteristic
57#1481C2211yescharacteristic
58#1480C2211yescharacteristic
59#1452C2211yescharacteristic
60#1452C2211yescharacteristic
61#1303C2211yescharacteristic
62#1303C2211yescharacteristic
63#1301C2211yescharacteristic
64#1301C2211yescharacteristic
65#1276C2211yescharacteristic
66#1276C2211yescharacteristic
67C2×#131—21no—
68#533C2311yescharacteristic
69#532C2311yescharacteristic
70C2×C4×Q8C2311yescharacteristic
71#488—21no—
72#488—21no—
73#487C2×C411yescharacteristic
74#487C2×C411yescharacteristic
75#486C2×C411yescharacteristic
76#486C2×C411yescharacteristic
77#484—21no—
78#481—21no—
79#477C2×C412yes—
80#475C2×C412yes—
81#462C2×C412yes—
82#460C2×C412yes—
83#451—21no—
84#448—21no—
85#446—21no—
86#446—21no—
87#446—21no—
88#446—21no—
89#446D412yes—
90#446D412yes—
91C8×Q8—21no—
92C8×Q8—21no—
93C8×Q8—21no—
94C8×Q8—21no—
95C8×Q8C2×C412yes—
96C8×Q8C2×C412yes—
97C4×Q16—22no—
98C4×SD16—22no—
99#436—21no—
100#436—21no—
101C8×D4—21no—
102C8×D4—21no—
103#431C2311yescharacteristic
104#431C2311yescharacteristic
105#431C2311yescharacteristic
106#431C2311yescharacteristic
107#424—21no—
108#423C2311yescharacteristic
109#423C2311yescharacteristic
110C2×C4⋊C8—21no—
111C2×C4⋊C8—21no—
112C2×C4⋊C8C2311yescharacteristic
113C2×C4⋊C8C2311yescharacteristic
114C2×#104—21no—
115C2×#104—21no—
116#416—21no—
117#416—21no—
118#408—21no—
119#405—21no—
120C2×#97—21no—
121C2×#97—21no—
122C2×#97—21no—
123C2×C4×C8—21no—
124C2×C4×C8—21no—
125C2×C4×C8D411yescharacteristic
126C2×C4×C8D411yescharacteristic
127C2×C4×C8D411yescharacteristic
128C2×C4×C8D411yescharacteristic
129C2×C4×C8C2311yescharacteristic
130C2×C4×C8C2311yescharacteristic
131#391—21no—
132#389—21no—
133#387—22no—
134C4×C4⋊C4—21no—
135C4×C4⋊C4C2311yescharacteristic
136#376—21no—
137C43C2311yescharacteristic
138#341—21no—
139#341—21no—
140#339—21no—
141#339—21no—
142#336—21no—
143#336—21no—
144#336—21no—
145#335—22no—
146#335—21no—
147#335C2×C412yes—
148#334—22no—
149#334—21no—
150#334C2×C412yes—
151#333C2×C412yes—
152#332C2×C412yes—
153#332C2×C412yes—
154#331C2×C412yes—
155#328—22no—
156#327—22no—
157Q8⋊C8—22no—
158Q8⋊C8—22no—
159Q8⋊C8—22no—
160D4⋊C8—22no—
161#322—21no—
162#322—21no—
163#322—21no—
164#322—21no—
165C2×#41—21no—
166C22×Q8#4111yescharacteristic
167C2×Q16—22no—
168C2×SD16—22no—
169#131—42no—
170#131—42no—
171C2×M16—21no—
172C2×M16—21no—
173C2×M16—41no—
174C22×C8—21no—
175C22×C8—21no—
176C22×C8—41no—
177C22×C8C2×D411yescharacteristic
178C22×C8C2×D411yescharacteristic
179C22×C8C2×D411yescharacteristic
180C22×C8C2×D411yescharacteristic
181C4⋊Q8C22×C411yescharacteristic
182C4⋊Q8C22×C411yescharacteristic
183C4⋊Q8C22×C411yescharacteristic
184#127C22×C411yescharacteristic
