Finite Group Databasebeta

← #237807

Subgroups — #237807

Order 512.

1272 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35002C211yesmaximal, characteristic
2C2×#1823C212yesmaximal
3#13904C211yesmaximal, characteristic
4#10342C218yesmaximal, complemented
5#8023C212yesmaximal
6#7907C211yesmaximal, characteristic
7C2×#543—21no—
8C23×C4⋊C4C2211yescharacteristic
9C22×C22⋊C8C2212yes—
10#2857—22no—
11#2659C2214yes—
12#2549—24no—
13#2440—24no—
14#2315—21no—
15C2×#381—21no—
16C2×#381—21no—
17C2×#380C2211yescharacteristic
18C2×#380C2212yes—
19C2×C4×C22⋊C4C2211yescharacteristic
20#1823—24no—
21#1823C2218yes—
22C2×#340—24no—
23#1445C2214yes—
24C4⋊#95—22no—
25#1294C2218yes—
26#1280C2214yes—
27C24×C4—21no—
28C24×C4C2311yescharacteristic
29C23×C8—22no—
30#543—44no—
31#543—44no—
32C2×C4⋊Q8—22no—
33C2×C22⋊Q8—28no—
34C2×C22⋊Q8—24no—
35C22×C4⋊C4—22no—
36C22×C4⋊C4—22no—
37C22×C4⋊C4—24no—
38C22×C4⋊C4—21no—
39C22×C4⋊C4—22no—
40C22×C4⋊C4—21no—
41C22×C4⋊C4D414yes—
42C22×C4⋊C4Q814yescomplemented
43C22×C4⋊C4C2312yes—
44C22×C22⋊C4—22no—
45C22×C22⋊C4—21no—
46C22×C22⋊C4D411yescharacteristic
47C22×C22⋊C4D411yescharacteristic
48C22×C42D412yes—
49C2×C22⋊C8—24no—
50C2×C22⋊C8—24no—
51C2×C22⋊C8—22no—
52C2×C22⋊C8C2314yes—
53#398—28no—
54#395—28no—
55#391—24no—
56#389—24no—
57#389—24no—
58#385—28no—
59#384—22no—
60#381—44no—
61#381—44no—
62#380—24no—
63#380—22no—
64#380—22no—
65#380—24no—
66#380D414yes—
67#380C2314yes—
68C4×C22⋊C4—22no—
69C4×C22⋊C4—22no—
70C2×#95—22no—
71C2×#95—24no—
72C2×#95—24no—
73C2×#95C2312yes—
74C2×#95C2312yes—
75#340—216no—
76C25C2×D411yescharacteristic
77C22×Q8—24no—
78C23×C4—24no—
79C23×C4—22no—
80C23×C4—24no—
81C23×C4—22no—
82C23×C4—44no—
83C23×C4—42no—
84C23×C4—42no—
85C23×C4—42no—
86C23×C4—42no—
87C23×C4—42no—
88C23×C4—42no—
89C23×C4C2×D411yescharacteristic
90C23×C4C2×D412yes—
91C23×C4C2×D412yes—
92C23×C4C2×D412yes—
93C23×C4C2×D412yes—
94C23×C4C2×Q812yes—
95C23×C4#4114yes—
96C23×C4#4112yes—
97C23×C4C2411yescharacteristic
98C22×C8—24no—
99C22×C8—48no—
100C22×C8—44no—
101C4⋊Q8—84no—
102C22⋊Q8—432no—
103C22⋊Q8—48no—
104C22⋊Q8—48no—
105C2×C4⋊C4—28no—
106C2×C4⋊C4—28no—
107C2×C4⋊C4—24no—
108C2×C4⋊C4—24no—
109C2×C4⋊C4—24no—
110C2×C4⋊C4—28no—
111C2×C4⋊C4—24no—
112C2×C4⋊C4—24no—
113C2×C4⋊C4—22no—
114C2×C4⋊C4—24no—
115C2×C4⋊C4—24no—
116C2×C4⋊C4—41no—
117C2×C4⋊C4—41no—
118C2×C4⋊C4—44no—
119C2×C4⋊C4—44no—
120C2×C4⋊C4—42no—
121C2×C4⋊C4—48no—
122C2×C4⋊C4—44no—
123C2×C4⋊C4—42no—
124C2×C4⋊C4C2×D418yes—
125C2×C4⋊C4C2×Q814yes—
126C2×C22⋊C4—22no—
127C2×C22⋊C4—22no—
128C2×C22⋊C4—22no—
129C2×C22⋊C4—22no—
130C2×C22⋊C4—24no—
131C2×C22⋊C4—24no—
132C2×C22⋊C4—44no—
133C2×C22⋊C4—44no—
134C2×C22⋊C4—42no—
135C2×C22⋊C4—42no—
136C2×C22⋊C4—42no—
137C2×C22⋊C4D814yes—
138C2×C22⋊C4SD1614yes—
139C2×C42—22no—
140C2×C42—22no—
141C2×C42—44no—
142C22⋊C8—24no—
143C22⋊C8—24no—
144C22⋊C8—44no—
145#95—28no—
146#95—24no—
147#95—24no—
148#95—24no—
149#95—24no—
150#95—48no—
151#95—48no—
152C24—42no—
153C24—42no—
154C24#12012yes—
155C24C22⋊Q812yes—
156C24#12412yes—
157C24C2×D812yes—
158C24C2×SD1612yes—
159C24AGL(1,ℤ/8)12yes—
160C24#13712yes—
161C24C22×D411yescharacteristic
162C2×Q8—48no—
163C2×Q8—84no—
164C2×Q8—84no—
165C22×C4—24no—
166C22×C4—28no—
167C22×C4—22no—
168C22×C4—22no—
169C22×C4—22no—
170C22×C4—22no—
171C22×C4—28no—
172C22×C4—24no—
173C22×C4—44no—
174C22×C4—48no—
