Finite Group Databasebeta

← #6483504

Subgroups — #6483504

Order 512.

1622 conjugacy classes of subgroups.

The subgroup lattice has 306 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35167C212yesmaximal
2#33209C214yesmaximal
3#30291C214yesmaximal
4#28827C214yesmaximal, complemented
5#28813C214yesmaximal, complemented
6#28789C218yesmaximal, complemented
7#21961C212yesmaximal
8#20817C212yesmaximal
9C2×C8×C4⋊C4C211yesmaximal, characteristic
10C2×#544—22no—
11C2×#500C2214yes—
12C22×C4⋊C8C2211yescharacteristic
13C22×C4×C8C2212yes—
14C22×C4×C8C2211yescharacteristic
15#2857C2218yes—
16#2659—24no—
17#2552C2218yes—
18#2551C2218yes—
19#2550—28no—
20#2316—22no—
21#2285C2214yes—
22C2×C4×C4⋊C4C2211yescharacteristic
23#2083C22116yes—
24#2057C22116yes—
25#1990C2218yes—
26#1947C2218yes—
27#1947C2218yes—
28#1942C2218yes—
29#1942C2218yes—
30#1845C2212yes—
31#1845C2212yes—
32#1845C2214yes—
33#1834C2214yes—
34#1808C2214yes—
35#1808C2214yes—
36#1808C2218yes—
37C4×#107C2214yes—
38C4×#107C2214yes—
39C8×C4⋊C4C2214yes—
40C8×C4⋊C4C2214yes—
41C2×#336C2212yes—
42C23×C8C2311yescharacteristic
43C23×C8C2312yes—
44C23×C8C2311yescharacteristic
45#544—416no—
46C2×C4⋊Q8—28no—
47C2×C4⋊Q8C2318yes—
48C2×#125—28no—
49C2×C22⋊Q8—28no—
50C2×#117—24no—
51C22×C4⋊C4D411yescharacteristic
52C22×C4⋊C4D411yescharacteristic
53C22×C42C2312yes—
54C22×C42C2311yescharacteristic
55#500—48no—
56C2×#107C2314yes—
57C2×#107C2314yes—
58C2×#107C2314yes—
59C2×#107C2318yes—
60C2×#107C2314yes—
61C2×#107C2318yes—
62C2×C4⋊C8Q818yescomplemented
63C2×C4⋊C8C2314yes—
64C2×C4×C8—44no—
65C2×C4×C8Q818yescomplemented
66C2×C4×C8C2314yes—
67C2×C4×C8C2314yes—
68C2×C4×C8C2312yes—
69C2×C4×C8C2312yes—
70#398—216no—
71#395—216no—
72#391—28no—
73#389—28no—
74#389—28no—
75#389—28no—
76#389C23116yes—
77#384C2318yes—
78#384C2314yes—
79#384C2314yes—
80#382—24no—
81#380—28no—
82#380—24no—
83#380C2318yes—
84C4×C4⋊C4—24no—
85C4×C4⋊C4D414yes—
86C4×C4⋊C4C2318yes—
87C4×C4⋊C4C2314yes—
88C2×#95D412yes—
89#340C2318yes—
90#340C23116yes—
91#340C2318yes—
92#336C2318yes—
93C22×Q8—28no—
94C23×C4C2×D411yescharacteristic
95C23×C4C2×D412yes—
96C23×C4C2×D412yes—
97C23×C4C2×D411yescharacteristic
98C23×C4C2411yescharacteristic
99C22×C8—22no—
100C22×C8—22no—
101C22×C8—41no—
102C22×C8—42no—
103C22×C8—41no—
104C22×C8—42no—
105C22×C8C2×Q814yes—
106C22×C8C2×Q814yes—
107C22×C8C2×Q818yes—
108C22×C8C2412yes—
109C22×C8C2414yes—
110C22×C8C2411yescharacteristic
111C22×C8C2411yescharacteristic
112C4⋊Q8—48no—
113C4⋊Q8—48no—
114C4⋊Q8—432no—
115#125—432no—
116C22⋊Q8—432no—
117#117—416no—
118C2×C4⋊C4—24no—
119C2×C4⋊C4—24no—
120C2×C4⋊C4—28no—
121C2×C4⋊C4—216no—
122C2×C4⋊C4—28no—
123C2×C4⋊C4—24no—
124C2×C4⋊C4—24no—
125C2×C4⋊C4—28no—
126C2×C4⋊C4D818yes—
127C2×C4⋊C4Q1618yes—
128C2×C4⋊C4C2×D414yes—
129C2×C4⋊C4C2×D414yes—
130C2×C4⋊C4C2418yes—
131C2×C4⋊C4C2418yes—
132C2×C22⋊C4—28no—
133C2×C42—28no—
134C2×C42—42no—
135C2×C42—42no—
136C2×C42C2×D414yes—
137C2×C42C2×D412yes—
138C2×C42C2×Q814yes—
139C2×C42C2×Q814yes—
140C2×C42C2414yes—
141C2×C42C2412yes—
142#107—28no—
143#107—28no—
144#107—44no—
145#107—48no—
146#107—44no—
147#107—48no—
148C4⋊C8C2×Q8116yes—
149C4×C8—28no—
150C4×C8—44no—
151C4×C8C2×Q818yes—
152C4×C8C2×Q818yes—
153#95—28no—
154#95—28no—
155#95—216no—
156#95C2×D418yes—
157C24C22×D411yescharacteristic
