Finite Group Databasebeta

← #480953

Subgroups — #480953

Order 512.

1214 conjugacy classes of subgroups.

The subgroup lattice has 449 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36179C211yesmaximal, characteristic
2#17416C212yesmaximal, complemented
3#11570C212yesmaximal, complemented
4#11549C214yesmaximal, complemented
5#11529C211yesmaximal, characteristic, complemented
6#11512C212yesmaximal, complemented
7#11475C211yesmaximal, characteristic, complemented
8C2×#1325C212yesmaximal, complemented
9C2×#547—22no—
10C2×#547C411yescharacteristic
11C2×#547C2211yescharacteristic
12#2880—22no—
13#2868C411yescharacteristic
14#2856C412yes—
15#2542—22no—
16#2542C411yescharacteristic
17#2542C2211yescharacteristic
18#2501—24no—
19#2501C412yes—
20#2501C2212yes—
21C4×#127C2211yescharacteristic
22C2×#384C2212yes—
23C42×D4C411yescharacteristic
24#1472—22no—
25#1468—24no—
26#1465C2212yes—
27#1463C2214yes—
28#1459C2212yes—
29C2×#324C2214yes—
30#1325C2212yes—
31#1325C2214yes—
32#1325C2214yes—
33#1325C2211yescharacteristic
34#1325C2211yescharacteristic
35#1318C2212yes—
36#1318C2212yes—
37C22×#41—21no—
38C22×#41D412yes—
39C2×#131—42no—
40#547—41no—
41#547—42no—
42#547—41no—
43#547—41no—
44#547—42no—
45#547—41no—
46#547—416no—
47C2×C4⋊Q8—22no—
48C2×C4⋊Q8C2×C411yescharacteristic
49C2×C4⋊Q8C2×C411yescharacteristic
50C2×#127C2×C411yescharacteristic
51C2×#123—24no—
52C2×#123—24no—
53C2×#123—22no—
54C2×#123D414yes—
55C2×C22⋊Q8—22no—
56C2×C22⋊Q8—22no—
57C2×C22⋊Q8—21no—
58C2×C22⋊Q8D412yes—
59C2×#121—24no—
60C2×#121C2×C412yes—
61C2×#121C2×C412yes—
62C2×C4×D4—21no—
63C2×C4×D4—22no—
64C2×C4×D4C2×C411yescharacteristic
65C2×C4×D4C2×C411yescharacteristic
66C2×C4×D4C2311yescharacteristic
67C22×C4⋊C4—24no—
68C22×C4⋊C4C2×C412yes—
69C22×C4⋊C4C2312yes—
70C22×C42C2×C412yes—
71#408—42no—
72#408—44no—
73#407—24no—
74#407—22no—
75C2×C22⋊C8—24no—
76C2×C22⋊C8—22no—
77#399—22no—
78#399—24no—
79#399—24no—
80#399—22no—
81#390—24no—
82#390—22no—
83#390C2×C411yescharacteristic
84#390C2×C412yes—
85#390C2×C412yes—
86#390C2311yescharacteristic
87#384—22no—
88#384—24no—
89#384C2×C412yes—
90#384C2×C412yes—
91#384C2×C412yes—
92#384C2311yescharacteristic
93#384C2312yes—
94C4×C4⋊C4—21no—
95C4×C4⋊C4C2×C411yescharacteristic
96C4×C22⋊C4—22no—
97C4×C22⋊C4C2×C412yes—
98C43C2×C411yescharacteristic
99#324—24no—
100#324—24no—
101#324C2312yes—
102#324C2314yes—
103#324C2312yes—
104#323—22no—
105#323—22no—
106C2×#41—22no—
107C2×#41—24no—
108C2×#41—22no—
109C2×#41—48no—
110C2×#41—48no—
111C2×#41—41no—
112C2×#41—42no—
113C2×#41—41no—
114C22×Q8—21no—
115C22×Q8C22⋊C412yes—
116C22×D4—22no—
117C22×D4—22no—
118C22×D4C22⋊C414yes—
119C22×D4C22⋊C411yescharacteristic
120C22×D4C22×C411yescharacteristic
121C22×D4C22×C411yescharacteristic
122C23×C4—22no—
123C23×C4—22no—
124C23×C4—21no—
125C23×C4C22×C412yes—
126C23×C4C2×D414yes—
127C23×C4C2×D412yes—
128#131—88no—
129C2×M16—44no—
130C2×M16—42no—
