Finite Group Databasebeta

← #381301

Subgroups — #381301

Order 512.

1338 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36625C211yesmaximal, characteristic
2#20682C211yesmaximal, characteristic
3#13753C211yesmaximal, characteristic
4#11386C212yesmaximal, complemented
5#11370C212yesmaximal, complemented
6#11201C214yesmaximal
7#10879C214yesmaximal, complemented
8C2×#533C2211yescharacteristic
9C23×C42C2211yescharacteristic
10C2×#407C2212yes—
11#2857C412yescomplemented
12#2857C412yescomplemented
13#2855—22no—
14#2851C414yescomplemented
15#2848—22no—
16C4×C22⋊Q8—22no—
17C4×C22⋊Q8—22no—
18#2296C2212yes—
19#2285C2211yescharacteristic
20#2280C2212yes—
21C2×C4×C22⋊C4C2212yes—
22C2×#376C2212yes—
23#1817—22no—
24#1817—22no—
25C2×#339C2218yes—
26C2×#339C2218yes—
27#1728C2212yes—
28#1728C2212yes—
29#1434—22no—
30C22⋊#95—22no—
31C4×#95C2212yes—
32C24×C4D412yes—
33C24×C4C2311yescharacteristic
34C22×M16C2314yes—
35#533—44no—
36C2×C4⋊Q8C2×C412yes—
37C2×#125C2×C412yes—
38C2×C22⋊Q8—22no—
39C2×C22⋊Q8—22no—
40C2×C22⋊Q8—22no—
41C2×C4×Q8—22no—
42C2×#117C2312yes—
43C2×#117C2312yes—
44C22×C22⋊C4—22no—
45C22×C42—44no—
46C22×C42C2×C412yes—
47C22×C42C2×C412yes—
48C22×C42D412yes—
49C22×C42D412yes—
50C22×C42D412yes—
51C22×C42C2314yes—
52C22×C42C2311yescharacteristic
53C22×C42C2311yescharacteristic
54#407—44no—
55C2×C22⋊C8—24no—
56#391—22no—
57#391—22no—
58#389—22no—
59#389—22no—
60#389C2×C414yes—
61#389C2×C414yes—
62#387—24no—
63#387—24no—
64#385—24no—
65#385—24no—
66#385—24no—
67#384C2×C411yescharacteristic
68#384C2×C411yescharacteristic
69#382C2×C412yes—
70#381—22no—
71#380—22no—
72#380—22no—
73#380—22no—
74#380C2×C414yes—
75#380C2×C414yes—
76#379—22no—
77C4×C4⋊C4—22no—
78C4×C4⋊C4—22no—
79C4×C4⋊C4—22no—
80C4×C22⋊C4—24no—
81C4×C22⋊C4—28no—
82#376—44no—
83C2×#95—22no—
84C2×#95—24no—
85C2×#95—24no—
86#339—216no—
87#339—48no—
88#336—24no—
89#336—22no—
90#336—22no—
91C25C2×D411yescharacteristic
92C22×Q8—22no—
93C23×C4—24no—
94C23×C4—24no—
95C23×C4—24no—
96C23×C4—24no—
97C23×C4—44no—
98C23×C4—44no—
99C23×C4—44no—
100C23×C4C22⋊C412yes—
101C23×C4C22⋊C412yes—
102C23×C4C22⋊C412yes—
103C23×C4C22⋊C412yes—
104C23×C4C22×C412yes—
105C23×C4C2×D411yescharacteristic
106C23×C4C2×D412yes—
107C23×C4C2×D412yes—
108C23×C4C2×D412yes—
109C23×C4C2411yescharacteristic
110C2×M16—28no—
111C2×M16—28no—
112C2×M16—24no—
113C2×M16—24no—
114C22×C8—24no—
115C4⋊Q8—44no—
116#125—44no—
117C22⋊Q8—48no—
118C22⋊Q8—48no—
119C22⋊Q8—48no—
120C4×Q8—48no—
121#117—28no—
122#117—28no—
123C2×C4⋊C4—24no—
124C2×C4⋊C4—22no—
125C2×C4⋊C4—22no—
126C2×C4⋊C4—22no—
127C2×C4⋊C4—24no—
128C2×C4⋊C4—24no—
129C2×C4⋊C4C22×C414yes—
130C2×C4⋊C4C22×C412yes—
131C2×C4⋊C4C22×C412yes—
132C2×C22⋊C4—28no—
133C2×C22⋊C4—24no—
134C2×C22⋊C4—24no—
135C2×C22⋊C4—22no—
136C2×C22⋊C4—22no—
137C2×C22⋊C4—48no—
138C2×C42—24no—
139C2×C42—24no—
140C2×C42—28no—
141C2×C42—28no—
142C2×C42—22no—
143C2×C42—22no—
144C2×C42—24no—
145C2×C42—44no—
146C2×C42—44no—
147C2×C42—44no—
148C2×C42—44no—
149C2×C42—44no—
150C2×C42C22⋊C412yes—
151C2×C42C22⋊C412yes—
152C2×C42C22⋊C412yes—
153C2×C42C22⋊C412yes—
154C2×C42C22×C411yescharacteristic
155C2×C42C22×C411yescharacteristic
156C2×C42C22×C412yes—
157C2×C42C2×D412yes—
158C2×C42C2×D411yescharacteristic
159C2×C42C2×D411yescharacteristic
160C2×C42#4118yes—
161C22⋊C8—48no—
162#95—28no—
163#95—24no—
164#95—24no—
165#95—24no—
166#95—24no—
167#95—24no—
168#95—48no—
169#95—48no—
170C24—22no—
171C24—24no—
172C24—22no—
173C24—22no—
174C24—42no—
175C24C2×C22⋊C412yes—
