Finite Group Databasebeta

← #201497

Subgroups — #201497

Order 512.

1279 conjugacy classes of subgroups.

The subgroup lattice has 392 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35249C211yesmaximal, characteristic
2#34059C211yesmaximal, characteristic, complemented
3#21815C212yesmaximal
4#17365C211yesmaximal, characteristic, complemented
5#9480C218yesmaximal, complemented
6#7941C212yesmaximal, complemented
7C2×#532C2211yescharacteristic
8C22×C4×Q8C2211yescharacteristic
9#2920C2212yes—
10#2854—22no—
11#2632C414yescomplemented
12#2580—22no—
13#2567C412yescomplemented
14#2548C412yescomplemented
15#2501—21no—
16#2436—21no—
17#2406C2214yes—
18#2326—22no—
19#2320—24no—
20#2316—21no—
21#2313C2214yes—
22#2310—21no—
23#2295—24no—
24#2283—22no—
25#2283C2211yescharacteristic
26C4×#117C2212yes—
27#1980—24no—
28#1962—24no—
29#1789C2218yes—
30#1467—22no—
31#1463C2214yes—
32#1461—22no—
33#1284C2218yes—
34C22×#41C2312yes—
35C23×Q8D411yescharacteristic
36#532—22no—
37#532—22no—
38#532—22no—
39#532—22no—
40C2×C4⋊Q8C2×C411yescharacteristic
41C2×#127C2×C411yescharacteristic
42C2×#124—22no—
43C2×#124C2×C412yes—
44C2×#123—24no—
45C2×#121—21no—
46C2×#121—21no—
47C2×C4×Q8—22no—
48C2×C4×Q8—22no—
49C2×C4×Q8—22no—
50C2×C4×Q8—22no—
51C2×C4×Q8C2×C412yes—
52C2×C4×Q8C2×C412yes—
53C2×C4×Q8D412yes—
54C2×C4×Q8D412yes—
55C2×C4×D4—21no—
56C2×C4×D4—21no—
57C2×#117—22no—
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—
67C2×#104—216no—
68#407C2314yes—
69C4×M16—22no—
70#399—24no—
71#395C2×C418yes—
72#394C2×C418yes—
73#393—24no—
74#391—28no—
75#391—22no—
76#390—22no—
77#390—24no—
78#389—28no—
79#389—24no—
80#388—28no—
81#387—216no—
82#386—24no—
83#385—28no—
84#384—22no—
85#383—24no—
86#383C2×C412yes—
87#382—24no—
88#382—21no—
89#381—24no—
90#381C2312yes—
91#380—24no—
92#380—21no—
93C4×C4⋊C4—24no—
94C4×C4⋊C4—22no—
95C4×C4⋊C4C2×C412yes—
96C4×C4⋊C4D414yes—
97C4×C22⋊C4—21no—
98C4×C22⋊C4—22no—
99C4×C22⋊C4C2314yes—
100#376—24no—
101#376—22no—
102#376—21no—
103C43D414yes—
104#324—28no—
105C2×#41—24no—
106C2×#41—24no—
107C2×#41—84no—
108C22×Q8—22no—
109C22×Q8—22no—
110C22×Q8—44no—
111C22×Q8C22⋊C412yes—
112C22×Q8C22⋊C412yes—
113C22×Q8C22×C412yes—
114C22×D4—22no—
115C22×D4—21no—
116C22×D4—21no—
117C22×D4C22×C412yes—
118C23×C4—22no—
119C23×C4C2×D411yescharacteristic
120C23×C4C2×D412yes—
121C23×C4C2×D412yes—
122C23×C4C2×D411yescharacteristic
123C23×C4C2411yescharacteristic
124C2×M16—24no—
125C4⋊Q8—22no—
126C4⋊Q8—22no—
127#127—22no—
128#127—22no—
