Finite Group Databasebeta

← #204645

Subgroups — #204645

Order 512.

1184 conjugacy classes of subgroups.

The subgroup lattice has 433 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35103C211yesmaximal, characteristic
2#20534C211yesmaximal, characteristic, complemented
3C2×#1512C212yesmaximal, complemented
4C2×#1509C212yesmaximal, complemented
5#13954C211yesmaximal, characteristic, complemented
6#9682C214yesmaximal, complemented
7#9680C212yesmaximal
8#9679C212yesmaximal, complemented
9C2×#529C2212yes—
10C2×#432C2212yes—
11#2857C2211yescharacteristic
12#2856C2211yescharacteristic
13#2850C2212yes—
14#2618—22no—
15#2601—24no—
16#2566C412yescomplemented
17#2565C412yescomplemented
18#2545C412yescomplemented
19#2544C412yescomplemented
20#2325—22no—
21C2×#389—22no—
22C2×#382C2211yescharacteristic
23C2×C4×C4⋊C4C2211yescharacteristic
24C2×C4×C4⋊C4C2211yescharacteristic
25#1771—24no—
26#1749C2212yes—
27C4×C22⋊C8C2212yes—
28#1512—28no—
29#1509—28no—
30C2×#328C2214yes—
31C2×#328C2214yes—
32C2×#327C2214yes—
33C2×#327C2214yes—
34#1444—22no—
35C4⋊#95C2212yes—
36C4⋊#95C2212yes—
37C2×C4⋊Q8C2×C411yescharacteristic
38C2×#127C2×C411yescharacteristic
39#529—24no—
40#529—24no—
41#529—24no—
42#529—24no—
43C2×#126—22no—
44C2×#125C2×C412yes—
45C2×#124C2×C412yes—
46C2×#124C2×C412yes—
47C2×#123—22no—
48C2×C22⋊Q8C2312yes—
49C2×#121—22no—
50C2×#121C2312yes—
51C2×C4×Q8C2×C412yes—
52C2×C4×D4—22no—
53C2×C4×D4C2×C412yes—
54C2×#117—22no—
55C22×C4⋊C4—21no—
56C22×C4⋊C4—21no—
57C22×C4⋊C4—22no—
58C22×C4⋊C4—22no—
59C22×C4⋊C4D412yes—
60C22×C42—21no—
61C22×C42C2311yescharacteristic
62C22×C42C2311yescharacteristic
63C22×C42C2311yescharacteristic
64#432—24no—
65#432D418yescomplemented
66C2×C4⋊C8—22no—
67C2×C22⋊C8C2314yes—
68C2×#97—22no—
69C2×#97—22no—
70C2×C4×C8—22no—
71#398—24no—
72#396—24no—
73#396—24no—
74#394—24no—
75#391C2×C412yes—
76#391C2×C412yes—
77#390—22no—
78#390C2×C412yes—
79#390C2×C412yes—
80#389—22no—
81#389—48no—
82#388—24no—
83#388—24no—
84#387—24no—
85#386—24no—
86#386—24no—
87#385—24no—
88#384C2×C411yescharacteristic
89#384C2×C411yescharacteristic
90#382—24no—
91#382C2×C412yes—
92#380—22no—
93C4×C4⋊C4—24no—
94C4×C4⋊C4C2×C412yes—
95C4×C4⋊C4C2×C412yes—
96C4×C4⋊C4D414yes—
97C2×#95—22no—
98C2×#95—22no—
99C2×#95—21no—
100C2×#95D412yes—
101C2×#95C2312yes—
102C2×#95C2312yes—
103#336—22no—
104#336—22no—
105#328—28no—
106#328—28no—
107#327—28no—
108#327—28no—
109C22×Q8C22×C412yes—
110C22×D4—22no—
111C22×D4—22no—
112C22×D4C22×C412yes—
113C23×C4—22no—
114C23×C4—22no—
115C23×C4—22no—
116C23×C4C2×D411yescharacteristic
117C23×C4C2×D412yes—
118C23×C4C2×D411yescharacteristic
119C23×C4C2×D412yes—
120C23×C4C2411yescharacteristic
121C22×C8—24no—
122C4⋊Q8—81no—
123#127—81no—
124#126—44no—
125#126—44no—
126#125—24no—
127#125—24no—
128#124—24no—