185#125—21no—
186#124—21no—
187C22⋊Q8—21no—
188C22⋊Q8—21no—
189#121—21no—
190#121—21no—
191C4×Q8—21no—
192C4×Q8—21no—
193C4×Q8—21no—
194C4×Q8—21no—
195C4×Q8C22×C412yes—
196C4×Q8C2×D412yes—
197C4×D4—21no—
198C4×D4—21no—
199#117—21no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4#4111yescharacteristic
204C2×C42—21no—
205C2×C42—21no—
206C2×C42—21no—
207C2×C42—21no—
208C2×C42C2×D411yescharacteristic
209C2×C42C2×D411yescharacteristic
210C2×C42C2411yescharacteristic
211#107—22no—
212#106—22no—
213C4⋊C8—22no—
214C4⋊C8—22no—
215C4⋊C8—22no—
216C4⋊C8—22no—
217C4⋊C8—21no—
218C4⋊C8—21no—
219C4⋊C8—21no—
220C4⋊C8—21no—
221C4⋊C8—21no—
222C4⋊C8—21no—
223C4⋊C8—21no—
224C4⋊C8—21no—
225C4⋊C8—41no—
226C4⋊C8—41no—
227C4⋊C8C22×C412yes—
228C4⋊C8C22×C412yes—
229C4⋊C8C22×C412yes—
230C4⋊C8C2×D412yes—
231#104—42no—
232#104—42no—
233#103—22no—
234#103—22no—
235#103—21no—
236#103—21no—
237#103—21no—
238#103—21no—
239#103—22no—
240#102—21no—
241#102—21no—
242#102—21no—
243#102—21no—
244#102—22no—
245C22⋊C8—21no—
246C22⋊C8—21no—
247C22⋊C8—21no—
248C22⋊C8—21no—
249C22⋊C8—41no—
250C22⋊C8—41no—
251#97—22no—
252#97—22no—
253#97—42no—
254#97—42no—
255#97—41no—
256C4×C8—22no—
257C4×C8—22no—
258C4×C8—22no—
259C4×C8—21no—
260C4×C8—21no—
261C4×C8—21no—
262C4×C8—21no—
263C4×C8—41no—
264C4×C8—41no—
265C4×C8—41no—
266C4×C8—42no—
267C4×C8—41no—
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—22no—
278#95—22no—
279#95#4112yes—
280#41—41no—
281#41—41no—
282#41—41no—
283#41—41no—
284C2×Q8—21no—
285C2×Q8—21no—
286C2×Q8—21no—
287C2×Q8—21no—
288C2×Q8—42no—
289C2×Q8C4×D412yes—
290C2×D4—21no—
291C2×D4—21no—
292C2×D4—21no—
293C2×D4—41no—
294C22×C4—21no—
295C22×C4—21no—
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—22no—
299C22×C4—41no—
300C22×C4C22×D411yescharacteristic
301C22×C4C2×#4111yescharacteristic
302C22×C4C2×#4111yescharacteristic
303Q16—82no—
304SD16—82no—
305M16—41no—
306M16—41no—
307M16—41no—
308M16—41no—
309M16—82no—
310C2×C8—21no—
311C2×C8—21no—
312C2×C8—21no—
313C2×C8—21no—
314C2×C8—21no—
315C2×C8—21no—
316C2×C8—21no—
317C2×C8—21no—
318C2×C8—22no—
319C2×C8—22no—
320C2×C8—22no—
321C2×C8—22no—
322C2×C8—41no—
323C2×C8—41no—
324C2×C8—41no—
325C2×C8—41no—
326C2×C8—41no—
327C2×C8—41no—
328C2×C8—41no—
329C2×C8—41no—
330C2×C8—82no—
331C2×C8C4×D411yescharacteristic
332C2×C8C4×D411yescharacteristic
333C2×C8C4×D411yescharacteristic
334C2×C8C4×D411yescharacteristic
335C2×C8#12111yescharacteristic
336C2×C8#12111yescharacteristic
337C2×C8#12111yescharacteristic
338C2×C8#12111yescharacteristic
339C4⋊C4—22no—
340C4⋊C4—21no—
341C4⋊C4—22no—
342C4⋊C4—21no—
343C4⋊C4—21no—
344C4⋊C4—21no—
345C4⋊C4—21no—
346C4⋊C4—21no—
