Finite Group Databasebeta

← #7231924

Subgroups — #7231924

Order 512.

1247 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#53132C212yesmaximal, complemented
2#33284C212yesmaximal, complemented
3#32857C211yesmaximal, characteristic, complemented
4#32814C211yesmaximal, characteristic
5#32253C214yesmaximal, complemented
6#31830C214yesmaximal, complemented
7#31658C214yesmaximal, complemented
8#29080C214yesmaximal, complemented
9#28902C214yesmaximal, complemented
10#27829C214yesmaximal, complemented
11#21677C211yesmaximal, characteristic, complemented
12#2827—24no—
13#2823—24no—
14#2718—24no—
15#2695—24no—
16#2647C2212yes—
17#2641C2212yes—
18#2566C2214yes—
19#2552C2214yes—
20#2512C2214yes—
21#2512C2214yes—
22#2454C2214yes—
23#2445C2214yes—
24#2402C2211yescharacteristic
25#2101C2214yes—
26#2099C2214yes—
27#2088C2214yes—
28#2084C2214yes—
29#2071C2214yes—
30#2065C2214yes—
31#2054C2214yes—
32#2054C2214yes—
33#2025C2214yes—
34#2025C2214yes—
35#2025C2214yes—
36#2011C2214yes—
37#1992C2212yes—
38#1992C2214yes—
39#1991C2212yes—
40#1989C2214yes—
41#1972C2214yes—
42#1961C2212yes—
43#1957C2212yes—
44#1940C2212yes—
45#1936C2214yes—
46#1936C2214yes—
47#1936C2214yes—
48#1934C2214yes—
49#1934C2214yes—
50#1926C2214yes—
51#1919C2214yes—
52#1917C2214yes—
53#1879C2214yes—
54#1879C2214yes—
55#1832C2211yescharacteristic
56#1802C2214yes—
57#1771C2214yes—
58#1770C2212yes—
59#1762C2214yes—
60#1726C2211yescharacteristic
61C22×D8C2314yes—
62C2×#127C2312yes—
63C2×#126—24no—
64C2×#126—24no—
65C2×#125C2314yes—
66C2×#125C2312yes—
67C2×#123—24no—
68C2×C4×D4C2314yes—
69C2×#106C2314yes—
70C2×C4⋊C8C2312yes—
71C2×C4⋊C8C2314yes—
72C2×#102C2314yes—
73C2×#102C2314yes—
74C2×#102C2314yes—
75C2×#102C2314yes—
76C2×#102C2314yes—
77C2×#102C2314yes—
78C2×#97C2312yes—
79C2×#97C2311yescharacteristic
80C2×#97C2314yes—
81C2×C4×C8C2311yescharacteristic
82C2×C4×C8C2312yes—
83#400—24no—
84#400D414yes—
85#399—24no—
86#398—24no—
87#398—24no—
88#398—24no—
89#398D414yes—
90#397—24no—
91#397—24no—
92#396—24no—
93#396—24no—
94#396—24no—
95#396—24no—
96#396—24no—
97#394—24no—
98#394—24no—
99#394—24no—
100#394—24no—
101#390C2314yes—
102#390C2314yes—
103#390C2314yes—
104#390C2314yes—
105#390C2314yes—
106#389—24no—
107#389C2314yes—
108#389C2314yes—
109#389C2314yes—
110#388—24no—
111#388—24no—
112#386—24no—
113#385—24no—
114#385—24no—
115#385—24no—
116#384—24no—
117#384C2314yes—
118#384C2312yes—
119#382C2312yes—
120#380—24no—
121C4×C4⋊C4C2314yes—
122C4×C4⋊C4C2314yes—
123C4×C4⋊C4C2312yes—
124#376C2311yescharacteristic
125#340C2314yes—
126#340C2314yes—
127#340C2314yes—
128#340C2314yes—
129#340C2314yes—
130#340C2314yes—
131#340C2314yes—
132#340C2314yes—
133#336C2314yes—
134#336C2314yes—
135#336C2314yes—
136C22×D4—24no—
137C22×D4C2414yes—
138C23×C4—24no—
139C2×D8—44no—
140C2×D8—44no—
141C2×D8—44no—
142C22×C8C2414yes—
143C22×C8C2411yescharacteristic
144C22×C8C2412yes—
145C22×C8C2411yescharacteristic
146#127—42no—
147#127—42no—
148#126—88no—
149#126—88no—
150#125—42no—
151#125—44no—
152#125—42no—
153#125—44no—
154#123—88no—
155C4×D4—44no—
156C4×D4—44no—
157C2×C4⋊C4—24no—
158C2×C4⋊C4—24no—
159C2×C4⋊C4—24no—
160C2×C4⋊C4—24no—
161C2×C4⋊C4—24no—
162C2×C4⋊C4—24no—
163C2×C4⋊C4—24no—
164C2×C4⋊C4—24no—
165C2×C4⋊C4—24no—
166C2×C4⋊C4C2×D414yes—
167C2×C4⋊C4C2×D414yes—
168C2×C4⋊C4C2×D414yes—
169C2×C4⋊C4C2414yes—
170C2×C4⋊C4C2414yes—
171C2×C4⋊C4C2414yes—
