Finite Group Databasebeta

← #211697

Subgroups — #211697

Order 512.

635 conjugacy classes of subgroups.

The subgroup lattice has 466 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21658C211yesmaximal, characteristic
2C4×C8×Q8C211yesmaximal, characteristic, complemented
3#19123C211yesmaximal, characteristic
4#17903C211yesmaximal, characteristic, complemented
5#17901C211yesmaximal, characteristic
6#9751C211yesmaximal, characteristic
7#9629C211yesmaximal, characteristic, complemented
8#9579C211yesmaximal, characteristic, complemented
9#9577C211yesmaximal, characteristic
10#9234C211yesmaximal, characteristic
11#9232C211yesmaximal, characteristic, complemented
12#8661C211yesmaximal, characteristic, complemented
13#8266C211yesmaximal, characteristic, complemented
14#8252C211yesmaximal, characteristic, complemented
15#8251C211yesmaximal, characteristic
16#3131C2211yescharacteristic
17#3104C2211yescharacteristic
18#3103C2211yescharacteristic
19#2960C2211yescharacteristic
20C2×C8×Q8C2211yescharacteristic
21C2×C8×Q8C2211yescharacteristic
22#2931C2211yescharacteristic
23C42×Q8C2211yescharacteristic
24#1994C2211yescharacteristic
25#1962C2211yescharacteristic
26#1937—21no—
27#1937—21no—
28#1837C2211yescharacteristic
29C8×C4⋊C4—21no—
30C8×C4⋊C4—21no—
31C8×C4⋊C4C2211yescharacteristic
32C8×C4⋊C4C2211yescharacteristic
33C4×C4⋊C8—21no—
34C4×C4⋊C8C2211yescharacteristic
35#1764—21no—
36#1764—21no—
37C42×C8—21no—
38C42×C8C2211yescharacteristic
39#1684C412yescomplemented
40#1682C412yescomplemented
41#1680C412yescomplemented
42#1678C412yescomplemented
43#1584—22no—
44#1583—22no—
45C8×Q16—22no—
46C8×SD16—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#1450C2211yescharacteristic
60#1450C2211yescharacteristic
61#1303C2211yescharacteristic
62#1303C2211yescharacteristic
63#1301C2211yescharacteristic
64#1301C2211yescharacteristic
65#1276C2211yescharacteristic
66#1276C2211yescharacteristic
67C2×#131—21no—
68#533C2311yescharacteristic
69#532C2311yescharacteristic
70C2×C4×Q8—21no—
71C2×C4×Q8C2311yescharacteristic
72#488—21no—
73#488—21no—
74#487C2×C411yescharacteristic
75#487C2×C411yescharacteristic
76#486C2×C411yescharacteristic
77#486C2×C411yescharacteristic
78#484—21no—
79#481—21no—
80#477C2×C412yes—
81#475C2×C412yes—
82#462C2×C412yes—
83#460C2×C412yes—
84#451—21no—
85#448—21no—
86#446—21no—
87#446—21no—
88C8×Q8—21no—
89C8×Q8—21no—
90C8×Q8—21no—
91C8×Q8—21no—
92C8×Q8—21no—
93C8×Q8—21no—
94C8×Q8C2×C412yes—
95C8×Q8C2×C412yes—
96C8×Q8D412yes—
97C8×Q8D412yes—
98C4×Q16—22no—
99C4×SD16—22no—
100#436—21no—
101#436—21no—
102C8×D4—21no—
103C8×D4—21no—
104#431C2311yescharacteristic
105#431C2311yescharacteristic
106#431C2311yescharacteristic
107#431C2311yescharacteristic
108#424—21no—
109#423C2311yescharacteristic
110#423C2311yescharacteristic
111C2×C4⋊C8—21no—
112C2×C4⋊C8—21no—
113C2×C4⋊C8C2311yescharacteristic
114C2×C4⋊C8C2311yescharacteristic
115C2×#104—21no—
116C2×#104—21no—
