Finite Group Databasebeta

← #481658

Subgroups — #481658

Order 512.

931 conjugacy classes of subgroups.

The subgroup lattice has 415 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36031C211yesmaximal, characteristic
2#20393C211yesmaximal, characteristic
3#17372C211yesmaximal, characteristic
4#11789C214yesmaximal, complemented
5#11786C214yesmaximal, complemented
6#11718C212yesmaximal, complemented
7#11715C212yesmaximal, complemented
8C2×#554C411yescharacteristic, complemented
9C2×#520C2211yescharacteristic
10#2983—21no—
11C2×#410—21no—
12#2628C2211yescharacteristic
13#2627C412yescomplemented
14#2581C411yescharacteristic, complemented
15#2568C411yescharacteristic, complemented
16#2564C2212yes—
17#2487C2212yes—
18#2453—21no—
19#2450—22no—
20#2442—21no—
21#2438—22no—
22#2396C2211yescharacteristic
23C2×#400C412yescomplemented
24#2349—21no—
25#2349C411yescharacteristic, complemented
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#1352C2218yes—
37#1351C2214yes—
38C22×#41D411yescharacteristic
39#554—44no—
40C2×C4⋊Q8C2×C411yescharacteristic
41C2×#125—21no—
42C2×C22⋊Q8—22no—
43C2×C22⋊Q8C2×C412yes—
44#520—44no—
45C2×C4×Q8C2×C411yescharacteristic
46C2×C4×Q8C2×C411yescharacteristic
47C2×C4×Q8D411yescharacteristic
48C2×C4×D4—21no—
49C2×C4×D4—21no—
50C2×C4×D4C2×C411yescharacteristic
51C2×C4×D4D412yes—
52C2×C4×D4D411yescharacteristic
53C2×C4×D4C2312yes—
54C2×#117—22no—
55C2×#117D412yes—
56C2×#117C2311yescharacteristic
57C22×C4⋊C4C2×C412yes—
58C22×C4⋊C4C2×C412yes—
59C22×C4⋊C4D411yescharacteristic
60#446—44no—
61#432—24no—
62#431—22no—
63#410—28no—
64C2×#99—24no—
65#400—22no—
66#400—22no—
67#400C2×C412yes—
68#400C2×C412yes—
69#400C2×C412yes—
70#400C2×C412yes—
71#400C2×C411yescharacteristic
72#399C2312yes—
73#399C2311yescharacteristic
74#398—22no—
75#398—22no—
76#398—21no—
77#396—21no—
78#395—21no—
79#395C2×C412yes—
80#395C2×C411yescharacteristic
81#393—21no—
82#393C2311yescharacteristic
83#391—21no—
84#391—21no—
85#391—22no—
86#389—21no—
87#389—21no—
88#389—21no—
89#389—22no—
90#387—22no—
91#387—22no—
92#386—22no—
93#386—22no—
94#385—22no—
95#385—22no—
96#385—22no—
97#385—22no—
98#382—21no—
99#382C2×C411yescharacteristic
100#381—22no—
101#380—21no—
102#380—22no—
103#380—22no—
104#380C2×C412yes—
105C4×C4⋊C4—21no—
106C4×C4⋊C4—44no—
107C4×C4⋊C4C2×C411yescharacteristic
108C4×C22⋊C4—21no—
109C4×C22⋊C4—24no—
110C2×#95—22no—
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—21no—
122C22×Q8C22⋊C411yescharacteristic
123C22×Q8C22×C411yescharacteristic
124C22×D4C22⋊C411yescharacteristic
125C22×D4C2×D412yes—
126C23×C4—22no—
127C23×C4—22no—
128C23×C4C22⋊C412yes—
129C23×C4C22⋊C412yes—
130C23×C4C22×C412yes—
