Finite Group Databasebeta

← #480959

Subgroups — #480959

Order 512.

881 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36180C211yesmaximal, characteristic
2#17426C212yesmaximal
3#11574C212yesmaximal, complemented
4#11555C214yesmaximal, complemented
5#11532C211yesmaximal, characteristic, complemented
6#11512C212yesmaximal, complemented
7#11478C211yesmaximal, characteristic, complemented
8C2×#1325C212yesmaximal, complemented
9C2×#554C411yescharacteristic
10C2×#554C2211yescharacteristic
11#2974—22no—
12#2868C411yescharacteristic
13#2857C412yes—
14#2567C411yescharacteristic
15#2567C2211yescharacteristic
16#2552—22no—
17#2502C412yes—
18#2502C2212yes—
19#2440—24no—
20D4×C4⋊C4—22no—
21C4×C4⋊Q8C2211yescharacteristic
22C2×#384C2212yes—
23C42×D4C411yescharacteristic
24#1472—24no—
25#1470—22no—
26#1467C2212yes—
27#1463C2214yes—
28#1461C2212yes—
29C2×#324C2214yes—
30#1325C2212yes—
31#1325C2214yes—
32#1325C2214yes—
33#1325C2211yescharacteristic
34#1325C2211yescharacteristic
35#1319C2212yes—
36#1319C2212yes—
37#554—41no—
38#554—42no—
39#554—41no—
40#554—41no—
41#554—42no—
42#554—41no—
43C2×C4⋊Q8—22no—
44C2×C4⋊Q8C2×C411yescharacteristic
45C2×C4⋊Q8C2×C411yescharacteristic
46C2×C4⋊Q8C2×C411yescharacteristic
47C2×#125—21no—
48C2×#125D412yes—
49C2×C22⋊Q8—22no—
50C2×C22⋊Q8C2×C412yes—
51C2×C22⋊Q8C2×C412yes—
52C2×C22⋊Q8D414yes—
53C2×C4×Q8C2×C411yescharacteristic
54C2×C4×Q8C2311yescharacteristic
55C2×C4×D4—21no—
56C2×C4×D4—21no—
57C2×C4×D4C2×C411yescharacteristic
58C2×C4×D4D412yes—
59C22×C4⋊C4—24no—
60C22×C4⋊C4C2×C412yes—
61C22×C4⋊C4C2312yes—
62C22×C42C2×C412yes—
63#436—42no—
64#435—44no—
65C8×D4—42no—
66#432—22no—
67#431—22no—
68#407—24no—
69C2×C22⋊C8—24no—
70#398—24no—
71#398—24no—
72#395—22no—
73#395—22no—
74#391C2×C411yescharacteristic
75#391C2×C412yes—
76#391C2×C412yes—
77#391C2311yescharacteristic
78#389—22no—
79#389—24no—
80#389—22no—
81#389—22no—
82#385—24no—
83#385—24no—
84#384—22no—
85#384—24no—
86#384C2×C412yes—
87#384C2×C412yes—
88#384C2×C412yes—
89#384C2311yescharacteristic
90#384C2312yes—
91#380—24no—
92C4×C4⋊C4—21no—
93C4×C4⋊C4—22no—
94C4×C4⋊C4C2×C411yescharacteristic
95C4×C22⋊C4—22no—
96C4×C22⋊C4C2×C412yes—
97C43C2×C411yescharacteristic
98#324—24no—
99#324—22no—
100#324—24no—
101#324—22no—
102#324C2312yes—
103#324C2314yes—
104#324C2312yes—
105C22×Q8—21no—
106C22×Q8C22⋊C412yes—
107C22×Q8C22×C411yescharacteristic
108C22×Q8C22×C411yescharacteristic
109C22×D4C22⋊C411yescharacteristic
110C23×C4—22no—
111C23×C4—22no—
112C23×C4C22×C412yes—
113C23×C4C2×D414yes—
114C2×M16—44no—
115C22×C8—44no—
116C4⋊Q8—22no—
117C4⋊Q8—41no—
118C4⋊Q8—41no—
119C4⋊Q8—41no—
