Finite Group Databasebeta

← #326585

Subgroups — #326585

Order 512.

431 conjugacy classes of subgroups.

The subgroup lattice has 414 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22308C211yesmaximal, characteristic, complemented
2#22236C211yesmaximal, characteristic
3#10669C211yesmaximal, characteristic, complemented
4#10665C211yesmaximal, characteristic
5#10371C211yesmaximal, characteristic, complemented
6#10370C211yesmaximal, characteristic
7#9751C211yesmaximal, characteristic
8#9751C211yesmaximal, characteristic
9#9628C211yesmaximal, characteristic, complemented
10#9627C211yesmaximal, characteristic, complemented
11#8881C211yesmaximal, characteristic, complemented
12#8880C211yesmaximal, characteristic
13#8879C211yesmaximal, characteristic, complemented
14#8878C211yesmaximal, characteristic
15#8782C211yesmaximal, characteristic, complemented
16#3131C2211yescharacteristic
17#2961C2211yescharacteristic
18#2932C2211yescharacteristic
19#2399C2211yescharacteristic
20#1995C2211yescharacteristic
21#1984—21no—
22#1984—21no—
23#1963C2211yescharacteristic
24#1940C2211yescharacteristic
25#1940C2211yescharacteristic
26#1929—21no—
27#1929—21no—
28#1916—21no—
29#1916—21no—
30#1866—21no—
31#1866—21no—
32#1848C2211yescharacteristic
33#1832C2211yescharacteristic
34C8×C4⋊C4C2211yescharacteristic
35C8×C4⋊C4C2211yescharacteristic
36#1746C2211yescharacteristic
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#435—41no—
72#435—41no—
73#431C2311yescharacteristic
74#431C2311yescharacteristic
75#431C2311yescharacteristic
76#431C2311yescharacteristic
77C2×#108—21no—
78C2×#108—21no—
79#424—21no—
80#423C2311yescharacteristic
81#423C2311yescharacteristic
82C2×C4⋊C8—21no—
83C2×C4⋊C8—21no—
84C2×C4⋊C8C2311yescharacteristic
85C2×C4⋊C8C2311yescharacteristic
86C2×#104—21no—
87C2×#104—21no—
88#416—21no—
89#416—21no—
90#408—21no—
91#405—21no—
92C2×#97C2311yescharacteristic
93C2×C4×C8—21no—
94C2×C4×C8—21no—
95C2×C4×C8C2311yescharacteristic
96#398—21no—
97#398—21no—
98#395—21no—
99#395—21no—
100C4×C4⋊C4—21no—
101C4×C4⋊C4C2311yescharacteristic
102C4×C4⋊C4C2311yescharacteristic
103#376C2311yescharacteristic
104#339—21no—
105#339—21no—
106#336—21no—
107#336—21no—
108#336—21no—
109#336—21no—
110#336—21no—
111#336—21no—
112#336D411yescharacteristic
113#336D411yescharacteristic
114#336D411yescharacteristic
115#336D411yescharacteristic
116#335—21no—
117#335—21no—
118#335—41no—
119#335—41no—
120#334—21no—
121#334—21no—
122#334—41no—
123#334—41no—
124#328—21no—
125#328—21no—
126#328—21no—
127#328—21no—
128#327—21no—
129#327—21no—
130#327—21no—
131#327—21no—
132Q8⋊C8—41no—
133Q8⋊C8—41no—
134D4⋊C8—41no—
135D4⋊C8—41no—
136C2×#41—21no—
137C22×Q8—21no—
138#131—81no—
139#131—81no—
140C2×M16—21no—
141C2×M16—21no—
142C2×M16—41no—
143C22×C8—21no—
144C22×C8—21no—
145C22×C8—41no—
146C22×C8C2×D411yescharacteristic
147C22×C8C2×D411yescharacteristic
148C22×C8C2×D411yescharacteristic
149C22×C8C2×D411yescharacteristic
150C4⋊Q8C2×Q811yescharacteristic
151C4⋊Q8C2×Q811yescharacteristic
152C4⋊Q8C2×Q811yescharacteristic
153#127C2×Q811yescharacteristic
154#125—21no—
155#124—21no—
156C22⋊Q8—21no—
157C22⋊Q8—21no—
158#121—21no—
159#121—21no—
160C4×Q8—21no—
161C4×Q8—21no—
162C4×Q8—81no—
163C4×Q8—81no—
164C4×D4—21no—
165C4×D4—21no—
166#117—21no—
167C2×C4⋊C4—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×C42—21no—
175C2×C42—21no—
176C2×C42C2×D411yescharacteristic
177C2×C42C2×D411yescharacteristic
178C2×C42C2411yescharacteristic
179#108—81no—
180#108—81no—
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—21no—
193C4⋊C8—42no—
194C4⋊C8—81no—
195C4⋊C8—81no—
196#104—81no—
197#104—81no—
198#103—21no—
199#103—21no—
200#103—21no—
201#103—21no—
202#102—21no—
203#102—21no—
