Finite Group Databasebeta

← #413234

Subgroups — #413234

Order 512.

1226 conjugacy classes of subgroups.

The subgroup lattice has 353 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35167C211yesmaximal, characteristic
2C2×#2061C214yesmaximal
3#20800C212yesmaximal
4#13911C211yesmaximal, characteristic
5#11342C214yesmaximal
6#10851C212yesmaximal
7C4×#343C211yesmaximal, characteristic
8C2×#544C2212yes—
9C2×#428C2218yes—
10C2×C4×M16C2212yes—
11#2857—22no—
12#2659—22no—
13#2552—22no—
14#2551—22no—
15#2550—24no—
16#2316C2212yes—
17#2285C2212yes—
18C2×#384C2211yescharacteristic
19C2×#382C2212yes—
20C2×C4×C4⋊C4C2211yescharacteristic
21C2×C4×C4⋊C4C2211yescharacteristic
22#2061—216no—
23#1847—24no—
24#1845—22no—
25#1844—22no—
26C2×#343C2212yes—
27C2×#343C2212yes—
28C2×#339C2218yes—
29#1442—22no—
30C4⋊#95—22no—
31C4⋊#95C2212yes—
32C22×M16C2314yes—
33#544—28no—
34#544—44no—
35C2×C4⋊Q8—22no—
36C2×C4⋊Q8—24no—
37C2×#125—24no—
38C2×C22⋊Q8—24no—
39C2×#117C2314yes—
40C22×C4⋊C4—21no—
41C22×C4⋊C4—21no—
42C22×C4⋊C4—21no—
43C22×C4⋊C4C2312yes—
44C22×C4⋊C4C2311yescharacteristic
45C22×C4⋊C4C2311yescharacteristic
46C22×C42D412yes—
47C22×C42C2312yes—
48C22×C42C2311yescharacteristic
49#428—216no—
50#428—48no—
51C2×#107—28no—
52C2×#106—28no—
53C4×M16—24no—
54C4×M16Q818yescomplemented
55C2×#97—22no—
56C2×C4×C8—22no—
57#398—28no—
58#395—28no—
59#391—24no—
60#389—24no—
61#389—24no—
62#389—24no—
63#389—24no—
64#384—22no—
65#384—22no—
66#384—22no—
67#384D412yes—
68#384D412yes—
69#382—28no—
70#382D414yes—
71#380—24no—
72#380—22no—
73#380—22no—
74C4×C4⋊C4—22no—
75C4×C4⋊C4—22no—
76C4×C4⋊C4—22no—
77C4×C4⋊C4—22no—
78C4×C4⋊C4—22no—
79C4×C4⋊C4Q814yescomplemented
80C2×#95—21no—
81C2×#95—24no—
82C2×#95—24no—
83C2×#95D412yes—
84#343D418yes—
85#343Q818yescomplemented
86#339—216no—
87C22×Q8—24no—
88C23×C4—22no—
89C23×C4—22no—
90C23×C4C2×D411yescharacteristic
91C23×C4C2×D412yes—
92C23×C4C2×D412yes—
93C23×C4C2×D411yescharacteristic
94C23×C4C2411yescharacteristic
95C2×M16—24no—
96C2×M16—24no—
97C2×M16—24no—
98C2×M16C2×D418yes—
99C2×M16C2×Q818yes—
100C2×M16C2×Q818yes—
101C22×C8—24no—
102C4⋊Q8—416no—
103C4⋊Q8—84no—
104#125—416no—
105C22⋊Q8—48no—
106C22⋊Q8—48no—
107#117—28no—
108#117—28no—
109C2×C4⋊C4—22no—
110C2×C4⋊C4—22no—
111C2×C4⋊C4—22no—
112C2×C4⋊C4—22no—
113C2×C4⋊C4—24no—
114C2×C4⋊C4—28no—
115C2×C4⋊C4—22no—
116C2×C4⋊C4—22no—
117C2×C4⋊C4—24no—
118C2×C4⋊C4—24no—
119C2×C4⋊C4—24no—
120C2×C4⋊C4—24no—
121C2×C4⋊C4—24no—
122C2×C4⋊C4—24no—
123C2×C4⋊C4—22no—
124C2×C4⋊C4—22no—
125C2×C4⋊C4—42no—
126C2×C4⋊C4—42no—
127C2×C4⋊C4—44no—
128C2×C4⋊C4—41no—
129C2×C4⋊C4—42no—
130C2×C4⋊C4—41no—
131C2×C4⋊C4—82no—
132C2×C4⋊C4C2×D414yes—
133C2×C4⋊C4C2×D414yes—
134C2×C4⋊C4C2×D414yes—
135C2×C4⋊C4C2×Q814yes—
136C2×C22⋊C4—24no—
137C2×C42—24no—
138C2×C42—24no—
139C2×C42—22no—
140C2×C42—24no—
141C2×C42—22no—
142C2×C42—22no—
143C2×C42—44no—
144C2×C42C2×D412yes—
145C2×C42C2×D412yes—
146C2×C42C2×Q814yes—
147C2×C42C2×Q814yes—
148C2×C42C2×Q812yes—
149C2×C42#4112yes—
150#107—416no—
151#106—416no—
152#97—28no—
153C4×C8—28no—
154#95—28no—
155#95—24no—
156#95—24no—
157#95—24no—
158#95—24no—
159#95—48no—
160#95—84no—
161C24C22×D411yescharacteristic
162C2×Q8—48no—
163C2×Q8—84no—
164C2×Q8—84no—
165C22×C4—22no—
166C22×C4—22no—
167C22×C4—22no—
168C22×C4—22no—
169C22×C4—22no—
170C22×C4—22no—
171C22×C4—24no—
172C22×C4—24no—
173C22×C4—24no—
174C22×C4—41no—
175C22×C4—41no—
176C22×C4—42no—
177C22×C4—42no—
178C22×C4—42no—
179C22×C4—42no—
180C22×C4—42no—
181C22×C4—42no—
182C22×C4—42no—
183C22×C4—42no—
184C22×C4—42no—
185C22×C4—42no—
