Finite Group Databasebeta

← #381334

Subgroups — #381334

Order 512.

1092 conjugacy classes of subgroups.

The subgroup lattice has 365 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35574C211yesmaximal, characteristic
2#20695C211yesmaximal, characteristic
3#20397C211yesmaximal, characteristic
4#11382C212yesmaximal, complemented
5#11352C212yesmaximal, complemented
6#11175C212yesmaximal
7#11173C212yesmaximal
8#10919C212yesmaximal, complemented
9#10916C212yesmaximal, complemented
10C22×C4⋊Q8C2211yescharacteristic
11C2×#520C2211yescharacteristic
12C2×#413C2212yes—
13#2879C2212yes—
14#2873C2212yes—
15#2579C414yes—
16#2452—22no—
17#2451—22no—
18#2440—22no—
19#2439—22no—
20#2409C414yes—
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
38C23×Q8C2×C412yes—
39C22×M16C2314yes—
40C22×M16C2314yes—
41C2×C4⋊Q8—22no—
42C2×C4⋊Q8—22no—
43C2×C4⋊Q8—44no—
44C2×C22⋊Q8—24no—
45#520—44no—
46C2×C4×Q8D412yes—
47C2×C4×D4C2×C414yes—
48C2×C4×D4D412yes—
49C2×#117D412yes—
50C2×#117C2311yescharacteristic
51C2×#117C2311yescharacteristic
52C2×#117C2311yescharacteristic
53C22×C4⋊C4C2×C412yes—
54C22×C4⋊C4D412yes—
55C22×C4⋊C4C2311yescharacteristic
56C22×C4⋊C4C2311yescharacteristic
57C22×C42C2311yescharacteristic
58#413—28no—
59C2×#101—22no—
60C2×#100—22no—
61#408—44no—
62#407—24no—
63#405—44no—
64C4×M16—24no—
65#398—24no—
66#396—24no—
67#395—24no—
68#393—24no—
69#391—21no—
70#391—21no—
71#391—24no—
72#390—21no—
73#390—21no—
74#389—22no—
75#389—22no—
76#389—21no—
77#389—21no—
78#389—22no—
79#387—24no—
80#386C2×C414yes—
81#386C2×C412yes—
82#386C2×C412yes—
83#385—24no—
84#385C2×C414yes—
85#385C2×C412yes—
86#385C2×C412yes—
87#384—21no—
88#382—21no—
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×Q8—24no—
105C22×Q8—44no—
106C22×Q8—84no—
107C22×Q8C2×D411yescharacteristic
108C22×D4C22×C412yes—
109C22×D4C2×D411yescharacteristic
110C23×C4C22⋊C414yes—
111C23×C4C22×C411yescharacteristic
112C23×C4C22×C411yescharacteristic
113C23×C4C22×C412yes—
114C23×C4C22×C412yes—
115C23×C4C2×D412yes—
116C23×C4C2411yescharacteristic
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—24no—
126C2×M16—24no—
127C2×M16—24no—
128C2×M16#4118yes—
129C22×C8—24no—
130C22×C8—24no—
131C4⋊Q8—48no—
132C4⋊Q8—48no—
133C22⋊Q8—88no—
134C4×Q8—44no—
135C4×D4—44no—
136C4×D4—48no—
137#117—24no—
138#117—24no—
139#117—24no—
140#117—44no—
141C2×C4⋊C4—24no—
142C2×C4⋊C4—24no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—21no—
146C2×C4⋊C4—21no—
147C2×C4⋊C4—21no—
148C2×C4⋊C4—21no—
149C2×C4⋊C4—24no—
150C2×C4⋊C4—24no—
151C2×C4⋊C4—24no—
152C2×C4⋊C4—44no—
153C2×C4⋊C4—44no—
154C2×C4⋊C4C22⋊C414yes—
155C2×C4⋊C4C2×D411yescharacteristic
156C2×C4⋊C4C2×D412yes—
157C2×C4⋊C4C2×D412yes—
158C2×C4⋊C4C2×D411yescharacteristic
159C2×C22⋊C4—22no—
160C2×C22⋊C4—22no—
161C2×C22⋊C4—24no—
162C2×C22⋊C4C22⋊C414yes—
163C2×C22⋊C4C22×C412yes—
164C2×C22⋊C4C22×C412yes—
165C2×C22⋊C4C22×C412yes—
166C2×C42—21no—
167C2×C42—21no—
168C2×C42—21no—
169C2×C42—21no—
170C2×C42—24no—
171C2×C42C22⋊C414yes—
172C2×C42C2×D412yes—
173#101—28no—
174#100—28no—
175C22⋊C8—48no—
176#97—44no—
177C4×C8—44no—
178#95—22no—
179#95—22no—
180#95—22no—
181#95—22no—
182#95—24no—
