Finite Group Databasebeta

← #223100

Subgroups — #223100

Order 512.

1046 conjugacy classes of subgroups.

The subgroup lattice has 374 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36185C211yesmaximal, characteristic
2#20494C212yesmaximal, complemented
3#9890C212yesmaximal, complemented
4#9120C214yesmaximal, complemented
5#8721C212yesmaximal, complemented
6#8579C211yesmaximal, characteristic, complemented
7C4×#329C211yesmaximal, characteristic, complemented
8#8028C212yesmaximal, complemented
9C2×#556C411yescharacteristic
10C2×#555C411yescharacteristic
11C2×#436—22no—
12C2×C4×M16C2214yes—
13#2862C2211yescharacteristic
14#2850C2212yes—
15#2567C411yescharacteristic
16#2542C411yescharacteristic
17#2521—24no—
18#2500C412yes—
19#2499C412yes—
20#2427—22no—
21#2362—22no—
22C4×#124C2211yescharacteristic
23#2285C2212yes—
24C42×D4C2211yescharacteristic
25C8×C4⋊C4—22no—
26#1767C2212yes—
27#1753—24no—
28#1749C2214yes—
29C4×#97C2212yes—
30C2×#329C2212yes—
31C2×#329C2212yes—
32#1295C2211yescharacteristic
33#1295C2212yes—
34#1295C2214yes—
35#1295C2211yescharacteristic
36#1295C2214yes—
37C22×M16—24no—
38#556—44no—
39#555—44no—
40C2×C4⋊Q8C2×C411yescharacteristic
41C2×C4⋊Q8C2×C411yescharacteristic
42C2×#126—22no—
43C2×#124—21no—
44C2×#124—22no—
45C2×#124C2311yescharacteristic
46C2×C22⋊Q8—22no—
47C2×C22⋊Q8C2×C412yes—
48C2×#121—21no—
49C2×#121C2×C412yes—
50C2×C4×Q8—21no—
51C2×C4×Q8C2×C411yescharacteristic
52C2×C4×D4C2×C411yescharacteristic
53C2×C4×D4C2×C411yescharacteristic
54C2×C4×D4D412yes—
55C2×#117—21no—
56C2×#117C2×C412yes—
57C2×#117C2×C412yes—
58C22×C42C2312yes—
59#436—216no—
60C2×C4⋊C8—22no—
61C2×C22⋊C8—24no—
62C4×M16—28no—
63C4×M16—44no—
64C2×#97—22no—
65C2×#97C2314yes—
66C2×#97C2312yes—
67C2×#97C2312yes—
68C2×C4×C8—22no—
69C2×C4×C8—24no—
70#399—22no—
71#397—24no—
72#395—22no—
73#394—24no—
74#391—22no—
75#391—22no—
76#391C2×C412yes—
77#391C2×C411yescharacteristic
78#390C2×C412yes—
79#390C2×C411yescharacteristic
80#389—22no—
81#388—24no—
82#388—22no—
83#388—24no—
84#387—24no—
85#386—24no—
86#384C2×C412yes—
87#384C2×C412yes—
88#382C2×C412yes—
89#382C2312yes—
90#382C2311yescharacteristic
91#380—24no—
92C4×C4⋊C4—24no—
93C4×C4⋊C4C2×C411yescharacteristic
94C4×C4⋊C4D412yes—
95C4×C22⋊C4—22no—
96C4×C22⋊C4C2×C412yes—
97C4×C22⋊C4D414yes—
98C43C2311yescharacteristic
99#336—24no—
100#336—22no—
101#329—24no—
102#329—42no—
103C22×Q8—21no—
104C22×Q8—21no—
105C22×Q8C22×C411yescharacteristic
106C22×D4—22no—
107C22×D4C22⋊C411yescharacteristic