204#102—21no—
205#102—21no—
206C22⋊C8—21no—
207C22⋊C8—21no—
208C22⋊C8—21no—
209C22⋊C8—21no—
210C22⋊C8—42no—
211#97—21no—
212#97—21no—
213#97—41no—
214C4×C8—21no—
215C4×C8—21no—
216C4×C8—21no—
217C4×C8—21no—
218C4×C8—41no—
219C4×C8—81no—
220C4×C8—81no—
221C4×C8C2×Q811yescharacteristic
222C4×C8C2×Q811yescharacteristic
223C4×C8#4111yescharacteristic
224C4×C8#4111yescharacteristic
225#95—21no—
226#95—21no—
227#95—21no—
228#95—21no—
229#95#4111yescharacteristic
230#95#4111yescharacteristic
231#95#4111yescharacteristic
232#95#4111yescharacteristic
233#41—41no—
234#41—41no—
235#41—41no—
236#41—41no—
237C2×Q8—21no—
238C2×Q8—21no—
239C2×Q8—21no—
240C2×Q8—42no—
241C2×Q8—42no—
242C2×Q8—81no—
243C2×D4—21no—
244C2×D4—21no—
245C2×D4—21no—
246C2×D4—41no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—41no—
254C22×C4C22×D411yescharacteristic
255C22×C4C2×#4111yescharacteristic
256C22×C4C2×#4111yescharacteristic
257M16—41no—
258M16—41no—
259M16—41no—
260M16—41no—
261M16—82no—
262C2×C8—21no—
263C2×C8—21no—
264C2×C8—21no—
265C2×C8—21no—
266C2×C8—21no—
267C2×C8—21no—
268C2×C8—21no—
269C2×C8—21no—
270C2×C8—21no—
271C2×C8—21no—
272C2×C8—41no—
273C2×C8—41no—
274C2×C8—41no—
275C2×C8—41no—
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—82no—
285C2×C8C22⋊Q811yescharacteristic
286C2×C8C22⋊Q811yescharacteristic
287C2×C8C22⋊Q811yescharacteristic
288C2×C8C22⋊Q811yescharacteristic
289C4⋊C4—21no—
290C4⋊C4—21no—
291C4⋊C4—21no—
292C4⋊C4—21no—
293C4⋊C4—21no—
294C4⋊C4—21no—
295C4⋊C4—42no—
296C4⋊C4—42no—
297C4⋊C4—42no—
298C4⋊C4—42no—
299C4⋊C4—41no—
300C4⋊C4—41no—
301C4⋊C4—81no—
302C4⋊C4—81no—
303C4⋊C4—81no—
304C22⋊C4—41no—
305C22⋊C4—41no—
306C22⋊C4—41no—
307C42—21no—
308C42—21no—
309C42—21no—
310C42—21no—
311C42—41no—
312C42—41no—
313C42—41no—
314C42—41no—
315C42—41no—
316C42C22×Q811yescharacteristic
317C42C2×#4111yescharacteristic
318C42H(5,𝔽2)11yescharacteristic
319C42H(5,𝔽2)11yescharacteristic
320C23—41no—
321C23#52511yescharacteristic
322Q8—41no—
323Q8—41no—
324Q8—81no—
325Q8—81no—
326Q8—84no—
327D4—41no—
328D4—41no—
329D4—81no—
330D4—81no—
331D4—81no—
332D4—81no—
333C2×C4—22no—
334C2×C4—22no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—41no—
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—81no—
357C2×C4—81no—
358C2×C4C2×C22⋊Q811yescharacteristic
359C2×C4#53511yescharacteristic
360C2×C4#55411yescharacteristic
361C2×C4#55411yescharacteristic
362C2×C4#55911yescharacteristic
363C2×C4#55911yescharacteristic
364C2×C4#57611yescharacteristic
365C2×C4#57711yescharacteristic
366C2×C4#57711yescharacteristic
367C2×C4#57811yescharacteristic
368C8—21no—
369C8—21no—
370C8—21no—
371C8—21no—
372C8—41no—
373C8—41no—
374C8—81no—
375C8—81no—
376C8—81no—
377C8—81no—
378C8—81no—
379C8—81no—
380C8—81no—
381C8—81no—
382C22—21no—
383C22—21no—
384C22—81no—
385C22—81no—
386C22—81no—
387C22#258011yescharacteristic
388C22#311811yescharacteristic
389C22#312111yescharacteristic
390C4—21no—
391C4—21no—
392C4—21no—
393C4—21no—
394C4—42no—
395C4—42no—
396C4—81no—
397C4—81no—
398C4—81no—
399C4—81no—
400C4—81no—
401C4—81no—
402C4—81no—
403C4#308311yescharacteristic
404C4#308411yescharacteristic
405C4#322411yescharacteristic
406C4#322511yescharacteristic
407C2—21no—
408C2—21no—
409C2—161no—
410C2#2926711yescharacteristic
411C2#2927011yescharacteristic
412C2#2927011yescharacteristic
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. #22308C2semidirect
3. #10669C2semidirect
5. #10371C2semidirect
9. #9628C2semidirect
10. #9627C2semidirect
11. #8881C2semidirect
13. #8879C2semidirect
15. #8782C2semidirect
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
#223082
#222362
#106692
#106652
#103712
#103702
#97512
#96282
#96272
#88812
#88802
#88792
#88782
#87822