131C23×C4C2×D411yescharacteristic
132C23×C4C2×D412yes—
133C4⋊Q8—42no—
134#125—44no—
135C22⋊Q8—48no—
136C22⋊Q8—44no—
137C4×Q8—24no—
138C4×Q8—42no—
139C4×Q8—42no—
140C4×D4—24no—
141C4×D4—42no—
142C4×D4—44no—
143C4×D4—44no—
144C4×D4—44no—
145C4×D4—44no—
146#117—28no—
147#117—42no—
148#117—44no—
149#117—44no—
150C2×C4⋊C4—24no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—21no—
153C2×C4⋊C4—21no—
154C2×C4⋊C4—21no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—44no—
163C2×C4⋊C4—42no—
164C2×C4⋊C4—44no—
165C2×C4⋊C4C22⋊C412yes—
166C2×C4⋊C4C22⋊C412yes—
167C2×C4⋊C4C22⋊C411yescharacteristic
168C2×C4⋊C4C22⋊C411yescharacteristic
169C2×C4⋊C4C22×C411yescharacteristic
170C2×C4⋊C4C22×C412yes—
171C2×C4⋊C4C22×C411yescharacteristic
172C2×C4⋊C4C22×C412yes—
173C2×C4⋊C4C2×D411yescharacteristic
174C2×C22⋊C4—24no—
175C2×C22⋊C4—22no—
176C2×C22⋊C4—22no—
177C2×C22⋊C4—21no—
178C2×C22⋊C4—21no—
179C2×C22⋊C4—22no—
180C2×C22⋊C4C22⋊C412yes—
181C2×C22⋊C4C2×D411yescharacteristic
182C2×C22⋊C4C2×D412yes—
183C2×C22⋊C4C2411yescharacteristic
184C2×C42—21no—
185C2×C42—21no—
186C2×C42—42no—
187C2×C42—44no—
188C2×C42C22⋊C412yes—
189C2×C42C22×C411yescharacteristic
190C2×C42C2×D412yes—
191C2×C42C2×D411yescharacteristic
192C4⋊C8—42no—
193C4⋊C8—44no—
194#99—216no—
195C22⋊C8—24no—
196#97—44no—
197C4×C8—42no—
198#95—21no—
199#95—21no—
200#95—22no—
201#95—22no—
202#95—22no—
203#95—22no—
204#95—22no—
205#95—22no—
206#95—22no—
207#95—21no—
208#95—22no—
209#95—22no—
210#95—22no—
211#95—21no—
212#95—21no—
213#95—44no—
214#95C22×C411yescharacteristic
215#95C22×C412yes—
216#95C22×C411yescharacteristic
217C24C2×C22⋊C412yes—
218C24C22×D411yescharacteristic
219#41—48no—
220#41—48no—
221C2×Q8—22no—
222C2×Q8—22no—
223C2×Q8—22no—
224C2×Q8—42no—
225C2×Q8—41no—
226C2×Q8—81no—
227C2×Q8—82no—
228C2×D4—24no—
229C2×D4—24no—
230C2×D4—22no—
231C2×D4—22no—
232C2×D4—42no—
233C2×D4#10414yes—
234C22×C4—22no—
235C22×C4—22no—
236C22×C4—24no—
237C22×C4—24no—
238C22×C4—24no—
239C22×C4—24no—
240C22×C4—21no—
241C22×C4—22no—
242C22×C4—21no—
243C22×C4—22no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—42no—
247C22×C4—42no—
248C22×C4—44no—
249C22×C4—44no—
250C22×C4—44no—
251C22×C4—82no—
252C22×C4—82no—
253C22×C4—82no—
254C22×C4—82no—
255C22×C4—82no—
256C22×C4—82no—
257C22×C4—82no—
258C22×C4C2×C22⋊C412yes—
259C22×C4C2×C22⋊C411yescharacteristic
260C22×C4C2×C22⋊C412yes—
261C22×C4C2×C22⋊C411yescharacteristic
262C22×C4C2×C22⋊C411yescharacteristic
263C22×C4C2×C22⋊C412yes—
264C22×C4C2×C22⋊C411yescharacteristic
265C22×C4#13714yes—
266C22×C4C23×C411yescharacteristic
267C22×C4C22×D411yescharacteristic
268C2×C8—44no—
269C4⋊C4—24no—
270C4⋊C4—22no—
