Finite Group Databasebeta

← #381302

Subgroups — #381302

Order 512.

1204 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#34634C211yesmaximal, characteristic
2#20683C211yesmaximal, characteristic
3#13761C211yesmaximal, characteristic
4#11386C212yesmaximal, complemented
5#11370C212yesmaximal, complemented
6#11201C214yesmaximal
7#10879C212yesmaximal, complemented
8#10878C212yesmaximal, complemented
9C2×#533C2211yescharacteristic
10C22×#117C2211yescharacteristic
11C2×#407C2212yes—
12#2629C414yescomplemented
13#2628C414yescomplemented
14#2627—22no—
15#2626—22no—
16#2396—24no—
17#2296C2212yes—
18#2285C2211yescharacteristic
19#2280C2212yes—
20C2×C4×C22⋊C4C2212yes—
21C2×#376C2211yescharacteristic
22C2×C43C2211yescharacteristic
23#1818—24no—
24C2×#339C2214yes—
25C2×#339C2214yes—
26C2×#339C2214yes—
27C2×#339C2214yes—
28#1728C2212yes—
29#1728C2212yes—
30#1437—22no—
31#1436—22no—
32C4×#95C2212yes—
33C24×C4C2311yescharacteristic
34C22×M16C2314yes—
35#533—42no—
36#533—42no—
37C2×C4⋊Q8C2×C412yes—
38C2×#125C2×C412yes—
39C2×C22⋊Q8—22no—
40C2×C4×Q8—22no—
41C2×#117—42no—
42C2×#117—42no—
43C2×#117C2×C412yes—
44C2×#117C2×C412yes—
45C2×#117D414yes—
46C2×#117C2312yes—
47C2×#117C2312yes—
48C22×C4⋊C4D412yes—
49C22×C22⋊C4—22no—
50C22×C22⋊C4D412yes—
51C22×C42—22no—
52C22×C42C2312yes—
53C22×C42C2312yes—
54C22×C42C2311yescharacteristic
55C22×C42C2311yescharacteristic
56#407—42no—
57#407—42no—
58C2×C22⋊C8—24no—
59#400—24no—
60#399—24no—
61#398—28no—
62#396—28no—
63#395—24no—
64#393—24no—
65#389C2×C418yes—
66#385—24no—
67#384C2×C412yes—
68#382C2×C412yes—
69#381—22no—
70#380—22no—
71#380C2×C418yes—
72#379—22no—
73C4×C4⋊C4—22no—
74C4×C4⋊C4—24no—
75C4×C22⋊C4—24no—
76C4×C22⋊C4—24no—
77C4×C22⋊C4—24no—
78#376—42no—
79C2×#95—22no—
80C2×#95—24no—
81C2×#95—22no—
82C43—42no—
83#339—28no—
84#339—28no—
85#339—44no—
86#339—44no—
87#336—24no—
88#336—24no—
89C25C2×D411yescharacteristic
90C22×Q8—22no—
91C23×C4—22no—
92C23×C4—22no—
93C23×C4—22no—
94C23×C4—22no—
95C23×C4—22no—
96C23×C4—24no—
97C23×C4—22no—
98C23×C4C22×C412yes—
99C23×C4C2×D411yescharacteristic
100C23×C4C2×D412yes—
101C23×C4C2×D412yes—
102C23×C4C2×D412yes—
103C23×C4C2411yescharacteristic
104C2×M16—24no—
105C2×M16—24no—
106C2×M16—24no—
107C2×M16—24no—
108C2×M16—24no—
109C2×M16—24no—
110C22×C8—24no—
111C4⋊Q8—44no—
112#125—44no—
113C22⋊Q8—44no—
114C22⋊Q8—44no—
115C4×Q8—48no—
116#117—24no—
117#117—24no—
118#117—24no—
119#117—24no—
120#117—44no—
121#117—44no—
122#117—48no—
123C2×C4⋊C4—24no—
124C2×C4⋊C4—22no—
125C2×C4⋊C4—24no—
126C2×C4⋊C4—28no—
127C2×C4⋊C4—42no—
128C2×C4⋊C4—42no—
129C2×C4⋊C4C22⋊C414yes—
130C2×C4⋊C4C22⋊C414yes—
131C2×C4⋊C4C22×C414yes—
132C2×C4⋊C4C22×C414yes—
133C2×C22⋊C4—28no—
134C2×C22⋊C4—22no—
135C2×C22⋊C4—22no—
136C2×C22⋊C4—28no—
137C2×C22⋊C4—42no—
138C2×C22⋊C4—42no—
139C2×C22⋊C4—48no—
140C2×C22⋊C4C22⋊C414yes—
141C2×C22⋊C4C22⋊C414yes—
142C2×C42—22no—
143C2×C42—22no—
144C2×C42—22no—
145C2×C42—22no—
146C2×C42—24no—
147C2×C42—24no—
148C2×C42—24no—
149C2×C42—24no—
150C2×C42—22no—
151C2×C42—22no—
152C2×C42—22no—
153C2×C42—22no—
154C2×C42—44no—
155C2×C42—48no—
156C2×C42C22×C411yescharacteristic
157C2×C42C22×C411yescharacteristic
158C2×C42C22×C412yes—
159C2×C42C2×D412yes—
160C2×C42C2×D412yes—
161C2×C42#4114yes—
162C2×C42#4114yes—
163C22⋊C8—48no—
164#95—28no—
165#95—28no—
166#95—24no—
167#95—28no—
168#95—48no—
169#95—44no—
170C24—22no—
171C24—21no—
172C24—21no—
