Finite Group Databasebeta

← #237739

Subgroups — #237739

Order 512.

1100 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#34642C211yesmaximal, characteristic
2#20734C212yesmaximal
3#13850C211yesmaximal, characteristic
4#10343C214yesmaximal
5#10341C214yesmaximal
6#8014C212yesmaximal
7#7906C211yesmaximal, characteristic
8C2×#533—21no—
9C22×C4×Q8C2211yescharacteristic
10C2×#431C2212yes—
11#2660—24no—
12#2659—22no—
13#2659C2214yes—
14#2399—24no—
15#2285—21no—
16#2285C2212yes—
17C2×C4×C4⋊C4C2211yescharacteristic
18C2×#376C2211yescharacteristic
19#1824C2214yes—
20#1803—24no—
21#1787C2214yes—
22C4×#103—24no—
23#1446C2212yes—
24#1445C2212yes—
25#1441—22no—
26C4×#95—21no—
27C4×#95—21no—
28#1294C2218yes—
29#1280C2214yes—
30C23×Q8C2311yescharacteristic
31#533—44no—
32#533—44no—
33C2×C4⋊Q8—22no—
34C2×#125—22no—
35C2×C22⋊Q8—24no—
36C2×C4×Q8—24no—
37C2×C4×Q8—22no—
38C2×C4×Q8—22no—
39C2×C4×Q8—22no—
40C2×C4×Q8—24no—
41C2×C4×Q8Q814yescomplemented
42C2×#117—22no—
43C22×C4⋊C4—21no—
44C22×C4⋊C4—21no—
45C22×C4⋊C4C2311yescharacteristic
46C22×C42—21no—
47C22×C42—21no—
48C22×C42D411yescharacteristic
49C22×C42D411yescharacteristic
50C22×C42C2311yescharacteristic
51#431—24no—
52#431—24no—
53C2×C4⋊C8—22no—
54C2×#103—28no—
55C2×C22⋊C8C2314yes—
56C2×C4×C8—22no—
57#398—28no—
58#398—28no—
59#395—28no—
60#395—28no—
61#384—22no—
62#384D414yes—
63#382—22no—
64#382—22no—
65#380—22no—
66#380—24no—
67#380C2314yes—
68C4×C4⋊C4—24no—
69C4×C4⋊C4—22no—
70C4×C4⋊C4—24no—
71C4×C4⋊C4D414yes—
72#376—22no—
73#376—22no—
74C2×#95—22no—
75C2×#95—22no—
76C2×#95—22no—
77C2×#95—22no—
78C2×#95D412yes—
79C2×#95C2312yes—
80C2×#95C2312yes—
81#340—28no—
82C22×Q8—24no—
83C22×Q8—21no—
84C22×Q8—22no—
85C22×Q8—22no—
86C22×Q8—44no—
87C22×Q8C2×Q812yes—
88C23×C4—21no—
89C23×C4—21no—
90C23×C4—22no—
91C23×C4C2×D412yes—
92C23×C4C2×D412yes—
93C23×C4C2×D411yescharacteristic
94C23×C4C2×D411yescharacteristic
95C23×C4C2411yescharacteristic
96C22×C8—24no—
97C4⋊Q8—48no—
98#125—48no—
99C22⋊Q8—48no—
100C22⋊Q8—48no—
101C4×Q8—28no—
102C4×Q8—28no—
103C4×Q8—416no—
104C4×Q8—44no—
105C4×Q8—44no—
106#117—44no—
107#117—44no—
108C2×C4⋊C4—22no—
109C2×C4⋊C4—22no—
110C2×C4⋊C4—28no—
111C2×C4⋊C4—24no—
112C2×C4⋊C4—24no—
113C2×C4⋊C4—24no—
114C2×C4⋊C4—21no—
115C2×C4⋊C4—24no—
116C2×C4⋊C4—24no—
117C2×C4⋊C4—24no—
118C2×C4⋊C4—44no—
119C2×C4⋊C4—42no—
120C2×C4⋊C4—42no—
121C2×C4⋊C4—42no—
122C2×C4⋊C4C2×D418yes—
123C2×C4⋊C4C2×Q812yes—
124C2×C22⋊C4—24no—
125C2×C42—21no—
126C2×C42—21no—
127C2×C42—21no—
128C2×C42—21no—
129C2×C42—22no—
130C2×C42—22no—
131C2×C42—24no—
132C2×C42—24no—
133C2×C42—41no—
134C2×C42—41no—
135C2×C42—42no—
136C2×C42—42no—
137C2×C42—42no—
138C2×C42—42no—
139C2×C42SD1614yes—
140C2×C42Q1614yes—
141C2×C42C2×D412yes—
142C2×C42C2×D412yes—
143C2×C42C2×Q812yes—
144C2×C42#4114yes—
145C2×C42#4112yes—
146C4⋊C8—44no—
147#103—416no—
148C22⋊C8—24no—
149C22⋊C8—24no—
150C4×C8—44no—
151#95—28no—
152#95—24no—
153#95—24no—
154#95—24no—
155#95—24no—
156#95—24no—
157#95—44no—
158#95—44no—
159#95—44no—
160#95—44no—
161C24C22×D411yescharacteristic
162C2×Q8—28no—
163C2×Q8—24no—
164C2×Q8—44no—
165C2×Q8—44no—
166C2×Q8—44no—
167C2×Q8—48no—
168C2×Q8—48no—
169C2×Q8—42no—
170C2×Q8—84no—
171C2×Q8—84no—
172C22×C4—24no—
173C22×C4—22no—
174C22×C4—22no—
175C22×C4—22no—
176C22×C4—22no—
177C22×C4—22no—
178C22×C4—24no—
179C22×C4—28no—
180C22×C4—22no—
181C22×C4—42no—