271C4⋊C4—24no—
272C4⋊C4—41no—
273C4⋊C4—42no—
274C4⋊C4—41no—
275C4⋊C4—41no—
276C4⋊C4—42no—
277C4⋊C4—42no—
278C4⋊C4—48no—
279C4⋊C4—44no—
280C4⋊C4—42no—
281C4⋊C4—44no—
282C4⋊C4—42no—
283C4⋊C4—82no—
284C4⋊C4—81no—
285C4⋊C4—82no—
286C4⋊C4—81no—
287C4⋊C4—82no—
288C4⋊C4—81no—
289C4⋊C4#9914yes—
290C22⋊C4—24no—
291C22⋊C4—28no—
292C22⋊C4—42no—
293C22⋊C4—41no—
294C22⋊C4—41no—
295C22⋊C4—44no—
296C22⋊C4—44no—
297C22⋊C4—44no—
298C22⋊C4—81no—
299C22⋊C4—82no—
300C22⋊C4—81no—
301C42—22no—
302C42—24no—
303C42—22no—
304C42—42no—
305C42—48no—
306C42—42no—
307C42—42no—
308C42—164no—
309C23—22no—
310C23—21no—
311C23—22no—
312C23—21no—
313C23—42no—
314C23—42no—
315C23—41no—
316C23—41no—
317C23C2×#10414yes—
318C23C22×C22⋊C411yescharacteristic
319C23C2×#13712yes—
320Q8—44no—
321Q8—82no—
322Q8—162no—
323D4—28no—
324D4—48no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—24no—
328C2×C4—24no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—22no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—22no—
335C2×C4—22no—
336C2×C4—22no—
337C2×C4—42no—
338C2×C4—48no—
339C2×C4—41no—
340C2×C4—42no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—48no—
345C2×C4—41no—
346C2×C4—42no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—42no—
351C2×C4—41no—
352C2×C4—42no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—42no—
357C2×C4—41no—
358C2×C4—41no—
359C2×C4—42no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—42no—
363C2×C4—84no—
364C2×C4—84no—
365C2×C4—84no—
366C2×C4—84no—
367C2×C4—162no—
368C2×C4—164no—
369C2×C4—164no—
370C2×C4C2×#9914yes—
371C2×C4C2×#9912yes—
372C2×C4#41912yes—
373C2×C4#41912yes—
374C2×C4C2×#10412yes—
375C8—84no—
376C22—21no—
377C22—21no—
378C22—22no—
379C22—24no—
380C22—44no—
381C22—42no—
382C22—44no—
383C22—41no—
384C22—42no—
385C22—41no—
386C22—42no—
387C22—81no—
388C22C22×#9911yescharacteristic
389C22C2×#41911yescharacteristic
390C22C22×#10411yescharacteristic
391C4—24no—
392C4—22no—
393C4—41no—
394C4—42no—
395C4—41no—
396C4—41no—
397C4—41no—
398C4—42no—
399C4—81no—
400C4—82no—
401C4—81no—
402C4—88no—
403C4—81no—
404C4—81no—
405C4—82no—
406C4#150814yes—
407C2—21no—
408C2—21no—
409C2—42no—
410C2—44no—
411C2C2×#150811yescharacteristic
412C2#3335011yescharacteristic
413C2#3336211yescharacteristic
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. #11786C2semidirect
6. #11718C2semidirect
7. #11715C2semidirect
8. C2×#554C4semidirect
13. #2627C4semidirect
14. #2581C4semidirect
15. #2568C4semidirect
23. C2×#400C4semidirect
25. #2349C4semidirect

Maximal subgroups

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

subgroupindex
#360312
#203932
#173722
#117892
#117862
#117182
#117152