173C24—22no—
174C24—44no—
175C24C2×C22⋊C412yes—
176C24C22×D411yescharacteristic
177C2×Q8—44no—
178C2×Q8—84no—
179C22×C4—28no—
180C22×C4—22no—
181C22×C4—24no—
182C22×C4—22no—
183C22×C4—22no—
184C22×C4—21no—
185C22×C4—21no—
186C22×C4—22no—
187C22×C4—22no—
188C22×C4—22no—
189C22×C4—22no—
190C22×C4—22no—
191C22×C4—24no—
192C22×C4—24no—
193C22×C4—24no—
194C22×C4—24no—
195C22×C4—24no—
196C22×C4—24no—
197C22×C4—24no—
198C22×C4—24no—
199C22×C4—44no—
200C22×C4—44no—
201C22×C4—44no—
202C22×C4—42no—
203C22×C4—42no—
204C22×C4—44no—
205C22×C4—44no—
206C22×C4—44no—
207C22×C4—44no—
208C22×C4—44no—
209C22×C4—44no—
210C22×C4—42no—
211C22×C4—42no—
212C22×C4C2×C22⋊C412yes—
213C22×C4C2×C22⋊C414yes—
214C22×C4C2×C22⋊C412yes—
215C22×C4C2×C22⋊C412yes—
216C22×C4C23×C411yescharacteristic
217C22×C4C22×D411yescharacteristic
218C22×C4C2×#4112yes—
219C22×C4C2×#4112yes—
220C22×C4H(5,𝔽2)12yes—
221C22×C4H(5,𝔽2)12yes—
222C22×C42−1+414yes—
223M16—44no—
224M16—44no—
225M16—44no—
226M16—44no—
227C2×C8—44no—
228C2×C8—44no—
229C2×C8—44no—
230C4⋊C4—24no—
231C4⋊C4—24no—
232C4⋊C4—28no—
233C4⋊C4—44no—
234C4⋊C4—44no—
235C4⋊C4—48no—
236C4⋊C4—44no—
237C4⋊C4—84no—
238C4⋊C4—88no—
239C22⋊C4—44no—
240C22⋊C4—44no—
241C22⋊C4—416no—
242C22⋊C4—44no—
243C22⋊C4—42no—
244C22⋊C4—42no—
245C22⋊C4—88no—
246C42—24no—
247C42—22no—
248C42—22no—
249C42—24no—
250C42—24no—
251C42—28no—
252C42—28no—
253C42—416no—
254C42—44no—
255C42—44no—
256C42—44no—
257C42—44no—
258C23—24no—
259C23—22no—
260C23—22no—
261C23—28no—
262C23—22no—
263C23—44no—
264C23—44no—
265C23—42no—
266C23—42no—
267C23—82no—
268C23—82no—
269C23—84no—
270C23C22×C22⋊C411yescharacteristic
271C23#52512yes—
272C23#53411yescharacteristic
273C23#53411yescharacteristic
274C23#53612yes—
275Q8—88no—
276C2×C4—28no—
277C2×C4—22no—
278C2×C4—22no—
279C2×C4—24no—
280C2×C4—22no—
281C2×C4—22no—
282C2×C4—28no—
283C2×C4—24no—
284C2×C4—24no—
285C2×C4—22no—
286C2×C4—48no—
287C2×C4—44no—
288C2×C4—48no—
289C2×C4—44no—
290C2×C4—44no—
291C2×C4—44no—
292C2×C4—42no—
293C2×C4—42no—
294C2×C4—44no—
295C2×C4—44no—
296C2×C4—48no—
297C2×C4—44no—
298C2×C4—44no—
299C2×C4—44no—
300C2×C4—44no—
301C2×C4—48no—
302C2×C4—48no—
303C2×C4—44no—
304C2×C4—82no—
305C2×C4—82no—
306C2×C4—84no—
307C2×C4—84no—
308C2×C4C4×#4111yescharacteristic
309C2×C4C4×#4111yescharacteristic
310C2×C4#51812yes—
311C2×C4#51912yes—
312C2×C4#53212yes—
313C2×C4#53412yes—
314C2×C4#53612yes—
315C2×C4#53914yes—
316C2×C4#53914yes—
317C2×C4#54214yes—
318C2×C4#54214yes—
319C8—84no—
320C22—22no—
321C22—22no—
322C22—24no—
323C22—22no—
324C22—22no—
325C22—44no—
326C22—44no—
327C22—44no—
328C22—42no—
329C22—84no—
330C22—84no—
331C22#229711yescharacteristic
332C22#230811yescharacteristic
333C22#231011yescharacteristic
334C22#263812yes—
335C22#263812yes—
336C22#290114yes—
337C4—48no—
338C4—44no—
339C4—42no—
340C4—42no—
341C4—42no—
342C4—84no—
343C4—84no—
344C4—84no—
345C4#233212yes—
346C4#233212yes—
347C4#266314yes—
348C2—22no—
349C2—22no—
350C2—82no—
351C2#1429211yescharacteristic
352C2#2141912yes—
353C2#2177512yes—
354C2#2184612yes—
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 ◁ C22×#117 ◁ 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
8. #10878C2semidirect
12. #2629C4semidirect
13. #2628C4semidirect

Maximal subgroups

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

subgroupindex
#346342
#206832
#137612
#113862
#113702
#112012
#108792
#108782