Finite Group Databasebeta

← #326579

Subgroups — #326579

Order 512.

461 conjugacy classes of subgroups.

The subgroup lattice has 419 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21717C211yesmaximal, characteristic, complemented
2#21659C211yesmaximal, characteristic
3#10668C211yesmaximal, characteristic, complemented
4#10666C211yesmaximal, characteristic
5#10371C211yesmaximal, characteristic, complemented
6#10370C211yesmaximal, characteristic
7#9754C211yesmaximal, characteristic
8#9751C211yesmaximal, characteristic
9#9632C211yesmaximal, characteristic, complemented
10#9629C211yesmaximal, characteristic, complemented
11#8881C211yesmaximal, characteristic, complemented
12#8881C211yesmaximal, characteristic, complemented
13#8878C211yesmaximal, characteristic
14#8878C211yesmaximal, characteristic
15#8780C211yesmaximal, characteristic, complemented
16#3131C2211yescharacteristic
17#2961C2211yescharacteristic
18C2×#446—21no—
19#2932C2211yescharacteristic
20#2399C2211yescharacteristic
21#1995C2211yescharacteristic
22#1984—21no—
23#1963C2211yescharacteristic
24#1940C2211yescharacteristic
25#1940C2211yescharacteristic
26#1939—21no—
27#1938—21no—
28#1937—21no—
29#1843C2211yescharacteristic
30#1837C2211yescharacteristic
31#1771—21no—
32#1770C2211yescharacteristic
33C8×C4⋊C4C2211yescharacteristic
34#1765—21no—
35#1764—21no—
36#1726C2211yescharacteristic
37#1571C2211yescharacteristic
38#1571C2211yescharacteristic
39#1570C2211yescharacteristic
40#1570C2211yescharacteristic
41#1564C2211yescharacteristic
42#1564C2211yescharacteristic
43#1503C2211yescharacteristic
44#1503C2211yescharacteristic
45#1502C2211yescharacteristic
46#1502C2211yescharacteristic
47#1481C2211yescharacteristic
48#1481C2211yescharacteristic
49#1480C2211yescharacteristic
50#1480C2211yescharacteristic
51#1303C2211yescharacteristic
52#1303C2211yescharacteristic
53#1303C2211yescharacteristic
54#1303C2211yescharacteristic
55#1281C2211yescharacteristic
56#1281C2211yescharacteristic
57#1274C2211yescharacteristic
58#1274C2211yescharacteristic
59C2×#131—21no—
60#533C2311yescharacteristic
61#532C2311yescharacteristic
62C2×C4×Q8—21no—
63#488—21no—
64#488—21no—
65#487Q811yescharacteristic, complemented
66#487Q811yescharacteristic, complemented
67#486Q811yescharacteristic, complemented
68#486Q811yescharacteristic, complemented
69#446—41no—
70#446—41no—
71#446—44no—
72#435—41no—
73#435—41no—
74#431C2311yescharacteristic
75#431C2311yescharacteristic
76#431C2311yescharacteristic
77#431C2311yescharacteristic
78#424—21no—
79#423C2311yescharacteristic
80#423C2311yescharacteristic
81C2×C4⋊C8—21no—
82C2×C4⋊C8—21no—
83C2×C4⋊C8—21no—
84C2×C4⋊C8—21no—
85C2×C4⋊C8C2311yescharacteristic
86C2×C4⋊C8C2311yescharacteristic
87C2×#104—21no—
88C2×#104—21no—
89#416—21no—
90#416—21no—
91#408—21no—
92#405—21no—
93C2×#97—21no—
94C2×#97D411yescharacteristic
95C2×#97D411yescharacteristic
96C2×#97C2311yescharacteristic
97C2×C4×C8—21no—
98C2×C4×C8D411yescharacteristic
