Finite Group Databasebeta

← #481718

Subgroups — #481718

Order 512.

1035 conjugacy classes of subgroups.

The subgroup lattice has 434 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36030C211yesmaximal, characteristic
2#20355C211yesmaximal, characteristic, complemented
3#17371C211yesmaximal, characteristic, complemented
4#11758C214yesmaximal
5#11756C214yesmaximal, complemented
6#11717C212yesmaximal, complemented
7#11691C212yesmaximal
8C2×#556C411yescharacteristic, complemented
9C2×#519—21no—
10C2×#519C2211yescharacteristic
11#2983—21no—
12#2632C412yescomplemented
13#2628C2211yescharacteristic
14#2610C2212yes—
15#2599C2212yes—
16#2567C411yescharacteristic, complemented
17#2536C411yescharacteristic, complemented
18#2479—22no—
19#2469—22no—
20#2423—21no—
21#2409—21no—
22#2399C412yescomplemented
23#2392C2211yescharacteristic
24#2362—21no—
25#2356C411yescharacteristic, complemented
26#2354—21no—
27#1740—21no—
28#1740—22no—
29#1740C2212yes—
30#1740C2214yes—
31#1467—22no—
32#1464C2214yes—
33#1461—21no—
34#1458C2212yes—
35#1351C2218yes—
36#1344C2214yes—
37#1343C2214yes—
38C23×Q8D411yescharacteristic
39#556—44no—
40C2×C4⋊Q8C2×C411yescharacteristic
41C2×#126—22no—
42C2×#124C2×C412yes—
43C2×C22⋊Q8—21no—
44C2×C22⋊Q8—21no—
45#519—28no—
46#519—44no—
47#519—44no—
48C2×C4×Q8—44no—
49C2×C4×Q8C2×C412yes—
50C2×C4×Q8C2×C411yescharacteristic
51C2×C4×Q8C2×C412yes—
52C2×C4×Q8C2×C411yescharacteristic
53C2×C4×Q8D411yescharacteristic
54C2×C4×Q8D412yes—
55C2×C4×Q8D411yescharacteristic
56C2×C4×D4—21no—
57C2×C4×D4C2×C411yescharacteristic
58C2×#117—21no—
59C2×#117—24no—
60C2×#117—22no—
61C2×#117D411yescharacteristic
62C2×#117D412yes—
63C2×#117C2312yes—
64C2×#117C2311yescharacteristic
65#446—44no—
66#432—24no—
67#431—22no—
68#400—21no—
69#399—22no—
70#399—21no—
71#398—22no—
72#398C2×C412yes—
73#398C2×C411yescharacteristic
74#397C2311yescharacteristic
75#396—21no—
76#395—22no—
77#395C2×C412yes—
78#395C2×C412yes—
79#395C2×C412yes—
80#394—21no—
81#394C2×C412yes—
82#394C2×C411yescharacteristic
83#393—22no—
84#393C2312yes—
85#392C2311yescharacteristic
86#391—21no—
87#391—21no—
88#390—22no—
89#389—22no—
90#389—21no—
91#388—22no—
92#388—22no—
93#388—22no—
94#388—22no—
95#388—22no—
96#388—22no—
97#387—22no—
98#387—22no—
99#387—22no—
100#386—22no—
101#385—22no—
102#384—21no—
103#383C2×C412yes—
104#382C2×C411yescharacteristic
105#380—21no—
106#380—21no—
107C4×C4⋊C4—21no—
108C4×C4⋊C4—22no—
109C4×C4⋊C4—22no—
110C4×C4⋊C4C2×C411yescharacteristic
111C4×C22⋊C4—21no—
112#376C2×C412yes—
113#342—24no—
114#342—22no—
115#342C2314yes—
116#324—22no—
117#324—24no—
118#323C2314yes—
119C22×Q8—24no—
120C22×Q8—24no—
121C22×Q8—22no—
122C22×Q8—22no—
123C22×Q8C22⋊C412yes—
124C22×Q8C22⋊C411yescharacteristic
125C22×Q8C22⋊C411yescharacteristic
126C22×Q8C22×C412yes—
127C22×D4—21no—
128C22×D4C22×C411yescharacteristic
129C23×C4—21no—
130C23×C4C2×D412yes—
