Finite Group Databasebeta

← #164871

Subgroups — #164871

Order 512.

940 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36303C211yesmaximal, characteristic
2#22012C212yesmaximal, complemented
3#7730C214yesmaximal, complemented
4#7682C212yesmaximal, complemented
5#7643C212yesmaximal, complemented
6#7635C211yesmaximal, characteristic, complemented
7#7595C212yesmaximal, complemented
8#7567C211yesmaximal, characteristic, complemented
9C2×D4×Q8C411yescharacteristic
10C2×#547C411yescharacteristic
11#2862C2211yescharacteristic
12#2850C2212yes—
13#2568C411yescharacteristic
14#2547C411yescharacteristic
15#2503C412yes—
16#2492—22no—
17#2488—24no—
18#2436C412yes—
19#2339—22no—
20C4×#124C2211yescharacteristic
21C2×#391C2212yes—
22C42×D4C2211yescharacteristic
23#1954—22no—
24#1939—22no—
25#1924—24no—
26#1817C2214yes—
27#1797C2214yes—
28#1767C2212yes—
29#1744C2212yes—
30#1272C2214yes—
31#1272C2214yes—
32#1272C2211yescharacteristic
33#1272C2211yescharacteristic
34#1272C2212yes—
35#1270C2212yes—
36#1270C2212yes—
37C22×#41—21no—
38C23×Q8C2×C412yes—
39D4×Q8—44no—
40#547—44no—
41C2×C4⋊Q8—21no—
42C2×C4⋊Q8C2×C411yescharacteristic
43C2×C4⋊Q8C2×C411yescharacteristic
44C2×#124C2×C411yescharacteristic
45C2×#124C2×C412yes—
46C2×#123—22no—
47C2×C22⋊Q8—21no—
48C2×C22⋊Q8—22no—
49C2×C22⋊Q8C2×C412yes—
50C2×#121C2×C412yes—
51C2×C4×Q8C2×C411yescharacteristic
52C2×C4×D4—21no—
53C2×C4×D4C2×C411yescharacteristic
54C2×C4×D4C2×C411yescharacteristic
55C2×C4×D4D412yes—
56C22×C4⋊C4C2×C412yes—
57C22×C42C2312yes—
58C2×C4⋊C8—22no—
59C2×C4⋊C8—24no—
60C2×C22⋊C8—24no—
61C2×C22⋊C8—24no—
62C2×#97—22no—
63C2×C4×C8—22no—
64#400—22no—
65#400—24no—
66#397—22no—
67#393—24no—
68#391—22no—
69#391—22no—
70#391—24no—
71#391C2×C412yes—
72#391C2×C412yes—
73#391C2×C412yes—
74#391C2312yes—
75#391C2311yescharacteristic
76#390C2311yescharacteristic
77#388—24no—
78#388—22no—
79#386—24no—
80#386—24no—
81#382—22no—
82#382C2×C412yes—
83#382C2×C411yescharacteristic
84#381—24no—
85#380—22no—
86C4×C4⋊C4C2×C411yescharacteristic
87C4×C4⋊C4D412yes—
88C4×C22⋊C4—22no—
89C4×C22⋊C4C2×C412yes—
90C4×C22⋊C4D414yes—
91C2×#95—24no—
92C43C2311yescharacteristic
93#336—24no—
94#336—22no—
95#336C2312yes—
96#336C2314yes—
97#336C2312yes—
98C2×#41—48no—
99C2×#41—44no—
100C22×Q8—21no—
101C22×Q8—22no—
102C22×Q8—24no—
103C22×Q8—42no—
104C22×Q8—44no—
105C22×Q8—82no—
106C22×Q8C22×C412yes—
107C22×Q8C22×C411yescharacteristic
108C22×D4—22no—
