Finite Group Databasebeta

← #480880

Subgroups — #480880

Order 512.

879 conjugacy classes of subgroups.

The subgroup lattice has 410 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36406C211yesmaximal, characteristic
2#17428C212yesmaximal, complemented
3#11568C212yesmaximal, complemented
4#11556C214yesmaximal, complemented
5#11530C211yesmaximal, characteristic, complemented
6#11507C212yesmaximal, complemented
7#11476C211yesmaximal, characteristic, complemented
8#11431C212yesmaximal, complemented
9C2×#556C2211yescharacteristic
10C2×#547C411yescharacteristic
11#2984—22no—
12#2862C411yescharacteristic
13#2857C412yes—
14#2553—24no—
15#2546C2211yescharacteristic
16#2542C411yescharacteristic
17#2499C2212yes—
18#2432C412yes—
19#2427—22no—
20#2360—22no—
21C4×#124C2211yescharacteristic
22#2297C2212yes—
23C42×D4C411yescharacteristic
24#1473—24no—
25#1469—22no—
26#1467C2212yes—
27#1464C2214yes—
28#1461C2212yes—
29#1458C2214yes—
30#1324C2212yes—
31#1324C2212yes—
32#1319C2214yes—
33#1319C2214yes—
34#1318C2212yes—
35#1318C2211yescharacteristic
36#1318C2211yescharacteristic
37C22×#41C2×C412yes—
38#556—41no—
39#556—42no—
40#556—41no—
41#547—42no—
42#547—41no—
43#547—41no—
44C2×C4⋊Q8C2×C411yescharacteristic
45C2×C4⋊Q8C2×C412yes—
46C2×C4⋊Q8C2×C411yescharacteristic
47C2×#126D414yes—
48C2×#124C2×C411yescharacteristic
49C2×#124D412yes—
50C2×#123—22no—
51C2×C22⋊Q8—22no—
52C2×C22⋊Q8C2×C412yes—
53C2×C22⋊Q8C2×C412yes—
54C2×#121—21no—
55C2×C4×Q8D412yes—
56C2×C4×Q8C2311yescharacteristic
57C2×C4×D4—21no—
58C2×C4×D4C2×C411yescharacteristic
59C2×C4×D4C2×C411yescharacteristic
60C2×#117C2312yes—
61C22×C4⋊C4—21no—
62C22×C42C2×C412yes—
63#446—44no—
64#446—42no—
65C8×Q8—42no—
66#432—24no—
67#432—22no—
68#431—24no—
69#431—22no—
70#399—22no—
71#398—22no—
72#397—24no—
73#394—24no—
74#391—22no—
75#391C2×C411yescharacteristic
76#391C2×C412yes—
77#391C2×C412yes—
78#391C2312yes—
79#390C2×C412yes—
80#390C2311yescharacteristic
81#389—24no—
82#389—22no—
83#388—22no—
84#387—24no—
85#386—24no—
86#385—24no—
87#384C2×C412yes—
88#382—22no—
89#382C2311yescharacteristic
90#380—24no—
91C4×C4⋊C4—21no—
92C4×C4⋊C4C2×C411yescharacteristic
93C4×C22⋊C4—22no—
94C4×C22⋊C4—22no—
95C4×C22⋊C4—24no—
96C4×C22⋊C4C2×C412yes—
97C43C2×C411yescharacteristic
98#324—22no—
99#324—22no—
100#324C2312yes—
101#324C2314yes—
102#324C2312yes—
103#323—24no—
104#323—24no—
105C2×#41—24no—
106C2×#41—42no—
107C2×#41—42no—
108C2×#41—42no—
109C2×#41—82no—
110C22×Q8C22⋊C412yes—
111C22×Q8C22×C412yes—
112C22×Q8C22×C411yescharacteristic
113C22×D4—22no—
114C22×D4C22⋊C411yescharacteristic
115C22×D4C22×C411yescharacteristic
116C23×C4—22no—
117C23×C4—21no—
118C23×C4—22no—
119C23×C4C22×C412yes—