347C4⋊C4—21no—
348C4⋊C4—42no—
349C4⋊C4—44no—
350C4⋊C4—41no—
351C4⋊C4—41no—
352C4⋊C4C4×D412yes—
353C22⋊C4—41no—
354C22⋊C4—41no—
355C22⋊C4—41no—
356C42—22no—
357C42—22no—
358C42—21no—
359C42—21no—
360C42—21no—
361C42—42no—
362C42—42no—
363C42—41no—
364C42—41no—
365C42—41no—
366C42#13712yes—
367C42C23×C411yescharacteristic
368C42C22×D411yescharacteristic
369C42C2×#4111yescharacteristic
370C42C2×#4111yescharacteristic
371C23—41no—
372C23#52511yescharacteristic
373Q8—22no—
374Q8—22no—
375Q8—42no—
376Q8—41no—
377Q8—41no—
378Q8—42no—
379Q8—81no—
380D4—41no—
381D4—41no—
382D4—42no—
383D4—81no—
384D4—81no—
385D4—81no—
386C2×C4—22no—
387C2×C4—22no—
388C2×C4—22no—
389C2×C4—22no—
390C2×C4—22no—
391C2×C4—41no—
392C2×C4—41no—
393C2×C4—41no—
394C2×C4—41no—
395C2×C4—42no—
396C2×C4—42no—
397C2×C4—41no—
398C2×C4—42no—
399C2×C4—42no—
400C2×C4—42no—
401C2×C4—41no—
402C2×C4—41no—
403C2×C4—41no—
404C2×C4—81no—
405C2×C4—81no—
406C2×C4C2×C4×D411yescharacteristic
407C2×C4C2×C4×D411yescharacteristic
408C2×C4C4×#4111yescharacteristic
409C2×C4C2×#12111yescharacteristic
410C2×C4#52911yescharacteristic
411C2×C4#53211yescharacteristic
412C2×C4C2×#13711yescharacteristic
413C2×C4#57511yescharacteristic
414C2×C4#57511yescharacteristic
415C2×C4#57511yescharacteristic
416C8—21no—
417C8—21no—
418C8—21no—
419C8—21no—
420C8—21no—
421C8—21no—
422C8—21no—
423C8—21no—
424C8—42no—
425C8—42no—
426C8—42no—
427C8—42no—
428C8—81no—
429C8—81no—
430C8—81no—
431C8—81no—
432C22—21no—
433C22—21no—
434C22—81no—
435C22—81no—
436C22—81no—
437C22C4×#12111yescharacteristic
438C22#311111yescharacteristic
439C22#311711yescharacteristic
440C4—21no—
441C4—21no—
442C4—21no—
443C4—42no—
444C4—42no—
445C4—42no—
446C4—42no—
447C4—82no—
448C4—81no—
449C4—81no—
450C4—81no—
451C4#294311yescharacteristic
452C4C4×#13711yescharacteristic
453C4#295712yes—
454C4#303711yescharacteristic
455C4#304011yescharacteristic
456C2—21no—
457C2—21no—
458C2—161no—
459C2#2150811yescharacteristic
460C2#2154511yescharacteristic
461C2#2155811yescharacteristic
subgroupG/H
centerC2×C4C2×#121
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
socleC22C4×#121
prime radical#532C23
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
2. #20516C2semidirect
5. #17934C2semidirect
7. #9629C2semidirect
8. #9584C2semidirect
11. #9236C2semidirect
12. #8661C2semidirect
13. #8267C2semidirect
14. #8254C2semidirect
39. #1684C4semidirect
40. #1682C4semidirect
41. #1680C4semidirect
42. #1678C4semidirect

Maximal subgroups

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

subgroupindex
#216582
#205162
#191232
#179362
#179342
#97512
#96292
#95842
#95822
#92382
#92362
#86612
#82672
#82542
#82532