Finite Group Databasebeta

← #164337

Subgroups — #164337

Order 512.

1021 conjugacy classes of subgroups.

The subgroup lattice has 373 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36406C211yesmaximal, characteristic
2#20567C212yesmaximal, complemented
3#7687C214yesmaximal, complemented
4#7682C212yesmaximal, complemented
5#7635C211yesmaximal, characteristic, complemented
6#7631C212yesmaximal, complemented
7#7567C211yesmaximal, characteristic, complemented
8#7563C212yesmaximal, complemented
9C2×#556C411yescharacteristic
10C2×#547C411yescharacteristic
11C2×#436—22no—
12#2862C2211yescharacteristic
13#2857C2212yes—
14#2553—24no—
15#2546C411yescharacteristic
16#2542C411yescharacteristic
17#2499C412yes—
18#2432C412yes—
19#2427—22no—
20#2360—22no—
21C4×#124C2211yescharacteristic
22#2297C2212yes—
23C42×D4C2211yescharacteristic
24C8×C4⋊C4—22no—
25#1767C2212yes—
26#1753—24no—
27C4×C22⋊C8C2214yes—
28#1744C2212yes—
29#1728C2214yes—
30#1272C2214yes—
31#1272C2214yes—
32#1272C2212yes—
33#1272C2212yes—
34#1270C2211yescharacteristic
35#1270C2212yes—
36#1270C2211yescharacteristic
37C22×#41C2×C412yes—
38C22×M16—24no—
39#556—44no—
40#547—44no—
41C2×C4⋊Q8C2×C411yescharacteristic
42C2×C4⋊Q8C2×C412yes—
43C2×C4⋊Q8C2×C411yescharacteristic
44C2×#126—22no—
45C2×#124—21no—
46C2×#124C2×C411yescharacteristic
47C2×#123—22no—
48C2×C22⋊Q8—22no—
49C2×C22⋊Q8C2×C412yes—
50C2×C22⋊Q8C2×C412yes—
51C2×#121—21no—
52C2×C4×Q8—21no—
53C2×C4×Q8C2×C411yescharacteristic
54C2×C4×D4C2×C411yescharacteristic
55C2×C4×D4C2×C411yescharacteristic
56C2×C4×D4D412yes—
57C2×#117C2×C412yes—
58C22×C4⋊C4—21no—
59C22×C42C2312yes—
60#436—216no—
61C2×C4⋊C8—22no—
62C2×C22⋊C8—24no—
63C2×#97—22no—
64C2×C4×C8—22no—
65C2×C4×C8—24no—
66#399—22no—
67#398—22no—
68#397—24no—
69#394—24no—
70#391—22no—
71#391C2×C412yes—
72#391C2×C412yes—
73#391C2311yescharacteristic
74#391C2312yes—
75#390C2×C412yes—
76#390C2311yescharacteristic
77#389—22no—
78#389—24no—
79#388—22no—
80#387—24no—
81#386—24no—
82#385—24no—
83#384C2×C412yes—
84#382—22no—
85#382C2×C411yescharacteristic
86#380—24no—
87C4×C4⋊C4C2×C411yescharacteristic
88C4×C4⋊C4D412yes—
89C4×C22⋊C4—22no—
90C4×C22⋊C4—24no—
91C4×C22⋊C4C2×C412yes—
92C4×C22⋊C4D414yes—
93C43C2311yescharacteristic
94#336—24no—
95#336—22no—
96#336C2312yes—
97#336C2314yes—
98#336C2312yes—
99C2×#41—24no—
100C2×#41—42no—
101C2×#41—44no—
102C2×#41—82no—
103C22×Q8—21no—
104C22×Q8C22×C411yescharacteristic
105C22×Q8C22×C412yes—
