Finite Group Databasebeta

← #481661

Subgroups — #481661

Order 512.

1005 conjugacy classes of subgroups.

The subgroup lattice has 426 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35739C211yesmaximal, characteristic
2#20393C211yesmaximal, characteristic
3#17372C211yesmaximal, characteristic
4#11789C214yesmaximal, complemented
5#11785C214yesmaximal, complemented
6#11717C212yesmaximal, complemented
7#11715C212yesmaximal, complemented
8C2×D4×Q8C411yescharacteristic, complemented
9C2×#520C2211yescharacteristic
10#2983—21no—
11C2×#410—21no—
12#2628C2211yescharacteristic
13#2627C412yescomplemented
14#2581C411yescharacteristic, complemented
15#2567C411yescharacteristic, complemented
16#2564C2212yes—
17#2451—21no—
18#2449—22no—
19#2440—21no—
20#2438—22no—
21#2409C2212yes—
22#2396C2211yescharacteristic
23C2×#395C412yescomplemented
24#2349C411yescharacteristic, complemented
25D4×C4⋊C4—21no—
26#2339—21no—
27#2274—21no—
28C4×#99—22no—
29C4×#99C2214yes—
30#1740C2212yes—
31#1466C2214yes—
32#1464—22no—
33#1460—21no—
34#1458C2212yes—
35#1352C2214yes—
36#1351C2214yes—
37#1351C2218yes—
38C22×#41D411yescharacteristic
39C23×Q8C2×C412yes—
40D4×Q8—44no—
41C2×C4⋊Q8—21no—
42C2×C4⋊Q8C2×C411yescharacteristic
43C2×C22⋊Q8—22no—
44C2×C22⋊Q8—22no—
45C2×C22⋊Q8C2×C412yes—
46#520—44no—
47C2×C4×Q8C2×C411yescharacteristic
48C2×C4×Q8C2×C411yescharacteristic
49C2×C4×Q8D411yescharacteristic
50C2×C4×D4—21no—
51C2×C4×D4—21no—
52C2×C4×D4C2×C411yescharacteristic
53C2×C4×D4D412yes—
54C2×C4×D4D411yescharacteristic
55C2×C4×D4C2312yes—
56C2×#117—22no—
57C2×#117D412yes—
58C2×#117C2311yescharacteristic
59C22×C4⋊C4—22no—
60C22×C4⋊C4C2×C412yes—
61C22×C4⋊C4D411yescharacteristic
62#446—44no—
63#432—24no—
64#431—22no—
65#410—28no—
66C2×#99—24no—
67#400—21no—
68#400C2×C412yes—
69#400C2×C411yescharacteristic
70#399—21no—
71#399C2311yescharacteristic
72#398—22no—
73#398—22no—
74#398—21no—
75#396—21no—
76#395—22no—
77#395—22no—
78#395C2×C412yes—
79#395C2×C412yes—
80#395C2×C412yes—
81#395C2×C412yes—
82#395C2×C411yescharacteristic
83#393C2312yes—
84#393C2311yescharacteristic
85#391—21no—
86#391—21no—
87#391—22no—
88#389—21no—
89#389—21no—
90#389—22no—
91#389—21no—
92#387—22no—
93#387—22no—
94#386—22no—
95#386—22no—
96#385—22no—
97#385—22no—
98#385—22no—
99#385—22no—
100#384—21no—
101#382C2×C411yescharacteristic
102#381—22no—
103#380—21no—
104#380—22no—
105#380C2×C412yes—
106C4×C4⋊C4—21no—
107C4×C4⋊C4—44no—
108C4×C4⋊C4C2×C411yescharacteristic
109C4×C22⋊C4—21no—
110C4×C22⋊C4—24no—
111C2×#95—22no—
112C2×#95C2×C412yes—
113#342—22no—
114#342C2314yes—
115#324—22no—
116#323—24no—
117#323C2314yes—
118C2×#41—24no—
119C2×#41—24no—
120C2×#41—42no—
121C22×Q8—22no—
122C22×Q8—44no—
123C22×Q8—84no—
124C22×Q8C22⋊C411yescharacteristic
125C22×Q8C22×C411yescharacteristic
126C22×Q8C22×C412yes—
