Finite Group Databasebeta

← #481932

Subgroups — #481932

Order 512.

981 conjugacy classes of subgroups.

The subgroup lattice has 457 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35693C211yesmaximal, characteristic
2#20634C211yesmaximal, characteristic, complemented
3#17372C211yesmaximal, characteristic, complemented
4#11826C214yesmaximal, complemented
5#11820C214yesmaximal, complemented
6#11717C212yesmaximal, complemented
7#11691C212yesmaximal, complemented
8C2×#547C411yescharacteristic, complemented
9C2×#520C2211yescharacteristic
10#2983—21no—
11C2×#410—21no—
12#2638C2212yes—
13#2628C2211yescharacteristic
14#2583C412yescomplemented
15#2567C411yescharacteristic, complemented
16#2536C411yescharacteristic, complemented
17#2470—22no—
18#2455—22no—
19#2422—21no—
20#2407—21no—
21#2406C412yescomplemented
22#2392C2211yescharacteristic
23#2388C2212yes—
24#2360—21no—
25#2357C411yescharacteristic, complemented
26D4×C4⋊C4—21no—
27#2316—21no—
28#1799—22no—
29#1799C2214yes—
30#1740C2212yes—
31#1466C2214yes—
32#1464—22no—
33#1460—21no—
34#1458C2212yes—
35#1351C2218yes—
36#1344C2214yes—
37#1343C2214yes—
38C22×#41C2×C412yes—
39C22×#41D411yescharacteristic
40#547—44no—
41C2×C4⋊Q8C2×C411yescharacteristic
42C2×#123—22no—
43C2×#123—22no—
44C2×C22⋊Q8—21no—
45C2×C22⋊Q8C2×C412yes—
46C2×#121—21no—
47#520—44no—
48C2×C4×Q8C2×C411yescharacteristic
49C2×C4×Q8D411yescharacteristic
50C2×C4×D4—21no—
51C2×C4×D4—22no—
52C2×C4×D4C2×C411yescharacteristic
53C2×C4×D4C2×C411yescharacteristic
54C2×C4×D4D412yes—
55C2×C4×D4D411yescharacteristic
56C2×C4×D4C2312yes—
57C2×#117—22no—
58C2×#117D412yes—
59C2×#117C2311yescharacteristic
60C22×C4⋊C4—21no—
61C22×C4⋊C4C2×C412yes—
62C22×C4⋊C4D411yescharacteristic
63#446—44no—
64#432—24no—
65#431—22no—
66#410—44no—
67C2×#99—24no—
68#400—21no—
69#399—21no—
70#398C2×C411yescharacteristic
71#398C2×C412yes—
72#397—22no—
73#397C2311yescharacteristic
74#396—21no—
75#395C2×C412yes—
76#395C2×C412yes—
77#395C2×C412yes—
78#394—22no—
79#394—22no—
80#394—21no—
81#394C2×C412yes—
82#394C2×C411yescharacteristic
83#393C2312yes—
84#392—22no—
85#392C2311yescharacteristic
86#391—21no—
87#390—21no—
88#389—22no—
89#389—22no—
90#389—22no—
91#389—21no—
92#389—44no—
93#388—22no—
94#388—22no—
95#387—22no—
96#386—22no—
97#385—22no—
98#385—22no—
99#385—22no—
100#385—22no—
101#385—22no—
102#384—21no—
103#382C2×C411yescharacteristic
104#381C2×C412yes—
105#380—21no—
106#380—21no—
107#380—24no—
108C4×C4⋊C4—21no—
109C4×C4⋊C4C2×C411yescharacteristic
110C4×C22⋊C4—22no—
111C4×C22⋊C4—21no—
112C4×C22⋊C4C2×C412yes—
113#342—22no—
114#342C2314yes—
115#324—22no—
116#323—24no—
117#323C2314yes—
118C2×#41—22no—
119C2×#41—22no—
120C2×#41—42no—
121C2×#41—42no—
122C2×#41—44no—
123C2×#41—84no—
124C22×Q8C22⋊C411yescharacteristic
125C22×Q8C22×C412yes—
126C22×D4—21no—
127C22×D4—22no—
128C22×D4C22⋊C411yescharacteristic
129C22×D4C22×C411yescharacteristic
130C22×D4C2×D412yes—
131C23×C4—21no—
132C23×C4—22no—
133C23×C4C22⋊C412yes—
134C23×C4C22⋊C412yes—
135C23×C4C22×C412yes—