117#416—21no—
118#416—21no—
119#408—21no—
120#405—21no—
121C2×C4×C8—21no—
122C2×C4×C8—21no—
123C2×C4×C8—21no—
124C2×C4×C8—21no—
125C2×C4×C8—21no—
126C2×C4×C8D411yescharacteristic
127C2×C4×C8D411yescharacteristic
128C2×C4×C8D411yescharacteristic
129C2×C4×C8D411yescharacteristic
130C2×C4×C8C2311yescharacteristic
131C2×C4×C8C2311yescharacteristic
132C4×C4⋊C4—21no—
133C4×C4⋊C4—21no—
134C4×C4⋊C4—22no—
135C4×C4⋊C4C2311yescharacteristic
136C43—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—
163C82—21no—
164C82—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—28no—
192C4×Q8—21no—
193C4×Q8—21no—
194C4×Q8—21no—
195C4×Q8—21no—
196C4×Q8C22×C412yes—
197C4×Q8C2×D412yes—
198C4×D4—21no—
199C4×D4—21no—
200#117—21no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4#4111yescharacteristic
205C2×C42—22no—
206C2×C42—21no—
207C2×C42—21no—
208C2×C42—22no—
209C2×C42—21no—
210C2×C42—21no—
211C2×C42C2×D411yescharacteristic
212C2×C42C2×D411yescharacteristic
213C2×C42C2411yescharacteristic
214#107—22no—
215#106—22no—
216C4⋊C8—22no—
217C4⋊C8—22no—
218C4⋊C8—22no—
219C4⋊C8—22no—
220C4⋊C8—21no—
221C4⋊C8—21no—
222C4⋊C8—21no—
223C4⋊C8—21no—
224C4⋊C8—21no—
225C4⋊C8—21no—
226C4⋊C8—21no—
227C4⋊C8—21no—
228C4⋊C8—41no—
229C4⋊C8—41no—
230C4⋊C8C22×C412yes—
231C4⋊C8C22×C412yes—
232C4⋊C8C22×C412yes—
233C4⋊C8C2×D412yes—
234#104—42no—
235#104—42no—
236#103—22no—
237#103—22no—
238#103—21no—
239#103—21no—
240#103—21no—
241#103—21no—
242#103—22no—
243#102—21no—
244#102—21no—
245#102—21no—
246#102—21no—
247#102—22no—
248C22⋊C8—21no—
249C22⋊C8—21no—
250C22⋊C8—21no—
251C22⋊C8—21no—
252C22⋊C8—41no—
253C22⋊C8—41no—
254#97—41no—
255C4×C8—22no—
256C4×C8—22no—
257C4×C8—22no—
258C4×C8—22no—
259C4×C8—22no—
260C4×C8—22no—
261C4×C8—22no—
262C4×C8—21no—
263C4×C8—21no—
264C4×C8—21no—
265C4×C8—21no—
266C4×C8—41no—
267C4×C8—42no—
268C4×C8—41no—
269C4×C8—42no—
270C4×C8—42no—
271C4×C8C22×C411yescharacteristic
272C4×C8C22×C411yescharacteristic
273C4×C8C2×D411yescharacteristic
274C4×C8C2×D411yescharacteristic
275C4×C8#4111yescharacteristic
276C4×C8#4111yescharacteristic
277C4×C8#4111yescharacteristic
278C4×C8#4111yescharacteristic
279#95—22no—
280#95#4112yes—
281#41—41no—
282#41—41no—
283#41—41no—
284#41—41no—
285C2×Q8—24no—
286C2×Q8—21no—
287C2×Q8—21no—
288C2×Q8—21no—
289C2×Q8—21no—
290C2×Q8C4×D412yes—
291C2×D4—21no—
292C2×D4—21no—
293C2×D4—21no—
294C2×D4—41no—
295C22×C4—21no—
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—21no—
299C22×C4—22no—
300C22×C4—41no—
301C22×C4C22×D411yescharacteristic
302C22×C4C2×#4111yescharacteristic
303C22×C4C2×#4111yescharacteristic
304Q16—44no—
305SD16—44no—
306M16—41no—
307M16—41no—
308M16—41no—
309M16—41no—
310M16—82no—
311C2×C8—21no—
312C2×C8—21no—
313C2×C8—21no—
314C2×C8—21no—
315C2×C8—21no—
316C2×C8—21no—
317C2×C8—22no—
318C2×C8—22no—
319C2×C8—22no—
320C2×C8—22no—
321C2×C8—41no—
322C2×C8—41no—
323C2×C8—41no—
324C2×C8—41no—
325C2×C8—41no—
326C2×C8—41no—
327C2×C8—41no—