127C22×D4C22⋊C411yescharacteristic
128C22×D4C2×D412yes—
129C23×C4—22no—
130C23×C4—22no—
131C23×C4C22⋊C412yes—
132C23×C4C22⋊C412yes—
133C23×C4C22×C412yes—
134C23×C4C2×D411yescharacteristic
135C23×C4C2×D412yes—
136C4⋊Q8—42no—
137C4⋊Q8—44no—
138C22⋊Q8—48no—
139C22⋊Q8—44no—
140C22⋊Q8—84no—
141C4×Q8—24no—
142C4×Q8—42no—
143C4×Q8—42no—
144C4×D4—24no—
145C4×D4—42no—
146C4×D4—44no—
147C4×D4—44no—
148C4×D4—44no—
149C4×D4—44no—
150#117—28no—
151#117—42no—
152#117—44no—
153#117—44no—
154C2×C4⋊C4—24no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—21no—
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—22no—
168C2×C4⋊C4—44no—
169C2×C4⋊C4—42no—
170C2×C4⋊C4—84no—
171C2×C4⋊C4C22⋊C412yes—
172C2×C4⋊C4C22⋊C412yes—
173C2×C4⋊C4C22⋊C411yescharacteristic
174C2×C4⋊C4C22⋊C411yescharacteristic
175C2×C4⋊C4C22×C411yescharacteristic
176C2×C4⋊C4C22×C411yescharacteristic
177C2×C4⋊C4C22×C412yes—
178C2×C4⋊C4C2×D411yescharacteristic
179C2×C22⋊C4—24no—
180C2×C22⋊C4—22no—
181C2×C22⋊C4—22no—
182C2×C22⋊C4—21no—
183C2×C22⋊C4—21no—
184C2×C22⋊C4—22no—
185C2×C22⋊C4C22⋊C412yes—
186C2×C22⋊C4C2×D411yescharacteristic
187C2×C22⋊C4C2×D412yes—
188C2×C22⋊C4C2411yescharacteristic
189C2×C42—21no—
190C2×C42—21no—
191C2×C42—42no—
192C2×C42—44no—
193C2×C42C22⋊C412yes—
194C2×C42C22×C411yescharacteristic
195C2×C42C2×D412yes—
196C2×C42C2×D411yescharacteristic
197C4⋊C8—42no—
198C4⋊C8—44no—
199#99—216no—
200C22⋊C8—24no—
201#97—44no—
202C4×C8—42no—
203#95—21no—
204#95—22no—
205#95—22no—
206#95—22no—
207#95—22no—
208#95—21no—
209#95—21no—
210#95—22no—
211#95—22no—
212#95—22no—
213#95—21no—
214#95—21no—
215#95—22no—
216#95—44no—
217#95C22×C411yescharacteristic
218#95C22×C412yes—
219#95C22×C411yescharacteristic
220C24C2×C22⋊C412yes—
221C24C22×D411yescharacteristic
222#41—48no—
223#41—48no—
224C2×Q8—22no—
225C2×Q8—22no—
226C2×Q8—22no—
227C2×Q8—42no—
228C2×Q8—44no—
229C2×Q8—41no—
230C2×Q8—42no—
231C2×Q8—44no—
232C2×Q8—82no—
233C2×Q8—82no—
234C2×Q8—88no—
235C2×Q8—88no—
236C2×D4—24no—
237C2×D4—24no—
238C2×D4—22no—
239C2×D4—22no—
240C2×D4—42no—
241C2×D4#10414yes—
242C22×C4—22no—
243C22×C4—22no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—24no—
247C22×C4—24no—
248C22×C4—24no—
249C22×C4—24no—
250C22×C4—22no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—22no—
254C22×C4—42no—
255C22×C4—42no—
256C22×C4—44no—
257C22×C4—44no—
258C22×C4—44no—
259C22×C4—44no—
260C22×C4—44no—
261C22×C4—82no—
262C22×C4—82no—
263C22×C4—82no—
264C22×C4—82no—
265C22×C4—82no—
266C22×C4C2×C22⋊C412yes—
267C22×C4C2×C22⋊C411yescharacteristic
268C22×C4C2×C22⋊C411yescharacteristic
269C22×C4C2×C22⋊C412yes—
270C22×C4C2×C22⋊C411yescharacteristic
271C22×C4C2×C22⋊C412yes—
272C22×C4C2×C22⋊C411yescharacteristic
273C22×C4#13714yes—
274C22×C4C23×C411yescharacteristic
275C22×C4C22×D411yescharacteristic
276C2×C8—44no—
277C4⋊C4—24no—
278C4⋊C4—22no—
