Finite Group Databasebeta

← #237809

Subgroups — #237809

Order 512.

1244 conjugacy classes of subgroups.

The subgroup lattice has 344 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35002C211yesmaximal, characteristic
2C2×#1824C212yesmaximal
3#13924C211yesmaximal, characteristic
4#10343C218yesmaximal, 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×#380C2211yescharacteristic
16C2×#380C2212yes—
17C2×C4×C22⋊C4C2211yescharacteristic
18#1824—24no—
19#1824C2218yes—
20C2×#340—24no—
21#1445C2214yes—
22C4⋊#95—22no—
23C22⋊#95—21no—
24C22⋊#95—21no—
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#380—24no—
61#380—22no—
62#380—22no—
63#380—24no—
64#380D414yes—
65#380C2314yes—
66C4×C22⋊C4—22no—
67C4×C22⋊C4—22no—
68C2×#95—22no—
69C2×#95—24no—
70C2×#95—24no—
71C2×#95C2312yes—
72C2×#95C2312yes—
73#340—216no—
74C25C2×D411yescharacteristic
75C22×Q8—24no—
76C23×C4—22no—
77C23×C4—24no—
78C23×C4—24no—
79C23×C4—22no—
80C23×C4—44no—
81C23×C4—42no—
82C23×C4—42no—
83C23×C4—84no—
84C23×C4C2×D411yescharacteristic
85C23×C4C2×D412yes—
86C23×C4C2×D412yes—
87C23×C4C2×D412yes—
88C23×C4C2×D412yes—
89C23×C4C2×Q812yes—
90C23×C4#4114yes—
91C23×C4#4112yes—
92C23×C4C2411yescharacteristic
93C22×C8—24no—
94C22×C8—48no—
95C22×C8—44no—
96C4⋊Q8—84no—
97C22⋊Q8—432no—
98C22⋊Q8—48no—
99C22⋊Q8—48no—
100C2×C4⋊C4—28no—
101C2×C4⋊C4—28no—
102C2×C4⋊C4—24no—
103C2×C4⋊C4—24no—
104C2×C4⋊C4—24no—
105C2×C4⋊C4—28no—
106C2×C4⋊C4—24no—
107C2×C4⋊C4—24no—
108C2×C4⋊C4—22no—
109C2×C4⋊C4—24no—
110C2×C4⋊C4—24no—
111C2×C4⋊C4—41no—
112C2×C4⋊C4—41no—
113C2×C4⋊C4—44no—
114C2×C4⋊C4—44no—
115C2×C4⋊C4—42no—
116C2×C4⋊C4—48no—
117C2×C4⋊C4—44no—
118C2×C4⋊C4—42no—
119C2×C4⋊C4C2×D418yes—
120C2×C4⋊C4C2×Q814yes—
121C2×C22⋊C4—22no—
122C2×C22⋊C4—22no—
123C2×C22⋊C4—22no—
124C2×C22⋊C4—22no—
125C2×C22⋊C4—24no—
126C2×C22⋊C4—24no—
127C2×C22⋊C4—44no—
128C2×C22⋊C4—44no—
129C2×C22⋊C4—42no—
130C2×C22⋊C4—42no—
131C2×C22⋊C4—42no—
132C2×C22⋊C4SD1614yes—
133C2×C22⋊C4Q1614yes—
134C2×C42—22no—
135C2×C42—22no—
136C2×C42—44no—
137C22⋊C8—24no—
138C22⋊C8—24no—
139C22⋊C8—44no—
140#95—28no—
141#95—24no—
142#95—24no—
143#95—24no—
144#95—24no—
145#95—84no—
146#95—84no—
147C24—42no—
148C24—42no—
149C24#12012yes—
150C24C22⋊Q812yes—
151C24#12412yes—
152C24C2×SD1612yes—
153C24C2×Q1612yes—
154C24AGL(1,ℤ/8)12yes—
155C24#13712yes—
156C24C22×D411yescharacteristic
157C2×Q8—48no—
158C2×Q8—84no—
159C2×Q8—84no—
160C22×C4—24no—
161C22×C4—28no—
162C22×C4—22no—
163C22×C4—22no—
164C22×C4—22no—
165C22×C4—22no—
166C22×C4—28no—
167C22×C4—24no—
168C22×C4—44no—
169C22×C4—48no—
170C22×C4—44no—
171C22×C4—44no—
172C22×C4—42no—
173C22×C4—42no—
174C22×C4—42no—
175C22×C4—42no—
176C22×C4—42no—
177C22×C4—44no—
178C22×C4—44no—
179C22×C4—42no—
180C22×C4—42no—
181C22×C4—48no—
182C22×C4—48no—
183C22×C4—44no—
184C22×C4—44no—
185C22×C4—42no—
186C22×C4—42no—
187C22×C4—42no—
188C22×C4—88no—
189C22×C4—84no—
190C22×C4—84no—