108C22×D4C22×C411yescharacteristic
109C23×C4—21no—
110C23×C4C22×C412yes—
111C23×C4C2×D414yes—
112C2×M16—216no—
113C2×M16—48no—
114C2×M16—48no—
115C22×C8—22no—
116C22×C8—24no—
117C22×C8—22no—
118C4⋊Q8—42no—
119C4⋊Q8—42no—
120#126—48no—
121#124—22no—
122#124—21no—
123#124—21no—
124#124—48no—
125#124—44no—
126C22⋊Q8—48no—
127C22⋊Q8—44no—
128#121—44no—
129#121—44no—
130C4×Q8—42no—
131C4×Q8—44no—
132C4×D4—24no—
133C4×D4—28no—
134C4×D4—42no—
135#117—44no—
136#117—44no—
137#117—44no—
138C2×C4⋊C4—21no—
139C2×C4⋊C4—22no—
140C2×C4⋊C4—21no—
141C2×C4⋊C4—24no—
142C2×C4⋊C4—21no—
143C2×C4⋊C4—24no—
144C2×C4⋊C4C22⋊C411yescharacteristic
145C2×C4⋊C4C22⋊C412yes—
146C2×C4⋊C4C22×C411yescharacteristic
147C2×C4⋊C4C22×C411yescharacteristic
148C2×C4⋊C4C22×C412yes—
149C2×C4⋊C4C22×C412yes—
150C2×C22⋊C4—22no—
151C2×C22⋊C4—22no—
152C2×C22⋊C4—22no—
153C2×C22⋊C4—22no—
154C2×C22⋊C4—21no—
155C2×C22⋊C4—24no—
156C2×C22⋊C4—22no—
157C2×C22⋊C4—22no—
158C2×C22⋊C4C22⋊C412yes—
159C2×C22⋊C4C22⋊C414yes—
160C2×C42—24no—
161C2×C42—22no—
162C2×C42—22no—
163C2×C42—22no—
164C2×C42—22no—
165C2×C42—21no—
166C2×C42C22×C411yescharacteristic
167C2×C42C22×C412yes—
168C2×C42C22×C411yescharacteristic
169C2×C42C2×D412yes—
170C2×C42C2×D414yes—
171C2×C42C2×D412yes—
172C2×C42C2411yescharacteristic
173C4⋊C8—28no—
174C22⋊C8—216no—
175#97—22no—
176#97—24no—
177#97—22no—
178#97—24no—
179#97—22no—
180#97—44no—
181#97#4114yes—
182C4×C8—28no—
183C4×C8—48no—
184#95—22no—
185#95—24no—
186#95—24no—
187#95—22no—
188#95—24no—
189#95C22⋊C414yes—
190#95C22⋊C412yes—
191C24—21no—
192C24C2×C22⋊C412yes—
193C2×Q8—21no—
194C2×Q8—21no—
195C2×Q8—42no—
196C2×Q8—41no—
197C2×Q8—41no—
198C2×Q8—42no—
199C2×Q8—44no—
200C2×Q8—82no—
201C2×D4—21no—
202C2×D4—21no—
203C2×D4—22no—
204C2×D4—24no—
205C2×D4—41no—
206C2×D4—41no—
207C2×D4—44no—
208C2×D4—84no—
209C22×C4—22no—
210C22×C4—22no—
211C22×C4—24no—
212C22×C4—24no—
213C22×C4—24no—
214C22×C4—28no—
215C22×C4—21no—
216C22×C4—22no—
217C22×C4—21no—
218C22×C4—22no—
219C22×C4—21no—
220C22×C4—44no—
221C22×C4—42no—
222C22×C4—42no—
223C22×C4—44no—
224C22×C4C2×C22⋊C412yes—
225C22×C4C2×C22⋊C411yescharacteristic
226C22×C4C2×C22⋊C412yes—
227C22×C4C2×C22⋊C414yes—
228C22×C4C2×C22⋊C411yescharacteristic
229C22×C4C23×C411yescharacteristic
230C22×C4C22×D412yes—
231M16—432no—
232C2×C8—24no—
233C2×C8—24no—
234C2×C8—24no—
235C2×C8—24no—
236C2×C8—28no—
237C2×C8—42no—
238C2×C8—44no—
239C2×C8—44no—
240C2×C8—42no—
241C4⋊C4—41no—
242C4⋊C4—41no—
243C4⋊C4—42no—
244C4⋊C4—42no—
245C4⋊C4—42no—
246C4⋊C4—42no—
247C4⋊C4—42no—
