Finite Group Databasebeta

← #6376425

Subgroups — #6376425

Order 512.

2638 conjugacy classes of subgroups.

The subgroup lattice has 334 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3150C211yesmaximal, characteristic
2C4×#532C212yesmaximal, complemented
3#22273C218yesmaximal, complemented
4#22272C218yesmaximal, complemented
5#21199C212yesmaximal, complemented
6#20924C218yesmaximal, complemented
7#20762C211yesmaximal, characteristic, complemented
8C2×#1745C211yesmaximal, characteristic, complemented
9C22×#135C2214yes—
10C2×#532C2212yes—
11C2×C4×#41C2214yes—
12#3150C4116yescomplemented
13C2×#497C2214yes—
14C2×#496C2214yes—
15C2×#416C2218yes—
16C22×#97C2211yescharacteristic
17C22×#97C2211yescharacteristic
18C4×C4⋊Q8C2214yes—
19C4×#127C2214yes—
20C4×#124C2218yes—
21C4×#121—24no—
22C4×#121—24no—
23#2297C2214yes—
24#2293—22no—
25#2293—22no—
26#2285C2212yes—
27C42×D4—22no—
28C42×D4—22no—
29C2×C43C2211yescharacteristic
30#1961C2214yes—
31#1960C2214yes—
32#1959C2218yes—
33#1880C22116yes—
34#1879C22116yes—
35#1878C22116yes—
36#1877C22116yes—
37#1836C2218yes—
38#1745C2218yes—
39#1735C2214yes—
40C2×#336C2214yes—
41C22×#41C2314yes—
42C2×#135—416no—
43C2×#135C2×C4132yescomplemented
44C22×Q16C2314yes—
45C22×SD16C2318yes—
46C22×D8C2314yes—
47C23×C8C2312yes—
48C23×C8C2312yes—
49#532—28no—
50#532C2×C4116yescomplemented
51C2×C4⋊Q8C2312yes—
52C2×#127C2312yes—
53C2×#124C2314yes—
54C2×#121—22no—
55C2×#121—22no—
56C4×#41—416no—
57C2×C4×Q8C2318yes—
58C2×C4×D4—28no—
59C2×C4×D4—22no—
60C2×C4×D4—22no—
61C2×C4×D4—24no—
62C2×C4×D4—24no—
63C2×C4×D4C2318yes—
64C2×#117—24no—
65C2×#117C2314yes—
66C22×C42—24no—
67C22×C42D414yes—
68C22×C42C2312yes—
69C22×C42C2311yescharacteristic
70#497C2×C4132yescomplemented
71#496C2×C4132yescomplemented
72#416—416no—
73C2×#103C23116yes—
74C2×#102C23116yes—
75C2×#97—24no—
76C2×#97C2×C418yescomplemented
77C2×#97C2314yes—
78C2×#97C2314yes—
79#391C2318yes—
80#390—24no—
81#390—24no—
82#390C2318yes—
83#389—24no—
84#389—24no—
85#388—28no—
86#388—28no—
87#386—28no—
88#384C2314yes—
89#382C2314yes—
90#380—22no—
91#380—22no—
92C4×C4⋊C4—24no—
93C4×C4⋊C4—24no—
94C4×C4⋊C4—22no—
95C4×C4⋊C4—22no—
96C4×C22⋊C4—24no—
97C4×C22⋊C4—24no—
98C4×C22⋊C4—28no—
99C4×C22⋊C4—28no—
100C43D411yescharacteristic
101C43D412yes—
102C43D411yescharacteristic
103C43C2314yes—
104#340C23116yes—
105#336C23116yes—
106C2×#41—28no—
107C2×#41—216no—
108C2×#41—216no—
109C2×#41C22×C4116yes—
110C22×Q8C2414yes—
111C22×D4—24no—
112C22×D4—22no—
113C22×D4—22no—
114C22×D4C2414yes—
115C23×C4—24no—
116C23×C4—28no—
117C23×C4C2×D414yes—
118C23×C4C2×D412yes—
119C23×C4C2411yescharacteristic
120#135—464no—
121C2×Q16—48no—
122C2×Q16C22×C4116yes—
123C2×SD16—48no—
124C2×SD16—48no—
125C2×SD16C22×C4132yes—
126C2×D8—48no—
127C2×D8C22×C4116yes—
128C22×C8—28no—
129C22×C8—24no—
130C22×C8—28no—
131C22×C8C22×C418yes—
132C22×C8C2414yes—
133C22×C8C2414yes—
134C4⋊Q8—24no—
135C4⋊Q8C22×C418yes—
136#127—24no—
137#127C22×C418yes—
138#124—28no—
139#124C22×C4116yes—
140#121—216no—
141#121—216no—
142C4×Q8—416no—
143C4×D4—216no—
144C4×D4—216no—
145C4×D4—416no—
146C4×D4—416no—
147C4×D4—416no—
148C4×D4—432no—
149#117—216no—
150#117—416no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—28no—
154C2×C4⋊C4—24no—
155C2×C4⋊C4—24no—
156C2×C4⋊C4C2418yes—
157C2×C22⋊C4—24no—
158C2×C22⋊C4—24no—
159C2×C22⋊C4—24no—
160C2×C22⋊C4—28no—
161C2×C22⋊C4—28no—
162C2×C22⋊C4—28no—
163C2×C42—28no—
164C2×C42—28no—
165C2×C42—24no—
166C2×C42—48no—
167C2×C42—416no—
168C2×C42C22×C414yes—