182C22×C4—42no—
183C22×C4—42no—
184C22×C4—41no—
185C22×C4—41no—
186C22×C4—42no—
187C22×C4—42no—
188C22×C4—42no—
189C22×C4—42no—
190C22×C4—42no—
191C22×C4—42no—
192C22×C4—42no—
193C22×C4—42no—
194C22×C4—42no—
195C22×C4—41no—
196C22×C4—42no—
197C22×C4—41no—
198C22×C4—42no—
199C22×C4—42no—
200C22×C4#12012yes—
201C22×C4#12011yescharacteristic
202C22×C4#12011yescharacteristic
203C22×C4C22⋊Q812yes—
204C22×C4C22⋊Q812yes—
205C22×C4#12412yes—
206C22×C4#12412yes—
207C22×C4C2×SD1614yes—
208C22×C4C2×SD1612yes—
209C22×C4C2×Q1614yes—
210C22×C4C2×Q1612yes—
211C22×C4AGL(1,ℤ/8)12yes—
212C22×C4#13712yes—
213C22×C4C22×D412yes—
214C22×C4C22×Q811yescharacteristic
215C22×C4C2×#4112yes—
216C22×C4C2×#4111yescharacteristic
217C2×C8—48no—
218C2×C8—44no—
219C4⋊C4—24no—
220C4⋊C4—48no—
221C4⋊C4—416no—
222C4⋊C4—48no—
223C4⋊C4—48no—
224C4⋊C4—42no—
225C4⋊C4—48no—
226C4⋊C4—48no—
227C4⋊C4—84no—
228C4⋊C4—84no—
229C4⋊C4—84no—
230C22⋊C4—44no—
231C22⋊C4—44no—
232C42—24no—
233C42—24no—
234C42—24no—
235C42—24no—
236C42—44no—
237C42—44no—
238C42—48no—
239C42—84no—
240C42C22⋊Q814yes—
241C42C22⋊Q814yes—
242C42#12414yes—
243C42#12414yes—
244C23—22no—
245C23—22no—
246C23C2×#12011yescharacteristic
247C23C2×C22⋊Q811yescharacteristic
248C23C2×#12411yescharacteristic
249C23C22×SD1611yescharacteristic
250C23C22×Q1611yescharacteristic
251C23C2×AGL(1,ℤ/8)11yescharacteristic
252C23C2×#13711yescharacteristic
253Q8—28no—
254Q8—48no—
255Q8—44no—
256Q8—816no—
257C2×C4—24no—
258C2×C4—24no—
259C2×C4—24no—
260C2×C4—24no—
261C2×C4—24no—
262C2×C4—42no—
263C2×C4—42no—
264C2×C4—41no—
265C2×C4—41no—
266C2×C4—42no—
267C2×C4—42no—
268C2×C4—44no—
269C2×C4—42no—
270C2×C4—42no—
271C2×C4—42no—
272C2×C4—42no—
273C2×C4—44no—
274C2×C4—42no—
275C2×C4—44no—
276C2×C4—44no—
277C2×C4—44no—
278C2×C4—44no—
279C2×C4—44no—
280C2×C4—42no—
281C2×C4—42no—
282C2×C4—42no—
283C2×C4—48no—
284C2×C4—48no—
285C2×C4—48no—
286C2×C4—82no—
287C2×C4—82no—
288C2×C4—82no—
289C2×C4—82no—
290C2×C4—82no—
291C2×C4—82no—
292C2×C4—82no—
293C2×C4—82no—
294C2×C4#39314yes—
295C2×C4#39312yes—
296C2×C4#39312yes—
297C2×C4#45012yes—
298C2×C4#45012yes—
299C2×C4#45112yes—
300C2×C4#45112yes—
301C2×C4#47514yes—
302C2×C4#47714yes—
303C2×C4#48714yes—
304C2×C4#48914yes—
305C2×C4C2×C22⋊Q812yes—
306C2×C4C2×#12412yes—
307C8—84no—
308C22—21no—
309C22—21no—
310C22—21no—
311C22—21no—
312C22—22no—
313C22—22no—
314C22—22no—
315C22—22no—
316C22C2×#39311yescharacteristic
317C22C2×#45011yescharacteristic
318C22C2×#45111yescharacteristic
319C22#301812yes—
320C22#302512yes—
321C22C2×#47511yescharacteristic
322C22C2×#47711yescharacteristic
323C22C2×#48711yescharacteristic
324C22C2×#48911yescharacteristic
325C4—24no—
326C4—22no—
327C4—22no—
328C4—44no—
329C4—44no—
330C4—44no—
331C4—84no—
332C4—82no—
333C4—88no—
334C4#202414yes—
335C4#202414yes—
336C2—21no—
337C2—21no—
338C2—21no—
339C2—21no—
340C2#1812312yes—
341C2#1812512yes—
342C2C2×#202411yescharacteristic
343C2#2346111yescharacteristic
344C2#2346711yescharacteristic
subgroupG/H
centerC22×C4#120
commutator, Frattini, cosocleC23×C4C24
socleC23C2×#120
prime radicalC24C22×D4
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C4 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
41. C2×C4×Q8Q8semidirect

Maximal subgroups

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

subgroupindex
#346422
#207342
#138502
#103432
#103412
#80142
#79062