120C4⋊Q8—41no—
121C4⋊Q8—41no—
122C4⋊Q8—84no—
123#125—42no—
124#125—42no—
125#125—42no—
126#125—42no—
127C22⋊Q8—42no—
128C22⋊Q8—42no—
129C22⋊Q8—44no—
130C22⋊Q8—44no—
131C22⋊Q8—44no—
132C22⋊Q8—44no—
133C22⋊Q8—42no—
134C22⋊Q8—42no—
135C4×Q8—41no—
136C4×Q8—41no—
137C4×Q8—41no—
138C4×Q8—41no—
139C4×D4—28no—
140C4×D4—22no—
141C4×D4—21no—
142C4×D4—21no—
143C4×D4—22no—
144C4×D4—24no—
145C4×D4—22no—
146C4×D4—42no—
147C4×D4—41no—
148C4×D4—41no—
149C2×C4⋊C4—21no—
150C2×C4⋊C4—24no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—21no—
153C2×C4⋊C4—21no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—24no—
158C2×C4⋊C4—24no—
159C2×C4⋊C4—24no—
160C2×C4⋊C4—42no—
161C2×C4⋊C4—42no—
162C2×C4⋊C4—42no—
163C2×C4⋊C4—42no—
164C2×C4⋊C4—88no—
165C2×C4⋊C4C22⋊C411yescharacteristic
166C2×C4⋊C4C22⋊C412yes—
167C2×C4⋊C4C22⋊C414yes—
168C2×C4⋊C4C22×C411yescharacteristic
169C2×C4⋊C4C22×C412yes—
170C2×C4⋊C4C22×C412yes—
171C2×C4⋊C4C2×D414yes—
172C2×C4⋊C4C2×D412yes—
173C2×C4⋊C4C2411yescharacteristic
174C2×C22⋊C4—22no—
175C2×C22⋊C4—22no—
176C2×C22⋊C4C22⋊C412yes—
177C2×C22⋊C4C22⋊C414yes—
178C2×C42—21no—
179C2×C42—24no—
180C2×C42—21no—
181C2×C42—21no—
182C2×C42—22no—
183C2×C42—42no—
184C2×C42C22×C412yes—
185C2×C42C22×C411yescharacteristic
186C2×C42C22×C411yescharacteristic
187C2×C42C22×C411yescharacteristic
188C2×C42C2×D412yes—
189C4⋊C8—42no—
190C4⋊C8—42no—
191C22⋊C8—22no—
192C22⋊C8—24no—
193C22⋊C8—22no—
194C22⋊C8—44no—
195C22⋊C8—44no—
196#97—42no—
197C4×C8—42no—
198#95—21no—
199#95—22no—
200#95—22no—
201#95—24no—
202#95—24no—
203#95—22no—
204C24C2×C22⋊C412yes—
205C2×Q8—21no—
206C2×Q8—21no—
207C2×Q8—21no—
208C2×Q8—21no—
209C2×Q8—41no—
210C2×Q8—41no—
211C2×Q8—42no—
212C2×Q8—42no—
213C2×Q8—41no—
214C2×Q8—41no—
215C2×Q8—82no—
216C2×Q8—82no—
217C2×D4—24no—
218C2×D4#9912yes—
219C2×D4#9911yescharacteristic
220C2×D4#9911yescharacteristic
221C22×C4—21no—
222C22×C4—28no—
223C22×C4—21no—
224C22×C4—22no—
225C22×C4—22no—
226C22×C4—22no—
227C22×C4—24no—
228C22×C4—24no—
229C22×C4—22no—
230C22×C4—22no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—22no—
234C22×C4—42no—
235C22×C4—44no—
236C22×C4—44no—
237C22×C4—42no—
238C22×C4—42no—
239C22×C4—84no—
240C22×C4—84no—
241C22×C4C2×C22⋊C412yes—
242C22×C4C2×C22⋊C412yes—
243C22×C4C2×C22⋊C411yescharacteristic
244C22×C4C2×C22⋊C411yescharacteristic
245C22×C4C2×C22⋊C414yes—
246C22×C4C23×C411yescharacteristic
247C22×C4C22×D412yes—
248M16—164no—
249C2×C8—42no—
250C2×C8—42no—
251C2×C8—44no—
252C2×C8—164no—
253C4⋊C4—24no—
254C4⋊C4—21no—
255C4⋊C4—21no—
256C4⋊C4—22no—
257C4⋊C4—22no—
258C4⋊C4—41no—
259C4⋊C4—42no—
260C4⋊C4—44no—
261C4⋊C4—44no—
262C4⋊C4—42no—
263C4⋊C4—44no—
