Finite Group Databasebeta

← #482168

Subgroups — #482168

Order 512.

843 conjugacy classes of subgroups.

The subgroup lattice has 395 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35749C211yesmaximal, characteristic
2#20691C211yesmaximal, characteristic
3#17352C211yesmaximal, characteristic
4#11846C214yesmaximal
5#11842C214yesmaximal
6#11715C212yesmaximal
7#11694C212yesmaximal
8C2×#557C411yescharacteristic, complemented
9C2×#519C2211yescharacteristic
10#2982—21no—
11#2634C412yescomplemented
12#2621C2211yescharacteristic
13#2611C2212yes—
14#2599C2212yes—
15#2581C411yescharacteristic, complemented
16#2546C411yescharacteristic, complemented
17#2475—22no—
18#2462—22no—
19#2446—21no—
20#2434—21no—
21#2399C412yescomplemented
22#2396C2211yescharacteristic
23#2365—21no—
24#2362—21no—
25#2356C411yescharacteristic, complemented
26#2310—21no—
27#1820—22no—
28#1820C2214yes—
29#1740—21no—
30#1740C2212yes—
31#1461—21no—
32#1461—22no—
33#1458C2214yes—
34#1458C2212yes—
35#1352C2214yes—
36#1351C2214yes—
37#1344C2218yes—
38C23×Q8D411yescharacteristic
39#557—44no—
40C2×C4⋊Q8—21no—
41C2×C4⋊Q8—21no—
42C2×C4⋊Q8C2×C411yescharacteristic
43C2×#125—22no—
44C2×#125C2×C412yes—
45C2×C22⋊Q8—21no—
46#519—44no—
47C2×C4×Q8—21no—
48C2×C4×Q8C2×C411yescharacteristic
49C2×C4×Q8C2×C412yes—
50C2×C4×Q8C2×C411yescharacteristic
51C2×C4×Q8C2×C412yes—
52C2×C4×Q8C2×C411yescharacteristic
53C2×C4×Q8D411yescharacteristic
54C2×C4×Q8D412yes—
55C2×C4×Q8D411yescharacteristic
56C2×#117—22no—
57C2×#117D412yes—
58C2×#117D411yescharacteristic
59C2×#117C2312yes—
60C2×#117C2311yescharacteristic
61C8×Q8—44no—
62#431—22no—
63#431—24no—
64#400—22no—
65#400C2×C412yes—
66#400C2×C411yescharacteristic
67#399C2311yescharacteristic
68#398—22no—
69#398—21no—
70#398—21no—
71#398C2×C412yes—
72#398C2×C412yes—
73#398C2×C412yes—
74#397—21no—
75#397C2312yes—
76#396—21no—
77#395—22no—
78#395—22no—
79#395C2×C412yes—
80#395C2×C411yescharacteristic
81#393C2311yescharacteristic
82#391—21no—
83#391—21no—
84#391—21no—
85#391—21no—
86#391—44no—
87#389—21no—
88#389—22no—
89#389—22no—
90#388—22no—
91#388—22no—
92#387—22no—
93#387—22no—
94#387—22no—
95#387—22no—
96#387—22no—
97#387—22no—
98#387—22no—
99#385—22no—
100#385—22no—
101#383C2×C412yes—
102#382C2×C411yescharacteristic
103#381—24no—
104C4×C4⋊C4—21no—
105C4×C4⋊C4—22no—
106C4×C4⋊C4—22no—
107C4×C4⋊C4—21no—
108C4×C4⋊C4C2×C411yescharacteristic
109C4×C22⋊C4—21no—
110#376C2×C412yes—
111#342—22no—
112#342—24no—
113#342C2314yes—
114#324—24no—
115#324—22no—
116#323C2314yes—
117C22×Q8—24no—
118C22×Q8—21no—
119C22×Q8—24no—
120C22×Q8—42no—
121C22×Q8C22⋊C411yescharacteristic
122C22×Q8C22⋊C412yes—