129#124—24no—
130#124—82no—
131#123—44no—
132#123—44no—
133C22⋊Q8—22no—
134C22⋊Q8—22no—
135C22⋊Q8—22no—
136C22⋊Q8—22no—
137#121—22no—
138#121—22no—
139#121—22no—
140#121—22no—
141#121—44no—
142#121—44no—
143C4×Q8—24no—
144C4×Q8—24no—
145C4×D4—24no—
146C4×D4—24no—
147C4×D4—44no—
148C4×D4—44no—
149#117—44no—
150#117—44no—
151C2×C4⋊C4—24no—
152C2×C4⋊C4—24no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—44no—
161C2×C4⋊C4—42no—
162C2×C4⋊C4—44no—
163C2×C4⋊C4—44no—
164C2×C4⋊C4—44no—
165C2×C4⋊C4—44no—
166C2×C4⋊C4—42no—
167C2×C4⋊C4—44no—
168C2×C4⋊C4—82no—
169C2×C4⋊C4—82no—
170C2×C4⋊C4C22⋊C414yes—
171C2×C4⋊C4C22⋊C414yes—
172C2×C4⋊C4C22×C412yes—
173C2×C4⋊C4C22×C412yes—
174C2×C22⋊C4—22no—
175C2×C22⋊C4—22no—
176C2×C22⋊C4—22no—
177C2×C22⋊C4—22no—
178C2×C22⋊C4—24no—
179C2×C22⋊C4—22no—
180C2×C22⋊C4—24no—
181C2×C22⋊C4—24no—
182C2×C42—22no—
183C2×C42—44no—
184C2×C42—42no—
185C2×C42—81no—
186C2×C42—81no—
187C2×C42C22×C412yes—
188C2×C42C22×C411yescharacteristic
189C2×C42C22×C411yescharacteristic
190C2×C42C22×C412yes—
191C2×C42C2×D412yes—
192C2×C42C2×D412yes—
193C2×C42C2×D414yes—
194C2×C42C2×D412yes—
195C2×C42#4114yes—
196C4⋊C8—28no—
197C22⋊C8—24no—
198C22⋊C8C2×D418yes—
199#97—28no—
200#97—82no—
201C4×C8—82no—
202#95—24no—
203#95—24no—
204#95—24no—
205#95—24no—
206#95—24no—
207#95—44no—
208#95—42no—
209#95—44no—
210#95C22⋊C414yes—
211#95C22⋊C414yes—
212C24—22no—
213C24C22×D411yescharacteristic
214C2×Q8—22no—
215C2×Q8—22no—
216C2×Q8—42no—
217C2×Q8—42no—
218C2×D4—22no—
219C2×D4—22no—
220C2×D4—42no—
221C2×D4—42no—
222C2×D4—42no—
223C2×D4—42no—
224C2×D4—44no—
225C2×D4—44no—
226C2×D4—44no—
227C2×D4—82no—
228C2×D4—82no—
229C22×C4—22no—
230C22×C4—22no—
231C22×C4—22no—
232C22×C4—22no—
233C22×C4—24no—
234C22×C4—22no—
235C22×C4—22no—
236C22×C4—22no—
237C22×C4—41no—
238C22×C4—42no—
239C22×C4—42no—
240C22×C4—42no—
241C22×C4—42no—
242C22×C4—42no—
243C22×C4—41no—
244C22×C4—42no—
245C22×C4—42no—
246C22×C4—42no—
247C22×C4—42no—
248C22×C4—42no—
249C22×C4—42no—
250C22×C4—44no—
251C22×C4—42no—
252C22×C4—42no—
253C22×C4—42no—
254C22×C4—42no—
255C22×C4—42no—
256C22×C4—42no—
257C22×C4—42no—
258C22×C4—84no—
259C22×C4C2×C22⋊C414yes—
260C22×C4C2×C22⋊C412yes—
261C22×C4C2×C22⋊C412yes—
262C22×C4C2×C22⋊C412yes—
263C22×C4C2×C22⋊C412yes—
264C22×C4#12412yes—
265C22×C4#12412yes—
266C22×C4#12712yes—
267C22×C4#12712yes—
268C22×C4AGL(1,ℤ/8)14yes—
269C22×C4#13714yes—
270C22×C4C23×C411yescharacteristic
271C22×C4C22×D411yescharacteristic
272C22×C4C22×D412yes—
273C22×C4C2×#4112yes—
274C2×C8—28no—
275C2×C8—44no—
276C2×C8—44no—
277C4⋊C4—28no—
278C4⋊C4—28no—
279C4⋊C4—22no—
280C4⋊C4—22no—
281C4⋊C4—22no—
282C4⋊C4—22no—
283C4⋊C4—48no—
284C4⋊C4—42no—
285C4⋊C4—42no—
286C4⋊C4—44no—
287C4⋊C4—44no—
288C4⋊C4—44no—
289C4⋊C4—44no—
290C4⋊C4—82no—
291C4⋊C4—82no—
292C4⋊C4—84no—
