Finite Group Databasebeta

← #201500

Subgroups — #201500

Order 512.

1141 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35248C211yesmaximal, characteristic
2#34059C211yesmaximal, characteristic
3#21844C212yesmaximal
4#17365C211yesmaximal, characteristic
5#9482C218yesmaximal
6#7941C212yesmaximal
7C2×#533C2211yescharacteristic
8C22×C4×Q8C2211yescharacteristic
9#2920C2212yes—
10#2855—22no—
11#2634C414yescomplemented
12#2581—22no—
13#2568C412yescomplemented
14#2546C412yescomplemented
15#2503—21no—
16#2484C2214yes—
17#2434—21no—
18#2327—22no—
19#2320—24no—
20#2316—21no—
21#2313C2214yes—
22#2310—21no—
23#2295—24no—
24#2283—22no—
25#2283C2211yescharacteristic
26C4×#117C2212yes—
27#1994—24no—
28#1986—24no—
29#1826C2218yes—
30#1467—22no—
31#1463C2214yes—
32#1461—22no—
33#1284C2218yes—
34C23×Q8D411yescharacteristic
35#533—22no—
36#533—22no—
37#533—22no—
38#533—22no—
39C2×C4⋊Q8—22no—
40C2×C4⋊Q8C2×C411yescharacteristic
41C2×C4⋊Q8C2×C411yescharacteristic
42C2×#125C2×C412yes—
43C2×C22⋊Q8—21no—
44C2×C22⋊Q8—21no—
45C2×C4×Q8—22no—
46C2×C4×Q8—22no—
47C2×C4×Q8—21no—
48C2×C4×Q8—21no—
49C2×C4×Q8—22no—
50C2×C4×Q8—22no—
51C2×C4×Q8C2×C412yes—
52C2×C4×Q8C2×C412yes—
53C2×C4×Q8D412yes—
54C2×C4×Q8D412yes—
55C2×#117—22no—
56C2×#117C2312yes—
57C22×C4⋊C4D411yescharacteristic
58C22×C4⋊C4C2312yes—
59C22×C42D412yes—
60C22×C42C2311yescharacteristic
61#432—22no—
62#432—22no—
63#431—22no—
64#431—22no—
65#423—22no—
66#407C2314yes—
67C4×M16—22no—
68#400—24no—
69#400C2×C418yes—
70#398C2×C418yes—
71#395—24no—
72#391—28no—
73#391—22no—
74#391—22no—
75#391—24no—
76#389—28no—
77#389—24no—
78#387—28no—
79#387—28no—
80#387—216no—
81#385—24no—
82#383—24no—
83#383C2×C412yes—
84#382—24no—
85#382—22no—
86#382—21no—
87#381—24no—
88#381—24no—
89#381C2312yes—
90#380—24no—
91#380—21no—
92C4×C4⋊C4—24no—
93C4×C4⋊C4—22no—
94C4×C4⋊C4C2×C412yes—
95C4×C4⋊C4D414yes—
96C4×C22⋊C4—21no—
97C4×C22⋊C4—22no—
98C4×C22⋊C4C2314yes—
99#376—24no—
100#376—22no—
101#376—21no—
102C43D414yes—
103#339—216no—
104#324—28no—
105C22×Q8—21no—
106C22×Q8—21no—
107C22×Q8—22no—
108C22×Q8—22no—
109C22×Q8—44no—
110C22×Q8C22⋊C412yes—
111C22×Q8C22⋊C412yes—
112C23×C4C2×D411yescharacteristic
113C23×C4C2×D412yes—
114C23×C4C2×D412yes—
115C23×C4C2×D411yescharacteristic
116C23×C4C2411yescharacteristic
117C2×M16—24no—
118C4⋊Q8—22no—
119C4⋊Q8—22no—
120C4⋊Q8—22no—
121C4⋊Q8—22no—
122C4⋊Q8—48no—
123#125—24no—
124#125—24no—
125C22⋊Q8—41no—
126C22⋊Q8—42no—
127C22⋊Q8—41no—
128C22⋊Q8—41no—
129C22⋊Q8—42no—
130C22⋊Q8—41no—
131C4×Q8—24no—
132C4×Q8—24no—
133C4×Q8—24no—
134C4×Q8—24no—
135C4×Q8—24no—
136C4×Q8—24no—
137C4×Q8—24no—
138C4×Q8—24no—
139C4×Q8—42no—
140C4×Q8—42no—
141C4×Q8—42no—
142C4×Q8—42no—
143#117—24no—
144#117—24no—
145#117—48no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—24no—
149C2×C4⋊C4—24no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—21no—
153C2×C4⋊C4—21no—
154C2×C4⋊C4—28no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—24no—
157C2×C4⋊C4—28no—
158C2×C4⋊C4C22⋊C412yes—
159C2×C4⋊C4C22⋊C412yes—
160C2×C4⋊C4C22×C412yes—
161C2×C4⋊C4C22×C412yes—
162C2×C4⋊C4C22×C414yes—
163C2×C4⋊C4C2×D414yes—
164C2×C22⋊C4—24no—
165C2×C22⋊C4—22no—
166C2×C22⋊C4—22no—
167C2×C42—22no—
168C2×C42—22no—
169C2×C42—22no—
170C2×C42—22no—
171C2×C42—22no—
172C2×C42—24no—
173C2×C42—24no—
174C2×C42—22no—
175C2×C42C22⋊C414yes—
176C2×C42C22⋊C414yes—
177C2×C42C22×C411yescharacteristic
178C2×C42C22×C411yescharacteristic
179C2×C42C2×D418yes—
180C2×C42C2×D412yes—
181C2×C42C2×D412yes—
182C2×C42C2×D411yescharacteristic