191C22×C4—88no—
192C22×C4—88no—
193C22×C4#12011yescharacteristic
194C22×C4#12011yescharacteristic
195C22×C4C22⋊Q814yes—
196C22×C4C22⋊Q812yes—
197C22×C4C22⋊Q814yes—
198C22×C4#12414yes—
199C22×C4#12412yes—
200C22×C4#12414yes—
201C22×C4C2×SD1614yes—
202C22×C4C2×Q1614yes—
203C22×C4C22×D412yes—
204C22×C4C22×Q811yescharacteristic
205C22×C4C2×#4112yes—
206C22×C4C2×#4111yescharacteristic
207C2×C8—48no—
208C2×C8—44no—
209C2×C8—88no—
210C4⋊C4—28no—
211C4⋊C4—48no—
212C4⋊C4—48no—
213C4⋊C4—416no—
214C4⋊C4—44no—
215C4⋊C4—44no—
216C4⋊C4—44no—
217C4⋊C4—48no—
218C4⋊C4—84no—
219C4⋊C4—84no—
220C4⋊C4—84no—
221C22⋊C4—28no—
222C22⋊C4—28no—
223C22⋊C4—44no—
224C22⋊C4—44no—
225C22⋊C4—48no—
226C22⋊C4—84no—
227C42—44no—
228C42—82no—
229C23—22no—
230C23—24no—
231C23—22no—
232C23—24no—
233C23—42no—
234C23—42no—
235C23—42no—
236C23—42no—
237C23—44no—
238C23—44no—
239C23—44no—
240C23—44no—
241C23#39314yes—
242C23#45014yes—
243C23#45114yes—
244C23#47514yes—
245C23#47714yes—
246C23#48714yes—
247C23#48914yes—
248C23C2×#12011yescharacteristic
249C23C2×C22⋊Q811yescharacteristic
250C23C2×#12411yescharacteristic
251C23C22×SD1611yescharacteristic
252C23C22×Q1611yescharacteristic
253C23C2×AGL(1,ℤ/8)11yescharacteristic
254C23C2×#13711yescharacteristic
255Q8—168no—
256C2×C4—22no—
257C2×C4—24no—
258C2×C4—24no—
259C2×C4—28no—
260C2×C4—28no—
261C2×C4—24no—
262C2×C4—24no—
263C2×C4—24no—
264C2×C4—44no—
265C2×C4—44no—
266C2×C4—41no—
267C2×C4—41no—
268C2×C4—42no—
269C2×C4—42no—
270C2×C4—42no—
271C2×C4—42no—
272C2×C4—42no—
273C2×C4—44no—
274C2×C4—44no—
275C2×C4—44no—
276C2×C4—44no—
277C2×C4—44no—
278C2×C4—48no—
279C2×C4—48no—
280C2×C4—48no—
281C2×C4—82no—
282C2×C4—82no—
283C2×C4—84no—
284C2×C4—816no—
285C2×C4—84no—
286C2×C4—82no—
287C2×C4—84no—
288C2×C4—84no—
289C2×C4—82no—
290C2×C4—82no—
291C2×C4—168no—
292C2×C4#39312yes—
293C2×C4#39312yes—
294C2×C4#45814yes—
295C2×C4C2×C22⋊Q812yes—
296C2×C4C2×#12412yes—
297C8—84no—
298C22—22no—
299C22—22no—
300C22—28no—
301C22—22no—
302C22—22no—
303C22—28no—
304C22—21no—
305C22—21no—
306C22—24no—
307C22—21no—
308C22—21no—
309C22—24no—
310C22—44no—
311C22—44no—
312C22—44no—
313C22—44no—
314C22#202418yes—
315C22C2×#39311yescharacteristic
316C22C2×#45011yescharacteristic
317C22C2×#45111yescharacteristic
318C22#302112yes—
319C22C2×#45812yes—
320C22C2×#47511yescharacteristic
321C22C2×#47711yescharacteristic
322C22C2×#48711yescharacteristic
323C22C2×#48911yescharacteristic
324C4—22no—
325C4—22no—
326C4—22no—
327C4—44no—
328C4—44no—
329C4—88no—
330C4—84no—
331C4—162no—
332C4#203014yes—
333C2—21no—
334C2—21no—
335C2—21no—
336C2—21no—
337C2—28no—
338C2#1820312yes—
339C2#1820512yes—
340C2C2×#202411yescharacteristic
341C2#2346311yescharacteristic
342C2C2×#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. #10343C2semidirect
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×#18242
#139242
#103432
#80232
#79072