136C23×C4C2×D411yescharacteristic
137C23×C4C2×D412yes—
138C4⋊Q8—42no—
139#123—48no—
140#123—84no—
141C22⋊Q8—44no—
142C22⋊Q8—44no—
143#121—44no—
144C4×Q8—24no—
145C4×Q8—42no—
146C4×D4—24no—
147C4×D4—42no—
148C4×D4—44no—
149C4×D4—42no—
150C4×D4—44no—
151C4×D4—42no—
152C4×D4—42no—
153C4×D4—84no—
154#117—42no—
155#117—44no—
156#117—44no—
157#117—44no—
158C2×C4⋊C4—24no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—44no—
170C2×C4⋊C4—44no—
171C2×C4⋊C4—42no—
172C2×C4⋊C4—44no—
173C2×C4⋊C4C22⋊C412yes—
174C2×C4⋊C4C22⋊C412yes—
175C2×C4⋊C4C22⋊C411yescharacteristic
176C2×C4⋊C4C22⋊C411yescharacteristic
177C2×C4⋊C4C22×C411yescharacteristic
178C2×C4⋊C4C22×C411yescharacteristic
179C2×C4⋊C4C22×C412yes—
180C2×C4⋊C4C2×D411yescharacteristic
181C2×C22⋊C4—22no—
182C2×C22⋊C4—22no—
183C2×C22⋊C4—21no—
184C2×C22⋊C4—22no—
185C2×C22⋊C4—21no—
186C2×C22⋊C4—21no—
187C2×C22⋊C4—22no—
188C2×C22⋊C4—21no—
189C2×C22⋊C4—21no—
190C2×C22⋊C4—24no—
191C2×C22⋊C4—22no—
192C2×C22⋊C4—22no—
193C2×C22⋊C4C22⋊C412yes—
194C2×C22⋊C4C2×D411yescharacteristic
195C2×C22⋊C4C2×D412yes—
196C2×C22⋊C4C2411yescharacteristic
197C2×C42—21no—
198C2×C42—22no—
199C2×C42—42no—
200C2×C42C22⋊C412yes—
201C2×C42C22×C411yescharacteristic
202C2×C42C2×D412yes—
203C2×C42C2×D411yescharacteristic
204C4⋊C8—42no—
205C4⋊C8—44no—
206#99—48no—
207C22⋊C8—24no—
208#97—44no—
209C4×C8—42no—
210#95—21no—
211#95—22no—
212#95—22no—
213#95—22no—
214#95—22no—
215#95—21no—
216#95—22no—
217#95—21no—
218#95—22no—
219#95—44no—
220#95C22×C411yescharacteristic
221#95C22×C412yes—
222#95C22×C411yescharacteristic
223C24—21no—
224C24C2×C22⋊C412yes—
225C24C22×D411yescharacteristic
226#41—48no—
227#41—48no—
228#41—88no—
229#41—88no—
230C2×Q8—22no—
231C2×Q8—22no—
232C2×Q8—22no—
233C2×Q8—42no—
234C2×Q8—82no—
235C2×D4—22no—
236C2×D4—22no—
237C2×D4—22no—
238C2×D4—22no—
239C2×D4—22no—
240C2×D4—42no—
241C2×D4—42no—
242C2×D4—41no—
243C2×D4—42no—
244C2×D4—42no—
245C2×D4—41no—
246C2×D4—41no—
247C2×D4—42no—
248C2×D4—44no—
249C2×D4—44no—
250C2×D4—82no—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—21no—
254C22×C4—24no—
255C22×C4—24no—
256C22×C4—24no—
257C22×C4—22no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—22no—
261C22×C4—42no—
262C22×C4—42no—
263C22×C4—42no—
264C22×C4—44no—
265C22×C4—42no—
266C22×C4—44no—
267C22×C4—42no—
268C22×C4—42no—
269C22×C4—44no—
270C22×C4—82no—
271C22×C4—82no—
272C22×C4—82no—
273C22×C4—82no—
274C22×C4C2×C22⋊C412yes—
275C22×C4C2×C22⋊C411yescharacteristic
276C22×C4C2×C22⋊C411yescharacteristic
277C22×C4C2×C22⋊C412yes—
278C22×C4C2×C22⋊C411yescharacteristic
279C22×C4C2×C22⋊C412yes—
280C22×C4C2×C22⋊C411yescharacteristic
281C22×C4AGL(1,ℤ/8)12yes—
282C22×C4#13712yes—
283C22×C4C23×C411yescharacteristic
284C22×C4C22×D411yescharacteristic
285C2×C8—44no—
286C4⋊C4—24no—
287C4⋊C4—22no—
288C4⋊C4—24no—
289C4⋊C4—41no—
290C4⋊C4—42no—
291C4⋊C4—41no—
292C4⋊C4—41no—
293C4⋊C4—42no—
294C4⋊C4—44no—
295C4⋊C4—42no—
296C4⋊C4—42no—
297C4⋊C4—44no—
298C4⋊C4—42no—
299C4⋊C4—42no—
300C4⋊C4—84no—
301C4⋊C4—82no—
302C4⋊C4—82no—
303C4⋊C4—82no—