109C22×D4C22⋊C411yescharacteristic
110C22×D4C22×C411yescharacteristic
111C23×C4—22no—
112C23×C4—22no—
113C23×C4—21no—
114C23×C4C22×C412yes—
115C23×C4C2×D414yes—
116C22×C8—22no—
117C22×C8—22no—
118C22×C8—24no—
119C4⋊Q8—42no—
120C4⋊Q8—42no—
121C4⋊Q8—44no—
122#124—22no—
123#124—41no—
124#124—82no—
125#123—48no—
126C22⋊Q8—48no—
127C22⋊Q8—44no—
128C22⋊Q8—44no—
129#121—44no—
130C4×Q8—42no—
131C4×D4—24no—
132C4×D4—44no—
133C4×D4—42no—
134C4×D4—44no—
135C2×C4⋊C4—22no—
136C2×C4⋊C4—21no—
137C2×C4⋊C4—21no—
138C2×C4⋊C4—21no—
139C2×C4⋊C4—22no—
140C2×C4⋊C4—21no—
141C2×C4⋊C4—44no—
142C2×C4⋊C4C22⋊C411yescharacteristic
143C2×C4⋊C4C22⋊C412yes—
144C2×C4⋊C4C22×C411yescharacteristic
145C2×C4⋊C4C22×C412yes—
146C2×C4⋊C4C22×C411yescharacteristic
147C2×C22⋊C4—22no—
148C2×C22⋊C4—21no—
149C2×C22⋊C4—22no—
150C2×C22⋊C4—22no—
151C2×C22⋊C4—22no—
152C2×C22⋊C4—24no—
153C2×C22⋊C4C22⋊C412yes—
154C2×C22⋊C4C22⋊C414yes—
155C2×C42—21no—
156C2×C42—82no—
157C2×C42C22×C411yescharacteristic
158C2×C42C22×C411yescharacteristic
159C2×C42C22×C412yes—
160C2×C42C2×D412yes—
161C2×C42C2×D412yes—
162C2×C42C2×D414yes—
163C2×C42C2411yescharacteristic
164C4⋊C8—44no—
165C4⋊C8—84no—
166C22⋊C8—84no—
167C22⋊C8—84no—
168#97—82no—
169C4×C8—44no—
170#95—24no—
171#95—24no—
172#95—22no—
173#95—24no—
174#95—24no—
175#95—22no—
176#95—24no—
177#95C22⋊C414yes—
178#95C22⋊C412yes—
179C24—21no—
180C24C2×C22⋊C412yes—
181#41—816no—
182C2×Q8—21no—
183C2×Q8—22no—
184C2×Q8—22no—
185C2×Q8—44no—
186C2×Q8—44no—
187C2×Q8—41no—
188C2×Q8—41no—
189C2×Q8—42no—
190C2×Q8—42no—
191C2×Q8—42no—
192C2×Q8—44no—
193C2×Q8—82no—
194C2×Q8—88no—
195C2×Q8—88no—
196C2×Q8#13112yes—
197C2×D4—21no—
198C2×D4—21no—
199C2×D4—24no—
200C2×D4—41no—
201C2×D4—41no—
202C2×D4—48no—
203C2×D4—44no—
204C2×D4#10014yes—
205C22×C4—21no—
206C22×C4—24no—
207C22×C4—22no—
208C22×C4—22no—
209C22×C4—24no—
210C22×C4—24no—
211C22×C4—21no—
212C22×C4—22no—
213C22×C4—21no—
214C22×C4—22no—
215C22×C4—42no—
216C22×C4—44no—
217C22×C4—44no—
218C22×C4—44no—
219C22×C4—48no—
220C22×C4—44no—
221C22×C4—44no—
222C22×C4—42no—
223C22×C4—44no—
224C22×C4—82no—
225C22×C4C2×C22⋊C412yes—
226C22×C4C2×C22⋊C414yes—
227C22×C4C2×C22⋊C411yescharacteristic
228C22×C4C2×C22⋊C412yes—
229C22×C4C2×C22⋊C411yescharacteristic
230C22×C4C23×C411yescharacteristic
231C22×C4C22×D412yes—
232C2×C8—42no—
233C2×C8—42no—
234C2×C8—42no—
235C2×C8—44no—
236C2×C8—44no—
237C2×C8—42no—
238C2×C8—44no—
239C2×C8—42no—
240C2×C8—42no—