106C22×D4—22no—
107C22×D4C22⋊C411yescharacteristic
108C22×D4C22×C411yescharacteristic
109C23×C4—21no—
110C23×C4—22no—
111C23×C4C22×C412yes—
112C23×C4C2×D414yes—
113C2×M16—216no—
114C2×M16—84no—
115C2×M16—84no—
116C22×C8—22no—
117C22×C8—24no—
118C22×C8—22no—
119C4⋊Q8—42no—
120C4⋊Q8—42no—
121C4⋊Q8—82no—
122#126—48no—
123#124—22no—
124#124—41no—
125#124—44no—
126#123—48no—
127C22⋊Q8—48no—
128C22⋊Q8—44no—
129C22⋊Q8—44no—
130#121—44no—
131C4×Q8—42no—
132C4×Q8—44no—
133C4×D4—24no—
134C4×D4—28no—
135C4×D4—42no—
136#117—44no—
137C2×C4⋊C4—21no—
138C2×C4⋊C4—22no—
139C2×C4⋊C4—22no—
140C2×C4⋊C4—21no—
141C2×C4⋊C4—24no—
142C2×C4⋊C4—21no—
143C2×C4⋊C4—21no—
144C2×C4⋊C4—24no—
145C2×C4⋊C4—44no—
146C2×C4⋊C4C22⋊C411yescharacteristic
147C2×C4⋊C4C22⋊C412yes—
148C2×C4⋊C4C22×C411yescharacteristic
149C2×C4⋊C4C22×C412yes—
150C2×C4⋊C4C22×C411yescharacteristic
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—22no—
161C2×C42—21no—
162C2×C42—82no—
163C2×C42C22×C411yescharacteristic
164C2×C42C22×C411yescharacteristic
165C2×C42C22×C412yes—
166C2×C42C2×D412yes—
167C2×C42C2×D412yes—
168C2×C42C2×D414yes—
169C2×C42C2411yescharacteristic
170C4⋊C8—28no—
171C22⋊C8—216no—
172#97—82no—
173C4×C8—28no—
174C4×C8—84no—
175#95—22no—
176#95—24no—
177#95—24no—
178#95—22no—
179#95—24no—
180#95C22⋊C414yes—
181#95C22⋊C412yes—
182C24—21no—
183C24C2×C22⋊C412yes—
184#41—88no—
185#41—88no—
186C2×Q8—21no—
187C2×Q8—22no—
188C2×Q8—21no—
189C2×Q8—22no—
190C2×Q8—41no—
191C2×Q8—42no—
192C2×Q8—41no—
193C2×Q8—42no—
194C2×Q8—42no—
195C2×Q8—44no—
196C2×D4—21no—
197C2×D4—22no—
198C2×D4—24no—
199C2×D4—41no—
200C2×D4—41no—
201C2×D4—44no—
202C2×D4—44no—
203C2×D4—82no—
204C2×D4#13112yes—
205C22×C4—22no—
206C22×C4—22no—
207C22×C4—24no—
208C22×C4—24no—
209C22×C4—24no—
210C22×C4—28no—
211C22×C4—21no—
212C22×C4—22no—
213C22×C4—22no—
214C22×C4—21no—
215C22×C4—21no—
216C22×C4—44no—
217C22×C4—42no—
218C22×C4—44no—
219C22×C4—42no—
220C22×C4—44no—
221C22×C4—44no—
222C22×C4—82no—
223C22×C4C2×C22⋊C412yes—
224C22×C4C2×C22⋊C411yescharacteristic
225C22×C4C2×C22⋊C412yes—
226C22×C4C2×C22⋊C414yes—
227C22×C4C2×C22⋊C411yescharacteristic
228C22×C4C23×C411yescharacteristic
229C22×C4C22×D412yes—
230M16—816no—
231C2×C8—24no—
232C2×C8—24no—
233C2×C8—28no—
234C2×C8—42no—
235C2×C8—44no—
236C2×C8—42no—
237C2×C8—44no—
238C2×C8—42no—
239C2×C8—42no—
240C4⋊C4—41no—
241C4⋊C4—42no—
242C4⋊C4—41no—
243C4⋊C4—42no—
244C4⋊C4—42no—
245C4⋊C4—42no—
246C4⋊C4—42no—
247C4⋊C4—44no—
