Finite Group Databasebeta

← #207018

Subgroups — #207018

Order 512.

930 conjugacy classes of subgroups.

The subgroup lattice has 347 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35104C211yesmaximal, characteristic
2#20534C211yesmaximal, characteristic
3#13954C211yesmaximal, characteristic
4#9887C214yesmaximal
5#9885C212yesmaximal
6#9884C212yesmaximal
7#8021C212yesmaximal
8#8017C212yesmaximal
9C2×#432C2212yes—
10#2857C2212yes—
11#2851C2212yes—
12#2627—22no—
13#2606—22no—
14#2605—22no—
15#2570C414yescomplemented
16#2550C414yescomplemented
17#2329—22no—
18C2×#389—22no—
19#2283C2212yes—
20C2×#382C2211yescharacteristic
21C2×C4×C4⋊C4C2211yescharacteristic
22C2×C4×C4⋊C4C2211yescharacteristic
23#1771—22no—
24#1771—22no—
25#1749C2212yes—
26C4×C22⋊C8C2211yescharacteristic
27C4×C22⋊C8C2211yescharacteristic
28#1444—22no—
29C4⋊#95C2212yes—
30C4⋊#95C2212yes—
31#1294C2214yes—
32#1294C2214yes—
33#1294C2214yes—
34#1294C2214yes—
35C2×C4⋊Q8C2×C412yes—
36C2×#125C2×C412yes—
37C2×C22⋊Q8—22no—
38C22×C4⋊C4—21no—
39C22×C4⋊C4—21no—
40C22×C4⋊C4—22no—
41C22×C4⋊C4—22no—
42C22×C4⋊C4D412yes—
43C22×C42—21no—
44C22×C42C2311yescharacteristic
45C22×C42C2311yescharacteristic
46C22×C42C2311yescharacteristic
47#432—24no—
48#432Q818yescomplemented
49C2×C4⋊C8—22no—
50C2×C22⋊C8C2312yes—
51C2×C22⋊C8C2312yes—
52C2×#97—22no—
53C2×#97—22no—
54C2×C4×C8—21no—
55C2×C4×C8—21no—
56#400—24no—
57#398—24no—
58#398—24no—
59#395—24no—
60#391—22no—
61#391—24no—
62#389—22no—
63#389—24no—
64#389—48no—
65#389C2×C414yes—
66#389C2×C414yes—
67#387—24no—
68#387—24no—
69#387—24no—
70#385—24no—
71#384C2×C412yes—
72#383—22no—
73#382—24no—
74#382C2×C412yes—
75#382C2×C412yes—
76#382C2×C412yes—
77#381—22no—
78#380—22no—
79#380C2314yes—
80C4×C4⋊C4—24no—
81C4×C4⋊C4C2×C414yes—
82C4×C4⋊C4C2×C414yes—
83C4×C4⋊C4D412yes—
84C4×C4⋊C4D412yes—
85C4×C22⋊C4—22no—
86#376—22no—
87C2×#95—22no—
88C2×#95—22no—
89C2×#95—21no—
90C2×#95D412yes—
91C2×#95C2312yes—
92C2×#95C2312yes—
93#336—21no—
94#336—21no—
95#336—22no—
96C22×Q8—22no—
97C23×C4—22no—
98C23×C4—22no—
99C23×C4C2×D411yescharacteristic
100C23×C4C2×D412yes—
101C23×C4C2×D411yescharacteristic
102C23×C4C2×D412yes—
103C23×C4C2411yescharacteristic
104C22×C8—22no—
105C22×C8—22no—
106C4⋊Q8—82no—
107#125—82no—
108C22⋊Q8—44no—
109C22⋊Q8—44no—
110C2×C4⋊C4—24no—
111C2×C4⋊C4—24no—
112C2×C4⋊C4—22no—
113C2×C4⋊C4—24no—
114C2×C4⋊C4—24no—
115C2×C4⋊C4—22no—
116C2×C4⋊C4—22no—
117C2×C4⋊C4—24no—
118C2×C4⋊C4—44no—
119C2×C4⋊C4—42no—
120C2×C4⋊C4—44no—
121C2×C4⋊C4—44no—
122C2×C4⋊C4—44no—
123C2×C4⋊C4—44no—
124C2×C4⋊C4—42no—
125C2×C4⋊C4—44no—
126C2×C4⋊C4—82no—
127C2×C4⋊C4—82no—
128C2×C4⋊C4C22⋊C414yes—
129C2×C4⋊C4C22⋊C414yes—
130C2×C4⋊C4C22×C414yes—
131C2×C4⋊C4C22×C414yes—
132C2×C22⋊C4—24no—
133C2×C42—24no—
134C2×C42—44no—
135C2×C42—42no—
136C2×C42—81no—
137C2×C42—81no—
138C2×C42C22×C412yes—
139C2×C42C22×C412yes—
140C2×C42C22×C412yes—
141C2×C42C2×D412yes—
142C2×C42C2×D411yescharacteristic
143C2×C42C2×D411yescharacteristic
144C2×C42C2×D412yes—
145C2×C42C2×Q814yes—
146C2×C42#4114yes—
147C4⋊C8—28no—
148C22⋊C8—22no—
149C22⋊C8—22no—
150C22⋊C8C2×Q814yes—
151C22⋊C8C2×Q814yes—
152#97—28no—
153#97—82no—
154C4×C8—81no—
155C4×C8—81no—
156#95—22no—
157#95—22no—
158#95—22no—
159#95—22no—
160#95—24no—
161#95—24no—
162#95—24no—
163#95—24no—
164#95—24no—
165#95—24no—
166#95—24no—
167#95—44no—
168#95—42no—
