Finite Group Databasebeta

← #480950

Subgroups — #480950

Order 512.

924 conjugacy classes of subgroups.

The subgroup lattice has 412 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36164C211yesmaximal, characteristic
2#17416C212yesmaximal
3#11569C212yesmaximal, complemented
4#11551C214yesmaximal, complemented
5#11531C211yesmaximal, characteristic, complemented
6#11511C212yesmaximal, complemented
7#11477C211yesmaximal, characteristic, complemented
8C2×#1324C212yesmaximal, complemented
9C2×#554C411yescharacteristic
10C2×#547C2211yescharacteristic
11#2880—22no—
12#2862C411yescharacteristic
13#2850C412yes—
14#2568C411yescharacteristic
15#2563C2211yescharacteristic
16#2504C412yes—
17#2503C2212yes—
18#2489—22no—
19#2488—24no—
20#2339—22no—
21C4×#124C2211yescharacteristic
22C2×#382C2212yes—
23C42×D4C411yescharacteristic
24#1472—22no—
25#1468—24no—
26#1465C2212yes—
27#1463C2214yes—
28#1459C2212yes—
29C2×#324C2214yes—
30#1324C2212yes—
31#1324C2214yes—
32#1324C2214yes—
33#1324C2211yescharacteristic
34#1324C2211yescharacteristic
35#1319C2212yes—
36#1319C2212yes—
37C22×#41D412yes—
38C2×#131—42no—
39#554—41no—
40#554—42no—
41#554—41no—
42#547—41no—
43#547—42no—
44#547—41no—
45C2×C4⋊Q8C2×C411yescharacteristic
46C2×C4⋊Q8C2×C411yescharacteristic
47C2×#125—21no—
48C2×#124C2×C411yescharacteristic
49C2×#123D414yes—
50C2×C22⋊Q8—22no—
51C2×C22⋊Q8C2×C412yes—
52C2×C22⋊Q8D412yes—
53C2×#121C2×C412yes—
54C2×C4×Q8C2×C411yescharacteristic
55C2×C4×D4—21no—
56C2×C4×D4—21no—
57C2×C4×D4C2×C411yescharacteristic
58C2×C4×D4C2311yescharacteristic
59C22×C4⋊C4C2×C412yes—
60C22×C4⋊C4C2312yes—
61C22×C42C2×C412yes—
62#408—42no—
63#408—44no—
64#407—24no—
65#407—22no—
66C2×C22⋊C8—24no—
67C2×C22⋊C8—22no—
68#400—22no—
69#400—24no—
70#397—24no—
71#393—22no—
72#391—22no—
73#391—22no—
74#391—22no—
75#391C2×C412yes—
76#391C2311yescharacteristic
77#390C2×C412yes—
78#390C2×C411yescharacteristic
79#388—24no—
80#388—22no—
81#386—24no—
82#386—24no—
83#382—24no—
84#382C2×C412yes—
85#382C2×C412yes—
86#382C2×C412yes—
87#382C2312yes—
88#382C2311yescharacteristic
89#381—24no—
90#380—22no—
91C4×C4⋊C4—21no—
92C4×C4⋊C4C2×C411yescharacteristic
93C4×C22⋊C4—22no—
94C4×C22⋊C4—22no—
95C4×C22⋊C4C2×C412yes—
96C2×#95—24no—
97C43C2×C411yescharacteristic
98#324—24no—
99#324—24no—
100#324C2312yes—
101#324C2314yes—
102#324C2312yes—
103#323—22no—
104#323—22no—
105C2×#41—22no—
106C2×#41—24no—
107C2×#41—22no—
108C2×#41—48no—
109C22×Q8—21no—
110C22×Q8C22⋊C412yes—
111C22×Q8C22×C411yescharacteristic
112C22×D4C22⋊C414yes—
113C22×D4C22⋊C411yescharacteristic
114C22×D4C22×C411yescharacteristic
115C23×C4—22no—
116C23×C4—22no—
117C23×C4C22×C412yes—