123C22×Q8C22⋊C411yescharacteristic
124C22×Q8C22×C411yescharacteristic
125C23×C4C2×D412yes—
126C23×C4C2×D411yescharacteristic
127C4⋊Q8—42no—
128C4⋊Q8—44no—
129C4⋊Q8—44no—
130#125—44no—
131#125—48no—
132C22⋊Q8—44no—
133C4×Q8—24no—
134C4×Q8—24no—
135C4×Q8—44no—
136C4×Q8—42no—
137C4×Q8—42no—
138C4×Q8—44no—
139C4×Q8—44no—
140C4×Q8—42no—
141C4×Q8—44no—
142#117—44no—
143#117—42no—
144#117—42no—
145#117—44no—
146#117—44no—
147#117—44no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—21no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—21no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—21no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—42no—
162C2×C4⋊C4C22⋊C411yescharacteristic
163C2×C4⋊C4C22⋊C412yes—
164C2×C4⋊C4C22⋊C411yescharacteristic
165C2×C4⋊C4C22⋊C412yes—
166C2×C4⋊C4C22×C411yescharacteristic
167C2×C4⋊C4C22×C412yes—
168C2×C4⋊C4C22×C412yes—
169C2×C4⋊C4C22×C411yescharacteristic
170C2×C4⋊C4C2×D412yes—
171C2×C4⋊C4C2×D411yescharacteristic
172C2×C22⋊C4—22no—
173C2×C22⋊C4—21no—
174C2×C22⋊C4—21no—
175C2×C22⋊C4—24no—
176C2×C22⋊C4C2×D412yes—
177C2×C22⋊C4C2×D411yescharacteristic
178C2×C22⋊C4C2411yescharacteristic
179C2×C42—22no—
180C2×C42—21no—
181C2×C42—21no—
182C2×C42—22no—
183C2×C42—42no—
184C2×C42C22⋊C412yes—
185C2×C42C22⋊C412yes—
186C2×C42C22⋊C412yes—
187C2×C42C22×C412yes—
188C2×C42C22×C411yescharacteristic
189C2×C42C2×D411yescharacteristic
190C2×C42C2×D412yes—
191C4⋊C8—44no—
192C4⋊C8—42no—
193C22⋊C8—24no—
194C4×C8—44no—
195C4×C8—42no—
196#95—22no—
197#95—21no—
198#95—21no—
199#95—22no—
200#95—22no—
201#95—21no—
202#95—22no—
203#95—22no—
204#95—21no—
205#95—22no—
206#95—22no—
207#95—22no—
208#95—22no—
209#95—21no—
210#95—22no—
211#95—48no—
212#95C22×C411yescharacteristic
213#95C22×C412yes—
214#95C22×C411yescharacteristic
215C24C22×D411yescharacteristic
216C2×Q8—24no—
217C2×Q8—22no—
218C2×Q8—22no—
219C2×Q8—24no—
220C2×Q8—22no—
221C2×Q8—22no—
222C2×Q8—22no—
223C2×Q8—42no—
224C2×Q8—48no—
225C2×Q8—48no—
226C2×Q8—42no—
227C2×Q8—41no—
228C2×Q8—42no—
229C2×Q8—42no—
230C2×Q8—42no—
231C2×Q8#9914yes—
232C22×C4—22no—
233C22×C4—22no—
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—24no—
237C22×C4—24no—
238C22×C4—22no—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—22no—
242C22×C4—42no—
243C22×C4—44no—
244C22×C4C2×C22⋊C412yes—
245C22×C4C2×C22⋊C412yes—
246C22×C4C2×C22⋊C411yescharacteristic
247C22×C4C2×C22⋊C411yescharacteristic
248C22×C4C2×C22⋊C412yes—
249C22×C4C2×C22⋊C412yes—
250C22×C4C2×C22⋊C411yescharacteristic
251C22×C4C2×C22⋊C411yescharacteristic
252C22×C4#13514yes—
253C22×C4C23×C411yescharacteristic
254C22×C4C22×D411yescharacteristic
255C2×C8—44no—
