Finite Group Databasebeta

← #200629

Subgroups — #200629

Order 512.

1176 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#34953C211yesmaximal, characteristic
2#34059C211yesmaximal, characteristic
3#17545C212yesmaximal
4#17365C211yesmaximal, characteristic
5#9412C218yesmaximal
6#8048C212yesmaximal
7C2×#533C2211yescharacteristic
8C22×C4×Q8C2211yescharacteristic
9#2920C2212yes—
10#2584—22no—
11#2583C2214yes—
12#2567C412yescomplemented
13#2546C412yescomplemented
14#2502—21no—
15#2434—21no—
16#2399C414yescomplemented
17#2320—24no—
18#2316—21no—
19#2313C2214yes—
20#2310—21no—
21C4×C4⋊Q8—22no—
22#2296—22no—
23#2295—24no—
24#2283—22no—
25C4×#117C2211yescharacteristic
26C4×#117C2212yes—
27#1508C2218yes—
28#1506—24no—
29#1503—24no—
30#1467—22no—
31#1463C2214yes—
32#1461—22no—
33#1297C2218yes—
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×#125—21no—
43C2×C22⋊Q8—21no—
44C2×C22⋊Q8C2312yes—
45C2×C4×Q8—22no—
46C2×C4×Q8—22no—
47C2×C4×Q8—21no—
48C2×C4×Q8—22no—
49C2×C4×Q8—22no—
50C2×C4×Q8C2×C412yes—
51C2×C4×Q8C2×C412yes—
52C2×C4×Q8C2×C412yes—
53C2×C4×Q8D412yes—
54C2×C4×Q8D412yes—
55C2×#117—21no—
56C2×#117—22no—
57C2×#117—21no—
58C22×C4⋊C4D411yescharacteristic
59C22×C4⋊C4C2312yes—
60C22×C42D412yes—
61C22×C42C2311yescharacteristic
62#432—22no—
63#432—22no—
64#431—22no—
65#431—22no—
66#423—22no—
67#407C2314yes—
68C4×M16—22no—
69#398—24no—
70#398C2×C418yes—
71#395—24no—
72#395C2×C418yes—
73#391—28no—
74#391—22no—
75#391—22no—
76#389—28no—
77#389—28no—
78#389—24no—
79#389—24no—
80#387—216no—
81#385—24no—
82#385—24no—
83#384—22no—
84#383—24no—
85#382—24no—
86#381—24no—
87#380—24no—
88C4×C4⋊C4—24no—
89C4×C4⋊C4—28no—
90C4×C4⋊C4—22no—
91C4×C4⋊C4—21no—
92C4×C4⋊C4C2×C412yes—
93C4×C4⋊C4D414yes—
94C4×C22⋊C4—21no—
95C4×C22⋊C4—22no—
96C4×C22⋊C4C2312yes—
97C4×C22⋊C4C2314yes—
98#376—24no—
99#376C2×C412yes—
100#376D414yes—
101C43—21no—
102C43—22no—
103#328—216no—
104#324—28no—
105C22×Q8—21no—
106C22×Q8—22no—
107C22×Q8—22no—
108C22×Q8—44no—
109C22×Q8C22⋊C412yes—
110C22×Q8C22⋊C412yes—
111C22×Q8C22×C412yes—
112C23×C4C2×D411yescharacteristic
113C23×C4C2×D411yescharacteristic
114C23×C4C2×D412yes—
115C23×C4C2×D412yes—
116C23×C4C2411yescharacteristic
117C2×M16—24no—
118C4⋊Q8—22no—
119C4⋊Q8—22no—
120C4⋊Q8—22no—
121C4⋊Q8—22no—
122C4⋊Q8—48no—
123#125—42no—
124#125—42no—
125C22⋊Q8—24no—
126C22⋊Q8—24no—
127C22⋊Q8—42no—
128C22⋊Q8—42no—
129C4×Q8—24no—
130C4×Q8—24no—
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—
141#117—28no—
142#117—48no—
143#117—41no—
144#117—41no—
145#117—41no—
146#117—41no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—24no—
151C2×C4⋊C4—24no—
152C2×C4⋊C4—21no—
153C2×C4⋊C4—21no—
154C2×C4⋊C4—21no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—28no—
157C2×C4⋊C4—24no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—28no—
160C2×C4⋊C4—28no—
161C2×C4⋊C4C22⋊C412yes—
162C2×C4⋊C4C22⋊C412yes—
163C2×C4⋊C4C22×C412yes—
164C2×C4⋊C4C22×C414yes—
165C2×C4⋊C4C2×D414yes—
166C2×C22⋊C4—22no—
167C2×C22⋊C4—22no—
168C2×C22⋊C4—24no—
169C2×C42—22no—
170C2×C42—22no—
171C2×C42—22no—
172C2×C42—22no—
173C2×C42—22no—
174C2×C42—22no—
175C2×C42—22no—
176C2×C42—24no—
177C2×C42—24no—
178C2×C42—24no—
179C2×C42—22no—
180C2×C42C22⋊C414yes—
181C2×C42C22⋊C414yes—
182C2×C42C22×C411yescharacteristic
183C2×C42C22×C411yescharacteristic
184C2×C42C2×D412yes—
185C2×C42C2×D412yes—
186C2×C42C2×D411yescharacteristic