175C22×C4—44no—
176C22×C4—44no—
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—42no—
187C22×C4—42no—
188C22×C4—48no—
189C22×C4—48no—
190C22×C4—48no—
191C22×C4—48no—
192C22×C4—44no—
193C22×C4—44no—
194C22×C4—42no—
195C22×C4—42no—
196C22×C4—42no—
197C22×C4—84no—
198C22×C4—84no—
199C22×C4—84no—
200C22×C4—84no—
201C22×C4—84no—
202C22×C4—84no—
203C22×C4#12011yescharacteristic
204C22×C4#12011yescharacteristic
205C22×C4C22⋊Q814yes—
206C22×C4C22⋊Q812yes—
207C22×C4C22⋊Q814yes—
208C22×C4#12414yes—
209C22×C4#12412yes—
210C22×C4#12414yes—
211C22×C4C2×D814yes—
212C22×C4C2×SD1614yes—
213C22×C4C22×D412yes—
214C22×C4C22×Q811yescharacteristic
215C22×C4C2×#4112yes—
216C22×C4C2×#4111yescharacteristic
217C2×C8—48no—
218C2×C8—44no—
219C2×C8—88no—
220C4⋊C4—28no—
221C4⋊C4—48no—
222C4⋊C4—48no—
223C4⋊C4—416no—
224C4⋊C4—44no—
225C4⋊C4—44no—
226C4⋊C4—44no—
227C4⋊C4—48no—
228C4⋊C4—84no—
229C4⋊C4—84no—
230C4⋊C4—84no—
231C22⋊C4—28no—
232C22⋊C4—28no—
233C22⋊C4—44no—
234C22⋊C4—44no—
235C22⋊C4—48no—
236C22⋊C4—84no—
237C42—44no—
238C42—82no—
239C23—22no—
240C23—24no—
241C23—22no—
242C23—24no—
243C23—42no—
244C23—42no—
245C23—42no—
246C23—42no—
247C23—44no—
248C23—44no—
249C23—44no—
250C23—44no—
251C23#39314yes—
252C23#44714yes—
253C23#44814yes—
254C23#47414yes—
255C23#47614yes—
256C23#48614yes—
257C23#48914yes—
258C23C2×#12011yescharacteristic
259C23C2×C22⋊Q811yescharacteristic
260C23C2×#12411yescharacteristic
261C23C22×D811yescharacteristic
262C23C22×SD1611yescharacteristic
263C23C2×AGL(1,ℤ/8)11yescharacteristic
264C23C2×#13711yescharacteristic
265Q8—168no—
266C2×C4—22no—
267C2×C4—24no—
268C2×C4—24no—
269C2×C4—28no—
270C2×C4—28no—
271C2×C4—24no—
272C2×C4—24no—
273C2×C4—24no—
274C2×C4—44no—
275C2×C4—44no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—42no—
279C2×C4—42no—
280C2×C4—42no—
281C2×C4—42no—
282C2×C4—42no—
283C2×C4—44no—
284C2×C4—44no—
285C2×C4—44no—
286C2×C4—48no—
287C2×C4—48no—
288C2×C4—48no—
289C2×C4—44no—
290C2×C4—44no—
291C2×C4—82no—
292C2×C4—82no—
293C2×C4—82no—
294C2×C4—82no—
295C2×C4—82no—
296C2×C4—82no—
297C2×C4—816no—
298C2×C4—82no—
299C2×C4—84no—
300C2×C4—84no—
301C2×C4—82no—
302C2×C4—82no—
303C2×C4—168no—
304C2×C4#39312yes—
305C2×C4#39312yes—
306C2×C4#45814yes—
307C2×C4C2×C22⋊Q812yes—
308C2×C4C2×#12412yes—
309C8—84no—
310C22—22no—
311C22—22no—
312C22—28no—
313C22—22no—
314C22—22no—
315C22—28no—
316C22—21no—
317C22—21no—
318C22—24no—
319C22—21no—
320C22—21no—
321C22—24no—
322C22—44no—
323C22—44no—
324C22—44no—
325C22—44no—
326C22#202318yes—
327C22C2×#39311yescharacteristic
328C22C2×#44711yescharacteristic
329C22C2×#44811yescharacteristic
330C22#302212yes—
331C22C2×#45812yes—
332C22C2×#47411yescharacteristic
333C22C2×#47611yescharacteristic
334C22C2×#48611yescharacteristic
335C22C2×#48911yescharacteristic
336C4—22no—
337C4—22no—
338C4—22no—
339C4—44no—
340C4—44no—
341C4—88no—
342C4—84no—
343C4—162no—
344C4#203014yes—
345C2—21no—
346C2—21no—
347C2—28no—
348C2—21no—
349C2—21no—
350C2#1820212yes—
351C2#1820412yes—
352C2C2×#202311yescharacteristic
353C2#2346411yescharacteristic
354C2C2×#203011yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #10342C2semidirect
42. C22×C4⋊C4Q8semidirect

Maximal subgroups

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

subgroupindex
#350022
C2×#18232
#139042
#103422
#80232
#79072