169#95—44no—
170#95C22⋊C414yes—
171#95C22⋊C414yes—
172C24C22×D411yescharacteristic
173C2×Q8—44no—
174C2×Q8—82no—
175C2×Q8—82no—
176C22×C4—22no—
177C22×C4—22no—
178C22×C4—22no—
179C22×C4—22no—
180C22×C4—24no—
181C22×C4—24no—
182C22×C4—24no—
183C22×C4—41no—
184C22×C4—42no—
185C22×C4—42no—
186C22×C4—42no—
187C22×C4—42no—
188C22×C4—42no—
189C22×C4—41no—
190C22×C4—42no—
191C22×C4—42no—
192C22×C4—42no—
193C22×C4—42no—
194C22×C4—42no—
195C22×C4—42no—
196C22×C4—42no—
197C22×C4—42no—
198C22×C4—42no—
199C22×C4—42no—
200C22×C4—42no—
201C22×C4—42no—
202C22×C4—42no—
203C22×C4C2×C22⋊C414yes—
204C22×C4C2×C22⋊C412yes—
205C22×C4C2×C22⋊C412yes—
206C22×C4C2×C22⋊C412yes—
207C22×C4C2×C22⋊C412yes—
208C22×C4#12412yes—
209C22×C4#12412yes—
210C22×C4C4⋊Q812yes—
211C22×C4C4⋊Q812yes—
212C22×C4AGL(1,ℤ/8)14yes—
213C22×C4#13714yes—
214C22×C4C23×C411yescharacteristic
215C22×C4C22×D411yescharacteristic
216C22×C4C22×Q812yes—
217C22×C4C2×#4112yes—
218C2×C8—24no—
219C2×C8—24no—
220C2×C8—42no—
221C2×C8—42no—
222C2×C8—42no—
223C2×C8—42no—
224C4⋊C4—28no—
225C4⋊C4—28no—
226C4⋊C4—44no—
227C4⋊C4—44no—
228C4⋊C4—48no—
229C4⋊C4—44no—
230C4⋊C4—48no—
231C4⋊C4—82no—
232C4⋊C4—84no—
233C4⋊C4—84no—
234C4⋊C4—82no—
235C4⋊C4—84no—
236C4⋊C4—88no—
237C4⋊C4—88no—
238C22⋊C4—44no—
239C22⋊C4—44no—
240C42—41no—
241C42—41no—
242C42—42no—
243C42—42no—
244C42—42no—
245C42—84no—
246C42—84no—
247C42C4×Q818yes—
248C42C22⋊Q814yes—
249C42C22⋊Q814yes—
250C23—21no—
251C23—21no—
252C23—22no—
253C23C22×C22⋊C411yescharacteristic
254C23C2×#12411yescharacteristic
255C23C2×C4⋊Q811yescharacteristic
256C23C2×AGL(1,ℤ/8)12yes—
257C23C2×#13712yes—
258Q8—164no—
259C2×C4—22no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—22no—
263C2×C4—22no—
264C2×C4—22no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—24no—
268C2×C4—24no—
269C2×C4—22no—
270C2×C4—22no—
271C2×C4—24no—
272C2×C4—24no—
273C2×C4—24no—
274C2×C4—24no—
275C2×C4—42no—
276C2×C4—44no—
277C2×C4—42no—
278C2×C4—44no—
279C2×C4—42no—
280C2×C4—42no—
281C2×C4—44no—
282C2×C4—44no—
283C2×C4—44no—
284C2×C4—44no—
285C2×C4—44no—
286C2×C4—44no—
287C2×C4—44no—
288C2×C4—44no—
289C2×C4—44no—
290C2×C4—44no—
291C2×C4—44no—
292C2×C4—82no—
293C2×C4—82no—
294C2×C4—82no—
295C2×C4—82no—
296C2×C4—82no—
297C2×C4—82no—
298C2×C4—82no—
299C2×C4—82no—
300C2×C4—82no—
301C2×C4—82no—
302C2×C4—82no—
303C2×C4#39114yes—
304C2×C4#39114yes—
305C2×C4#41714yes—
306C2×C4#41714yes—
307C2×C4C2×C4×Q812yes—
308C2×C4C2×C22⋊Q812yes—
309C8—44no—
310C8—44no—
311C22—21no—
312C22—21no—
313C22—21no—
314C22—21no—
315C22—22no—
316C22—22no—
317C22—22no—
318C22—22no—
319C22C2×#39111yescharacteristic
320C22#288311yescharacteristic
321C22#288411yescharacteristic
322C22C2×#41712yes—
323C22#290112yes—
324C22#313811yescharacteristic
325C22#313911yescharacteristic
326C22#316512yes—
327C4—24no—
328C4—24no—
329C4—22no—
330C4—22no—
331C4—24no—
332C4—24no—
333C4—44no—
334C4—84no—
335C4—84no—
336C4—84no—
337C4—162no—
338C2—21no—
339C2—21no—
340C2—22no—
341C2#1762212yes—
342C2#1762812yes—
343C2#2130711yescharacteristic
344C2#2132211yescharacteristic
345C2#2133411yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
15. #2570C4semidirect
16. #2550C4semidirect
48. #432Q8semidirect

Maximal subgroups

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

subgroupindex
#351042
#205342
#139542
#98872
#98852
#98842
#80212
#80172