176C24C22×D411yescharacteristic
177C2×Q8—44no—
178C2×Q8—44no—
179C2×Q8—44no—
180C22×C4—22no—
181C22×C4—24no—
182C22×C4—24no—
183C22×C4—22no—
184C22×C4—22no—
185C22×C4—24no—
186C22×C4—24no—
187C22×C4—24no—
188C22×C4—24no—
189C22×C4—24no—
190C22×C4—24no—
191C22×C4—24no—
192C22×C4—24no—
193C22×C4—24no—
194C22×C4—24no—
195C22×C4—24no—
196C22×C4—22no—
197C22×C4—22no—
198C22×C4—24no—
199C22×C4—28no—
200C22×C4—42no—
201C22×C4—42no—
202C22×C4—42no—
203C22×C4—42no—
204C22×C4—42no—
205C22×C4—42no—
206C22×C4—42no—
207C22×C4—42no—
208C22×C4—42no—
209C22×C4—42no—
210C22×C4—42no—
211C22×C4—432no—
212C22×C4—48no—
213C22×C4—48no—
214C22×C4—48no—
215C22×C4—48no—
216C22×C4—44no—
217C22×C4—44no—
218C22×C4C2×C22⋊C412yes—
219C22×C4C2×C22⋊C412yes—
220C22×C4C2×C22⋊C412yes—
221C22×C4C2×C22⋊C412yes—
222C22×C4C2×C22⋊C412yes—
223C22×C4C23×C411yescharacteristic
224C22×C4C22×D411yescharacteristic
225C22×C4C2×#4114yes—
226C22×C4H(5,𝔽2)14yes—
227C22×C42−1+414yes—
228M16—48no—
229M16—48no—
230C2×C8—44no—
231C2×C8—44no—
232C2×C8—44no—
233C4⋊C4—24no—
234C4⋊C4—24no—
235C4⋊C4—28no—
236C4⋊C4—48no—
237C4⋊C4—44no—
238C4⋊C4—44no—
239C4⋊C4—48no—
240C4⋊C4—48no—
241C4⋊C4—44no—
242C22⋊C4—416no—
243C22⋊C4—48no—
244C22⋊C4—44no—
245C22⋊C4—44no—
246C22⋊C4—48no—
247C42—24no—
248C42—22no—
249C42—22no—
250C42—22no—
251C42—22no—
252C42—24no—
253C42—216no—
254C42—42no—
255C42—42no—
256C42—42no—
257C42—42no—
258C42—44no—
259C42—44no—
260C42—48no—
261C23—22no—
262C23—22no—
263C23—28no—
264C23—22no—
265C23—22no—
266C23—22no—
267C23—22no—
268C23—22no—
269C23—22no—
270C23—48no—
271C23—48no—
272C23—44no—
273C23—44no—
274C23#42114yes—
275C23C22×C22⋊C411yescharacteristic
276C23#52512yes—
277C23#53412yes—
278C23#53612yes—
279Q8—88no—
280C2×C4—22no—
281C2×C4—22no—
282C2×C4—22no—
283C2×C4—22no—
284C2×C4—22no—
285C2×C4—22no—
286C2×C4—22no—
287C2×C4—22no—
288C2×C4—22no—
289C2×C4—22no—
290C2×C4—22no—
291C2×C4—22no—
292C2×C4—22no—
293C2×C4—22no—
294C2×C4—48no—
295C2×C4—48no—
296C2×C4—48no—
297C2×C4—48no—
298C2×C4—48no—
299C2×C4—48no—
300C2×C4—44no—
301C2×C4—48no—
302C2×C4—48no—
303C2×C4—48no—
304C2×C4—48no—
305C2×C4—44no—
306C2×C4—48no—
307C2×C4—44no—
308C2×C4—44no—
309C2×C4—44no—
310C2×C4—44no—
311C2×C4—42no—
312C2×C4—42no—
313C2×C4—88no—
314C2×C4—88no—
315C2×C4#42114yes—
316C2×C4C4×#4112yes—
317C2×C4#51812yes—
318C2×C4#51912yes—
319C2×C4#53212yes—
320C2×C4#53412yes—
321C2×C4#53612yes—
322C2×C4#53918yes—
323C2×C4#54218yes—
324C8—84no—
325C22—24no—
326C22—24no—
327C22—22no—
328C22—22no—
329C22—24no—
330C22—24no—
331C22—24no—
332C22—416no—
333C22—42no—
334C22—42no—
335C22—42no—
336C22—42no—
337C22#229711yescharacteristic
338C22#230811yescharacteristic
339C22#231011yescharacteristic
340C22#263814yes—
341C22C2×#42112yes—
342C22C2×#42112yes—
343C4—24no—
344C4—42no—
345C4—42no—
346C4—42no—
347C4—42no—
348C4—88no—
349C4—84no—
350C4—82no—
351C4—82no—
352C4#233214yes—
353C4#266314yes—
354C2—24no—
355C2—44no—
356C2#1429211yescharacteristic
357C2#2141711yescharacteristic
358C2#2141811yescharacteristic
359C2#2177412yes—
360C2#2184512yes—
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
socleC23C22×C22⋊C4
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C23×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #11386C2semidirect
5. #11370C2semidirect
7. #10879C2semidirect
11. #2857C4semidirect
12. #2857C4semidirect
14. #2851C4semidirect

Maximal subgroups

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

subgroupindex
#366252
#206822
#137532
#113862
#113702
#112012
#108792