Finite Group Databasebeta

← #207246

Subgroups — #207246

Order 512.

1244 conjugacy classes of subgroups.

The subgroup lattice has 369 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35002C211yesmaximal, characteristic
2#21930C211yesmaximal, characteristic
3#13802C211yesmaximal, characteristic
4#9912C214yesmaximal, complemented
5#9909C214yesmaximal, complemented
6#8023C212yesmaximal
7C2×#1294C212yesmaximal
8C2×#543—21no—
9C22×C22⋊C8C2212yes—
10#2857C2214yes—
11#2659—22no—
12#2549—24no—
13#2440C418yescomplemented
14#2315—21no—
15C2×#380C2211yescharacteristic
16C2×#380C2212yes—
17C2×C4×C22⋊C4—21no—
18C2×C4×C22⋊C4C2211yescharacteristic
19C22×#95C2211yescharacteristic
20#1850—24no—
21#1797C2214yes—
22#1436—22no—
23#1434C2212yes—
24C22⋊#95—21no—
25C4×#95C2212yes—
26#1294C2214yes—
27#1294C2214yes—
28#1294C2218yes—
29C24×C4—21no—
30C24×C4C2311yescharacteristic
31C23×C8—22no—
32#543—44no—
33#543—44no—
34C2×C4⋊Q8C2×C414yescomplemented
35C2×C22⋊Q8—24no—
36C2×C22⋊Q8—28no—
37C22×C4⋊C4—21no—
38C22×C4⋊C4C2×C414yes—
39C22×C22⋊C4—22no—
40C22×C22⋊C4—21no—
41C22×C22⋊C4D411yescharacteristic
42C22×C22⋊C4D411yescharacteristic
43C22×C42—22no—
44C22×C42C2312yes—
45C2×C4⋊C8—24no—
46C2×C22⋊C8—24no—
47C2×C22⋊C8—24no—
48C2×C22⋊C8—22no—
49C2×C22⋊C8C2314yes—
50#398—28no—
51#395—28no—
52#391—24no—
53#389—24no—
54#389C2×C418yescomplemented
55#385—28no—
56#384C2×C414yescomplemented
57#380—24no—
58#380—22no—
59#380—22no—
60#380—22no—
61#380C2×C414yes—
62#380Q814yescomplemented
63#380C2314yes—
64C4×C22⋊C4—44no—
65C4×C22⋊C4C2×C414yes—
66C4×C22⋊C4D414yes—
67C2×#95—21no—
68C2×#95—22no—
69C2×#95—22no—
70C2×#95—22no—
71C2×#95—22no—
72C2×#95—22no—
73C2×#95D412yes—
74C2×#95Q814yescomplemented
75C2×#95C2311yescharacteristic
76C2×#95C2312yes—
77C2×#95C2311yescharacteristic
78#336—24no—
79C25C2×D411yescharacteristic
80C22×Q8—24no—
81C23×C4—21no—
82C23×C4—22no—
83C23×C4—21no—
84C23×C4—24no—
85C23×C4—24no—
86C23×C4—44no—
87C23×C4—42no—
88C23×C4—42no—
89C23×C4—82no—
90C23×C4—82no—
91C23×C4C22×C412yes—
92C23×C4C2×D412yes—
93C23×C4C2×D411yescharacteristic
94C23×C4C2×D412yes—
95C23×C4C2×D412yes—
96C23×C4C2×Q812yes—
97C23×C4C2×Q812yes—
98C23×C4#4112yes—
99C23×C4#4112yes—
100C23×C4C2411yescharacteristic
101C22×C8—24no—
102C22×C8—44no—
103C22×C8—84no—
104C4⋊Q8—84no—
105C22⋊Q8—48no—
106C22⋊Q8—48no—
107C22⋊Q8—432no—
108C2×C4⋊C4—24no—
109C2×C4⋊C4—28no—
110C2×C4⋊C4—24no—
111C2×C4⋊C4—24no—
112C2×C4⋊C4—24no—