293C4⋊C4—88no—
294C4⋊C4—88no—
295C22⋊C4—42no—
296C22⋊C4—42no—
297C22⋊C4—42no—
298C22⋊C4—42no—
299C22⋊C4—44no—
300C22⋊C4—44no—
301C22⋊C4—44no—
302C22⋊C4—48no—
303C22⋊C4—84no—
304C22⋊C4—84no—
305C42—42no—
306C42—42no—
307C42—42no—
308C42—41no—
309C42—41no—
310C42—44no—
311C42—84no—
312C42C4×D414yes—
313C42C4×D414yes—
314C42#12114yes—
315C42#12114yes—
316C23—21no—
317C23—21no—
318C23—22no—
319C23—42no—
320C23—42no—
321C23—44no—
322C23—44no—
323C23—42no—
324C23C22×C22⋊C411yescharacteristic
325C23C2×#12411yescharacteristic
326C23C2×#12711yescharacteristic
327C23C2×AGL(1,ℤ/8)12yes—
328C23C2×#13712yes—
329Q8—82no—
330Q8—82no—
331D4—82no—
332D4—82no—
333D4—84no—
334D4—82no—
335D4—82no—
336D4—164no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—22no—
340C2×C4—22no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—24no—
344C2×C4—24no—
345C2×C4—22no—
346C2×C4—22no—
347C2×C4—22no—
348C2×C4—24no—
349C2×C4—22no—
350C2×C4—24no—
351C2×C4—24no—
352C2×C4—24no—
353C2×C4—42no—
354C2×C4—44no—
355C2×C4—42no—
356C2×C4—44no—
357C2×C4—42no—
358C2×C4—42no—
359C2×C4—44no—
360C2×C4—44no—
361C2×C4—44no—
362C2×C4—44no—
363C2×C4—44no—
364C2×C4—42no—
365C2×C4—42no—
366C2×C4—44no—
367C2×C4—44no—
368C2×C4—42no—
369C2×C4—42no—
370C2×C4—42no—
371C2×C4—88no—
372C2×C4—82no—
373C2×C4—82no—
374C2×C4—82no—
375C2×C4—82no—
376C2×C4—82no—
377C2×C4—82no—
378C2×C4—82no—
379C2×C4—82no—
380C2×C4—82no—
381C2×C4—82no—
382C2×C4—82no—
383C2×C4#39014yes—
384C2×C4#39014yes—
385C2×C4#41714yes—
386C2×C4#41714yes—
387C2×C4C2×C4×D412yes—
388C2×C4C2×#12112yes—
389C8—48no—
390C22—21no—
391C22—21no—
392C22—22no—
393C22—22no—
394C22—22no—
395C22—22no—
396C22—21no—
397C22—21no—
398C22—82no—
399C22—82no—
400C22—84no—
401C22—84no—
402C22—82no—
403C22C2×#39011yescharacteristic
404C22#288311yescharacteristic
405C22#288411yescharacteristic
406C22C2×#41712yes—
407C22#290112yes—
408C22#314112yes—
409C22#315111yescharacteristic
410C22#315211yescharacteristic
411C4—24no—
412C4—24no—
413C4—24no—
414C4—24no—
415C4—22no—
416C4—22no—
417C4—44no—
418C4—84no—
419C4—82no—
420C4—82no—
421C4—82no—
422C4—162no—
423C2—21no—
424C2—22no—
425C2—21no—
426C2—162no—
427C2#1762212yes—
428C2#1762812yes—
429C2#2120011yescharacteristic
430C2#2121411yescharacteristic
431C2#2125311yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radical#2856C22
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #20534C2semidirect
3. C2×#1512C2semidirect
4. C2×#1509C2semidirect
5. #13954C2semidirect
6. #9682C2semidirect
8. #9679C2semidirect
16. #2566C4semidirect
17. #2565C4semidirect
18. #2545C4semidirect
19. #2544C4semidirect
65. #432D4semidirect

Maximal subgroups

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

subgroupindex
#351032
#205342
C2×#15122
C2×#15092
#139542
#96822
#96802
#96792