Finite Group Databasebeta

← #381335

Subgroups — #381335

Order 512.

1018 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#35639C211yesmaximal, characteristic
2#20695C211yesmaximal, characteristic
3#20397C211yesmaximal, characteristic
4#11383C212yesmaximal, complemented
5#11352C212yesmaximal, complemented
6#11175C212yesmaximal
7#11173C212yesmaximal
8#10919C212yesmaximal, complemented
9#10916C212yesmaximal, complemented
10C22×#125C2211yescharacteristic
11C2×#520C2211yescharacteristic
12C2×#413C2212yes—
13#2879C2212yes—
14#2873C2212yes—
15#2579C414yes—
16#2487C414yes—
17#2454—22no—
18#2453—22no—
19#2442—22no—
20#2441—22no—
21#2317C2211yescharacteristic
22#2316C2211yescharacteristic
23#2311C2211yescharacteristic
24#2297C2211yescharacteristic
25#2296C2211yescharacteristic
26#1822—22no—
27#1821—22no—
28C4×#101—22no—
29C4×#100—22no—
30C2×#344C2214yes—
31C2×#344C2214yes—
32#1742C2212yes—
33C2×#343C2212yes—
34C2×#343C2212yes—
35C2×#343C2214yes—
36#1728C2214yes—
37C22×#41C2311yescharacteristic
38C22×M16C2314yes—
39C22×M16C2314yes—
40C2×#125—22no—
41C2×#125—22no—
42C2×#125—44no—
43#520—44no—
44C2×C4×Q8D412yes—
45C2×C4×D4C2×C414yes—
46C2×C4×D4D412yes—
47C2×#117D412yes—
48C2×#117C2311yescharacteristic
49C2×#117C2311yescharacteristic
50C2×#117C2311yescharacteristic
51C22×C4⋊C4C2×C412yes—
52C22×C4⋊C4C2×C412yes—
53C22×C4⋊C4D412yes—
54C22×C4⋊C4C2311yescharacteristic
55C22×C4⋊C4C2311yescharacteristic
56C22×C42C2311yescharacteristic
57#413—28no—
58C2×#101—22no—
59C2×#100—22no—
60#408—44no—
61#407—24no—
62#405—44no—
63C4×M16—24no—
64#400—24no—
65#399—24no—
66#398—24no—
67#396—24no—
68#391—21no—
69#391—21no—
70#390—21no—
71#390—21no—
72#389—22no—
73#389—22no—
74#389—21no—
75#389—21no—
76#389—24no—
77#389—22no—
78#387—24no—
79#386C2×C414yes—
80#386C2×C412yes—
81#386C2×C412yes—
82#385—24no—
83#385C2×C414yes—
84#385C2×C412yes—
85#385C2×C412yes—
86#384—21no—
87#382—21no—
88#380—24no—
89#380—22no—
90#380C2×C414yes—
91C4×C4⋊C4—22no—
92C4×C22⋊C4—22no—
93#344—28no—
94#344—28no—
95#343—22no—
96#343—24no—
97#343—22no—
98#343—24no—
99#343—44no—
100#336—44no—
101C2×#41—24no—
102C2×#41—24no—
103C2×#41—24no—
104C22×Q8C2×D411yescharacteristic
105C22×D4C22×C412yes—
106C22×D4C2×D411yescharacteristic
107C23×C4C22⋊C414yes—
108C23×C4C22×C412yes—
109C23×C4C22×C411yescharacteristic
110C23×C4C22×C411yescharacteristic
111C23×C4C22×C412yes—
112C23×C4C2×D412yes—
113C23×C4C2411yescharacteristic
114C2×M16—24no—
115C2×M16—24no—
116C2×M16—24no—
117C2×M16—24no—
118C2×M16—24no—
119C2×M16—24no—
120C2×M16—24no—
121C2×M16—24no—
122C2×M16—24no—
123C2×M16—24no—
124C2×M16—24no—
125C2×M16#4118yes—
126C22×C8—24no—
127C22×C8—24no—
128#125—48no—
129#125—48no—
130C4×Q8—44no—
131C4×D4—44no—
132C4×D4—48no—
133#117—24no—
134#117—24no—
135#117—24no—
136#117—44no—
137C2×C4⋊C4—24no—
138C2×C4⋊C4—24no—
139C2×C4⋊C4—24no—
140C2×C4⋊C4—21no—
141C2×C4⋊C4—21no—
142C2×C4⋊C4—21no—
143C2×C4⋊C4—21no—
144C2×C4⋊C4—24no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—24no—
148C2×C4⋊C4—24no—
149C2×C4⋊C4—44no—
150C2×C4⋊C4—44no—
151C2×C4⋊C4—44no—
152C2×C4⋊C4C22⋊C414yes—
153C2×C4⋊C4C2×D411yescharacteristic
154C2×C4⋊C4C2×D412yes—
155C2×C4⋊C4C2×D412yes—
156C2×C4⋊C4C2×D411yescharacteristic
157C2×C22⋊C4—22no—
158C2×C22⋊C4—22no—
159C2×C22⋊C4—24no—
160C2×C22⋊C4C22⋊C414yes—
161C2×C22⋊C4C22×C412yes—
162C2×C22⋊C4C22×C412yes—
163C2×C22⋊C4C22×C412yes—
164C2×C42—21no—
165C2×C42—21no—
166C2×C42—21no—
167C2×C42—21no—
168C2×C42—24no—
