Finite Group Databasebeta

← #201491

Subgroups — #201491

Order 512.

1429 conjugacy classes of subgroups.

The subgroup lattice has 375 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35248C211yesmaximal, characteristic
2C4×#519C211yesmaximal, characteristic
3#21844C212yesmaximal
4#17365C211yesmaximal, characteristic
5#9478C218yesmaximal
6#7941C212yesmaximal
7C2×#533C2211yescharacteristic
8C2×#519—21no—
9C22×C4×Q8C2211yescharacteristic
10#2920C2212yes—
11#2855—22no—
12#2634C414yescomplemented
13#2581—22no—
14#2568C412yescomplemented
15#2546C412yescomplemented
16#2503—21no—
17#2484C2214yes—
18#2434—21no—
19#2327—22no—
20#2283C2211yescharacteristic
21#2276—24no—
22#2274—21no—
23#2274C2214yes—
24C42×Q8—24no—
25C4×#117—22no—
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—
45#519—216no—
46C2×C4×Q8—28no—
47C2×C4×Q8—22no—
48C2×C4×Q8—22no—
49C2×C4×Q8—21no—
50C2×C4×Q8—21no—
51C2×C4×Q8—22no—
52C2×C4×Q8—22no—
53C2×C4×Q8C2×C412yes—
54C2×C4×Q8C2×C412yes—
55C2×C4×Q8D412yes—
56C2×C4×Q8D412yes—
57C2×#117—24no—
58C2×#117—22no—
59C2×#117C2312yes—
60C22×C4⋊C4D411yescharacteristic
61C22×C4⋊C4C2312yes—
62C22×C42D412yes—
63C22×C42C2311yescharacteristic
64#432—22no—
65#432—22no—
66#431—22no—
67#431—22no—
68#423—22no—
69#407C2314yes—
70C4×M16—22no—
71#400—24no—
72#400C2×C418yes—
73#398C2×C418yes—
74#395—24no—
75#391—22no—
76#391—22no—
77#391—24no—
78#389—24no—
79#387—28no—
80#387—28no—
81#385—24no—
82#383C2×C412yes—
83#382—22no—
84#382—21no—
85#381—24no—
86#381C2312yes—
87#380—21no—
88C4×C4⋊C4—24no—
89C4×C4⋊C4—24no—
90C4×C4⋊C4—28no—
91C4×C4⋊C4—22no—
92C4×C4⋊C4—216no—
93C4×C4⋊C4C2×C412yes—
94C4×C4⋊C4D414yes—
95C4×C22⋊C4—21no—
96C4×C22⋊C4—22no—
97C4×C22⋊C4—24no—
98C4×C22⋊C4C2314yes—
99#376—24no—
100#376—21no—
101#376—22no—
102C43—24no—
103C43D414yes—
104#339—216no—
105#324—28no—
106C22×Q8—28no—
107C22×Q8—21no—
108C22×Q8—21no—
109C22×Q8—22no—
110C22×Q8—22no—
111C22×Q8C22⋊C412yes—
112C22×Q8C22⋊C412yes—
113C23×C4C2×D412yes—
114C23×C4C2×D411yescharacteristic
115C23×C4C2×D412yes—
116C23×C4C2×D411yescharacteristic
117C23×C4C2411yescharacteristic
118C2×M16—24no—
119C4⋊Q8—22no—
120C4⋊Q8—22no—
121C4⋊Q8—22no—
122C4⋊Q8—22no—
123C4⋊Q8—48no—
124#125—24no—
125#125—24no—
126C22⋊Q8—41no—
127C22⋊Q8—42no—
128C22⋊Q8—41no—
129C22⋊Q8—41no—
130C22⋊Q8—42no—
131C22⋊Q8—41no—
132C4×Q8—264no—
133C4×Q8—24no—
134C4×Q8—24no—
135C4×Q8—24no—
136C4×Q8—24no—
137C4×Q8—24no—
138C4×Q8—24no—
139C4×Q8—24no—
140C4×Q8—24no—
141C4×Q8—42no—
142C4×Q8—42no—
143C4×Q8—42no—
144C4×Q8—42no—
145#117—232no—
146#117—216no—
147#117—24no—
148#117—24no—
149C2×C4⋊C4—28no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—24no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—24no—
154C2×C4⋊C4—24no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—28no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4C22⋊C412yes—
162C2×C4⋊C4C22⋊C412yes—
163C2×C4⋊C4C22×C412yes—
164C2×C4⋊C4C22×C412yes—
165C2×C4⋊C4C22×C414yes—
166C2×C4⋊C4C2×D414yes—
167C2×C22⋊C4—24no—
168C2×C22⋊C4—22no—
169C2×C22⋊C4—22no—
170C2×C42—28no—
171C2×C42—24no—
172C2×C42—22no—
173C2×C42—22no—
174C2×C42—22no—
175C2×C42—22no—
176C2×C42—22no—
177C2×C42—24no—
178C2×C42—22no—
179C2×C42—28no—
180C2×C42C22⋊C414yes—
181C2×C42C22⋊C414yes—
182C2×C42C22×C411yescharacteristic
183C2×C42C22×C411yescharacteristic
184C2×C42C2×D418yes—
185C2×C42C2×D412yes—
186C2×C42C2×D412yes—
187C2×C42C2×D411yescharacteristic
188C2×C42C2×D411yescharacteristic
189C2×C42#4112yes—
