Finite Group Databasebeta

← #326567

Subgroups — #326567

Order 512.

424 conjugacy classes of subgroups.

The subgroup lattice has 409 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21677C211yesmaximal, characteristic, complemented
2#21659C211yesmaximal, characteristic
3#10669C211yesmaximal, characteristic, complemented
4#10665C211yesmaximal, characteristic
5#10373C211yesmaximal, characteristic, complemented
6#10372C211yesmaximal, characteristic
7#9754C211yesmaximal, characteristic
8#9754C211yesmaximal, characteristic
9#9632C211yesmaximal, characteristic, complemented
10#9632C211yesmaximal, characteristic, complemented
11#8881C211yesmaximal, characteristic, complemented
12#8880C211yesmaximal, characteristic
13#8879C211yesmaximal, characteristic, complemented
14#8878C211yesmaximal, characteristic
15#8781C211yesmaximal, characteristic, complemented
16#3131C2211yescharacteristic
17#2961C2211yescharacteristic
18#2932C2211yescharacteristic
19#2402C2211yescharacteristic
20#1995C2211yescharacteristic
21#1963C2211yescharacteristic
22#1940C2211yescharacteristic
23#1940C2211yescharacteristic
24#1938—21no—
25#1937—21no—
26#1917—21no—
27#1917—21no—
28#1837C2211yescharacteristic
29#1832C2211yescharacteristic
30#1771—21no—
31#1771—21no—
32#1770C2211yescharacteristic
33#1770C2211yescharacteristic
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
62#488—21no—
63#488—21no—
64#487Q811yescharacteristic
65#487Q811yescharacteristic
66#486Q811yescharacteristic
67#486Q811yescharacteristic
68#446—41no—
69#446—41no—
70#435—41no—
71#435—41no—
72#431C2311yescharacteristic
73#431C2311yescharacteristic
74#431C2311yescharacteristic
75#431C2311yescharacteristic
76#424—21no—
77#423C2311yescharacteristic
78#423C2311yescharacteristic
79C2×C4⋊C8—21no—
80C2×C4⋊C8—21no—
81C2×C4⋊C8C2311yescharacteristic
82C2×C4⋊C8C2311yescharacteristic
83C2×#104—21no—
84C2×#104—21no—
85#416—21no—
86#416—21no—
87#408—21no—
88#405—21no—
89C2×#97—21no—
90C2×#97—21no—
91C2×#97D411yescharacteristic
92C2×#97D411yescharacteristic
93C2×#97C2311yescharacteristic
94C2×C4×C8D411yescharacteristic
95C2×C4×C8D411yescharacteristic
96C2×C4×C8C2311yescharacteristic
97#400—21no—
98#400—21no—
99#398—21no—
100#398—21no—
101C4×C4⋊C4—21no—
102C4×C4⋊C4—21no—
103C4×C4⋊C4C2311yescharacteristic
104C4×C4⋊C4C2311yescharacteristic
105#376C2311yescharacteristic
106#341—21no—
107#341—21no—
108#339—21no—
109#339—21no—
110#336—21no—
111#336—21no—
112#336—21no—
113#336—21no—
114#336—21no—
115#336—21no—
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—
137#131—42no—
138#131—42no—
139C2×M16—21no—
140C2×M16—21no—
141C2×M16—41no—
142C22×C8—21no—
143C22×C8—21no—
144C22×C8—41no—
145C22×C8C2×D411yescharacteristic
146C22×C8C2×D411yescharacteristic
147C22×C8C2×D411yescharacteristic
148C22×C8C2×D411yescharacteristic
149C4⋊Q8C2×Q811yescharacteristic
150C4⋊Q8C2×Q811yescharacteristic
151C4⋊Q8C2×Q811yescharacteristic
152#127C2×Q811yescharacteristic
153#125—21no—
154#124—21no—
155C22⋊Q8—21no—
156C22⋊Q8—21no—
157#121—21no—
158#121—21no—
159C4×Q8—21no—
160C4×Q8—21no—
161C4×D4—21no—
162C4×D4—21no—
163#117—21no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—21no—
170C2×C42—21no—
171C2×C42—21no—
172C2×C42C2×D411yescharacteristic
173C2×C42C2×D411yescharacteristic
174C2×C42C2411yescharacteristic
175C4⋊C8—21no—
176C4⋊C8—21no—
177C4⋊C8—21no—
178C4⋊C8—21no—
179C4⋊C8—21no—
180C4⋊C8—21no—
181C4⋊C8—21no—
182C4⋊C8—21no—
183C4⋊C8—21no—
184C4⋊C8—21no—
185C4⋊C8—21no—
186C4⋊C8—21no—
187C4⋊C8—42no—
188C4⋊C8—42no—
189C4⋊C8—42no—
190#104—42no—
191#104—42no—