158C2×Q8—416no—
159C2×Q8—88no—
160C2×Q8—88no—
161C22×C4—21no—
162C22×C4—21no—
163C22×C4—21no—
164C22×C4—21no—
165C22×C4—28no—
166C22×C4—28no—
167C22×C4—42no—
168C22×C4—42no—
169C22×C4—41no—
170C22×C4—41no—
171C22×C4—42no—
172C22×C4—42no—
173C22×C4C4⋊Q814yes—
174C22×C4C4⋊Q814yes—
175C22×C4C2×D818yes—
176C22×C4C2×D814yes—
177C22×C4C2×Q1618yes—
178C22×C4C2×Q1614yes—
179C22×C4C22×D411yescharacteristic
180C22×C4C22×D412yes—
181C22×C4C22×D414yes—
182C22×C4C22×D414yes—
183C22×C4C22×D412yes—
184C22×C4C22×Q812yes—
185C22×C4C22×Q812yes—
186C22×C4C2511yescharacteristic
187C2×C8—24no—
188C2×C8—21no—
189C2×C8—21no—
190C2×C8—22no—
191C2×C8—22no—
192C2×C8—24no—
193C2×C8—28no—
194C2×C8—44no—
195C2×C8—48no—
196C2×C8C22×Q812yes—
197C2×C8C22×Q814yes—
198C2×C8C22×Q818yes—
199C2×C8C22×Q812yes—
200C2×C8H(5,𝔽2)12yes—
201C2×C82−1+412yes—
202C4⋊C4—28no—
203C4⋊C4—28no—
204C4⋊C4—28no—
205C4⋊C4—28no—
206C4⋊C4—48no—
207C4⋊C4—48no—
208C4⋊C4—416no—
209C4⋊C4—432no—
210C4⋊C4—84no—
211C4⋊C4—88no—
212C4⋊C4—84no—
213C4⋊C4—88no—
214C4⋊C4C2×D8116yes—
215C4⋊C4C2×Q16116yes—
216C22⋊C4—416no—
217C42—24no—
218C42—24no—
219C42—44no—
220C42—416no—
221C42C4⋊Q818yes—
222C42C22×Q814yes—
223C42C22×Q814yes—
224C23—21no—
225C23—21no—
226C23—21no—
227C23—21no—
228C23C2×C4⋊Q812yes—
229C23C22×D812yes—
230C23C22×Q1612yes—
231C23C23×D411yescharacteristic
232Q8—1616no—
233C2×C4—24no—
234C2×C4—22no—
235C2×C4—22no—
236C2×C4—24no—
237C2×C4—24no—
238C2×C4—24no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—22no—
242C2×C4—22no—
243C2×C4—22no—
244C2×C4—22no—
245C2×C4—44no—
246C2×C4—48no—
247C2×C4—48no—
248C2×C4—48no—
249C2×C4—48no—
250C2×C4—48no—
251C2×C4—48no—
252C2×C4—48no—
253C2×C4—48no—
254C2×C4#500116yescomplemented
255C2×C4C2×C4⋊Q812yes—
256C2×C4C2×C4⋊Q814yes—
257C2×C4C2×C4⋊Q814yes—
258C2×C4C22×D818yes—
259C2×C4C22×D814yes—
260C2×C4C22×Q1618yes—
261C2×C4C22×Q1614yes—
262C2×C4C23×Q811yescharacteristic
263C2×C4C23×Q811yescharacteristic
264C2×C4C2×H(5,𝔽2)11yescharacteristic
265C2×C4C2×2−1+411yescharacteristic
266C2×C4#58511yescharacteristic
267C2×C4#58511yescharacteristic
268C8—22no—
269C8—24no—
270C8—28no—
271C8#54414yescomplemented
272C22—22no—
273C22—22no—
274C22—22no—
275C22—22no—
276C22—22no—
277C22—22no—
278C22C2×#50014yes—
279C22C22×C4⋊Q811yescharacteristic
280C22#349611yescharacteristic
281C22#349711yescharacteristic
282C22C23×D811yescharacteristic
283C22C23×Q1611yescharacteristic
284C22#358411yescharacteristic
285C22#358411yescharacteristic
286C4—28no—
287C4—28no—
288C4—22no—
289C4—24no—
290C4—88no—
291C4—88no—
292C4C2×#50018yes—
293C4C2×#54411yescharacteristic
294C4#345811yescharacteristic
295C4#350111yescharacteristic
296C4#350211yescharacteristic
297C2—24no—
298C2#3516711yescharacteristic
299C2C22×#50011yescharacteristic
300C2#6066011yescharacteristic
301C2#6224511yescharacteristic
302C2#6224611yescharacteristic
303C2#6235911yescharacteristic
304C2#6236011yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #28827C2semidirect
5. #28813C2semidirect
6. #28789C2semidirect
62. C2×C4⋊C8Q8semidirect
65. C2×C4×C8Q8semidirect
254. C2×C4#500semidirect
271. C8#544semidirect

Maximal subgroups

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

subgroupindex
#351672
#332092
#302912
#288272
#288132
#287892
#219612
#208172
C2×C8×C4⋊C42