264C4⋊C4—41no—
265C4⋊C4—42no—
266C4⋊C4—44no—
267C4⋊C4—44no—
268C4⋊C4—48no—
269C4⋊C4—42no—
270C4⋊C4—44no—
271C4⋊C4—41no—
272C4⋊C4—41no—
273C4⋊C4—84no—
274C4⋊C4—84no—
275C4⋊C4—84no—
276C22⋊C4—28no—
277C22⋊C4—24no—
278C22⋊C4—24no—
279C22⋊C4—22no—
280C22⋊C4—22no—
281C22⋊C4—42no—
282C22⋊C4—42no—
283C42—24no—
284C42—21no—
285C42—22no—
286C42—22no—
287C42—21no—
288C42—21no—
289C42—22no—
290C42—22no—
291C42—44no—
292C42—41no—
293C42—42no—
294C42—41no—
295C42—41no—
296C42#13112yes—
297C23—22no—
298C23—22no—
299C23—24no—
300C23—42no—
301C23C2×#9912yes—
302C23C2×#9912yes—
303C23C22×C22⋊C411yescharacteristic
304Q8—81no—
305Q8—81no—
306Q8—81no—
307Q8—81no—
308Q8—162no—
309Q8—162no—
310D4—24no—
311D4—22no—
312D4#35614yes—
313C2×C4—22no—
314C2×C4—22no—
315C2×C4—22no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—22no—
319C2×C4—22no—
320C2×C4—24no—
321C2×C4—22no—
322C2×C4—22no—
323C2×C4—21no—
324C2×C4—22no—
325C2×C4—22no—
326C2×C4—22no—
327C2×C4—24no—
328C2×C4—22no—
329C2×C4—22no—
330C2×C4—42no—
331C2×C4—42no—
332C2×C4—42no—
333C2×C4—42no—
334C2×C4—416no—
335C2×C4—41no—
336C2×C4—42no—
337C2×C4—42no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—42no—
341C2×C4—44no—
342C2×C4—42no—
343C2×C4—44no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—42no—
348C2×C4—41no—
349C2×C4—84no—
350C2×C4—84no—
351C2×C4—162no—
352C2×C4—162no—
353C2×C4C2×#9911yescharacteristic
354C2×C4C2×#9911yescharacteristic
355C2×C4C2×#13111yescharacteristic
356C2×C4#56811yescharacteristic
357C8—82no—
358C8—84no—
359C8—82no—
360C22—22no—
361C22—21no—
362C22—22no—
363C22—21no—
364C22—22no—
365C22—22no—
366C22—22no—
367C22—44no—
368C22—44no—
369C22C2×#35614yes—
370C22#288011yescharacteristic
371C22C22×#9911yescharacteristic
372C22#288911yescharacteristic
373C4—24no—
374C4—21no—
375C4—21no—
376C4—21no—
377C4—42no—
378C4—44no—
379C4—42no—
380C4—42no—
381C4—42no—
382C4—44no—
383C4—41no—
384C4—81no—
385C4—81no—
386C4—82no—
387C4C2×#35612yes—
388C2—21no—
389C2—21no—
390C2—24no—
391C2—42no—
392C2#1741611yescharacteristic
393C2#3335911yescharacteristic
394C2C22×#35611yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC22×C4C23×C4
Frattini, cosocleC2×C4⋊C4C24
prime radicalC22×D4C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×D4 ◁ C2×#554 ◁ G
LECSG ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #11574C2semidirect
4. #11555C2semidirect
5. #11532C2semidirect
6. #11512C2semidirect
7. #11478C2semidirect
8. C2×#1325C2semidirect

Maximal subgroups

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

subgroupindex
#361802
#174262
#115742
#115552
#115322
#115122
#114782
C2×#13252