192#103—21no—
193#103—21no—
194#103—21no—
195#103—21no—
196#102—21no—
197#102—21no—
198#102—21no—
199#102—21no—
200C22⋊C8—21no—
201C22⋊C8—21no—
202C22⋊C8—21no—
203C22⋊C8—21no—
204C22⋊C8—42no—
205#97—22no—
206#97—21no—
207#97—21no—
208#97—41no—
209#97—42no—
210#97—42no—
211#97—41no—
212C4×C8—21no—
213C4×C8—21no—
214C4×C8—21no—
215C4×C8—21no—
216C4×C8—41no—
217C4×C8—41no—
218C4×C8—41no—
219C4×C8C2×Q811yescharacteristic
220C4×C8C2×Q811yescharacteristic
221C4×C8#4111yescharacteristic
222C4×C8#4111yescharacteristic
223#95—21no—
224#95—21no—
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×D4—21no—
241C2×D4—21no—
242C2×D4—21no—
243C2×D4—41no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—41no—
251C22×C4C22×D411yescharacteristic
252C22×C4C2×#4111yescharacteristic
253C22×C4C2×#4111yescharacteristic
254M16—41no—
255M16—41no—
256M16—41no—
257M16—41no—
258M16—82no—
259C2×C8—21no—
260C2×C8—21no—
261C2×C8—21no—
262C2×C8—21no—
263C2×C8—21no—
264C2×C8—21no—
265C2×C8—21no—
266C2×C8—21no—
267C2×C8—21no—
268C2×C8—41no—
269C2×C8—41no—
270C2×C8—41no—
271C2×C8—41no—
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—82no—
281C2×C8C22⋊Q811yescharacteristic
282C2×C8C22⋊Q811yescharacteristic
283C2×C8C22⋊Q811yescharacteristic
284C2×C8C22⋊Q811yescharacteristic
285C2×C8#13512yes—
286C4⋊C4—21no—
287C4⋊C4—21no—
288C4⋊C4—21no—
289C4⋊C4—21no—
290C4⋊C4—21no—
291C4⋊C4—21no—
292C4⋊C4—41no—
293C4⋊C4—41no—
294C4⋊C4—81no—
295C4⋊C4—81no—
296C4⋊C4—81no—
297C4⋊C4—81no—
298C4⋊C4—81no—
299C4⋊C4—81no—
300C22⋊C4—41no—
301C22⋊C4—41no—
302C22⋊C4—41no—
303C42—21no—
304C42—21no—
305C42—21no—
306C42—21no—
307C42—41no—
308C42—41no—
309C42—41no—
310C42—41no—
311C42—41no—
312C42C22×Q811yescharacteristic
313C42C2×#4111yescharacteristic
314C42H(5,𝔽2)11yescharacteristic
315C42H(5,𝔽2)11yescharacteristic
316C23—41no—
317C23#52511yescharacteristic
318Q8—41no—
319Q8—41no—
320Q8—81no—
321Q8—81no—
322D4—41no—
323D4—41no—
324D4—81no—
325D4—81no—
326D4—81no—
327D4—81no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—41no—
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—81no—
352C2×C4—81no—
353C2×C4C2×C22⋊Q811yescharacteristic
354C2×C4#53511yescharacteristic
355C2×C4#55411yescharacteristic
356C2×C4#55411yescharacteristic
357C2×C4#55911yescharacteristic
358C2×C4#55911yescharacteristic
359C2×C4C2×#13511yescharacteristic
360C2×C4#57511yescharacteristic
361C2×C4#57511yescharacteristic
362C2×C4#57511yescharacteristic
363C8—21no—
364C8—21no—
365C8—21no—
366C8—21no—
367C8—22no—
368C8—41no—
369C8—81no—
370C8—81no—
371C8—81no—
372C8—81no—
373C8—81no—
374C8—81no—
375C8—81no—
376C8—81no—
377C22—21no—
378C22—21no—
379C22—81no—
380C22—81no—
381C22—81no—
382C22#258011yescharacteristic
383C22#310211yescharacteristic
384C22#311111yescharacteristic
385C4—21no—
386C4—21no—
387C4—21no—
388C4—21no—
389C4—81no—
390C4—81no—
391C4—81no—
392C4—81no—
393C4—81no—
394C4—81no—
395C4—81no—
396C4—81no—
397C4—81no—
398C4#307611yescharacteristic
399C4#308011yescharacteristic
400C4#321111yescharacteristic
401C4#321911yescharacteristic
402C2—21no—
403C2—21no—
404C2—161no—
405C2#2926111yescharacteristic
406C2#2927111yescharacteristic
407C2#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. #21677C2semidirect
3. #10669C2semidirect
5. #10373C2semidirect
9. #9632C2semidirect
10. #9632C2semidirect
11. #8881C2semidirect
13. #8879C2semidirect
15. #8781C2semidirect

Maximal subgroups

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

subgroupindex
#216772
#216592
#106692
#106652
#103732
#103722
#97542
#96322
#88812
#88802
#88792
#88782
#87812