Finite Group Databasebeta

← #480962

Subgroups — #480962

Order 512.

899 conjugacy classes of subgroups.

The subgroup lattice has 389 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#39488C211yesmaximal, characteristic
2#17416C212yesmaximal
3#11570C212yesmaximal, complemented
4#11555C214yesmaximal, complemented
5#11531C211yesmaximal, characteristic, complemented
6#11514C212yesmaximal, complemented
7#11477C211yesmaximal, characteristic, complemented
8#11448C212yesmaximal, complemented
9C2×#556C411yescharacteristic
10C2×#547C2211yescharacteristic
11#2880—22no—
12#2868C411yescharacteristic
13#2718—24no—
14#2620C412yes—
15#2568C411yescharacteristic
16#2542C2211yescharacteristic
17#2502C2212yes—
18#2500C412yes—
19#2432—22no—
20#2429—22no—
21#2317C2212yes—
22C4×#124C2211yescharacteristic
23#2293C411yescharacteristic
24#1472—22no—
25#1468—24no—
26#1465C2212yes—
27#1463C2214yes—
28#1459C2212yes—
29C2×#324C2214yes—
30#1325C2214yes—
31#1325C2214yes—
32#1324C2212yes—
33#1324C2211yescharacteristic
34#1324C2211yescharacteristic
35#1319C2212yes—
36#1319C2212yes—
37C22×#41D412yes—
38C2×#131—42no—
39#556—42no—
40#556—42no—
41#547—42no—
42#547—42no—
43C2×C4⋊Q8—22no—
44C2×C4⋊Q8C2×C411yescharacteristic
45C2×C4⋊Q8C2×C411yescharacteristic
46C2×#126—22no—
47C2×#124—21no—
48C2×#124C2×C411yescharacteristic
49C2×#123D414yes—
50C2×C22⋊Q8—22no—
51C2×C22⋊Q8C2×C412yes—
52C2×C22⋊Q8D412yes—
53C2×#121C2×C412yes—
54C2×C4×Q8—21no—
55C2×C4×Q8C2×C411yescharacteristic
56C2×C4×D4C2×C411yescharacteristic
57C2×C4×D4C2311yescharacteristic
58C2×#117C2×C412yes—
59C2×#117C2×C412yes—
60C22×C4⋊C4C2312yes—
61#408—42no—
62#408—44no—
63#407—24no—
64#407—22no—
65C2×C22⋊C8—24no—
66C2×C22⋊C8—22no—
67#400—22no—
68#399—22no—
69#398—24no—
70#396—24no—
71#394—24no—
72#394—24no—
73#391—22no—
74#391C2×C412yes—
75#391C2311yescharacteristic
76#390—21no—
77#390C2×C412yes—
78#390C2×C411yescharacteristic
79#389—21no—
80#389—24no—
81#388—24no—
82#388—22no—
83#388—22no—
84#385—24no—
85#385—24no—
86#384—22no—
87#384C2×C412yes—
88#384C2×C412yes—
89#384C2312yes—
90#382C2×C412yes—
91#382C2311yescharacteristic
92C4×C4⋊C4C2×C411yescharacteristic
93C4×C22⋊C4—22no—
94C4×C22⋊C4C2×C412yes—
95C43C2×C411yescharacteristic
96#324—24no—
97#324—24no—
98#324C2312yes—
99#324C2314yes—
100#324C2312yes—
101#323—22no—
102#323—22no—
103C2×#41—22no—
104C2×#41—24no—
105C2×#41—22no—
106C2×#41—84no—
107C22×Q8—21no—
108C22×Q8C22⋊C412yes—
109C22×Q8C22×C411yescharacteristic
110C22×D4C22⋊C414yes—
111C22×D4C22⋊C411yescharacteristic
112C22×D4C22×C411yescharacteristic
113C23×C4C22×C412yes—