131C22×C8—44no—
132C22×C8—42no—
133C4⋊Q8—41no—
134C4⋊Q8—41no—
135C4⋊Q8—41no—
136C4⋊Q8—41no—
137C4⋊Q8—48no—
138#127—22no—
139#127—41no—
140#123—44no—
141#123—44no—
142#123—416no—
143#123—416no—
144#123—44no—
145#123—44no—
146C22⋊Q8—42no—
147C22⋊Q8—42no—
148C22⋊Q8—48no—
149C22⋊Q8—48no—
150C22⋊Q8—42no—
151C22⋊Q8—42no—
152#121—42no—
153#121—42no—
154#121—42no—
155#121—42no—
156#121—416no—
157C4×D4—28no—
158C4×D4—22no—
159C4×D4—21no—
160C4×D4—21no—
161C4×D4—41no—
162C4×D4—41no—
163C4×D4—48no—
164C4×D4—41no—
165C4×D4—41no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—24no—
168C2×C4⋊C4—24no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—24no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—24no—
174C2×C4⋊C4—42no—
175C2×C4⋊C4—42no—
176C2×C4⋊C4—42no—
177C2×C4⋊C4—42no—
178C2×C4⋊C4—416no—
179C2×C4⋊C4C22⋊C411yescharacteristic
180C2×C4⋊C4C22⋊C412yes—
181C2×C4⋊C4C22×C411yescharacteristic
182C2×C4⋊C4C22×C412yes—
183C2×C4⋊C4C22×C412yes—
184C2×C4⋊C4C2×D414yes—
185C2×C4⋊C4C2411yescharacteristic
186C2×C22⋊C4—22no—
187C2×C22⋊C4—22no—
188C2×C22⋊C4—24no—
189C2×C22⋊C4—21no—
190C2×C22⋊C4—24no—
191C2×C22⋊C4—22no—
192C2×C22⋊C4—22no—
193C2×C22⋊C4C22⋊C412yes—
194C2×C22⋊C4C22⋊C414yes—
195C2×C22⋊C4C2×D412yes—
196C2×C42—21no—
197C2×C42—24no—
198C2×C42—21no—
199C2×C42—22no—
200C2×C42—42no—
201C2×C42C22×C412yes—
202C2×C42C22×C411yescharacteristic
203C2×C42C22×C411yescharacteristic
204C2×C42C22×C411yescharacteristic
205C22⋊C8—22no—
206C22⋊C8—22no—
207C22⋊C8—24no—
208C22⋊C8—44no—
209C22⋊C8—42no—
210C22⋊C8—44no—
211C22⋊C8—42no—
212#95—21no—
213#95—22no—
214C24—21no—
215C24C2×C22⋊C412yes—
216C24C2×C22⋊C412yes—
217#41—48no—
218#41—88no—
219#41—84no—
220#41—88no—
221#41—84no—
222#41—84no—
223C2×Q8—22no—
224C2×Q8—22no—
225C2×Q8—44no—
226C2×Q8—44no—
227C2×Q8—41no—
228C2×Q8—41no—
229C2×D4—24no—
230C2×D4—24no—
231C2×D4—24no—
232C2×D4—21no—
233C2×D4—21no—
234C2×D4—21no—
235C2×D4—21no—
236C2×D4—41no—
237C2×D4—41no—
238C2×D4—44no—
239C2×D4—41no—
240C2×D4—41no—
241C2×D4—44no—
242C2×D4—44no—
243C2×D4—48no—
244C2×D4—48no—
245C2×D4—42no—
246C2×D4—42no—
247C2×D4#9912yes—
248C2×D4#9911yescharacteristic
249C2×D4#9911yescharacteristic
250C22×C4—21no—
251C22×C4—28no—
252C22×C4—21no—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—22no—
256C22×C4—24no—
257C22×C4—24no—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—21no—
263C22×C4—22no—
264C22×C4—42no—
265C22×C4—44no—
266C22×C4—44no—
267C22×C4—44no—
268C22×C4—44no—
269C22×C4—48no—
270C22×C4—48no—
271C22×C4—42no—
272C22×C4—42no—
273C22×C4—41no—
274C22×C4—41no—
275C22×C4C2×C22⋊C412yes—
276C22×C4C2×C22⋊C411yescharacteristic
277C22×C4C2×C22⋊C411yescharacteristic
278C22×C4C2×C22⋊C414yes—
279C22×C4C23×C411yescharacteristic
280C22×C4C22×D412yes—
281M16—88no—
282M16—84no—
283C2×C8—42no—
284C2×C8—42no—
285C2×C8—44no—
286C2×C8—88no—
287C2×C8—84no—
288C4⋊C4—24no—
289C4⋊C4—21no—
290C4⋊C4—21no—
291C4⋊C4—41no—
292C4⋊C4—42no—