99C2×C4×C8D411yescharacteristic
100C2×C4×C8C2311yescharacteristic
101#398—21no—
102#398—21no—
103#395—21no—
104#395—21no—
105C4×C4⋊C4—21no—
106C4×C4⋊C4C2311yescharacteristic
107C4×C4⋊C4C2311yescharacteristic
108#376C2311yescharacteristic
109#341—21no—
110#341—21no—
111#339—21no—
112#339—21no—
113#336—21no—
114#336—21no—
115#336—21no—
116#336—21no—
117#335—21no—
118#335—21no—
119#335—41no—
120#335—41no—
121#334—21no—
122#334—21no—
123#334—41no—
124#334—41no—
125#328—21no—
126#328—21no—
127#328—21no—
128#328—21no—
129#327—21no—
130#327—21no—
131#327—21no—
132#327—21no—
133Q8⋊C8—41no—
134Q8⋊C8—41no—
135D4⋊C8—41no—
136D4⋊C8—41no—
137C2×#41—21no—
138C22×Q8—21no—
139#131—42no—
140#131—42no—
141C2×M16—21no—
142C2×M16—21no—
143C2×M16—41no—
144C22×C8—21no—
145C22×C8—21no—
146C22×C8—41no—
147C22×C8C2×D411yescharacteristic
148C22×C8C2×D411yescharacteristic
149C22×C8C2×D411yescharacteristic
150C22×C8C2×D411yescharacteristic
151C4⋊Q8C2×Q811yescharacteristic
152C4⋊Q8C2×Q811yescharacteristic
153C4⋊Q8C2×Q811yescharacteristic
154#127C2×Q811yescharacteristic
155#125—21no—
156#124—21no—
157C22⋊Q8—21no—
158C22⋊Q8—21no—
159#121—21no—
160#121—21no—
161C4×Q8—21no—
162C4×Q8—21no—
163C4×Q8—42no—
164C4×Q8—42no—
165C4×D4—21no—
166C4×D4—21no—
167#117—21no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—21no—
175C2×C42—21no—
176C2×C42—21no—
177C2×C42C2×D411yescharacteristic
178C2×C42C2×D411yescharacteristic
179C2×C42C2411yescharacteristic
180C4⋊C8—21no—
181C4⋊C8—21no—
182C4⋊C8—21no—
183C4⋊C8—21no—
184C4⋊C8—21no—
185C4⋊C8—21no—
186C4⋊C8—21no—
187C4⋊C8—21no—
188C4⋊C8—21no—
189C4⋊C8—21no—
190C4⋊C8—21no—
191C4⋊C8—21no—
192C4⋊C8—42no—
193C4⋊C8—42no—
194C4⋊C8—42no—
195C4⋊C8—42no—
196C4⋊C8—42no—
197#104—42no—
198#104—42no—
199#103—21no—
200#103—21no—
201#103—21no—
202#103—21no—
203#102—21no—
204#102—21no—
205#102—21no—
206#102—21no—
207C22⋊C8—21no—
208C22⋊C8—21no—
209C22⋊C8—21no—
210C22⋊C8—21no—
211C22⋊C8—42no—
212#97—22no—
213#97—22no—
214#97—21no—
215#97—21no—
216#97—42no—
217#97—41no—
218C4×C8—22no—
219C4×C8—21no—
220C4×C8—21no—
221C4×C8—21no—
222C4×C8—21no—
223C4×C8—41no—
224C4×C8—42no—
225C4×C8—41no—
226C4×C8C2×Q811yescharacteristic
227C4×C8C2×Q811yescharacteristic
228C4×C8#4111yescharacteristic
229C4×C8#4111yescharacteristic
230#95—21no—
231#95—21no—
232#95—21no—
233#95—21no—
234#95#4111yescharacteristic
235#95#4111yescharacteristic
236#95#4111yescharacteristic
237#95#4111yescharacteristic
238#41—41no—
239#41—41no—
240#41—41no—
241#41—41no—
242C2×Q8—21no—
243C2×Q8—21no—
244C2×Q8—21no—
245C2×Q8—42no—
246C2×Q8—42no—
247C2×Q8—81no—
248C2×D4—21no—
249C2×D4—21no—
250C2×D4—21no—
251C2×D4—41no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—21no—
258C22×C4—41no—
259C22×C4C22×D411yescharacteristic
260C22×C4C2×#4111yescharacteristic
261C22×C4C2×#4111yescharacteristic
262M16—41no—
263M16—41no—
264M16—41no—
265M16—41no—
266M16—82no—
267C2×C8—21no—
268C2×C8—21no—
269C2×C8—21no—
270C2×C8—21no—