129#124—24no—
130#124—24no—
131#124—48no—
132#123—88no—
133#121—41no—
134#121—42no—
135#121—41no—
136#121—41no—
137#121—42no—
138#121—41no—
139C4×Q8—24no—
140C4×Q8—24no—
141C4×Q8—24no—
142C4×Q8—24no—
143C4×Q8—24no—
144C4×Q8—24no—
145C4×Q8—24no—
146C4×Q8—24no—
147C4×D4—42no—
148C4×D4—42no—
149C4×D4—42no—
150C4×D4—42no—
151#117—48no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—24no—
155C2×C4⋊C4—24no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—24no—
160C2×C4⋊C4—28no—
161C2×C4⋊C4—28no—
162C2×C4⋊C4C22⋊C412yes—
163C2×C4⋊C4C22⋊C412yes—
164C2×C4⋊C4C22×C414yes—
165C2×C4⋊C4C2×D414yes—
166C2×C22⋊C4—24no—
167C2×C22⋊C4—22no—
168C2×C22⋊C4—22no—
169C2×C22⋊C4—22no—
170C2×C22⋊C4—28no—
171C2×C22⋊C4—28no—
172C2×C42—22no—
173C2×C42—22no—
174C2×C42—22no—
175C2×C42—22no—
176C2×C42—22no—
177C2×C42—24no—
178C2×C42—24no—
179C2×C42C22⋊C414yes—
180C2×C42C22⋊C414yes—
181C2×C42C22×C411yescharacteristic
182C2×C42C22×C411yescharacteristic
183C2×C42C2×D418yes—
184C2×C42C2×D412yes—
185C2×C42C2×D412yes—
186C2×C42C2×D411yescharacteristic
187C2×C42C2×D411yescharacteristic
188C2×C42#4112yes—
189C2×C42#4112yes—
190C4⋊C8—42no—
191C4⋊C8—42no—
192#104—816no—
193C22⋊C8—24no—
194C22⋊C8—24no—
195#97—42no—
196C4×C8—42no—
197#95—28no—
198#95—28no—
199#95—28no—
200#95—22no—
201#95—22no—
202#95—28no—
203#95—24no—
204#95—24no—
205#95—22no—
206#95C22×C414yes—
207C24—22no—
208C24C22×D411yescharacteristic
209#41—416no—
210#41—88no—
211C2×Q8—24no—
212C2×Q8—24no—
213C2×Q8—24no—
214C2×Q8—24no—
215C2×Q8—24no—
216C2×Q8—44no—
217C2×Q8—48no—
218C2×Q8—48no—
219C2×Q8—82no—
220C2×D4—24no—
221C2×D4—44no—
222C2×D4—48no—
223C2×D4—42no—
224C2×D4—42no—
225C2×D4—41no—
226C2×D4—42no—
227C2×D4—41no—
228C2×D4—44no—
229C2×D4—82no—
230C22×C4—24no—
231C22×C4—22no—
232C22×C4—22no—
233C22×C4—22no—
234C22×C4—22no—
235C22×C4—24no—
236C22×C4—22no—
237C22×C4—28no—
238C22×C4—42no—
239C22×C4—44no—
240C22×C4—42no—
241C22×C4—44no—
242C22×C4—44no—
243C22×C4—84no—
244C22×C4C2×C22⋊C414yes—
245C22×C4C2×C22⋊C411yescharacteristic
246C22×C4C2×C22⋊C411yescharacteristic
247C22×C4C2×C22⋊C414yes—
248C22×C4C2×C22⋊C411yescharacteristic
249C22×C4C2×C22⋊C411yescharacteristic
250C22×C4#12011yescharacteristic
251C22×C4#12012yes—
252C22×C4#12011yescharacteristic
253C22×C4#12311yescharacteristic
254C22×C4#12312yes—
255C22×C4#12311yescharacteristic
256C22×C4C23×C411yescharacteristic
257C22×C4C22×D412yes—
258C22×C4C22×D411yescharacteristic
259C22×C4C2×#4112yes—
260M16—48no—
261C2×C8—44no—
262C4⋊C4—28no—