114C23×C4C2×D412yes—
115C23×C4C2×D414yes—
116#131—88no—
117C2×M16—44no—
118C2×M16—42no—
119C22×C8—44no—
120C22×C8—42no—
121C4⋊Q8—41no—
122C4⋊Q8—41no—
123C4⋊Q8—41no—
124C4⋊Q8—41no—
125C4⋊Q8—84no—
126#126—48no—
127#124—22no—
128#124—41no—
129#124—42no—
130#124—42no—
131#123—44no—
132#123—44no—
133C22⋊Q8—42no—
134C22⋊Q8—42no—
135C22⋊Q8—44no—
136C22⋊Q8—84no—
137#121—44no—
138C4×Q8—41no—
139C4×Q8—41no—
140C4×Q8—42no—
141C4×Q8—42no—
142C4×D4—22no—
143C4×D4—22no—
144C4×D4—41no—
145C4×D4—41no—
146#117—24no—
147#117—42no—
148#117—44no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—21no—
152C2×C4⋊C4—21no—
153C2×C4⋊C4—21no—
154C2×C4⋊C4—24no—
155C2×C4⋊C4—24no—
156C2×C4⋊C4—24no—
157C2×C4⋊C4—44no—
158C2×C4⋊C4C22⋊C411yescharacteristic
159C2×C4⋊C4C22⋊C412yes—
160C2×C4⋊C4C22×C411yescharacteristic
161C2×C4⋊C4C22×C412yes—
162C2×C4⋊C4C22×C412yes—
163C2×C4⋊C4C2×D414yes—
164C2×C4⋊C4C2411yescharacteristic
165C2×C22⋊C4—22no—
166C2×C22⋊C4—22no—
167C2×C22⋊C4—22no—
168C2×C22⋊C4—22no—
169C2×C22⋊C4—22no—
170C2×C22⋊C4—22no—
171C2×C22⋊C4—24no—
172C2×C22⋊C4C22⋊C412yes—
173C2×C22⋊C4C22⋊C414yes—
174C2×C22⋊C4C2×D412yes—
175C2×C42—22no—
176C2×C42—21no—
177C2×C42—21no—
178C2×C42—21no—
179C2×C42C22×C412yes—
180C2×C42C22×C411yescharacteristic
181C2×C42C22×C411yescharacteristic
182C2×C42C22×C411yescharacteristic
183C22⋊C8—22no—
184C22⋊C8—22no—
185C22⋊C8—24no—
186C22⋊C8—44no—
187C22⋊C8—42no—
188C22⋊C8—44no—
189C22⋊C8—42no—
190#95—22no—
191#95—21no—
192#95—24no—
193#95—24no—
194#95—22no—
195#95—22no—
196C24C2×C22⋊C412yes—
197C24C2×C22⋊C412yes—
198#41—48no—
199#41—84no—
200#41—88no—
201C2×Q8—22no—
202C2×Q8—22no—
203C2×Q8—21no—
204C2×Q8—21no—
205C2×Q8—44no—
206C2×Q8—44no—
207C2×Q8—41no—
208C2×Q8—41no—
209C2×Q8—41no—
210C2×Q8—41no—
211C2×D4—24no—
212C2×D4—24no—
213C2×D4—21no—
214C2×D4—21no—
215C2×D4—42no—
216C2×D4—41no—
217C2×D4—41no—
218C2×D4—84no—
219C2×D4#9912yes—
220C2×D4#9912yes—
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—24no—
224C22×C4—24no—
225C22×C4—24no—
226C22×C4—22no—
227C22×C4—22no—
228C22×C4—22no—
229C22×C4—22no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—22no—
233C22×C4—22no—
234C22×C4—42no—
235C22×C4—44no—
236C22×C4—84no—
237C22×C4C2×C22⋊C412yes—
238C22×C4C2×C22⋊C411yescharacteristic
239C22×C4C2×C22⋊C414yes—
240C22×C4C2×C22⋊C411yescharacteristic
241C22×C4C23×C411yescharacteristic
242C22×C4C22×D412yes—
243M16—88no—
244M16—84no—
245C2×C8—42no—
246C2×C8—42no—
247C2×C8—44no—
248C2×C8—88no—
249C2×C8—84no—
250C4⋊C4—21no—
251C4⋊C4—21no—
252C4⋊C4—41no—
253C4⋊C4—42no—
254C4⋊C4—44no—
255C4⋊C4—42no—
256C4⋊C4—44no—
257C4⋊C4—41no—