131C23×C4C2×D411yescharacteristic
132C4⋊Q8—42no—
133#126—48no—
134#124—44no—
135C22⋊Q8—44no—
136C22⋊Q8—44no—
137C4×Q8—24no—
138C4×Q8—24no—
139C4×Q8—42no—
140C4×Q8—44no—
141C4×Q8—42no—
142C4×Q8—44no—
143C4×Q8—44no—
144C4×Q8—432no—
145C4×D4—42no—
146C4×D4—42no—
147C4×D4—42no—
148#117—216no—
149#117—28no—
150#117—42no—
151#117—44no—
152#117—42no—
153#117—44no—
154#117—44no—
155#117—48no—
156#117—44no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—44no—
168C2×C4⋊C4—42no—
169C2×C4⋊C4C22⋊C411yescharacteristic
170C2×C4⋊C4C22⋊C412yes—
171C2×C4⋊C4C22⋊C411yescharacteristic
172C2×C4⋊C4C22⋊C412yes—
173C2×C4⋊C4C22×C411yescharacteristic
174C2×C4⋊C4C22×C411yescharacteristic
175C2×C4⋊C4C22×C412yes—
176C2×C4⋊C4C2×D412yes—
177C2×C4⋊C4C2×D411yescharacteristic
178C2×C22⋊C4—24no—
179C2×C22⋊C4—21no—
180C2×C22⋊C4—22no—
181C2×C22⋊C4—21no—
182C2×C22⋊C4—21no—
183C2×C22⋊C4—22no—
184C2×C22⋊C4—22no—
185C2×C22⋊C4—21no—
186C2×C22⋊C4—22no—
187C2×C22⋊C4—21no—
188C2×C22⋊C4—22no—
189C2×C22⋊C4C2×D412yes—
190C2×C22⋊C4C2×D411yescharacteristic
191C2×C22⋊C4C2411yescharacteristic
192C2×C42—22no—
193C2×C42—21no—
194C2×C42—22no—
195C2×C42—44no—
196C2×C42—42no—
197C2×C42C22⋊C412yes—
198C2×C42C22⋊C412yes—
199C2×C42C22⋊C412yes—
200C2×C42C22×C411yescharacteristic
201C2×C42C22×C412yes—
202C2×C42C2×D411yescharacteristic
203C2×C42C2×D412yes—
204C4⋊C8—42no—
205C4⋊C8—44no—
206C22⋊C8—24no—
207#97—44no—
208C4×C8—42no—
209#95—22no—
210#95—21no—
211#95—22no—
212#95—22no—
213#95—22no—
214#95—22no—
215#95—21no—
216#95—22no—
217#95—22no—
218#95—21no—
219#95—22no—
220#95C22×C411yescharacteristic
221#95C22×C412yes—
222#95C22×C411yescharacteristic
223C24—21no—
224C24C22×D411yescharacteristic
225C2×Q8—216no—
226C2×Q8—216no—
227C2×Q8—24no—
228C2×Q8—22no—
229C2×Q8—24no—
230C2×Q8—22no—
231C2×Q8—22no—
232C2×Q8—24no—
233C2×Q8—22no—
234C2×Q8—42no—
235C2×Q8—82no—
236C2×Q8#9914yes—
237C2×Q8#10414yes—
238C2×D4—41no—
239C2×D4—42no—
240C2×D4—41no—
241C2×D4—41no—
242C2×D4—81no—
243C2×D4—81no—
244C22×C4—22no—
245C22×C4—22no—
246C22×C4—24no—
247C22×C4—22no—
248C22×C4—24no—
249C22×C4—22no—
250C22×C4—21no—
251C22×C4—22no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—22no—
255C22×C4—44no—
256C22×C4—42no—
257C22×C4—42no—
258C22×C4—42no—
259C22×C4C2×C22⋊C412yes—
260C22×C4C2×C22⋊C411yescharacteristic
261C22×C4C2×C22⋊C412yes—
262C22×C4C2×C22⋊C412yes—
263C22×C4C2×C22⋊C411yescharacteristic
264C22×C4C2×C22⋊C411yescharacteristic
265C22×C4C2×C22⋊C412yes—
266C22×C4C2×C22⋊C411yescharacteristic
267C22×C4AGL(1,ℤ/8)12yes—
268C22×C4#13712yes—
269C22×C4C23×C411yescharacteristic
270C22×C4C22×D411yescharacteristic
271C2×C8—44no—
272C4⋊C4—22no—
273C4⋊C4—24no—
274C4⋊C4—22no—
275C4⋊C4—24no—
276C4⋊C4—41no—
277C4⋊C4—42no—
278C4⋊C4—42no—
279C4⋊C4—41no—
280C4⋊C4—41no—
281C4⋊C4—42no—
282C4⋊C4—42no—
283C4⋊C4—416no—
284C4⋊C4—48no—
285C4⋊C4—42no—
286C4⋊C4—44no—
287C4⋊C4—42no—
288C4⋊C4—42no—
289C4⋊C4—44no—
