Finite Group Databasebeta

← #241194

Subgroups — #241194

Order 512.

1090 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35167C211yesmaximal, characteristic
2#21981C212yesmaximal
3#13910C211yesmaximal, characteristic
4#10570C218yesmaximal
5#8014C212yesmaximal
6#7908C211yesmaximal, characteristic
7C2×#544—21no—
8C2×#431C2212yes—
9#2857—22no—
10#2659C2214yes—
11#2552—22no—
12#2551—22no—
13#2550—24no—
14#2316—21no—
15C2×#389—22no—
16#2285C2212yes—
17C2×#384—21no—
18C2×#384C2211yescharacteristic
19C2×C4×C4⋊C4C2211yescharacteristic
20C2×C4×C4⋊C4C2211yescharacteristic
21#1948—24no—
22#1853C2218yes—
23#1844—22no—
24#1834—22no—
25#1444C2214yes—
26C4⋊#95—21no—
27#1294C2218yes—
28#1280C2214yes—
29#544—44no—
30#544—44no—
31C2×C4⋊Q8—24no—
32C2×C4⋊Q8—22no—
33C2×#125—24no—
34C2×C22⋊Q8—24no—
35C2×#117—22no—
36C22×C4⋊C4—21no—
37C22×C4⋊C4—21no—
38C22×C4⋊C4—21no—
39C22×C4⋊C4—24no—
40C22×C4⋊C4D411yescharacteristic
41C22×C4⋊C4D411yescharacteristic
42C22×C4⋊C4C2312yes—
43C22×C42—21no—
44C22×C42D412yes—
45C22×C42C2311yescharacteristic
46#431—24no—
47#431Q818yescomplemented
48C2×#106—28no—
49C2×C4⋊C8—22no—
50C2×C22⋊C8C2314yes—
51C2×C4×C8—22no—
52#398—28no—
53#395—28no—
54#391—24no—
55#389—24no—
56#389—24no—
57#389—24no—
58#389—24no—
59#389—48no—
60#384—22no—
61#384—22no—
62#384—22no—
63#384—44no—
64#384D412yes—
65#384D412yes—
66#382—22no—
67#380—24no—
68#380—22no—
69#380C2314yes—
70C4×C4⋊C4—22no—
71C4×C4⋊C4—22no—
72C4×C4⋊C4—22no—
73C4×C4⋊C4—22no—
74C4×C4⋊C4D414yes—
75C4×C4⋊C4Q814yescomplemented
76C2×#95—24no—
77C2×#95C2312yes—
78C2×#95C2312yes—
79#340—28no—
80C22×Q8—24no—
81C23×C4—22no—
82C23×C4—22no—
83C23×C4C2×D411yescharacteristic
84C23×C4C2×D412yes—
85C23×C4C2×D412yes—
86C23×C4C2×D411yescharacteristic
87C23×C4C2411yescharacteristic
88C22×C8—24no—
89C4⋊Q8—416no—
90C4⋊Q8—84no—
91#125—416no—
92C22⋊Q8—48no—
93C22⋊Q8—48no—
94#117—44no—
95#117—44no—
96C2×C4⋊C4—24no—
97C2×C4⋊C4—24no—
98C2×C4⋊C4—22no—
99C2×C4⋊C4—22no—
100C2×C4⋊C4—24no—
101C2×C4⋊C4—22no—
102C2×C4⋊C4—22no—
103C2×C4⋊C4—24no—
104C2×C4⋊C4—28no—
105C2×C4⋊C4—22no—
106C2×C4⋊C4—22no—
107C2×C4⋊C4—22no—
108C2×C4⋊C4—22no—
109C2×C4⋊C4—24no—
110C2×C4⋊C4—24no—
111C2×C4⋊C4—24no—
112C2×C4⋊C4—44no—
113C2×C4⋊C4—48no—
114C2×C4⋊C4—41no—
115C2×C4⋊C4—42no—
116C2×C4⋊C4—41no—
117C2×C4⋊C4—48no—
118C2×C4⋊C4—42no—
119C2×C4⋊C4—42no—
120C2×C4⋊C4—82no—
121C2×C4⋊C4—84no—
122C2×C4⋊C4SD1614yes—
123C2×C4⋊C4SD1614yes—
124C2×C4⋊C4C2×D414yes—
125C2×C4⋊C4C2×D414yes—
126C2×C22⋊C4—24no—
127C2×C42—24no—
128C2×C42—22no—
129C2×C42—24no—
130C2×C42—22no—
131C2×C42—42no—
132C2×C42—42no—
133C2×C42—44no—
134C2×C42C2×D412yes—
135C2×C42C2×D412yes—
136C2×C42C2×Q814yes—
137C2×C42C2×Q812yes—
138C2×C42#4112yes—
139#106—88no—
140C4⋊C8—28no—
141C22⋊C8—24no—
142C22⋊C8C2×Q818yes—
143C4×C8—28no—
144#95—28no—
145#95—24no—
146#95—24no—
147#95—24no—
148#95—22no—
149#95—48no—
150#95C2×Q814yes—
151C24C22×D411yescharacteristic
152C2×Q8—48no—
153C2×Q8—84no—
154C2×Q8—84no—
155C22×C4—22no—
156C22×C4—22no—
157C22×C4—24no—
158C22×C4—24no—
159C22×C4—22no—
160C22×C4—22no—
161C22×C4—24no—
162C22×C4—22no—
163C22×C4—22no—
164C22×C4—24no—
165C22×C4—24no—
166C22×C4—41no—
167C22×C4—41no—
168C22×C4—42no—
169C22×C4—42no—
170C22×C4—42no—
171C22×C4—42no—
172C22×C4—42no—
173C22×C4—42no—
174C22×C4—42no—
175C22×C4—42no—
176C22×C4—44no—
177C22×C4—44no—
178C22×C4—42no—
