Finite Group Databasebeta

← #481715

Subgroups — #481715

Order 512.

1042 conjugacy classes of subgroups.

The subgroup lattice has 436 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35738C211yesmaximal, characteristic
2#20355C211yesmaximal, characteristic, complemented
3#17371C211yesmaximal, characteristic, complemented
4#11760C214yesmaximal, complemented
5#11754C214yesmaximal
6#11718C212yesmaximal, complemented
7#11691C212yesmaximal
8C2×#554C411yescharacteristic, complemented
9C2×#519—21no—
10C2×#519C2211yescharacteristic
11#2983—21no—
12#2634C412yescomplemented
13#2628C2211yescharacteristic
14#2611C2212yes—
15#2598C2212yes—
16#2568C411yescharacteristic, complemented
17#2536C411yescharacteristic, complemented
18#2477—22no—
19#2465—22no—
20#2421—21no—
21#2409—21no—
22#2398C412yescomplemented
23#2392C2211yescharacteristic
24#2360—21no—
25#2356C411yescharacteristic, complemented
26#2352—21no—
27#1740—21no—
28#1740—22no—
29#1740C2212yes—
30#1740C2214yes—
31#1467—22no—
32#1464C2214yes—
33#1461—21no—
34#1458C2212yes—
35#1352C2218yes—
36#1344C2214yes—
37#1343C2214yes—
38C23×Q8D411yescharacteristic
39#554—44no—
40C2×C4⋊Q8C2×C411yescharacteristic
41C2×#125C2×C412yes—
42C2×C22⋊Q8—21no—
43C2×C22⋊Q8—21no—
44C2×C22⋊Q8—22no—
45#519—28no—
46#519—44no—
47#519—44no—
48C2×C4×Q8—44no—
49C2×C4×Q8C2×C411yescharacteristic
50C2×C4×Q8C2×C412yes—
51C2×C4×Q8C2×C411yescharacteristic
52C2×C4×Q8D411yescharacteristic
53C2×C4×Q8D412yes—
54C2×C4×Q8D411yescharacteristic
55C2×C4×D4—21no—
56C2×C4×D4C2×C412yes—
57C2×C4×D4C2×C411yescharacteristic
58C2×#117—24no—
59C2×#117—22no—
60C2×#117D411yescharacteristic
61C2×#117D412yes—
62C2×#117C2312yes—
63C2×#117C2311yescharacteristic
64C22×C4⋊C4—21no—
65#446—44no—
66#432—24no—
67#431—22no—
68#400—22no—
69#400C2×C412yes—
70#400C2×C412yes—
71#400C2×C412yes—
72#399C2312yes—
73#398C2×C412yes—
74#398C2×C411yescharacteristic
75#397—22no—
76#397C2311yescharacteristic
77#396—21no—
78#395—22no—
79#395—21no—
80#394—21no—
81#394C2×C412yes—
82#394C2×C411yescharacteristic
83#393—22no—
84#393—21no—
85#392C2311yescharacteristic
86#391—21no—
87#391—21no—
88#391—22no—
89#389—22no—
90#389—21no—
91#388—22no—
92#388—22no—
93#388—22no—
94#387—22no—
95#387—22no—
96#387—22no—
97#387—22no—
98#386—22no—
99#386—22no—
100#385—22no—
101#385—22no—
102#383C2×C412yes—
103#382—21no—
104#382C2×C411yescharacteristic
105#380—21no—
106#380—21no—
107C4×C4⋊C4—21no—
108C4×C4⋊C4C2×C411yescharacteristic
109C4×C22⋊C4—22no—
110C4×C22⋊C4—22no—
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—21no—
122C22×Q8—22no—
123C22×Q8—22no—
124C22×Q8C22⋊C412yes—
125C22×Q8C22⋊C411yescharacteristic
126C22×Q8C22⋊C411yescharacteristic
127C22×D4—21no—
128C22×D4C22×C411yescharacteristic
129C23×C4—21no—
130C23×C4—22no—
131C23×C4C2×D412yes—
132C23×C4C2×D411yescharacteristic
133C4⋊Q8—42no—
134#125—44no—
135C22⋊Q8—48no—
136C22⋊Q8—44no—
137C22⋊Q8—44no—
138C4×Q8—24no—
139C4×Q8—24no—
140C4×Q8—44no—
141C4×Q8—42no—
142C4×Q8—42no—
143C4×Q8—44no—
144C4×Q8—432no—
145C4×D4—42no—
146C4×D4—44no—
147C4×D4—42no—
148C4×D4—42no—
149#117—216no—
150#117—28no—
151#117—42no—
152#117—44no—
153#117—42no—
154#117—44no—
155#117—48no—
156#117—44no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—44no—
166C2×C4⋊C4—44no—
167C2×C4⋊C4—42no—
168C2×C4⋊C4C22⋊C411yescharacteristic
169C2×C4⋊C4C22⋊C412yes—
170C2×C4⋊C4C22⋊C411yescharacteristic
171C2×C4⋊C4C22⋊C412yes—
172C2×C4⋊C4C22×C411yescharacteristic
173C2×C4⋊C4C22×C412yes—
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—21no—
181C2×C22⋊C4—22no—
182C2×C22⋊C4—21no—
183C2×C22⋊C4—22no—
184C2×C22⋊C4—21no—
185C2×C22⋊C4—22no—
186C2×C22⋊C4—22no—
187C2×C22⋊C4—21no—
188C2×C22⋊C4C2×D412yes—
189C2×C22⋊C4C2×D411yescharacteristic
190C2×C22⋊C4C2411yescharacteristic
191C2×C42—21no—
192C2×C42—22no—
193C2×C42—44no—
194C2×C42—42no—