118C23×C4C2×D414yes—
119C23×C4C2×D412yes—
120#131—88no—
121C2×M16—44no—
122C2×M16—42no—
123C22×C8—44no—
124C22×C8—42no—
125C4⋊Q8—41no—
126C4⋊Q8—41no—
127C4⋊Q8—41no—
128C4⋊Q8—41no—
129#125—42no—
130#125—42no—
131#124—22no—
132#124—41no—
133#123—44no—
134#123—44no—
135C22⋊Q8—44no—
136C22⋊Q8—44no—
137C22⋊Q8—42no—
138C22⋊Q8—42no—
139C22⋊Q8—42no—
140C22⋊Q8—42no—
141#121—42no—
142#121—42no—
143C4×Q8—41no—
144C4×Q8—41no—
145C4×D4—22no—
146C4×D4—21no—
147C4×D4—21no—
148C4×D4—41no—
149C4×D4—41no—
150C4×D4—44no—
151C4×D4—42no—
152C4×D4—41no—
153C4×D4—41no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—42no—
160C2×C4⋊C4—42no—
161C2×C4⋊C4—42no—
162C2×C4⋊C4—42no—
163C2×C4⋊C4C22⋊C411yescharacteristic
164C2×C4⋊C4C22⋊C412yes—
165C2×C4⋊C4C22×C411yescharacteristic
166C2×C4⋊C4C22×C412yes—
167C2×C4⋊C4C22×C412yes—
168C2×C4⋊C4C2×D414yes—
169C2×C4⋊C4C2411yescharacteristic
170C2×C22⋊C4—22no—
171C2×C22⋊C4—22no—
172C2×C22⋊C4—22no—
173C2×C22⋊C4—22no—
174C2×C22⋊C4—24no—
175C2×C22⋊C4C22⋊C412yes—
176C2×C22⋊C4C22⋊C414yes—
177C2×C22⋊C4C2×D412yes—
178C2×C42—24no—
179C2×C42—21no—
180C2×C42—21no—
181C2×C42—22no—
182C2×C42—21no—
183C2×C42—42no—
184C2×C42C22×C412yes—
185C2×C42C22×C411yescharacteristic
186C2×C42C22×C411yescharacteristic
187C2×C42C22×C411yescharacteristic
188C22⋊C8—22no—
189C22⋊C8—22no—
190C22⋊C8—24no—
191C22⋊C8—44no—
192C22⋊C8—42no—
193C22⋊C8—44no—
194C22⋊C8—42no—
195#95—24no—
196#95—24no—
197#95—21no—
198#95—22no—
199#95—24no—
200#95—24no—
201#95—22no—
202#95—24no—
203#95—22no—
204C24C2×C22⋊C412yes—
205C24C2×C22⋊C412yes—
206#41—48no—
207#41—88no—
208#41—84no—
209C2×Q8—22no—
210C2×Q8—22no—
211C2×Q8—21no—
212C2×Q8—21no—
213C2×Q8—44no—
214C2×Q8—42no—
215C2×Q8—41no—
216C2×Q8—41no—
217C2×Q8—82no—
218C2×D4—24no—
219C2×D4—24no—
220C2×D4—24no—
221C2×D4—21no—
222C2×D4—21no—
223C2×D4—41no—
224C2×D4—41no—
225C2×D4—48no—
226C2×D4#9912yes—
227C2×D4#9911yescharacteristic
228C2×D4#9911yescharacteristic
229C22×C4—21no—
230C22×C4—22no—
231C22×C4—22no—
232C22×C4—24no—
233C22×C4—24no—
234C22×C4—22no—
235C22×C4—22no—
236C22×C4—22no—
237C22×C4—22no—
238C22×C4—21no—
239C22×C4—22no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—22no—
243C22×C4—42no—
244C22×C4—44no—
245C22×C4—44no—
246C22×C4—48no—
247C22×C4—44no—
248C22×C4—42no—
249C22×C4—42no—
250C22×C4—82no—
251C22×C4—84no—
252C22×C4C2×C22⋊C412yes—
253C22×C4C2×C22⋊C411yescharacteristic
254C22×C4C2×C22⋊C414yes—
255C22×C4C2×C22⋊C411yescharacteristic
256C22×C4C23×C411yescharacteristic
257C22×C4C22×D412yes—
258M16—88no—
259M16—84no—
260C2×C8—42no—
261C2×C8—42no—
262C2×C8—44no—
263C2×C8—88no—
264C2×C8—84no—
265C4⋊C4—21no—
266C4⋊C4—21no—
267C4⋊C4—41no—