258C4⋊C4—42no—
259C4⋊C4—44no—
260C4⋊C4—42no—
261C4⋊C4—42no—
262C4⋊C4—42no—
263C4⋊C4—84no—
264C4⋊C4—84no—
265C4⋊C4—84no—
266C22⋊C4—24no—
267C22⋊C4—44no—
268C22⋊C4—42no—
269C22⋊C4—44no—
270C22⋊C4—44no—
271C22⋊C4—44no—
272C22⋊C4—44no—
273C22⋊C4—44no—
274C22⋊C4—44no—
275C22⋊C4—84no—
276C42—21no—
277C42—21no—
278C42—21no—
279C42—22no—
280C42—22no—
281C42—41no—
282C42—42no—
283C42—44no—
284C42—42no—
285C42—41no—
286C42—41no—
287C42#13112yescomplemented
288C23—22no—
289C23—22no—
290C23—22no—
291C23—22no—
292C23—22no—
293C23—42no—
294C23—42no—
295C23—42no—
296C23—42no—
297C23C2×#9914yes—
298C23C22×C22⋊C411yescharacteristic
299Q8—44no—
300Q8—82no—
301Q8—81no—
302Q8—81no—
303Q8—82no—
304Q8—82no—
305D4—24no—
306D4—42no—
307D4—48no—
308D4—81no—
309D4—81no—
310D4—84no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—22no—
314C2×C4—22no—
315C2×C4—22no—
316C2×C4—22no—
317C2×C4—21no—
318C2×C4—22no—
319C2×C4—22no—
320C2×C4—24no—
321C2×C4—22no—
322C2×C4—44no—
323C2×C4—44no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—48no—
328C2×C4—41no—
329C2×C4—44no—
330C2×C4—41no—
331C2×C4—42no—
332C2×C4—44no—
333C2×C4—42no—
334C2×C4—42no—
335C2×C4—42no—
336C2×C4—44no—
337C2×C4—42no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—42no—
342C2×C4—41no—
343C2×C4—82no—
344C2×C4—84no—
345C2×C4C2×#9911yescharacteristic
346C2×C4C2×#9911yescharacteristic
347C2×C4C2×#13111yescharacteristic
348C2×C4#56811yescharacteristic
349C8—82no—
350C8—84no—
351C8—82no—
352C22—21no—
353C22—21no—
354C22—24no—
355C22—24no—
356C22—24no—
357C22—42no—
358C22—42no—
359C22—42no—
360C22—82no—
361C22—82no—
362C22—82no—
363C22—82no—
364C22#288011yescharacteristic
365C22C22×#9911yescharacteristic
366C22#288911yescharacteristic
367C4—21no—
368C4—21no—
369C4—21no—
370C4—42no—
371C4—42no—
372C4—42no—
373C4—42no—
374C4—44no—
375C4—41no—
376C4—82no—
377C4—81no—
378C4—82no—
379C4—81no—
380C4#212712yes—
381C2—21no—
382C2—21no—
383C2—44no—
384C2—82no—
385C2#1741611yescharacteristic
386C2C2×#212711yescharacteristic
387C2#3337211yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC22×C4C23×C4
Frattini, cosocleC2×C4⋊C4C24
prime radicalC2×#547C22
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×D4 ◁ C2×#547 ◁ G
LECSG ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #11570C2semidirect
4. #11555C2semidirect
5. #11531C2semidirect
6. #11514C2semidirect
7. #11477C2semidirect
8. #11448C2semidirect
287. C42#131semidirect

Maximal subgroups

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

subgroupindex
#394882
#174162
#115702
#115552
#115312
#115142
#114772
#114482