190C2×C42#4112yes—
191C4⋊C8—42no—
192C4⋊C8—42no—
193C22⋊C8—24no—
194C22⋊C8—24no—
195#97—42no—
196C4×C8—42no—
197#95—28no—
198#95—22no—
199#95—22no—
200#95—28no—
201#95—28no—
202#95—28no—
203#95—24no—
204#95—24no—
205#95—22no—
206#95C22×C414yes—
207C24C22×D411yescharacteristic
208C2×Q8—216no—
209C2×Q8—216no—
210C2×Q8—24no—
211C2×Q8—24no—
212C2×Q8—28no—
213C2×Q8—24no—
214C2×Q8—44no—
215C2×Q8—41no—
216C2×Q8—42no—
217C2×Q8—41no—
218C2×Q8—44no—
219C2×Q8#10418yes—
220C22×C4—22no—
221C22×C4—22no—
222C22×C4—28no—
223C22×C4—24no—
224C22×C4—28no—
225C22×C4—22no—
226C22×C4—22no—
227C22×C4—24no—
228C22×C4—22no—
229C22×C4—22no—
230C22×C4—42no—
231C22×C4—42no—
232C22×C4—44no—
233C22×C4C2×C22⋊C411yescharacteristic
234C22×C4C2×C22⋊C411yescharacteristic
235C22×C4C2×C22⋊C414yes—
236C22×C4C2×C22⋊C414yes—
237C22×C4C2×C22⋊C411yescharacteristic
238C22×C4C2×C22⋊C411yescharacteristic
239C22×C4#12011yescharacteristic
240C22×C4#12012yes—
241C22×C4#12011yescharacteristic
242C22×C4#12311yescharacteristic
243C22×C4#12312yes—
244C22×C4#12311yescharacteristic
245C22×C4C23×C411yescharacteristic
246C22×C4C22×D412yes—
247C22×C4C22×D411yescharacteristic
248C22×C4C2×#4112yes—
249M16—48no—
250C2×C8—44no—
251C4⋊C4—232no—
252C4⋊C4—216no—
253C4⋊C4—28no—
254C4⋊C4—28no—
255C4⋊C4—24no—
256C4⋊C4—24no—
257C4⋊C4—24no—
258C4⋊C4—24no—
259C4⋊C4—44no—
260C4⋊C4—42no—
261C4⋊C4—42no—
262C4⋊C4—44no—
263C4⋊C4—41no—
264C4⋊C4—42no—
265C4⋊C4—41no—
266C4⋊C4—416no—
267C22⋊C4—216no—
268C22⋊C4—44no—
269C22⋊C4—42no—
270C22⋊C4—42no—
271C42—232no—
272C42—216no—
273C42—24no—
274C42—24no—
275C42—24no—
276C42—24no—
277C42—48no—
278C42—48no—
279C42—44no—
280C42—44no—
281C42—88no—
282C42C4×D412yes—
283C42C4×D412yes—
284C42C4×D412yes—
285C42C4×D412yes—
286C42#12112yes—
287C42#12112yes—
288C42#12112yes—
289C42#12112yes—
290C23—22no—
291C23—22no—
292C23—41no—
293C23C22×C22⋊C411yescharacteristic
294C23C2×#12011yescharacteristic
295C23C2×#12311yescharacteristic
296Q8—216no—
297Q8—216no—
298Q8—82no—
299Q8—84no—
300Q8—82no—
301C2×C4—28no—
302C2×C4—28no—
303C2×C4—22no—
304C2×C4—24no—
305C2×C4—24no—
306C2×C4—24no—
307C2×C4—24no—
308C2×C4—24no—
309C2×C4—22no—
310C2×C4—22no—
311C2×C4—22no—
312C2×C4—216no—
313C2×C4—24no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—42no—
320C2×C4—48no—
321C2×C4—48no—
322C2×C4—44no—
323C2×C4—44no—
324C2×C4—44no—
325C2×C4—44no—
326C2×C4—44no—
327C2×C4—44no—
328C2×C4—44no—
329C2×C4—42no—
330C2×C4—42no—
331C2×C4—82no—
332C2×C4#38611yescharacteristic
333C2×C4#38611yescharacteristic
334C2×C4#38611yescharacteristic
335C2×C4#38612yes—
336C2×C4#38612yes—
337C2×C4#38611yescharacteristic
338C2×C4C2×#10418yes—
339C2×C4C2×#10414yes—
340C2×C4C2×C4×D412yes—
341C2×C4C2×#12112yes—
342C8—84no—
343C22—21no—
344C22—21no—
345C22—21no—
346C22—21no—
347C22—21no—
348C22—21no—
349C22—21no—
350C22—44no—
351C22C2×#38611yescharacteristic
352C22C22×#10412yes—
353C22#302812yes—
354C4—21no—
355C4—21no—
356C4—21no—
357C4—22no—
358C4—22no—
359C4—22no—
360C4—28no—
361C4—416no—
362C4—44no—
363C4—84no—
364C4—82no—
365C4—82no—
366C4#189614yes—
367C4#191212yes—
368C2—22no—
369C2—22no—
370C2—41no—
371C2#2096211yescharacteristic
372C2C2×#189611yescharacteristic
373C2C2×#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
12. #2634C4semidirect
14. #2568C4semidirect
15. #2546C4semidirect

Maximal subgroups

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

subgroupindex
#352482
C4×#5192
#218442
#173652
#94782
#79412