113C2×C4⋊C4—24no—
114C2×C4⋊C4—84no—
115C2×C4⋊C4—82no—
116C2×C4⋊C4C22×C418yes—
117C2×C22⋊C4—22no—
118C2×C22⋊C4—22no—
119C2×C22⋊C4—24no—
120C2×C22⋊C4—24no—
121C2×C22⋊C4—24no—
122C2×C22⋊C4—44no—
123C2×C22⋊C4—42no—
124C2×C22⋊C4—44no—
125C2×C22⋊C4—42no—
126C2×C22⋊C4C22⋊C414yes—
127C2×C22⋊C4C22⋊C414yes—
128C2×C22⋊C4D814yes—
129C2×C22⋊C4SD1614yes—
130C2×C42—44no—
131C2×C42—44no—
132C2×C42—82no—
133C2×C42—82no—
134C2×C42C22×C414yes—
135C2×C42C2×D414yes—
136C4⋊C8—84no—
137C22⋊C8—24no—
138C22⋊C8—24no—
139C22⋊C8—44no—
140#95—24no—
141#95—28no—
142#95—24no—
143#95—24no—
144#95—21no—
145#95—21no—
146#95—22no—
147#95—44no—
148#95—44no—
149#95—82no—
150#95C22⋊C418yes—
151#95C2×Q812yes—
152#95C2×Q812yes—
153#95C2×Q814yes—
154C24—42no—
155C24—42no—
156C24C2×C22⋊C412yes—
157C24C22⋊Q812yes—
158C24C22⋊Q812yes—
159C24C2×D812yes—
160C24C2×SD1612yes—
161C24#13712yes—
162C24#13712yes—
163C24C22×D411yescharacteristic
164C2×Q8—48no—
165C2×Q8—84no—
166C2×Q8—84no—
167C22×C4—24no—
168C22×C4—22no—
169C22×C4—22no—
170C22×C4—22no—
171C22×C4—28no—
172C22×C4—22no—
173C22×C4—24no—
174C22×C4—42no—
175C22×C4—42no—
176C22×C4—42no—
177C22×C4—44no—
178C22×C4—42no—
179C22×C4—44no—
180C22×C4—42no—
181C22×C4—42no—
182C22×C4—48no—
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—41no—
193C22×C4—42no—
194C22×C4—42no—
195C22×C4—41no—
196C22×C4—48no—
197C22×C4—48no—
198C22×C4—82no—
199C22×C4—82no—
200C22×C4—82no—
201C22×C4—82no—
202C22×C4—84no—
203C22×C4—84no—
204C22×C4—84no—
205C22×C4—88no—
206C22×C4—88no—
207C22×C4C2×C22⋊C414yes—
208C22×C4C2×C22⋊C412yes—
209C22×C4C2×C22⋊C414yes—
210C22×C4C4×Q814yes—
211C22×C4C4×Q814yes—
212C22×C4C22⋊Q812yes—
213C22×C4C22⋊Q812yes—
214C22×C4#12414yes—
215C22×C4C4⋊Q814yes—
216C22×C4C2×D814yes—
217C22×C4C2×SD1614yes—
218C22×C4C23×C411yescharacteristic
219C22×C4C22×D411yescharacteristic
220C22×C4C22×Q811yescharacteristic
221C22×C4C22×Q811yescharacteristic
222C22×C4C2×#4111yescharacteristic
223C22×C4C2×#4111yescharacteristic
224C2×C8—48no—
225C2×C8—44no—
226C2×C8—88no—
227C4⋊C4—48no—
228C4⋊C4—48no—
229C4⋊C4—416no—
230C4⋊C4—84no—
231C4⋊C4—84no—
232C4⋊C4—84no—
233C4⋊C4—84no—
234C22⋊C4—28no—
235C22⋊C4—44no—
236C22⋊C4—44no—
237C22⋊C4—48no—
238C22⋊C4—48no—
239C22⋊C4#102116yes—
240C42—44no—
241C42—44no—
242C42—88no—
243C23—22no—
244C23—24no—
245C23—22no—
246C23—24no—
247C23—42no—
248C23—42no—
249C23—42no—
250C23—44no—
251C23—44no—
252C23—42no—
253C23—44no—
254C23—44no—