172C2×C22⋊C4—24no—
173C2×C22⋊C4—24no—
174C2×C22⋊C4—24no—
175C2×C22⋊C4—24no—
176C2×C22⋊C4—24no—
177C2×C22⋊C4—24no—
178C2×C22⋊C4—24no—
179C2×C22⋊C4—24no—
180C2×C22⋊C4—24no—
181C2×C22⋊C4—24no—
182C2×C22⋊C4—24no—
183C2×C42—24no—
184C2×C42C2414yes—
185C2×C42C2412yes—
186C2×C42C2411yescharacteristic
187#106—44no—
188C4⋊C8—42no—
189C4⋊C8—44no—
190#102—28no—
191#102—44no—
192#102—44no—
193#102—44no—
194#102—44no—
195#102—44no—
196#97—41no—
197#97—44no—
198#97—42no—
199C4×C8—22no—
200C4×C8—42no—
201#95—24no—
202#95—24no—
203#95—24no—
204#95—24no—
205#95—24no—
206#95C2×D414yes—
207#95C2×D412yes—
208#95C2×D414yes—
209#95C2×D414yes—
210#95C2×D412yes—
211C24—24no—
212C2×D4—24no—
213C2×D4—24no—
214C2×D4—24no—
215C2×D4—24no—
216C2×D4—24no—
217C2×D4—24no—
218C2×D4—84no—
219C2×D4—84no—
220C2×D4—84no—
221C22×C4—24no—
222C22×C4—24no—
223C22×C4—24no—
224C22×C4—84no—
225C22×C4—84no—
226C22×C4—84no—
227C22×C4C22×D412yes—
228C22×C4C22×D414yes—
229C22×C4C22×D414yes—
230C22×C4C22×D414yes—
231C22×C4C2511yescharacteristic
232D8—88no—
233C2×C8—22no—
234C2×C8—21no—
235C2×C8—22no—
236C2×C8—22no—
237C2×C8—22no—
238C2×C8—24no—
239C2×C8—22no—
240C2×C8—24no—
241C2×C8—24no—
242C2×C82−1+412yes—
243C4⋊C4—24no—
244C4⋊C4—24no—
245C4⋊C4—24no—
246C4⋊C4—24no—
247C4⋊C4—24no—
248C4⋊C4—24no—
249C4⋊C4—44no—
250C4⋊C4—44no—
251C4⋊C4—44no—
252C4⋊C4—84no—
253C4⋊C4—84no—
254C4⋊C4—84no—
255C4⋊C4—84no—
256C4⋊C4—84no—
257C4⋊C4—84no—
258C4⋊C4—84no—
259C4⋊C4—84no—
260C4⋊C4—84no—
261C22⋊C4—48no—
262C22⋊C4—48no—
263C22⋊C4—84no—
264C22⋊C4—84no—
265C22⋊C4—84no—
266C22⋊C4—84no—
267C22⋊C4—84no—
268C22⋊C4—84no—
269C22⋊C4—84no—
270C22⋊C4—84no—
271C22⋊C4—84no—
272C42—21no—
273C42—22no—
274C42—21no—
275C42—24no—
276C42—22no—
277C42—24no—
278C42—48no—
279C23—44no—
280C23—44no—
281C23—44no—
282C23—44no—
283C23—44no—
284C23—44no—
285C23—44no—
286C23C23×D411yescharacteristic
287D4—44no—
288D4—44no—
289D4—44no—
290D4—44no—
291D4—168no—
292C2×C4—24no—
293C2×C4—24no—
294C2×C4—24no—
295C2×C4—24no—
296C2×C4—24no—
297C2×C4—24no—
298C2×C4—24no—
299C2×C4—22no—
300C2×C4—24no—
301C2×C4—24no—
302C2×C4—24no—
303C2×C4—44no—
304C2×C4—44no—
305C2×C4—44no—
306C2×C4—44no—
307C2×C4—44no—
308C2×C4—44no—
309C2×C4—44no—
310C2×C4—44no—
311C2×C4—44no—
312C2×C4—168no—
313C2×C4C2×2−1+411yescharacteristic
314C2×C4#58512yes—
315C2×C4#58512yes—
316C2×C4#58511yescharacteristic
317C8—22no—
318C8—41no—
319C8—42no—
320C8—44no—
321C22—84no—
322C22—84no—
323C22—84no—
324C22—84no—
325C22—84no—
326C22—84no—
327C22—84no—
328C22#349711yescharacteristic
329C22#350012yes—
330C22#358514yes—
331C4—44no—
332C4—44no—
333C4—44no—
334C4—42no—
335C4—84no—
336C4—84no—
337C4—84no—
338C4#353111yescharacteristic
339C4#355311yescharacteristic
340C4#356812yes—
341C2—164no—
342C2#4623911yescharacteristic
343C2#6238712yes—
344C2#6257314yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#32814C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ #2402 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #53132C2semidirect
2. #33284C2semidirect
3. #32857C2semidirect
5. #32253C2semidirect
6. #31830C2semidirect
7. #31658C2semidirect
8. #29080C2semidirect
9. #28902C2semidirect
10. #27829C2semidirect
11. #21677C2semidirect

Maximal subgroups

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

subgroupindex
#531322
#332842
#328572
#328142
#322532
#318302
#316582
#290802
#289022
#278292
#216772