183#95—24no—
184#95—24no—
185C24C2×C22⋊C412yes—
186C24C23×C411yescharacteristic
187#41—44no—
188#41—44no—
189#41—44no—
190#41—44no—
191C2×Q8—22no—
192C2×Q8—22no—
193C2×Q8—22no—
194C2×Q8—48no—
195C2×Q8—48no—
196C2×Q8—88no—
197C2×Q8—84no—
198C2×Q8—88no—
199C2×D4—24no—
200C2×D4—24no—
201C2×D4—24no—
202C2×D4—22no—
203C2×D4—22no—
204C2×D4—22no—
205C22×C4—21no—
206C22×C4—21no—
207C22×C4—22no—
208C22×C4—22no—
209C22×C4—24no—
210C22×C4—24no—
211C22×C4—24no—
212C22×C4—24no—
213C22×C4—22no—
214C22×C4—22no—
215C22×C4—22no—
216C22×C4—24no—
217C22×C4—42no—
218C22×C4—42no—
219C22×C4—44no—
220C22×C4—82no—
221C22×C4—82no—
222C22×C4—84no—
223C22×C4C2×C22⋊C412yes—
224C22×C4C2×C22⋊C412yes—
225C22×C4C2×C22⋊C414yes—
226C22×C4C2×C22⋊C412yes—
227C22×C4C22×D411yescharacteristic
228C22×C4C22×D411yescharacteristic
229C22×C4C2×#4114yes—
230C22×C4H(5,𝔽2)14yes—
231C22×C42−1+414yes—
232M16—28no—
233M16—28no—
234M16—44no—
235M16—44no—
236M16—44no—
237M16—44no—
238M16—44no—
239M16—44no—
240C2×C8—28no—
241C2×C8—44no—
242C2×C8—44no—
243C2×C8—44no—
244C2×C8—44no—
245C2×C8—44no—
246C4⋊C4—24no—
247C4⋊C4—24no—
248C4⋊C4—22no—
249C4⋊C4—22no—
250C4⋊C4—44no—
251C4⋊C4—44no—
252C4⋊C4—48no—
253C4⋊C4—44no—
254C4⋊C4—42no—
255C4⋊C4—42no—
256C4⋊C4—42no—
257C4⋊C4—42no—
258C4⋊C4—84no—
259C4⋊C4—84no—
260C22⋊C4—24no—
261C22⋊C4—24no—
262C22⋊C4—24no—
263C22⋊C4—44no—
264C22⋊C4—44no—
265C22⋊C4—44no—
266C22⋊C4—84no—
267C42—24no—
268C42—44no—
269C42—42no—
270C42—42no—
271C42—42no—
272C42—42no—
273C23—22no—
274C23—22no—
275C23—22no—
276C23—41no—
277C23—41no—
278C23—42no—
279C23—42no—
280C23C22×C22⋊C411yescharacteristic
281C23C4×#4112yes—
282C23#51812yes—
283C23#51912yes—
284Q8—44no—
285Q8—816no—
286D4—48no—
287D4—44no—
288C2×C4—22no—
289C2×C4—22no—
290C2×C4—24no—
291C2×C4—24no—
292C2×C4—22no—
293C2×C4—22no—
294C2×C4—22no—
295C2×C4—24no—
296C2×C4—22no—
297C2×C4—22no—
298C2×C4—41no—
299C2×C4—42no—
300C2×C4—41no—
301C2×C4—42no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—48no—
305C2×C4—42no—
306C2×C4—42no—
307C2×C4—44no—
308C2×C4—42no—
309C2×C4—42no—
310C2×C4—42no—
311C2×C4—44no—
312C2×C4—42no—
313C2×C4—42no—
314C2×C4—42no—
315C2×C4—42no—
316C2×C4—84no—
317C2×C4—84no—
318C2×C4—164no—
319C2×C4#52512yes—
320C2×C4#53212yes—
321C2×C4#53412yes—
322C2×C4#53412yes—
323C2×C4#53612yes—
324C2×C4#53612yes—
325C2×C4#53914yes—
326C2×C4#53914yes—
327C2×C4#54214yes—
328C2×C4#54214yes—
329C8—48no—
330C8—84no—
331C22—21no—
332C22—22no—
333C22—21no—
334C22—22no—
335C22—42no—
336C22—42no—
337C22—44no—
338C22—42no—
339C22—42no—
340C22—42no—
341C22—82no—
342C22#229711yescharacteristic
343C22#230811yescharacteristic
344C22#231011yescharacteristic
345C22#233214yes—
346C4—42no—
347C4—44no—
348C4—42no—
349C4—42no—
350C4—42no—
351C4—42no—
352C4—82no—
353C4—82no—
354C4—82no—
355C4#263814yes—
356C4#266314yes—
357C2—21no—
358C2—21no—
359C2—24no—
360C2—82no—
361C2#1429211yescharacteristic
362C2#2177811yescharacteristic
363C2#2178011yescharacteristic
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. #11382C2semidirect
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
#355742
#206952
#203972
#113822
#113522
#111752
#111732
#109192
#109162