187C2×C42C2×D411yescharacteristic
188C2×C42#4112yes—
189C2×C42#4112yes—
190C4⋊C8—42no—
191C4⋊C8—42no—
192C22⋊C8—24no—
193C22⋊C8—24no—
194#97—42no—
195C4×C8—42no—
196#95—28no—
197#95—28no—
198#95—28no—
199#95—22no—
200#95—28no—
201#95—28no—
202#95—24no—
203#95C22×C414yes—
204#95C2×D418yes—
205C24C22×D411yescharacteristic
206C2×Q8—24no—
207C2×Q8—24no—
208C2×Q8—24no—
209C2×Q8—24no—
210C2×Q8—24no—
211C2×Q8—44no—
212C2×Q8—48no—
213C2×Q8—48no—
214C2×Q8—42no—
215C2×Q8—44no—
216C2×Q8—82no—
217C22×C4—24no—
218C22×C4—24no—
219C22×C4—24no—
220C22×C4—22no—
221C22×C4—22no—
222C22×C4—22no—
223C22×C4—22no—
224C22×C4—22no—
225C22×C4—22no—
226C22×C4—28no—
227C22×C4—44no—
228C22×C4—81no—
229C22×C4—82no—
230C22×C4C2×C22⋊C414yes—
231C22×C4C2×C22⋊C411yescharacteristic
232C22×C4C2×C22⋊C411yescharacteristic
233C22×C4C2×C22⋊C411yescharacteristic
234C22×C4C2×C22⋊C414yes—
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—84no—
247C2×C8—44no—
248C4⋊C4—28no—
249C4⋊C4—28no—
250C4⋊C4—28no—
251C4⋊C4—24no—
252C4⋊C4—24no—
253C4⋊C4—24no—
254C4⋊C4—48no—
255C4⋊C4—416no—
256C4⋊C4—44no—
257C4⋊C4—41no—
258C4⋊C4—41no—
259C4⋊C4—41no—
260C4⋊C4—41no—
261C4⋊C4—44no—
262C4⋊C4—42no—
263C4⋊C4—88no—
264C4⋊C4—88no—
265C22⋊C4—28no—
266C22⋊C4—48no—
267C22⋊C4—42no—
268C22⋊C4—42no—
269C42—28no—
270C42—24no—
271C42—24no—
272C42—24no—
273C42—24no—
274C42—48no—
275C42—42no—
276C42—42no—
277C42—44no—
278C42—48no—
279C42—82no—
280C42—84no—
281C42C4×D412yes—
282C42C4×D412yes—
283C42C4×D412yes—
284C42C4×D412yes—
285C42#12112yes—
286C42#12112yes—
287C42#12112yes—
288C42#12112yes—
289C23—22no—
290C23—22no—
291C23—41no—
292C23C22×C22⋊C411yescharacteristic
293C23C2×#12011yescharacteristic
294C23C2×#12311yescharacteristic
295Q8—48no—
296Q8—48no—
297Q8—84no—
298Q8—84no—
299C2×C4—22no—
300C2×C4—22no—
301C2×C4—22no—
302C2×C4—22no—
303C2×C4—22no—
304C2×C4—22no—
305C2×C4—24no—
306C2×C4—24no—
307C2×C4—28no—
308C2×C4—24no—
309C2×C4—24no—
310C2×C4—24no—
311C2×C4—24no—
312C2×C4—41no—
313C2×C4—42no—
314C2×C4—42no—
315C2×C4—41no—
316C2×C4—44no—
317C2×C4—44no—
318C2×C4—44no—
319C2×C4—44no—
320C2×C4—44no—
321C2×C4—44no—
322C2×C4—48no—
323C2×C4—48no—
324C2×C4—44no—
325C2×C4—44no—
326C2×C4—48no—
327C2×C4—42no—
328C2×C4—42no—
329C2×C4—82no—
330C2×C4#38611yescharacteristic
331C2×C4#38611yescharacteristic
332C2×C4#38612yes—
333C2×C4#38611yescharacteristic
334C2×C4#38612yes—
335C2×C4#38611yescharacteristic
336C2×C4#42114yes—
337C2×C4#45614yes—
338C2×C4C2×C4×D412yes—
339C2×C4C2×#12112yes—
340C8—84no—
341C22—21no—
342C22—21no—
343C22—21no—
344C22—21no—
345C22—21no—
346C22—21no—
347C22—21no—
348C22—44no—
349C22C2×#38611yescharacteristic
350C22C2×#42112yes—
351C22C2×#45612yes—
352C4—22no—
353C4—22no—
354C4—21no—
355C4—21no—
356C4—22no—
357C4—48no—
358C4—44no—
359C4—44no—
360C4—88no—
361C4—82no—
362C4—82no—
363C4#188612yes—
364C4#188612yes—
365C4#189714yes—
366C2—22no—
367C2—22no—
368C2—41no—
369C2C2×#188611yescharacteristic
370C2C2×#189711yescharacteristic
371C2#2101411yescharacteristic
subgroupG/H
centerC2×C4#386
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
socleC22C2×#386
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
12. #2567C4semidirect
13. #2546C4semidirect
16. #2399C4semidirect

Maximal subgroups

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

subgroupindex
#349532
#340592
#175452
#173652
#94122
#80482