293C4⋊C4—44no—
294C4⋊C4—42no—
295C4⋊C4—41no—
296C4⋊C4—42no—
297C4⋊C4—44no—
298C4⋊C4—48no—
299C4⋊C4—48no—
300C4⋊C4—48no—
301C4⋊C4—44no—
302C4⋊C4—48no—
303C4⋊C4—42no—
304C22⋊C4—28no—
305C22⋊C4—22no—
306C22⋊C4—22no—
307C22⋊C4—44no—
308C22⋊C4—42no—
309C22⋊C4—44no—
310C22⋊C4—44no—
311C22⋊C4—42no—
312C22⋊C4—48no—
313C22⋊C4—48no—
314C22⋊C4—44no—
315C42—24no—
316C42—21no—
317C42—21no—
318C42—21no—
319C42—22no—
320C42—22no—
321C42—44no—
322C42—41no—
323C42—42no—
324C42#13112yescomplemented
325C23—22no—
326C23—22no—
327C23—24no—
328C23—22no—
329C23—22no—
330C23—22no—
331C23—42no—
332C23—42no—
333C23—42no—
334C23—42no—
335C23—42no—
336C23—41no—
337C23—41no—
338C23—41no—
339C23C2×#9912yes—
340C23C2×#9912yes—
341C23C22×C22⋊C411yescharacteristic
342Q8—44no—
343Q8—82no—
344Q8—82no—
345Q8—82no—
346D4—24no—
347D4—22no—
348D4—48no—
349D4—81no—
350D4—81no—
351D4—81no—
352D4—81no—
353D4—88no—
354D4—84no—
355D4—84no—
356D4—84no—
357D4#35314yes—
358C2×C4—22no—
359C2×C4—22no—
360C2×C4—22no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—22no—
364C2×C4—22no—
365C2×C4—24no—
366C2×C4—22no—
367C2×C4—21no—
368C2×C4—22no—
369C2×C4—22no—
370C2×C4—24no—
371C2×C4—22no—
372C2×C4—42no—
373C2×C4—42no—
374C2×C4—42no—
375C2×C4—42no—
376C2×C4—416no—
377C2×C4—41no—
378C2×C4—42no—
379C2×C4—42no—
380C2×C4—41no—
381C2×C4—48no—
382C2×C4—44no—
383C2×C4—41no—
384C2×C4—42no—
385C2×C4—44no—
386C2×C4—42no—
387C2×C4—44no—
388C2×C4—41no—
389C2×C4—42no—
390C2×C4—82no—
391C2×C4—82no—
392C2×C4—84no—
393C2×C4—84no—
394C2×C4—84no—
395C2×C4—82no—
396C2×C4C2×#9911yescharacteristic
397C2×C4C2×#9911yescharacteristic
398C2×C4C2×#13111yescharacteristic
399C2×C4#56811yescharacteristic
400C8—82no—
401C8—84no—
402C8—82no—
403C22—22no—
404C22—21no—
405C22—21no—
406C22—22no—
407C22—22no—
408C22—22no—
409C22—22no—
410C22—44no—
411C22—44no—
412C22—42no—
413C22—42no—
414C22—42no—
415C22—82no—
416C22—82no—
417C22—82no—
418C22—82no—
419C22—81no—
420C22—81no—
421C22—81no—
422C22C2×#35314yes—
423C22#288011yescharacteristic
424C22C22×#9911yescharacteristic
425C22#288911yescharacteristic
426C4—24no—
427C4—21no—
428C4—21no—
429C4—21no—
430C4—42no—
431C4—44no—
432C4—42no—
433C4—42no—
434C4—44no—
435C4—41no—
436C4—81no—
437C4—82no—
438C4C2×#35312yes—
439C2—21no—
440C2—21no—
441C2—24no—
442C2—42no—
443C2—82no—
444C2—161no—
445C2#1741611yescharacteristic
446C2C22×#35311yescharacteristic
447C2#3337211yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC22×C4C23×C4
Frattini, cosocleC2×C4⋊C4C24
prime radical#36179C2
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
2. #17416C2semidirect
3. #11570C2semidirect
4. #11549C2semidirect
5. #11529C2semidirect
6. #11512C2semidirect
7. #11475C2semidirect
8. C2×#1325C2semidirect
324. C42#131semidirect

Maximal subgroups

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

subgroupindex
#361792
#174162
#115702
#115492
#115292
#115122
#114752
C2×#13252