271C2×C8—21no—
272C2×C8—21no—
273C2×C8—21no—
274C2×C8—21no—
275C2×C8—21no—
276C2×C8—41no—
277C2×C8—41no—
278C2×C8—41no—
279C2×C8—41no—
280C2×C8—41no—
281C2×C8—41no—
282C2×C8—41no—
283C2×C8—41no—
284C2×C8—41no—
285C2×C8—41no—
286C2×C8—41no—
287C2×C8—41no—
288C2×C8—82no—
289C2×C8C22⋊Q811yescharacteristic
290C2×C8C22⋊Q811yescharacteristic
291C2×C8C22⋊Q811yescharacteristic
292C2×C8C22⋊Q811yescharacteristic
293C2×C8AGL(1,ℤ/8)12yes—
294C4⋊C4—21no—
295C4⋊C4—21no—
296C4⋊C4—21no—
297C4⋊C4—21no—
298C4⋊C4—21no—
299C4⋊C4—21no—
300C4⋊C4—42no—
301C4⋊C4—42no—
302C4⋊C4—42no—
303C4⋊C4—42no—
304C4⋊C4—42no—
305C4⋊C4—41no—
306C4⋊C4—41no—
307C4⋊C4—81no—
308C4⋊C4—81no—
309C22⋊C4—41no—
310C22⋊C4—41no—
311C22⋊C4—41no—
312C42—21no—
313C42—21no—
314C42—41no—
315C42—41no—
316C42—41no—
317C42—41no—
318C42—41no—
319C42#13512yes—
320C42#13712yes—
321C42C22×Q811yescharacteristic
322C42C2×#4111yescharacteristic
323C42H(5,𝔽2)11yescharacteristic
324C42H(5,𝔽2)11yescharacteristic
325C23—41no—
326C23#52511yescharacteristic
327Q8—41no—
328Q8—41no—
329Q8—81no—
330Q8—81no—
331Q8—84no—
332D4—41no—
333D4—41no—
334D4—81no—
335D4—81no—
336D4—81no—
337D4—81no—
338C2×C4—22no—
339C2×C4—22no—
340C2×C4—22no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—81no—
362C2×C4—81no—
363C2×C4C2×C22⋊Q811yescharacteristic
364C2×C4#53511yescharacteristic
365C2×C4#55411yescharacteristic
366C2×C4#55411yescharacteristic
367C2×C4#55911yescharacteristic
368C2×C4#55911yescharacteristic
369C2×C4C2×#13511yescharacteristic
370C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
371C2×C4C2×#13711yescharacteristic
372C2×C4#57511yescharacteristic
373C8—21no—
374C8—21no—
375C8—21no—
376C8—21no—
377C8—22no—
378C8—41no—
379C8—81no—
380C8—81no—
381C8—81no—
382C8—81no—
383C8—81no—
384C8—81no—
385C8—81no—
386C8—81no—
387C22—21no—
388C22—21no—
389C22—81no—
390C22—81no—
391C22—81no—
392C22#258011yescharacteristic
393C22#310511yescharacteristic
394C22#311711yescharacteristic
395C4—21no—
396C4—21no—
397C4—21no—
398C4—21no—
399C4—42no—
400C4—42no—
401C4—42no—
402C4—81no—
403C4—81no—
404C4—81no—
405C4—81no—
406C4—81no—
407C4—81no—
408C4#307611yescharacteristic
409C4#307911yescharacteristic
410C4#321011yescharacteristic
411C4#321311yescharacteristic
412C2—21no—
413C2—21no—
414C2—161no—
415C2#2926011yescharacteristic
416C2#2927011yescharacteristic
417C2#2927111yescharacteristic
subgroupG/H
centerC2×C4#535
commutator, Frattini, cosocleC2×C42C24
socleC22#2580
prime radical#532C23
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #21717C2semidirect
3. #10668C2semidirect
5. #10371C2semidirect
9. #9632C2semidirect
10. #9629C2semidirect
11. #8881C2semidirect
12. #8881C2semidirect
15. #8780C2semidirect
65. #487Q8semidirect
66. #487Q8semidirect
67. #486Q8semidirect
68. #486Q8semidirect

Maximal subgroups

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

subgroupindex
#217172
#216592
#106682
#106662
#103712
#103702
#97542
#97512
#96322
#96292
#88812
#88782
#87802