120C4⋊Q8—41no—
121C4⋊Q8—41no—
122C4⋊Q8—41no—
123C4⋊Q8—41no—
124C4⋊Q8—82no—
125#126—44no—
126#126—44no—
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—48no—
136C22⋊Q8—42no—
137C22⋊Q8—42no—
138#121—42no—
139#121—42no—
140C4×Q8—22no—
141C4×Q8—24no—
142C4×Q8—22no—
143C4×Q8—41no—
144C4×Q8—41no—
145C4×D4—22no—
146C4×D4—21no—
147C4×D4—21no—
148C4×D4—41no—
149C4×D4—41no—
150C4×D4—44no—
151#117—42no—
152#117—42no—
153C2×C4⋊C4—21no—
154C2×C4⋊C4—21no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—24no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—24no—
159C2×C4⋊C4—42no—
160C2×C4⋊C4—41no—
161C2×C4⋊C4—41no—
162C2×C4⋊C4C22⋊C414yes—
163C2×C4⋊C4C22⋊C412yes—
164C2×C4⋊C4C22⋊C411yescharacteristic
165C2×C4⋊C4C22×C411yescharacteristic
166C2×C4⋊C4C22×C412yes—
167C2×C4⋊C4C2×D412yes—
168C2×C4⋊C4C2411yescharacteristic
169C2×C22⋊C4—22no—
170C2×C22⋊C4—22no—
171C2×C22⋊C4—21no—
172C2×C22⋊C4—24no—
173C2×C22⋊C4—22no—
174C2×C22⋊C4—22no—
175C2×C22⋊C4C22⋊C412yes—
176C2×C22⋊C4C22⋊C414yes—
177C2×C22⋊C4C2×D414yes—
178C2×C42—21no—
179C2×C42—21no—
180C2×C42—22no—
181C2×C42—82no—
182C2×C42C22×C411yescharacteristic
183C2×C42C22×C411yescharacteristic
184C2×C42C22×C412yes—
185C2×C42C22×C411yescharacteristic
186C2×C42C2×D414yes—
187C2×C42C2×D412yes—
188C4⋊C8—44no—
189C4⋊C8—42no—
190C4⋊C8—44no—
191C4⋊C8—42no—
192C22⋊C8—22no—
193C22⋊C8—24no—
194C22⋊C8—22no—
195#97—44no—
196#97—42no—
197C4×C8—44no—
198C4×C8—42no—
199#95—21no—
200#95—22no—
201#95—22no—
202#95—24no—
203#95—24no—
204#95—22no—
205#95—24no—
206C24—21no—
207C24C2×C22⋊C412yes—
208#41—84no—
209#41—84no—
210#41—84no—
211#41—84no—
212C2×Q8—21no—
213C2×Q8—21no—
214C2×Q8—22no—
215C2×Q8—22no—
216C2×Q8—22no—
217C2×Q8—22no—
218C2×Q8—41no—
219C2×Q8—41no—
220C2×Q8—44no—
221C2×Q8—42no—
222C2×Q8—42no—
223C2×D4—21no—
224C2×D4—21no—
225C2×D4—21no—
226C2×D4—41no—
227C2×D4—41no—
228C2×D4—42no—
229C2×D4—44no—
230C2×D4—44no—
231C2×D4—82no—
232C2×D4#13112yes—
233C22×C4—21no—
234C22×C4—22no—
235C22×C4—22no—
236C22×C4—22no—
237C22×C4—22no—
238C22×C4—22no—
239C22×C4—21no—
240C22×C4—22no—
241C22×C4—21no—
242C22×C4—42no—
243C22×C4—44no—
244C22×C4—44no—
245C22×C4—44no—
246C22×C4—44no—
247C22×C4—41no—
248C22×C4—41no—
249C22×C4—42no—
250C22×C4—42no—
251C22×C4—82no—
252C22×C4—82no—
253C22×C4C2×C22⋊C412yes—
254C22×C4C2×C22⋊C412yes—
255C22×C4C2×C22⋊C411yescharacteristic
256C22×C4C2×C22⋊C414yes—
257C22×C4C2×C22⋊C411yescharacteristic
258C22×C4C23×C411yescharacteristic
259C22×C4C22×D412yes—
260C2×C8—42no—
261C2×C8—44no—
262C2×C8—42no—
263C4⋊C4—24no—
264C4⋊C4—24no—
265C4⋊C4—24no—
266C4⋊C4—42no—
267C4⋊C4—42no—
268C4⋊C4—42no—
269C4⋊C4—41no—
270C4⋊C4—41no—
271C4⋊C4—48no—
272C4⋊C4—42no—
273C4⋊C4—41no—
274C4⋊C4—41no—