328C2×C8—41no—
329C2×C8—82no—
330C2×C8C4×D411yescharacteristic
331C2×C8C4×D411yescharacteristic
332C2×C8C4×D411yescharacteristic
333C2×C8C4×D411yescharacteristic
334C2×C8#12111yescharacteristic
335C2×C8#12111yescharacteristic
336C2×C8#12111yescharacteristic
337C2×C8#12111yescharacteristic
338C2×C8#13512yes—
339C2×C8#13712yes—
340C4⋊C4—28no—
341C4⋊C4—24no—
342C4⋊C4—22no—
343C4⋊C4—21no—
344C4⋊C4—22no—
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⋊C4C4×D412yes—
354C22⋊C4—41no—
355C22⋊C4—41no—
356C22⋊C4—41no—
357C42—28no—
358C42—24no—
359C42—22no—
360C42—22no—
361C42—21no—
362C42—21no—
363C42—21no—
364C42—42no—
365C42—44no—
366C42—41no—
367C42—41no—
368C42—41no—
369C42#13512yes—
370C42C23×C411yescharacteristic
371C42C22×D411yescharacteristic
372C42C2×#4111yescharacteristic
373C42C2×#4111yescharacteristic
374C23—41no—
375C23#52511yescharacteristic
376Q8—24no—
377Q8—22no—
378Q8—41no—
379Q8—41no—
380Q8—42no—
381Q8—81no—
382Q8#44314yes—
383D4—41no—
384D4—41no—
385D4—42no—
386D4—81no—
387D4—81no—
388D4—81no—
389C2×C4—22no—
390C2×C4—24no—
391C2×C4—22no—
392C2×C4—22no—
393C2×C4—22no—
394C2×C4—22no—
395C2×C4—22no—
396C2×C4—24no—
397C2×C4—22no—
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—81no—
408C2×C4—81no—
409C2×C4C2×C4×D411yescharacteristic
410C2×C4C2×C4×D411yescharacteristic
411C2×C4C4×#4111yescharacteristic
412C2×C4C2×#12111yescharacteristic
413C2×C4#52911yescharacteristic
414C2×C4#53211yescharacteristic
415C2×C4C2×#13511yescharacteristic
416C2×C4C2×#13511yescharacteristic
417C2×C4C2×#13711yescharacteristic
418C2×C4#57511yescharacteristic
419C8—21no—
420C8—21no—
421C8—21no—
422C8—21no—
423C8—21no—
424C8—21no—
425C8—42no—
426C8—42no—
427C8—42no—
428C8—42no—
429C8—81no—
430C8—81no—
431C8—81no—
432C8—81no—
433C8#46412yes—
434C8#46412yes—
435C22—21no—
436C22—21no—
437C22—81no—
438C22—81no—
439C22—81no—
440C22C4×#12111yescharacteristic
441C22#310211yescharacteristic
442C22#310811yescharacteristic
443C4—24no—
444C4—24no—
445C4—21no—
446C4—21no—
447C4—21no—
448C4—42no—
449C4—44no—
450C4—42no—
451C4—81no—
452C4—81no—
453C4—81no—
454C4C4×#13511yescharacteristic
455C4C4×#13711yescharacteristic
456C4C2×#44312yes—
457C4C2×#46411yescharacteristic
458C4#303511yescharacteristic
459C2—21no—
460C2—21no—
461C2—161no—
462C2C4×#46411yescharacteristic
463C2#2154411yescharacteristic
464C2#2155811yescharacteristic
subgroupG/H
centerC2×C8#121
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
socleC22C4×#121
prime radical#532C23
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C2×C8 ◁ C2×C4×C8 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C4×C8×Q8C2semidirect
4. #17903C2semidirect
7. #9629C2semidirect
8. #9579C2semidirect
11. #9232C2semidirect
12. #8661C2semidirect
13. #8266C2semidirect
14. #8252C2semidirect
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
C4×C8×Q82
#191232
#179032
#179012
#97512
#96292
#95792
#95772
#92342
#92322
#86612
#82662
#82522
#82512