279C4⋊C4—24no—
280C4⋊C4—41no—
281C4⋊C4—42no—
282C4⋊C4—41no—
283C4⋊C4—41no—
284C4⋊C4—42no—
285C4⋊C4—42no—
286C4⋊C4—42no—
287C4⋊C4—44no—
288C4⋊C4—42no—
289C4⋊C4—42no—
290C4⋊C4—48no—
291C4⋊C4—42no—
292C4⋊C4—44no—
293C4⋊C4—42no—
294C4⋊C4—82no—
295C4⋊C4—82no—
296C4⋊C4—82no—
297C4⋊C4—82no—
298C4⋊C4#9914yes—
299C22⋊C4—24no—
300C22⋊C4—28no—
301C22⋊C4—42no—
302C22⋊C4—41no—
303C22⋊C4—41no—
304C22⋊C4—42no—
305C22⋊C4—42no—
306C22⋊C4—44no—
307C22⋊C4—44no—
308C22⋊C4—44no—
309C22⋊C4—82no—
310C42—22no—
311C42—24no—
312C42—22no—
313C42—42no—
314C42—48no—
315C42—42no—
316C42—42no—
317C42—164no—
318C23—22no—
319C23—21no—
320C23—22no—
321C23—21no—
322C23—42no—
323C23—42no—
324C23—41no—
325C23—41no—
326C23C2×#10414yes—
327C23C22×C22⋊C411yescharacteristic
328C23C2×#13712yes—
329Q8—42no—
330Q8—42no—
331Q8—44no—
332Q8—88no—
333Q8—84no—
334D4—28no—
335D4—48no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—24no—
339C2×C4—24no—
340C2×C4—22no—
341C2×C4—22no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—22no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—22no—
348C2×C4—22no—
349C2×C4—22no—
350C2×C4—42no—
351C2×C4—48no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—42no—
357C2×C4—41no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—48no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—42no—
365C2×C4—42no—
366C2×C4—41no—
367C2×C4—41no—
368C2×C4—42no—
369C2×C4—42no—
370C2×C4—41no—
371C2×C4—42no—
372C2×C4—41no—
373C2×C4—42no—
374C2×C4—84no—
375C2×C4—84no—
376C2×C4—88no—
377C2×C4—88no—
378C2×C4—84no—
379C2×C4—84no—
380C2×C4—162no—
381C2×C4C2×#9914yes—
382C2×C4C2×#9912yes—
383C2×C4#41912yes—
384C2×C4#41912yes—
385C2×C4C2×#10412yes—
386C8—84no—
387C22—21no—
388C22—21no—
389C22—22no—
390C22—24no—
391C22—44no—
392C22—42no—
393C22—44no—
394C22—41no—
395C22—42no—
396C22—41no—
397C22—42no—
398C22—81no—
399C22C22×#9911yescharacteristic
400C22C2×#41911yescharacteristic
401C22C22×#10411yescharacteristic
402C4—24no—
403C4—22no—
404C4—42no—
405C4—42no—
406C4—41no—
407C4—42no—
408C4—41no—
409C4—41no—
410C4—41no—
411C4—42no—
412C4—81no—
413C4—82no—
414C4—88no—
415C4—82no—
416C4—81no—
417C4#150814yes—
418C2—21no—
419C2—21no—
420C2—42no—
421C2—44no—
422C2C2×#150811yescharacteristic
423C2#3334811yescharacteristic
424C2#3336011yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC22×C4C23×C4
Frattini, cosocleC2×C22⋊C4C24
prime radicalC22×#41D4
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
4. #11789C2semidirect
5. #11785C2semidirect
6. #11717C2semidirect
7. #11715C2semidirect
8. C2×D4×Q8C4semidirect
13. #2627C4semidirect
14. #2581C4semidirect
15. #2567C4semidirect
23. C2×#395C4semidirect
24. #2349C4semidirect

Maximal subgroups

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

subgroupindex
#357392
#203932
#173722
#117892
#117852
#117172
#117152