183C2×C42C2×D411yescharacteristic
184C2×C42#4112yes—
185C2×C42#4112yes—
186C4⋊C8—42no—
187C4⋊C8—42no—
188C22⋊C8—24no—
189C22⋊C8—24no—
190#97—42no—
191C4×C8—42no—
192#95—28no—
193#95—28no—
194#95—28no—
195#95—22no—
196#95—22no—
197#95—28no—
198#95—28no—
199#95—28no—
200#95—24no—
201#95—24no—
202#95—22no—
203#95C22×C414yes—
204C24C22×D411yescharacteristic
205C2×Q8—24no—
206C2×Q8—24no—
207C2×Q8—24no—
208C2×Q8—24no—
209C2×Q8—44no—
210C2×Q8—48no—
211C2×Q8—48no—
212C2×Q8—44no—
213C2×Q8—41no—
214C2×Q8—42no—
215C2×Q8—41no—
216C2×Q8—44no—
217C22×C4—24no—
218C22×C4—22no—
219C22×C4—22no—
220C22×C4—22no—
221C22×C4—22no—
222C22×C4—24no—
223C22×C4—22no—
224C22×C4—22no—
225C22×C4—28no—
226C22×C4—42no—
227C22×C4—44no—
228C22×C4—42no—
229C22×C4—44no—
230C22×C4C2×C22⋊C414yes—
231C22×C4C2×C22⋊C411yescharacteristic
232C22×C4C2×C22⋊C411yescharacteristic
233C22×C4C2×C22⋊C414yes—
234C22×C4C2×C22⋊C411yescharacteristic
235C22×C4C2×C22⋊C411yescharacteristic
236C22×C4#12011yescharacteristic
237C22×C4#12012yes—
238C22×C4#12011yescharacteristic
239C22×C4#12311yescharacteristic
240C22×C4#12312yes—
241C22×C4#12311yescharacteristic
242C22×C4C23×C411yescharacteristic
243C22×C4C22×D412yes—
244C22×C4C22×D411yescharacteristic
245C22×C4C2×#4112yes—
246M16—48no—
247C2×C8—44no—
248C4⋊C4—28no—
249C4⋊C4—28no—
250C4⋊C4—24no—
251C4⋊C4—24no—
252C4⋊C4—24no—
253C4⋊C4—24no—
254C4⋊C4—44no—
255C4⋊C4—48no—
256C4⋊C4—416no—
257C4⋊C4—42no—
258C4⋊C4—42no—
259C4⋊C4—44no—
260C4⋊C4—41no—
261C4⋊C4—42no—
262C4⋊C4—41no—
263C4⋊C4—88no—
264C22⋊C4—44no—
265C22⋊C4—48no—
266C22⋊C4—42no—
267C22⋊C4—42no—
268C42—24no—
269C42—24no—
270C42—24no—
271C42—24no—
272C42—48no—
273C42—48no—
274C42—48no—
275C42—44no—
276C42—44no—
277C42C4×D412yes—
278C42C4×D412yes—
279C42C4×D412yes—
280C42C4×D412yes—
281C42#12112yes—
282C42#12112yes—
283C42#12112yes—
284C42#12112yes—
285C23—22no—
286C23—22no—
287C23—41no—
288C23C22×C22⋊C411yescharacteristic
289C23C2×#12011yescharacteristic
290C23C2×#12311yescharacteristic
291Q8—48no—
292Q8—48no—
293Q8—82no—
294Q8—84no—
295Q8—82no—
296C2×C4—22no—
297C2×C4—24no—
298C2×C4—24no—
299C2×C4—24no—
300C2×C4—22no—
301C2×C4—22no—
302C2×C4—22no—
303C2×C4—24no—
304C2×C4—24no—
305C2×C4—24no—
306C2×C4—24no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—42no—
313C2×C4—44no—
314C2×C4—44no—
315C2×C4—44no—
316C2×C4—44no—
317C2×C4—48no—
318C2×C4—48no—
319C2×C4—44no—
320C2×C4—44no—
321C2×C4—44no—
322C2×C4—44no—
323C2×C4—44no—
324C2×C4—48no—
325C2×C4—42no—
326C2×C4—42no—
327C2×C4—82no—
328C2×C4#38611yescharacteristic
329C2×C4#38611yescharacteristic
330C2×C4#38611yescharacteristic
331C2×C4#38612yes—
332C2×C4#38612yes—
333C2×C4#38611yescharacteristic
334C2×C4#42114yes—
335C2×C4C2×C4×D412yes—
336C2×C4C2×#12112yes—
337C8—84no—
338C22—21no—
339C22—21no—
340C22—21no—
341C22—21no—
342C22—21no—
343C22—21no—
344C22—21no—
345C22—44no—
346C22C2×#38611yescharacteristic
347C22C2×#42112yes—
348C22#302812yes—
349C4—21no—
350C4—21no—
351C4—21no—
352C4—22no—
353C4—22no—
354C4—22no—
355C4—44no—
356C4—44no—
357C4—88no—
358C4—84no—
359C4—82no—
360C4—82no—
361C4#189714yes—
362C4#191212yes—
363C2—22no—
364C2—22no—
365C2—41no—
366C2#2096211yescharacteristic
367C2C2×#189711yescharacteristic
368C2C2×#191211yescharacteristic
subgroupG/H
center, socleC22C2×#386
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
11. #2634C4semidirect
13. #2568C4semidirect
14. #2546C4semidirect

Maximal subgroups

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

subgroupindex
#352482
#340592
#218442
#173652
#94822
#79412