255C23#39114yes—
256C23C2×#10214yes—
257C23#41914yes—
258C23#47414yes—
259C23#47414yes—
260C23#47514yes—
261C23#47514yes—
262C23C22×C22⋊C411yescharacteristic
263C23C2×C22⋊Q811yescharacteristic
264C23C2×C22⋊Q811yescharacteristic
265C23C22×D811yescharacteristic
266C23C22×SD1611yescharacteristic
267C23C2×#13711yescharacteristic
268C23C2×#13711yescharacteristic
269Q8—168no—
270C2×C4—22no—
271C2×C4—22no—
272C2×C4—24no—
273C2×C4—24no—
274C2×C4—24no—
275C2×C4—24no—
276C2×C4—24no—
277C2×C4—24no—
278C2×C4—24no—
279C2×C4—44no—
280C2×C4—42no—
281C2×C4—44no—
282C2×C4—44no—
283C2×C4—44no—
284C2×C4—42no—
285C2×C4—44no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—44no—
291C2×C4—44no—
292C2×C4—42no—
293C2×C4—44no—
294C2×C4—42no—
295C2×C4—44no—
296C2×C4—44no—
297C2×C4—42no—
298C2×C4—42no—
299C2×C4—48no—
300C2×C4—48no—
301C2×C4—48no—
302C2×C4—44no—
303C2×C4—42no—
304C2×C4—82no—
305C2×C4—82no—
306C2×C4—816no—
307C2×C4—82no—
308C2×C4—82no—
309C2×C4—82no—
310C2×C4—82no—
311C2×C4—82no—
312C2×C4—82no—
313C2×C4—168no—
314C2×C4#39114yes—
315C2×C4C2×#10218yes—
316C2×C4C2×C4×Q811yescharacteristic
317C2×C4C2×C4×Q811yescharacteristic
318C2×C4C2×#12411yescharacteristic
319C2×C4C2×C4⋊Q811yescharacteristic
320C8—84no—
321C22—21no—
322C22—21no—
323C22—24no—
324C22—21no—
325C22—21no—
326C22—24no—
327C22—22no—
328C22—22no—
329C22—28no—
330C22—22no—
331C22—22no—
332C22—28no—
333C22—44no—
334C22—44no—
335C22—44no—
336C22—44no—
337C22#198818yes—
338C22C2×#39111yescharacteristic
339C22#288312yes—
340C22C22×#10211yescharacteristic
341C22C2×#41911yescharacteristic
342C22#290112yes—
343C22C2×#47411yescharacteristic
344C22C2×#47411yescharacteristic
345C22C2×#47511yescharacteristic
346C22C2×#47511yescharacteristic
347C4—21no—
348C4—21no—
349C4—21no—
350C4—21no—
351C4—24no—
352C4—28no—
353C4—44no—
354C4—44no—
355C4—84no—
356C4—82no—
357C4—88no—
358C2—21no—
359C2—21no—
360C2—21no—
361C2—21no—
362C2—28no—
363C2#1761312yes—
364C2#1762112yes—
365C2C2×#198811yescharacteristic
366C2#2133011yescharacteristic
367C2#2133311yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #9912C2semidirect
5. #9909C2semidirect
13. #2440C4semidirect
34. C2×C4⋊Q8C2×C4semidirect
54. #389C2×C4semidirect
56. #384C2×C4semidirect
62. #380Q8semidirect
74. C2×#95Q8semidirect

Maximal subgroups

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

subgroupindex
#350022
#219302
#138022
#99122
#99092
#80232
C2×#12942