263C4⋊C4—28no—
264C4⋊C4—24no—
265C4⋊C4—24no—
266C4⋊C4—44no—
267C4⋊C4—48no—
268C4⋊C4—41no—
269C4⋊C4—42no—
270C4⋊C4—41no—
271C4⋊C4—88no—
272C4⋊C4—88no—
273C22⋊C4—416no—
274C22⋊C4—48no—
275C22⋊C4—44no—
276C22⋊C4—42no—
277C22⋊C4—42no—
278C22⋊C4—44no—
279C22⋊C4—88no—
280C42—24no—
281C42—24no—
282C42—24no—
283C42—24no—
284C42—48no—
285C42—48no—
286C42—48no—
287C42—44no—
288C42C4×D412yes—
289C42C4×D412yes—
290C42C4×D412yes—
291C42C4×D412yes—
292C42#12112yes—
293C42#12112yes—
294C42#12112yes—
295C42#12112yes—
296C23—22no—
297C23—22no—
298C23—41no—
299C23—44no—
300C23—44no—
301C23—44no—
302C23—42no—
303C23C22×C22⋊C411yescharacteristic
304C23C2×#12011yescharacteristic
305C23C2×#12311yescharacteristic
306Q8—48no—
307Q8—48no—
308Q8—48no—
309D4—48no—
310D4—82no—
311D4—84no—
312D4—84no—
313D4—84no—
314D4—82no—
315C2×C4—22no—
316C2×C4—24no—
317C2×C4—24no—
318C2×C4—24no—
319C2×C4—22no—
320C2×C4—22no—
321C2×C4—22no—
322C2×C4—24no—
323C2×C4—24no—
324C2×C4—24no—
325C2×C4—24no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—42no—
332C2×C4—44no—
333C2×C4—44no—
334C2×C4—44no—
335C2×C4—44no—
336C2×C4—48no—
337C2×C4—48no—
338C2×C4—44no—
339C2×C4—44no—
340C2×C4—44no—
341C2×C4—44no—
342C2×C4—48no—
343C2×C4—42no—
344C2×C4—88no—
345C2×C4—82no—
346C2×C4#38611yescharacteristic
347C2×C4#38611yescharacteristic
348C2×C4#38611yescharacteristic
349C2×C4#38612yes—
350C2×C4#38612yes—
351C2×C4#38611yescharacteristic
352C2×C4#42114yes—
353C2×C4C2×C4×D412yes—
354C2×C4C2×#12112yes—
355C8—84no—
356C22—21no—
357C22—21no—
358C22—21no—
359C22—21no—
360C22—21no—
361C22—21no—
362C22—21no—
363C22—44no—
364C22—88no—
365C22—82no—
366C22—82no—
367C22—82no—
368C22C2×#38611yescharacteristic
369C22C2×#42112yes—
370C22#302712yes—
371C4—21no—
372C4—21no—
373C4—21no—
374C4—22no—
375C4—22no—
376C4—22no—
377C4—44no—
378C4—44no—
379C4—88no—
380C4—84no—
381C4—82no—
382C4#189714yes—
383C4#191212yes—
384C2—22no—
385C2—22no—
386C2—41no—
387C2—162no—
388C2#2096111yescharacteristic
389C2C2×#189711yescharacteristic
390C2C2×#191211yescharacteristic
subgroupG/H
center, socleC22C2×#386
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC2×#532C22
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
2. #34059C2semidirect
4. #17365C2semidirect
5. #9480C2semidirect
6. #7941C2semidirect
11. #2632C4semidirect
13. #2567C4semidirect
14. #2548C4semidirect

Maximal subgroups

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

subgroupindex
#352492
#340592
#218152
#173652
#94802
#79412