169C2×C42C22⋊C414yes—
170C2×C42C2×D412yes—
171#101—28no—
172#100—28no—
173C22⋊C8—48no—
174#97—44no—
175C4×C8—44no—
176#95—22no—
177#95—22no—
178#95—22no—
179#95—22no—
180#95—24no—
181#95—24no—
182#95—24no—
183C24C2×C22⋊C412yes—
184C24C23×C411yescharacteristic
185#41—44no—
186#41—44no—
187#41—44no—
188#41—44no—
189C2×Q8—22no—
190C2×Q8—22no—
191C2×Q8—22no—
192C2×D4—24no—
193C2×D4—24no—
194C2×D4—24no—
195C2×D4—22no—
196C2×D4—22no—
197C2×D4—22no—
198C22×C4—24no—
199C22×C4—24no—
200C22×C4—24no—
201C22×C4—24no—
202C22×C4—22no—
203C22×C4—22no—
204C22×C4—21no—
205C22×C4—21no—
206C22×C4—22no—
207C22×C4—22no—
208C22×C4—22no—
209C22×C4—24no—
210C22×C4—44no—
211C22×C4—81no—
212C22×C4—81no—
213C22×C4—82no—
214C22×C4—82no—
215C22×C4—84no—
216C22×C4C2×C22⋊C412yes—
217C22×C4C2×C22⋊C412yes—
218C22×C4C2×C22⋊C412yes—
219C22×C4C2×C22⋊C414yes—
220C22×C4C22×D411yescharacteristic
221C22×C4C22×D411yescharacteristic
222C22×C4C2×#4114yes—
223C22×C4H(5,𝔽2)14yes—
224C22×C42−1+414yes—
225M16—28no—
226M16—28no—
227M16—44no—
228M16—44no—
229M16—44no—
230M16—44no—
231M16—44no—
232M16—44no—
233C2×C8—28no—
234C2×C8—44no—
235C2×C8—44no—
236C2×C8—44no—
237C2×C8—44no—
238C2×C8—44no—
239C4⋊C4—24no—
240C4⋊C4—24no—
241C4⋊C4—22no—
242C4⋊C4—22no—
243C4⋊C4—44no—
244C4⋊C4—48no—
245C4⋊C4—48no—
246C4⋊C4—42no—
247C4⋊C4—42no—
248C4⋊C4—42no—
249C4⋊C4—42no—
250C4⋊C4—82no—
251C4⋊C4—84no—
252C4⋊C4—84no—
253C4⋊C4—82no—
254C22⋊C4—24no—
255C22⋊C4—24no—
256C22⋊C4—24no—
257C22⋊C4—44no—
258C22⋊C4—84no—
259C22⋊C4—82no—
260C22⋊C4—82no—
261C42—24no—
262C42—44no—
263C42—42no—
264C42—42no—
265C42—42no—
266C42—42no—
267C23—22no—
268C23—22no—
269C23—22no—
270C23—41no—
271C23—41no—
272C23—42no—
273C23—42no—
274C23C22×C22⋊C411yescharacteristic
275C23C4×#4112yes—
276C23#51812yes—
277C23#51912yes—
278Q8—44no—
279D4—48no—
280D4—44no—
281C2×C4—22no—
282C2×C4—22no—
283C2×C4—22no—
284C2×C4—24no—
285C2×C4—22no—
286C2×C4—22no—
287C2×C4—48no—
288C2×C4—42no—
289C2×C4—44no—
290C2×C4—42no—
291C2×C4—42no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—42no—
295C2×C4—42no—
296C2×C4—44no—
297C2×C4—42no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—44no—
301C2×C4—44no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—42no—
305C2×C4—42no—
306C2×C4—42no—
307C2×C4—84no—
308C2×C4—84no—
309C2×C4—164no—
310C2×C4#52512yes—
311C2×C4#53212yes—
312C2×C4#53412yes—
313C2×C4#53412yes—
314C2×C4#53612yes—
315C2×C4#53612yes—
316C2×C4#53914yes—
317C2×C4#53914yes—
318C2×C4#54214yes—
319C2×C4#54214yes—
320C8—48no—
321C8—84no—
322C22—21no—
323C22—22no—
324C22—21no—
325C22—22no—
326C22—42no—
327C22—42no—
328C22—44no—
329C22—42no—
330C22—42no—
331C22—42no—
332C22—82no—
333C22#229711yescharacteristic
334C22#230811yescharacteristic
335C22#231011yescharacteristic
336C22#233214yes—
337C4—44no—
338C4—42no—
339C4—42no—
340C4—42no—
341C4—82no—
342C4—82no—
343C4—81no—
344C4—81no—
345C4—82no—
346C4#263814yes—
347C4#266314yes—
348C2—21no—
349C2—21no—
350C2—24no—
351C2—82no—
352C2#1429211yescharacteristic
353C2#2177911yescharacteristic
354C2#2178111yescharacteristic
subgroupG/H
center, socleC22#2308
commutatorC24C23×C4
Frattini, cosocleC23×C4C24
prime radicalC22×#41C23
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #11383C2semidirect
5. #11352C2semidirect
8. #10919C2semidirect
9. #10916C2semidirect

Maximal subgroups

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

subgroupindex
#356392
#206952
#203972
#113832
#113522
#111752
#111732
#109192
#109162