304C4⋊C4—164no—
305C4⋊C4#9914yes—
306C22⋊C4—24no—
307C22⋊C4—42no—
308C22⋊C4—41no—
309C22⋊C4—41no—
310C22⋊C4—42no—
311C22⋊C4—44no—
312C22⋊C4—42no—
313C22⋊C4—44no—
314C22⋊C4—44no—
315C22⋊C4—44no—
316C22⋊C4—42no—
317C22⋊C4—41no—
318C22⋊C4—41no—
319C22⋊C4—42no—
320C22⋊C4—44no—
321C22⋊C4—44no—
322C22⋊C4—82no—
323C22⋊C4—82no—
324C42—22no—
325C42—24no—
326C42—22no—
327C42—42no—
328C42—42no—
329C42—84no—
330C42—82no—
331C23—21no—
332C23—21no—
333C23—22no—
334C23—22no—
335C23—22no—
336C23—22no—
337C23—41no—
338C23—41no—
339C23—41no—
340C23—41no—
341C23—41no—
342C23—42no—
343C23—42no—
344C23—41no—
345C23—41no—
346C23—41no—
347C23C22×C22⋊C411yescharacteristic
348C23C2×AGL(1,ℤ/8)11yescharacteristic
349C23C2×#13711yescharacteristic
350Q8—44no—
351Q8—84no—
352D4—44no—
353D4—44no—
354D4—42no—
355D4—44no—
356D4—44no—
357D4—42no—
358D4—44no—
359D4—88no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—24no—
363C2×C4—24no—
364C2×C4—22no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—22no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—22no—
371C2×C4—22no—
372C2×C4—22no—
373C2×C4—42no—
374C2×C4—48no—
375C2×C4—41no—
376C2×C4—41no—
377C2×C4—42no—
378C2×C4—41no—
379C2×C4—41no—
380C2×C4—41no—
381C2×C4—41no—
382C2×C4—42no—
383C2×C4—41no—
384C2×C4—42no—
385C2×C4—42no—
386C2×C4—41no—
387C2×C4—42no—
388C2×C4—41no—
389C2×C4—42no—
390C2×C4—41no—
391C2×C4—41no—
392C2×C4—42no—
393C2×C4—41no—
394C2×C4—84no—
395C2×C4—84no—
396C2×C4—88no—
397C2×C4—84no—
398C2×C4—84no—
399C2×C4—84no—
400C2×C4—84no—
401C2×C4—162no—
402C2×C4C2×#9914yes—
403C2×C4C2×#9912yes—
404C2×C4#41712yes—
405C2×C4#41712yes—
406C2×C4#42112yes—
407C8—84no—
408C22—21no—
409C22—21no—
410C22—21no—
411C22—21no—
412C22—41no—
413C22—41no—
414C22—42no—
415C22—42no—
416C22—41no—
417C22—41no—
418C22—42no—
419C22—42no—
420C22—42no—
421C22—42no—
422C22—81no—
423C22—82no—
424C22—82no—
425C22—81no—
426C22—81no—
427C22—81no—
428C22—81no—
429C22C22×#9911yescharacteristic
430C22C2×#41711yescharacteristic
431C22C2×#42111yescharacteristic
432C4—24no—
433C4—22no—
434C4—42no—
435C4—41no—
436C4—42no—
437C4—41no—
438C4—41no—
439C4—41no—
440C4—42no—
441C4—81no—
442C4—82no—
443C4—82no—
444C4—81no—
445C4—164no—
446C4#151314yes—
447C2—21no—
448C2—21no—
449C2—41no—
450C2—41no—
451C2—82no—
452C2—82no—
453C2C2×#151311yescharacteristic
454C2#3334811yescharacteristic
455C2#3336011yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC22×C4C23×C4
Frattini, cosocleC2×C22⋊C4C24
prime radical#35693C2
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C4⋊C4 ◁ C2×#520 ◁ G
LECSG ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #20634C2semidirect
3. #17372C2semidirect
4. #11826C2semidirect
5. #11820C2semidirect
6. #11717C2semidirect
7. #11691C2semidirect
8. C2×#547C4semidirect
14. #2583C4semidirect
15. #2567C4semidirect
16. #2536C4semidirect
21. #2406C4semidirect
25. #2357C4semidirect

Maximal subgroups

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

subgroupindex
#356932
#206342
#173722
#118262
#118202
#117172
#116912