268C4⋊C4—42no—
269C4⋊C4—44no—
270C4⋊C4—42no—
271C4⋊C4—41no—
272C4⋊C4—42no—
273C4⋊C4—44no—
274C4⋊C4—42no—
275C4⋊C4—42no—
276C4⋊C4—44no—
277C4⋊C4—41no—
278C4⋊C4—41no—
279C22⋊C4—22no—
280C22⋊C4—22no—
281C22⋊C4—44no—
282C22⋊C4—42no—
283C22⋊C4—44no—
284C22⋊C4—44no—
285C22⋊C4—42no—
286C22⋊C4—42no—
287C22⋊C4—82no—
288C22⋊C4—84no—
289C42—21no—
290C42—21no—
291C42—21no—
292C42—22no—
293C42—22no—
294C42—44no—
295C42—41no—
296C42—42no—
297C42—41no—
298C42—41no—
299C42—81no—
300C42#13112yescomplemented
301C23—22no—
302C23—22no—
303C23—24no—
304C23—22no—
305C23—22no—
306C23—22no—
307C23—42no—
308C23—42no—
309C23—42no—
310C23C2×#9912yes—
311C23C2×#9912yes—
312C23C22×C22⋊C411yescharacteristic
313Q8—44no—
314Q8—82no—
315Q8—81no—
316Q8—81no—
317Q8—162no—
318D4—24no—
319D4—22no—
320D4—48no—
321D4—81no—
322D4—81no—
323D4—84no—
324D4#35414yes—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—22no—
328C2×C4—22no—
329C2×C4—22no—
330C2×C4—21no—
331C2×C4—22no—
332C2×C4—22no—
333C2×C4—24no—
334C2×C4—22no—
335C2×C4—42no—
336C2×C4—42no—
337C2×C4—42no—
338C2×C4—42no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—48no—
342C2×C4—44no—
343C2×C4—41no—
344C2×C4—42no—
345C2×C4—44no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—42no—
349C2×C4—42no—
350C2×C4—42no—
351C2×C4—44no—
352C2×C4—42no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—42no—
357C2×C4—41no—
358C2×C4—41no—
359C2×C4—82no—
360C2×C4—84no—
361C2×C4—88no—
362C2×C4—162no—
363C2×C4—162no—
364C2×C4C2×#9911yescharacteristic
365C2×C4C2×#9911yescharacteristic
366C2×C4C2×#13111yescharacteristic
367C2×C4#56811yescharacteristic
368C8—82no—
369C8—84no—
370C8—82no—
371C22—22no—
372C22—21no—
373C22—21no—
374C22—22no—
375C22—22no—
376C22—22no—
377C22—22no—
378C22—44no—
379C22—44no—
380C22—42no—
381C22—42no—
382C22—42no—
383C22—82no—
384C22—82no—
385C22C2×#35414yes—
386C22#288011yescharacteristic
387C22C22×#9911yescharacteristic
388C22#288911yescharacteristic
389C4—21no—
390C4—21no—
391C4—21no—
392C4—42no—
393C4—42no—
394C4—42no—
395C4—44no—
396C4—41no—
397C4—81no—
398C4—82no—
399C4—81no—
400C4—81no—
401C4—82no—
402C4C2×#35412yes—
403C2—21no—
404C2—21no—
405C2—24no—
406C2—42no—
407C2—82no—
408C2#1741611yescharacteristic
409C2C22×#35411yescharacteristic
410C2#3337011yescharacteristic
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. #11569C2semidirect
4. #11551C2semidirect
5. #11531C2semidirect
6. #11511C2semidirect
7. #11477C2semidirect
8. C2×#1324C2semidirect
300. C42#131semidirect

Maximal subgroups

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

subgroupindex
#361642
#174162
#115692
#115512
#115312
#115112
#114772
C2×#13242