248C4⋊C4—44no—
249C4⋊C4—48no—
250C4⋊C4—84no—
251C4⋊C4#10014yes—
252C4⋊C4#10418yes—
253C22⋊C4—24no—
254C22⋊C4—28no—
255C22⋊C4—42no—
256C22⋊C4—44no—
257C22⋊C4—44no—
258C22⋊C4—44no—
259C22⋊C4—42no—
260C22⋊C4—42no—
261C22⋊C4—44no—
262C22⋊C4—48no—
263C22⋊C4—82no—
264C42—22no—
265C42—22no—
266C42—24no—
267C42—24no—
268C42—22no—
269C42—24no—
270C42—42no—
271C42—41no—
272C42—44no—
273C42—44no—
274C42—42no—
275C42C2×#4112yes—
276C42H(5,𝔽2)11yescharacteristic
277C422−1+411yescharacteristic
278C23—22no—
279C23—22no—
280C23—22no—
281C23—24no—
282C23—24no—
283C23—41no—
284C23—41no—
285C23—41no—
286C23—42no—
287C23—42no—
288C23C22×C22⋊C411yescharacteristic
289Q8—81no—
290Q8—81no—
291Q8—84no—
292Q8—162no—
293D4—44no—
294D4—81no—
295D4—81no—
296D4—164no—
297C2×C4—22no—
298C2×C4—24no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—22no—
302C2×C4—22no—
303C2×C4—22no—
304C2×C4—24no—
305C2×C4—21no—
306C2×C4—22no—
307C2×C4—22no—
308C2×C4—22no—
309C2×C4—22no—
310C2×C4—48no—
311C2×C4—44no—
312C2×C4—44no—
313C2×C4—48no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—42no—
319C2×C4—42no—
320C2×C4—41no—
321C2×C4—44no—
322C2×C4—42no—
323C2×C4—42no—
324C2×C4—44no—
325C2×C4—84no—
326C2×C4C2×#10014yes—
327C2×C4C2×#10012yes—
328C2×C4C2×#10418yes—
329C2×C4C2×#10414yes—
330C2×C4C4×#4111yescharacteristic
331C2×C4#52011yescharacteristic
332C2×C4#54612yes—
333C2×C4#54712yes—
334C8—28no—
335C8—44no—
336C8—48no—
337C22—42no—
338C22—42no—
339C22—42no—
340C22—44no—
341C22—44no—
342C22—81no—
343C22—81no—
344C22—81no—
345C22—82no—
346C22—82no—
347C22#233911yescharacteristic
348C22C22×#10011yescharacteristic
349C22#288911yescharacteristic
350C22C22×#10412yes—
351C22#290112yes—
352C4—21no—
353C4—21no—
354C4—24no—
355C4—22no—
356C4—22no—
357C4—28no—
358C4—41no—
359C4—42no—
360C4—81no—
361C4—82no—
362C4—81no—
363C4—81no—
364C4—82no—
365C4#152514yes—
366C2—82no—
367C2—161no—
368C2C2×#152511yescharacteristic
369C2#1765412yes—
370C2#1768811yescharacteristic
371C2#2247911yescharacteristic
372C2#2250512yes—
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC2×#555C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C42×D4 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #20494C2semidirect
3. #9890C2semidirect
4. #9120C2semidirect
5. #8721C2semidirect
6. #8579C2semidirect
7. C4×#329C2semidirect
8. #8028C2semidirect

Maximal subgroups

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

subgroupindex
#361852
#204942
#98902
#91202
#87212
#85792
C4×#3292
#80282