195C2×C42C22⋊C412yes—
196C2×C42C22⋊C412yes—
197C2×C42C22⋊C412yes—
198C2×C42C22×C412yes—
199C2×C42C22×C411yescharacteristic
200C2×C42C2×D411yescharacteristic
201C2×C42C2×D412yes—
202C4⋊C8—42no—
203C4⋊C8—44no—
204C22⋊C8—24no—
205#97—44no—
206C4×C8—42no—
207#95—22no—
208#95—22no—
209#95—22no—
210#95—22no—
211#95—21no—
212#95—22no—
213#95—22no—
214#95—22no—
215#95—22no—
216#95—21no—
217#95—22no—
218#95—22no—
219#95—21no—
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—42no—
241C2×D4—41no—
242C2×D4—41no—
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—21no—
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—21no—
255C22×C4—44no—
256C22×C4—44no—
257C22×C4—42no—
258C22×C4—42no—
259C22×C4—42no—
260C22×C4—82no—
261C22×C4—82no—
262C22×C4C2×C22⋊C412yes—
263C22×C4C2×C22⋊C411yescharacteristic
264C22×C4C2×C22⋊C412yes—
265C22×C4C2×C22⋊C412yes—
266C22×C4C2×C22⋊C411yescharacteristic
267C22×C4C2×C22⋊C412yes—
268C22×C4C2×C22⋊C411yescharacteristic
269C22×C4C2×C22⋊C411yescharacteristic
270C22×C4AGL(1,ℤ/8)12yes—
271C22×C4#13712yes—
272C22×C4C23×C411yescharacteristic
273C22×C4C22×D411yescharacteristic
274C2×C8—44no—
275C4⋊C4—22no—
276C4⋊C4—24no—
277C4⋊C4—22no—
278C4⋊C4—24no—
279C4⋊C4—41no—
280C4⋊C4—42no—
281C4⋊C4—41no—
282C4⋊C4—42no—
283C4⋊C4—41no—
284C4⋊C4—42no—
285C4⋊C4—44no—
286C4⋊C4—416no—
287C4⋊C4—48no—
288C4⋊C4—42no—
289C4⋊C4—44no—
290C4⋊C4—42no—
291C4⋊C4—42no—
292C4⋊C4—44no—
293C4⋊C4—82no—
294C4⋊C4—81no—
295C22⋊C4—28no—
296C22⋊C4—41no—
297C22⋊C4—42no—
298C22⋊C4—41no—
299C22⋊C4—42no—
300C22⋊C4—42no—
301C22⋊C4—44no—
302C22⋊C4—44no—
303C22⋊C4—41no—
304C22⋊C4—41no—
305C22⋊C4—44no—
306C22⋊C4—42no—
307C22⋊C4—44no—
308C22⋊C4—81no—
309C22⋊C4—82no—
310C42—22no—
311C42—24no—
312C42—24no—
313C42—22no—
314C42—42no—
315C42—42no—
316C42—42no—
317C42—416no—
318C42—48no—
319C42—42no—
320C42—82no—
321C23—21no—
322C23—21no—
323C23—22no—
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—162no—
339D4—42no—
340D4—42no—
341D4—82no—
342D4—82no—
343C2×C4—28no—
344C2×C4—28no—
345C2×C4—21no—
346C2×C4—22no—
347C2×C4—22no—
348C2×C4—22no—
349C2×C4—22no—
350C2×C4—21no—
351C2×C4—24no—
352C2×C4—24no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—22no—
357C2×C4—21no—
358C2×C4—22no—
359C2×C4—42no—
360C2×C4—41no—
361C2×C4—42no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—48no—
366C2×C4—41no—
367C2×C4—42no—
368C2×C4—41no—
369C2×C4—42no—
370C2×C4—41no—
371C2×C4—42no—
372C2×C4—42no—
373C2×C4—41no—
374C2×C4—41no—
375C2×C4—42no—
376C2×C4—41no—
377C2×C4—41no—
378C2×C4—41no—
379C2×C4—42no—
380C2×C4—41no—
381C2×C4—84no—
382C2×C4—84no—
383C2×C4—84no—
384C2×C4—164no—
385C2×C4C2×#9914yes—
386C2×C4C2×#9912yes—
387C2×C4#41712yes—
388C2×C4#41712yes—
389C2×C4C2×#10414yes—
390C2×C4C2×#10412yes—
391C8—84no—
392C22—21no—
393C22—21no—
394C22—21no—
395C22—21no—
396C22—41no—
397C22—41no—
398C22—42no—
399C22—42no—
400C22—41no—
401C22—41no—
402C22—42no—
403C22—81no—
404C22—81no—
405C22—81no—
406C22—81no—
407C22—81no—
408C22C22×#9911yescharacteristic
409C22C2×#41711yescharacteristic
410C22C22×#10411yescharacteristic
411C4—24no—
412C4—22no—
413C4—24no—
414C4—41no—
415C4—42no—
416C4—41no—
417C4—41no—
418C4—42no—
419C4—41no—
420C4—81no—
421C4—82no—
422C4—81no—
423C4—88no—
424C4—82no—
425C4—81no—
426C4#150414yes—
427C2—21no—
428C2—21no—
429C2—41no—
430C2—41no—
431C2—82no—
432C2C2×#150411yescharacteristic
433C2#3334911yescharacteristic
434C2#3336211yescharacteristic
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
4. #11760C2semidirect
6. #11718C2semidirect
8. C2×#554C4semidirect
12. #2634C4semidirect
16. #2568C4semidirect
17. #2536C4semidirect
22. #2398C4semidirect
25. #2356C4semidirect

Maximal subgroups

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

subgroupindex
#357382
#203552
#173712
#117602
#117542
#117182
#116912