275C4⋊C4—42no—
276C4⋊C4—42no—
277C4⋊C4—84no—
278C4⋊C4#9912yes—
279C4⋊C4#9911yescharacteristic
280C4⋊C4#9911yescharacteristic
281C22⋊C4—22no—
282C22⋊C4—22no—
283C22⋊C4—44no—
284C22⋊C4—44no—
285C22⋊C4—44no—
286C22⋊C4—44no—
287C22⋊C4—44no—
288C22⋊C4—42no—
289C22⋊C4—48no—
290C22⋊C4—44no—
291C42—24no—
292C42—22no—
293C42—24no—
294C42—22no—
295C42—21no—
296C42—21no—
297C42—21no—
298C42—21no—
299C42—41no—
300C42—42no—
301C42—42no—
302C42—42no—
303C42—82no—
304C23—22no—
305C23—22no—
306C23—22no—
307C23—22no—
308C23—42no—
309C23—42no—
310C23—42no—
311C23—42no—
312C23—41no—
313C23—41no—
314C23C22×C22⋊C411yescharacteristic
315Q8—44no—
316Q8—81no—
317Q8—81no—
318Q8—82no—
319Q8—82no—
320Q8—82no—
321D4—42no—
322D4—42no—
323D4—42no—
324D4—42no—
325D4—88no—
326C2×C4—22no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—22no—
330C2×C4—22no—
331C2×C4—22no—
332C2×C4—24no—
333C2×C4—22no—
334C2×C4—22no—
335C2×C4—21no—
336C2×C4—22no—
337C2×C4—24no—
338C2×C4—22no—
339C2×C4—42no—
340C2×C4—42no—
341C2×C4—42no—
342C2×C4—42no—
343C2×C4—41no—
344C2×C4—42no—
345C2×C4—41no—
346C2×C4—42no—
347C2×C4—41no—
348C2×C4—44no—
349C2×C4—42no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—42no—
353C2×C4—41no—
354C2×C4—42no—
355C2×C4—41no—
356C2×C4—42no—
357C2×C4—42no—
358C2×C4—84no—
359C2×C4—84no—
360C2×C4—82no—
361C2×C4—88no—
362C2×C4—82no—
363C2×C4C2×#9911yescharacteristic
364C2×C4C2×#9912yes—
365C2×C4C2×#9912yes—
366C2×C4C2×#9911yescharacteristic
367C2×C4C2×#13111yescharacteristic
368C2×C4#56811yescharacteristic
369C8—82no—
370C8—84no—
371C8—82no—
372C22—21no—
373C22—21no—
374C22—44no—
375C22—42no—
376C22—42no—
377C22—42no—
378C22—42no—
379C22—82no—
380C22—82no—
381C22—82no—
382C22—81no—
383C22—81no—
384C22#288011yescharacteristic
385C22C22×#9911yescharacteristic
386C22#288811yescharacteristic
387C4—21no—
388C4—21no—
389C4—22no—
390C4—22no—
391C4—21no—
392C4—44no—
393C4—42no—
394C4—42no—
395C4—42no—
396C4—41no—
397C4—44no—
398C4—81no—
399C4—82no—
400C4—81no—
401C4#213212yes—
402C2—21no—
403C2—21no—
404C2—82no—
405C2—82no—
406C2#1741111yescharacteristic
407C2#3336011yescharacteristic
408C2C2×#213211yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC22×C4C23×C4
Frattini, cosocleC2×C4⋊C4C24
prime radicalC2×#547C4
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C4⋊C4 ◁ C2×#556 ◁ G
LECSG ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #17428C2semidirect
3. #11568C2semidirect
4. #11556C2semidirect
5. #11530C2semidirect
6. #11507C2semidirect
7. #11476C2semidirect
8. #11431C2semidirect

Maximal subgroups

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

subgroupindex
#364062
#174282
#115682
#115562
#115302
#115072
#114762
#114312