248C4⋊C4—44no—
249C4⋊C4—48no—
250C4⋊C4—84no—
251C4⋊C4#10014yes—
252C4⋊C4#10418yes—
253C22⋊C4—24no—
254C22⋊C4—28no—
255C22⋊C4—44no—
256C22⋊C4—42no—
257C22⋊C4—44no—
258C22⋊C4—44no—
259C22⋊C4—44no—
260C22⋊C4—42no—
261C22⋊C4—42no—
262C22⋊C4—48no—
263C42—21no—
264C42—21no—
265C42—22no—
266C42—22no—
267C42—22no—
268C42—24no—
269C42—44no—
270C42—41no—
271C42—42no—
272C42—44no—
273C42—42no—
274C23—22no—
275C23—22no—
276C23—22no—
277C23—22no—
278C23—24no—
279C23—24no—
280C23—41no—
281C23—41no—
282C23—42no—
283C23—42no—
284C23C22×C22⋊C411yescharacteristic
285Q8—81no—
286Q8—81no—
287Q8—82no—
288Q8—82no—
289Q8—84no—
290D4—42no—
291D4—42no—
292D4—44no—
293D4—88no—
294C2×C4—22no—
295C2×C4—22no—
296C2×C4—21no—
297C2×C4—24no—
298C2×C4—22no—
299C2×C4—21no—
300C2×C4—24no—
301C2×C4—22no—
302C2×C4—22no—
303C2×C4—21no—
304C2×C4—22no—
305C2×C4—22no—
306C2×C4—22no—
307C2×C4—48no—
308C2×C4—48no—
309C2×C4—44no—
310C2×C4—44no—
311C2×C4—41no—
312C2×C4—42no—
313C2×C4—41no—
314C2×C4—42no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—42no—
318C2×C4—41no—
319C2×C4—42no—
320C2×C4—42no—
321C2×C4—44no—
322C2×C4—42no—
323C2×C4—88no—
324C2×C4—84no—
325C2×C4C2×#10014yes—
326C2×C4C2×#10012yes—
327C2×C4C2×#10418yes—
328C2×C4C2×#10414yes—
329C2×C4C2×#13111yescharacteristic
330C2×C4#56811yescharacteristic
331C2×C4#57712yes—
332C2×C4#57812yes—
333C8—44no—
334C8—48no—
335C8—44no—
336C22—42no—
337C22—42no—
338C22—42no—
339C22—44no—
340C22—44no—
341C22—44no—
342C22—42no—
343C22—81no—
344C22—82no—
345C22—81no—
346C22#288011yescharacteristic
347C22#288311yescharacteristic
348C22C22×#10011yescharacteristic
349C22#289812yes—
350C22C22×#10412yes—
351C4—21no—
352C4—21no—
353C4—24no—
354C4—22no—
355C4—22no—
356C4—28no—
357C4—41no—
358C4—42no—
359C4—82no—
360C4—81no—
361C4—81no—
362C4—81no—
363C4—82no—
364C4#152514yes—
365C2—82no—
366C2—82no—
367C2#1742111yescharacteristic
368C2#1751812yes—
369C2#1754812yes—
370C2C2×#152511yescharacteristic
371C2#1777311yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC2×#547C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C42×D4 ◁ G
LECSG ▷ C2×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #20567C2semidirect
3. #7687C2semidirect
4. #7682C2semidirect
5. #7635C2semidirect
6. #7631C2semidirect
7. #7567C2semidirect
8. #7563C2semidirect

Maximal subgroups

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

subgroupindex
#364062
#205672
#76872
#76822
#76352
#76312
#75672
#75632