179C22×C4—42no—
180C22×C4—42no—
181C22×C4#12011yescharacteristic
182C22×C4#12012yes—
183C22×C4#12011yescharacteristic
184C22×C4C22⋊Q812yes—
185C22×C4C22⋊Q812yes—
186C22×C4C4⋊Q812yes—
187C22×C4C4⋊Q812yes—
188C22×C4C2×SD1614yes—
189C22×C4C2×SD1614yes—
190C22×C4C2×SD1612yes—
191C22×C4C2×SD1612yes—
192C22×C4AGL(1,ℤ/8)12yes—
193C22×C4#13712yes—
194C22×C4C22×D412yes—
195C22×C4C22×Q812yes—
196C22×C4C22×Q811yescharacteristic
197C22×C4C2×#4111yescharacteristic
198C2×C8—28no—
199C2×C8—44no—
200C2×C8—44no—
201C4⋊C4—216no—
202C4⋊C4—28no—
203C4⋊C4—28no—
204C4⋊C4—44no—
205C4⋊C4—44no—
206C4⋊C4—48no—
207C4⋊C4—416no—
208C4⋊C4—48no—
209C4⋊C4—44no—
210C4⋊C4—44no—
211C4⋊C4—44no—
212C4⋊C4—44no—
213C4⋊C4—82no—
214C4⋊C4—84no—
215C4⋊C4—84no—
216C4⋊C4—816no—
217C4⋊C4—84no—
218C4⋊C4—82no—
219C22⋊C4—44no—
220C22⋊C4—44no—
221C42—48no—
222C42—44no—
223C42—48no—
224C42—82no—
225C42C22⋊Q814yes—
226C42C22⋊Q814yes—
227C42C4⋊Q814yes—
228C42C4⋊Q814yes—
229C23—21no—
230C23—21no—
231C23—21no—
232C23—21no—
233C23C2×#12011yescharacteristic
234C23C2×C22⋊Q811yescharacteristic
235C23C2×C4⋊Q811yescharacteristic
236C23C22×SD1611yescharacteristic
237C23C22×SD1611yescharacteristic
238C23C2×AGL(1,ℤ/8)11yescharacteristic
239C23C2×#13711yescharacteristic
240Q8—168no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—22no—
244C2×C4—22no—
245C2×C4—24no—
246C2×C4—24no—
247C2×C4—28no—
248C2×C4—24no—
249C2×C4—24no—
250C2×C4—42no—
251C2×C4—41no—
252C2×C4—41no—
253C2×C4—44no—
254C2×C4—44no—
255C2×C4—44no—
256C2×C4—44no—
257C2×C4—44no—
258C2×C4—42no—
259C2×C4—42no—
260C2×C4—42no—
261C2×C4—42no—
262C2×C4—44no—
263C2×C4—44no—
264C2×C4—44no—
265C2×C4—44no—
266C2×C4—44no—
267C2×C4—44no—
268C2×C4—44no—
269C2×C4—44no—
270C2×C4—44no—
271C2×C4—82no—
272C2×C4—82no—
273C2×C4—84no—
274C2×C4—84no—
275C2×C4—82no—
276C2×C4—82no—
277C2×C4—82no—
278C2×C4#39512yes—
279C2×C4#39514yes—
280C2×C4#39512yes—
281C2×C4#44814yes—
282C2×C4#45014yes—
283C2×C4#45312yes—
284C2×C4#45612yes—
285C2×C4#47514yes—
286C2×C4#47614yes—
287C2×C4#49814yes—
288C2×C4#50114yes—
289C2×C4C2×C22⋊Q812yes—
290C2×C4C2×C4⋊Q812yes—
291C8—48no—
292C22—21no—
293C22—21no—
294C22—21no—
295C22—21no—
296C22—21no—
297C22—21no—
298C22—21no—
299C22—21no—
300C22—21no—
301C22—21no—
302C22—21no—
303C22—21no—
304C22C2×#39511yescharacteristic
305C22C2×#45011yescharacteristic
306C22C2×#44811yescharacteristic
307C22C2×#45311yescharacteristic
308C22C2×#45611yescharacteristic
309C22#302711yescharacteristic
310C22#302811yescharacteristic
311C22C2×#47611yescharacteristic
312C22C2×#47511yescharacteristic
313C22C2×#49811yescharacteristic
314C22C2×#50111yescharacteristic
315C4—22no—
316C4—22no—
317C4—22no—
318C4—48no—
319C4—44no—
320C4—44no—
321C4—84no—
322C4—84no—
323C4—162no—
324C4#205818yes—
325C4#206014yes—
326C2—21no—
327C2—21no—
328C2—21no—
329C2—21no—
330C2#1838911yescharacteristic
331C2#1839511yescharacteristic
332C2#1840211yescharacteristic
333C2#1841011yescharacteristic
334C2C2×#205811yescharacteristic
335C2C2×#206011yescharacteristic
336C2#2356611yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC23×C4C24
prime radicalC24C22×D4
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
47. #431Q8semidirect
75. C4×C4⋊C4Q8semidirect

Maximal subgroups

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

subgroupindex
#351672
#219812
#139102
#105702
#80142
#79082