256C4⋊C4—24no—
257C4⋊C4—22no—
258C4⋊C4—22no—
259C4⋊C4—24no—
260C4⋊C4—41no—
261C4⋊C4—42no—
262C4⋊C4—41no—
263C4⋊C4—42no—
264C4⋊C4—41no—
265C4⋊C4—42no—
266C4⋊C4—42no—
267C4⋊C4—42no—
268C4⋊C4—42no—
269C4⋊C4—44no—
270C4⋊C4—44no—
271C4⋊C4—42no—
272C4⋊C4—42no—
273C4⋊C4—44no—
274C4⋊C4—44no—
275C4⋊C4—44no—
276C4⋊C4—44no—
277C4⋊C4—42no—
278C4⋊C4—44no—
279C4⋊C4—84no—
280C22⋊C4—42no—
281C22⋊C4—41no—
282C22⋊C4—41no—
283C22⋊C4—42no—
284C22⋊C4—42no—
285C22⋊C4—44no—
286C22⋊C4—44no—
287C22⋊C4—44no—
288C42—24no—
289C42—22no—
290C42—24no—
291C42—22no—
292C42—42no—
293C42—42no—
294C42—42no—
295C42—44no—
296C42—44no—
297C42—42no—
298C42—42no—
299C42—84no—
300C23—21no—
301C23—21no—
302C23—41no—
303C23—41no—
304C23C22×C22⋊C411yescharacteristic
305C23C2×#13512yes—
306Q8—28no—
307Q8—48no—
308Q8—42no—
309Q8—42no—
310Q8—44no—
311Q8—84no—
312C2×C4—21no—
313C2×C4—22no—
314C2×C4—21no—
315C2×C4—22no—
316C2×C4—24no—
317C2×C4—24no—
318C2×C4—22no—
319C2×C4—22no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—22no—
323C2×C4—21no—
324C2×C4—22no—
325C2×C4—21no—
326C2×C4—22no—
327C2×C4—22no—
328C2×C4—22no—
329C2×C4—48no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—42no—
335C2×C4—41no—
336C2×C4—42no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—42no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—42no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—42no—
348C2×C4—42no—
349C2×C4—42no—
350C2×C4—41no—
351C2×C4—42no—
352C2×C4—84no—
353C2×C4—84no—
354C2×C4—84no—
355C2×C4C2×#9914yes—
356C2×C4C2×#9912yes—
357C2×C4#41612yes—
358C2×C4#41612yes—
359C2×C4#42112yes—
360C8—84no—
361C22—21no—
362C22—21no—
363C22—22no—
364C22—42no—
365C22—41no—
366C22—41no—
367C22—81no—
368C22C22×#9911yescharacteristic
369C22C2×#41611yescharacteristic
370C22C2×#42111yescharacteristic
371C4—22no—
372C4—24no—
373C4—42no—
374C4—42no—
375C4—41no—
376C4—42no—
377C4—41no—
378C4—41no—
379C4—42no—
380C4—41no—
381C4—42no—
382C4—81no—
383C4—82no—
384C4—81no—
385C4—82no—
386C4—164no—
387C4#151614yes—
388C2—21no—
389C2—21no—
390C2—42no—
391C2C2×#151611yescharacteristic
392C2#3334911yescharacteristic
393C2#3336011yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC22×C4C23×C4
Frattini, cosocleC2×C22⋊C4C24
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×Q8 ◁ C2×#519 ◁ G
LECSG ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
8. C2×#557C4semidirect
11. #2634C4semidirect
15. #2581C4semidirect
16. #2546C4semidirect
21. #2399C4semidirect
25. #2356C4semidirect

Maximal subgroups

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

subgroupindex
#357492
#206912
#173522
#118462
#118422
#117152
#116942