186C22×C4—44no—
187C22×C4—44no—
188C22×C4—42no—
189C22×C4—42no—
190C22×C4—42no—
191C22×C4#12011yescharacteristic
192C22×C4#12012yes—
193C22×C4#12011yescharacteristic
194C22×C4C22⋊Q812yes—
195C22×C4C22⋊Q812yes—
196C22×C4#12714yes—
197C22×C4C4⋊Q812yes—
198C22×C4C4⋊Q814yes—
199C22×C4C4⋊Q812yes—
200C22×C4C22×D412yes—
201C22×C4C22×D412yes—
202C22×C4C22×Q812yes—
203C22×C4C22×Q812yes—
204C22×C4C22×Q811yescharacteristic
205C22×C4C2×#4111yescharacteristic
206C22×C4H(5,𝔽2)12yes—
207C22×C42−1+412yes—
208M16—28no—
209M16—28no—
210M16—28no—
211M16C4⋊Q8116yescomplemented
212C2×C8—28no—
213C2×C8—28no—
214C2×C8—28no—
215C4⋊C4—28no—
216C4⋊C4—28no—
217C4⋊C4—28no—
218C4⋊C4—28no—
219C4⋊C4—48no—
220C4⋊C4—48no—
221C4⋊C4—416no—
222C4⋊C4—44no—
223C4⋊C4—44no—
224C4⋊C4—48no—
225C4⋊C4—82no—
226C4⋊C4—82no—
227C4⋊C4—82no—
228C4⋊C4—84no—
229C4⋊C4—84no—
230C4⋊C4—84no—
231C4⋊C4—82no—
232C22⋊C4—44no—
233C22⋊C4—44no—
234C42—28no—
235C42—48no—
236C42—44no—
237C42—44no—
238C42—82no—
239C42C22⋊Q814yes—
240C42C22⋊Q814yes—
241C42C4⋊Q814yes—
242C42C4⋊Q814yes—
243C23—21no—
244C23—21no—
245C23—21no—
246C23—21no—
247C23C2×#12011yescharacteristic
248C23C2×C22⋊Q811yescharacteristic
249C23C2×#12711yescharacteristic
250C23C2×C4⋊Q811yescharacteristic
251C23C2×C4⋊Q811yescharacteristic
252C23#53511yescharacteristic
253C23#53611yescharacteristic
254Q8—168no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—22no—
258C2×C4—22no—
259C2×C4—24no—
260C2×C4—24no—
261C2×C4—24no—
262C2×C4—24no—
263C2×C4—42no—
264C2×C4—42no—
265C2×C4—42no—
266C2×C4—42no—
267C2×C4—42no—
268C2×C4—42no—
269C2×C4—44no—
270C2×C4—44no—
271C2×C4—44no—
272C2×C4—44no—
273C2×C4—44no—
274C2×C4—44no—
275C2×C4—44no—
276C2×C4—44no—
277C2×C4—44no—
278C2×C4—44no—
279C2×C4—44no—
280C2×C4—82no—
281C2×C4—82no—
282C2×C4—82no—
283C2×C4—82no—
284C2×C4—82no—
285C2×C4—82no—
286C2×C4—84no—
287C2×C4—84no—
288C2×C4—82no—
289C2×C4—82no—
290C2×C4—82no—
291C2×C4#39512yes—
292C2×C4#39514yes—
293C2×C4#39512yes—
294C2×C4#45412yes—
295C2×C4#45512yes—
296C2×C4C2×C22⋊Q812yes—
297C2×C4C2×C4⋊Q814yes—
298C2×C4C2×C4⋊Q812yes—
299C2×C4#54414yes—
300C2×C4D4214yes—
301C2×C4#54712yes—
302C2×C4D4×Q814yes—
303C2×C4D4×Q812yes—
304C2×C4#55412yes—
305C2×C4Q8212yes—
306C8—216no—
307C22—21no—
308C22—21no—
309C22—21no—
310C22—21no—
311C22—21no—
312C22—21no—
313C22—21no—
314C22—21no—
315C22—21no—
316C22—21no—
317C22—21no—
318C22—21no—
319C22C2×#39511yescharacteristic
320C22#243311yescharacteristic
321C22#243411yescharacteristic
322C22#254211yescharacteristic
323C22#256711yescharacteristic
324C22#266011yescharacteristic
325C22#286811yescharacteristic
326C22C2×#45511yescharacteristic
327C22C2×#45411yescharacteristic
328C22#302111yescharacteristic
329C22#302211yescharacteristic
330C4—22no—
331C4—24no—
332C4—44no—
333C4—44no—
334C4—84no—
335C4—84no—
336C4—84no—
337C4—162no—
338C4#206114yescomplemented
339C4#244014yes—
340C4#255114yes—
341C2—21no—
342C2—21no—
343C2—21no—
344C2—21no—
345C2#1514311yescharacteristic
346C2C2×#206111yescharacteristic
347C2#2356211yescharacteristic
348C2#2573811yescharacteristic
349C2#2574111yescharacteristic
350C2#2582311yescharacteristic
351C2#2582611yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC23×C4C24
prime radicalC24C22×D4
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
54. C4×M16Q8semidirect
79. C4×C4⋊C4Q8semidirect
85. #343Q8semidirect
211. M16C4⋊Q8semidirect
338. C4#2061semidirect

Maximal subgroups

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

subgroupindex
#351672
C2×#20612
#208002
#139112
#113422
#108512
C4×#3432