290C4⋊C4—82no—
291C4⋊C4—82no—
292C4⋊C4—82no—
293C22⋊C4—28no—
294C22⋊C4—42no—
295C22⋊C4—41no—
296C22⋊C4—41no—
297C22⋊C4—42no—
298C22⋊C4—44no—
299C22⋊C4—42no—
300C22⋊C4—44no—
301C22⋊C4—41no—
302C22⋊C4—41no—
303C22⋊C4—44no—
304C22⋊C4—42no—
305C22⋊C4—44no—
306C22⋊C4—82no—
307C22⋊C4—81no—
308C22⋊C4—82no—
309C42—22no—
310C42—24no—
311C42—22no—
312C42—24no—
313C42—42no—
314C42—42no—
315C42—42no—
316C42—44no—
317C42—44no—
318C42—416no—
319C42—48no—
320C42—42no—
321C23—21no—
322C23—21no—
323C23—41no—
324C23—41no—
325C23—41no—
326C23—41no—
327C23—41no—
328C23—41no—
329C23—41no—
330C23—41no—
331C23—41no—
332C23C22×C22⋊C411yescharacteristic
333C23C2×AGL(1,ℤ/8)11yescharacteristic
334C23C2×#13711yescharacteristic
335Q8—216no—
336Q8—28no—
337Q8—28no—
338Q8—84no—
339D4—82no—
340D4—82no—
341D4—82no—
342C2×C4—28no—
343C2×C4—28no—
344C2×C4—21no—
345C2×C4—22no—
346C2×C4—22no—
347C2×C4—22no—
348C2×C4—22no—
349C2×C4—21no—
350C2×C4—24no—
351C2×C4—24no—
352C2×C4—22no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—22no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—22no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—42no—
362C2×C4—41no—
363C2×C4—42no—
364C2×C4—41no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—48no—
368C2×C4—42no—
369C2×C4—42no—
370C2×C4—41no—
371C2×C4—42no—
372C2×C4—41no—
373C2×C4—42no—
374C2×C4—41no—
375C2×C4—42no—
376C2×C4—41no—
377C2×C4—41no—
378C2×C4—42no—
379C2×C4—41no—
380C2×C4—84no—
381C2×C4—84no—
382C2×C4—84no—
383C2×C4C2×#9914yes—
384C2×C4C2×#9912yes—
385C2×C4#41712yes—
386C2×C4#41712yes—
387C2×C4C2×#10414yes—
388C2×C4C2×#10412yes—
389C8—84no—
390C22—21no—
391C22—21no—
392C22—21no—
393C22—21no—
394C22—41no—
395C22—41no—
396C22—41no—
397C22—41no—
398C22—81no—
399C22—81no—
400C22—81no—
401C22—81no—
402C22—81no—
403C22—81no—
404C22—81no—
405C22—81no—
406C22C22×#9911yescharacteristic
407C22C2×#41711yescharacteristic
408C22C22×#10411yescharacteristic
409C4—24no—
410C4—22no—
411C4—24no—
412C4—42no—
413C4—41no—
414C4—42no—
415C4—41no—
416C4—41no—
417C4—42no—
418C4—41no—
419C4—81no—
420C4—82no—
421C4—88no—
422C4—82no—
423C4—81no—
424C4#150414yes—
425C2—21no—
426C2—21no—
427C2—41no—
428C2—41no—
429C2—161no—
430C2C2×#150411yescharacteristic
431C2#3335111yescharacteristic
432C2#3336011yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC22×C4C23×C4
Frattini, cosocleC2×C22⋊C4C24
prime radical#392C23
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×Q8 ◁ C2×#519 ◁ G
LECSG ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #20355C2semidirect
3. #17371C2semidirect
5. #11756C2semidirect
6. #11717C2semidirect
8. C2×#556C4semidirect
12. #2632C4semidirect
16. #2567C4semidirect
17. #2536C4semidirect
22. #2399C4semidirect
25. #2356C4semidirect

Maximal subgroups

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

subgroupindex
#360302
#203552
#173712
#117582
#117562
#117172
#116912