169C2×C42C2×D414yes—
170C2×C42C2×D414yes—
171C2×C42C2×D412yes—
172C2×C42C2×D412yes—
173C2×C42C2×D414yes—
174C2×C42C2×D414yes—
175C2×C42C2×D411yescharacteristic
176C2×C42C2×D411yescharacteristic
177C2×C42#4114yes—
178C2×C42C2414yes—
179C2×C42C2412yes—
180#103—416no—
181#102—416no—
182#97—28no—
183#97C22×C4116yes—
184#95—24no—
185#95—24no—
186#95—28no—
187#95—28no—
188C24—24no—
189C24C22×D411yescharacteristic
190#41—264no—
191#41—432no—
192C2×Q8—24no—
193C2×Q8—28no—
194C2×Q8—28no—
195C2×Q8C23×C418yes—
196C2×D4—216no—
197C2×D4—28no—
198C2×D4—28no—
199C2×D4—24no—
200C2×D4—28no—
201C2×D4—28no—
202C2×D4—48no—
203C2×D4—48no—
204C2×D4—48no—
205C2×D4—48no—
206C2×D4C23×C418yes—
207C22×C4—28no—
208C22×C4—24no—
209C22×C4—216no—
210C22×C4—28no—
211C22×C4—28no—
212C22×C4—44no—
213C22×C4—44no—
214C22×C4—416no—
215C22×C4—416no—
216C22×C4—48no—
217C22×C4—48no—
218C22×C4—416no—
219C22×C4C4×D418yes—
220C22×C4C23×C412yes—
221C22×C4C22×D414yes—
222C22×C4C22×D412yes—
223C22×C4C22×D412yes—
224C22×C4C22×D414yes—
225C22×C4C22×D411yescharacteristic
226C22×C4C2×#4112yes—
227C22×C4H(5,𝔽2)18yes—
228C22×C4C2511yescharacteristic
229Q16—416no—
230SD16—432no—
231D8—416no—
232C2×C8—216no—
233C2×C8—28no—
234C2×C8—24no—
235C2×C8—28no—
236C2×C8—48no—
237C2×C8C23×C418yes—
238C4⋊C4—28no—
239C4⋊C4—28no—
240C4⋊C4—216no—
241C4⋊C4—416no—
242C4⋊C4—48no—
243C4⋊C4—48no—
244C22⋊C4—216no—
245C22⋊C4—216no—
246C22⋊C4—416no—
247C22⋊C4—416no—
248C22⋊C4—48no—
249C22⋊C4—416no—
250C42—24no—
251C42—24no—
252C42—28no—
253C42—28no—
254C42—28no—
255C42—44no—
256C42—44no—
257C42—416no—
258C42—48no—
259C42C4×D414yes—
260C42C4×D414yes—
261C42C23×C414yes—
262C42C2×#4114yes—
263C23—24no—
264C23—28no—
265C23—44no—
266C23—44no—
267C23—48no—
268C23—48no—
269C23C2×C4×D412yes—
270C23#53514yes—
271C23C23×D411yescharacteristic
272Q8—216no—
273Q8—48no—
274D4—216no—
275D4—416no—
276D4—416no—
277D4—432no—
278D4—48no—
279C2×C4—24no—
280C2×C4—24no—
281C2×C4—24no—
282C2×C4—24no—
283C2×C4—28no—
284C2×C4—24no—
285C2×C4—24no—
286C2×C4—24no—
287C2×C4—48no—
288C2×C4—416no—
289C2×C4—44no—
290C2×C4—44no—
291C2×C4—432no—
292C2×C4—48no—
293C2×C4—48no—
294C2×C4—48no—
295C2×C4—832no—
296C2×C4C2×C4×D414yes—
297C2×C4C2×C4×D414yes—
298C2×C4C2×C4×D412yes—
299C2×C4#51818yes—
300C2×C4#53514yes—
301C2×C4#54018yes—
302C2×C4#57518yes—
303C2×C4C24×C411yescharacteristic
304C2×C4C22×#4111yescharacteristic
305C2×C4C2×H(5,𝔽2)14yes—
306C8—28no—
307C8—44no—
308C22—28no—
309C22—24no—
310C22—48no—
311C22—48no—
312C22—48no—
313C22—88no—
314C22—88no—
315C22#232814yes—
316C22C22×C4×D411yescharacteristic
317C22C2×#53512yes—
318C22C2×#57514yes—
319C4—28no—
320C4—24no—
321C4—24no—
322C4—48no—
323C4—44no—
324C4—88no—
325C4#232814yes—
326C4C2×#51812yes—
327C4C2×#54012yes—
328C2—24no—
329C2—88no—
330C2C2×#232811yescharacteristic
331C2C4×#57512yes—
332C2#6079014yes—
subgroupG/H
centerC22×C4C22×D4
commutatorC2×C4C24×C4
Frattini, cosocleC22×C4C25
socleC23C23×D4
prime radicalC2×#3150C2
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C2×C43 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C4×#532C2semidirect
3. #22273C2semidirect
4. #22272C2semidirect
5. #21199C2semidirect
6. #20924C2semidirect
7. #20762C2semidirect
8. C2×#1745C2semidirect
12. #3150C4semidirect
43. C2×#135C2×C4semidirect
50. #532C2×C4semidirect
70. #497C2×C4semidirect
71. #496C2×C4semidirect
76. C2×#97C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#31502
C4×#5322
#222732
#222722
#211992
#209242
#207622
C2×#17452