241C4⋊C4—22no—
242C4⋊C4—24no—
243C4⋊C4—41no—
244C4⋊C4—42no—
245C4⋊C4—41no—
246C4⋊C4—44no—
247C4⋊C4—42no—
248C4⋊C4—42no—
249C4⋊C4—44no—
250C4⋊C4—42no—
251C4⋊C4—42no—
252C22⋊C4—24no—
253C22⋊C4—44no—
254C22⋊C4—44no—
255C22⋊C4—42no—
256C22⋊C4—44no—
257C22⋊C4—44no—
258C22⋊C4—42no—
259C22⋊C4—42no—
260C22⋊C4—84no—
261C42—21no—
262C42—21no—
263C42—22no—
264C42—22no—
265C42—22no—
266C42—24no—
267C42—44no—
268C42—41no—
269C42—42no—
270C42—42no—
271C23—22no—
272C23—22no—
273C23—24no—
274C23—24no—
275C23—41no—
276C23—41no—
277C23—41no—
278C23—42no—
279C23—42no—
280C23C2×#10014yes—
281C23C22×C22⋊C411yescharacteristic
282Q8—42no—
283Q8—42no—
284Q8—82no—
285Q8—82no—
286Q8—88no—
287Q8—84no—
288D4—28no—
289D4—81no—
290D4—81no—
291D4—88no—
292C2×C4—22no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—22no—
296C2×C4—22no—
297C2×C4—22no—
298C2×C4—22no—
299C2×C4—24no—
300C2×C4—22no—
301C2×C4—21no—
302C2×C4—22no—
303C2×C4—22no—
304C2×C4—22no—
305C2×C4—24no—
306C2×C4—22no—
307C2×C4—24no—
308C2×C4—44no—
309C2×C4—44no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—42no—
313C2×C4—42no—
314C2×C4—41no—
315C2×C4—42no—
316C2×C4—41no—
317C2×C4—42no—
318C2×C4—42no—
319C2×C4—42no—
320C2×C4—44no—
321C2×C4—84no—
322C2×C4—84no—
323C2×C4—88no—
324C2×C4—88no—
325C2×C4—84no—
326C2×C4C2×#10012yes—
327C2×C4#42114yes—
328C2×C4C2×#13111yescharacteristic
329C2×C4#56811yescharacteristic
330C2×C4#57712yes—
331C2×C4#57812yes—
332C8—82no—
333C8—84no—
334C8—82no—
335C22—24no—
336C22—24no—
337C22—24no—
338C22—44no—
339C22—44no—
340C22—81no—
341C22—81no—
342C22—82no—
343C22—82no—
344C22—81no—
345C22#288011yescharacteristic
346C22#288311yescharacteristic
347C22C22×#10011yescharacteristic
348C22#289812yes—
349C22C2×#42112yes—
350C4—21no—
351C4—21no—
352C4—22no—
353C4—22no—
354C4—42no—
355C4—42no—
356C4—41no—
357C4—42no—
358C4—44no—
359C4—82no—
360C4—81no—
361C4—81no—
362C4—82no—
363C4#153314yes—
364C2—44no—
365C2—161no—
366C2#1742111yescharacteristic
367C2#1752512yes—
368C2#1760012yes—
369C2C2×#153311yescharacteristic
370C2#1777911yescharacteristic
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. #22012C2semidirect
3. #7730C2semidirect
4. #7682C2semidirect
5. #7643C2semidirect
6. #7635C2semidirect
7. #7595C2semidirect
8. #7567C2semidirect

Maximal subgroups

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

subgroupindex
#363032
#220122
#77302
#76822
#76432
#76352
#75952
#75672