Finite Group Databasebeta

← #480956

Subgroups — #480956

Order 512.

885 conjugacy classes of subgroups.

The subgroup lattice has 409 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36164C211yesmaximal, characteristic
2#17426C212yesmaximal, complemented
3#11573C212yesmaximal, complemented
4#11553C214yesmaximal, complemented
5#11530C211yesmaximal, characteristic, complemented
6#11511C212yesmaximal, complemented
7#11476C211yesmaximal, characteristic, complemented
8C2×#1324C212yesmaximal, complemented
9C2×#554C2211yescharacteristic
10C2×#547C411yescharacteristic
11#2974—22no—
12#2862C411yescharacteristic
13#2850C412yes—
14#2568C2211yescharacteristic
15#2563C411yescharacteristic
16#2504C2212yes—
17#2503C412yes—
18#2489—22no—
19#2488—24no—
20#2339—22no—
21C4×#124C2211yescharacteristic
22C2×#382C2212yes—
23C42×D4C411yescharacteristic
24#1472—24no—
25#1470—22no—
26#1467C2212yes—
27#1463C2214yes—
28#1461C2212yes—
29C2×#324C2214yes—
30#1324C2212yes—
31#1324C2214yes—
32#1324C2214yes—
33#1324C2211yescharacteristic
34#1324C2211yescharacteristic
35#1318C2212yes—
36#1318C2212yes—
37C22×#41—21no—
38#554—41no—
39#554—42no—
40#554—41no—
41#547—41no—
42#547—42no—
43#547—41no—
44C2×C4⋊Q8C2×C411yescharacteristic
45C2×C4⋊Q8C2×C411yescharacteristic
46C2×#125D412yes—
47C2×#124C2×C411yescharacteristic
48C2×#123—22no—
49C2×C22⋊Q8—21no—
50C2×C22⋊Q8C2×C412yes—
51C2×C22⋊Q8D414yes—
52C2×#121C2×C412yes—
53C2×C4×Q8C2311yescharacteristic
54C2×C4×D4—21no—
55C2×C4×D4C2×C411yescharacteristic
56C2×C4×D4C2×C411yescharacteristic
57C2×C4×D4D412yes—
58C22×C4⋊C4C2×C412yes—
59C22×C4⋊C4C2312yes—
60C22×C42C2×C412yes—
61#436—42no—
62#435—44no—
63C8×D4—42no—
64#432—22no—
65#431—22no—
66#407—24no—
67C2×C22⋊C8—24no—
68#400—24no—
69#400—22no—
70#397—24no—
71#393—22no—
72#391—22no—
73#391—22no—
74#391—22no—
75#391C2×C412yes—
76#391C2×C411yescharacteristic
77#390C2×C412yes—
78#390C2311yescharacteristic
79#388—24no—
80#388—22no—
81#386—24no—
82#386—24no—
83#382—24no—
84#382C2×C412yes—
85#382C2×C412yes—
86#382C2×C412yes—
87#382C2312yes—
88#382C2311yescharacteristic
89#381—24no—
90#380—22no—
91C4×C4⋊C4—21no—
92C4×C4⋊C4C2×C411yescharacteristic
93C4×C22⋊C4—22no—
94C4×C22⋊C4—22no—
95C4×C22⋊C4C2×C412yes—
96C2×#95—24no—
97C43C2×C411yescharacteristic
98#324—24no—
99#324—22no—
100#324—24no—
101#324—22no—
102#324C2312yes—
103#324C2314yes—
104#324C2312yes—
105C2×#41—48no—
106C2×#41—41no—
107C2×#41—42no—
108C2×#41—41no—
109C22×Q8—21no—
110C22×Q8C22⋊C412yes—
111C22×Q8C22×C411yescharacteristic
112C22×D4—22no—
113C22×D4C22⋊C411yescharacteristic
114C22×D4C22×C411yescharacteristic
115C23×C4—22no—
116C23×C4—22no—
117C23×C4—21no—
118C23×C4C22×C412yes—
119C23×C4C2×D414yes—
120C2×M16—44no—
121C22×C8—44no—
122C4⋊Q8—41no—
123C4⋊Q8—41no—
124C4⋊Q8—41no—
125C4⋊Q8—41no—
126#125—42no—
127#125—42no—
128#124—22no—
129#124—41no—
130#123—44no—
131#123—44no—
132C22⋊Q8—42no—
133C22⋊Q8—42no—
134C22⋊Q8—44no—
135C22⋊Q8—44no—
136C22⋊Q8—42no—
137C22⋊Q8—42no—
138#121—42no—
139#121—42no—
140C4×Q8—41no—
141C4×Q8—41no—
142C4×D4—22no—
143C4×D4—21no—
144C4×D4—21no—
145C4×D4—22no—
146C4×D4—24no—
147C4×D4—22no—
148C4×D4—41no—
149C4×D4—41no—
150C4×D4—44no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—21no—
153C2×C4⋊C4—21no—
154C2×C4⋊C4—42no—
155C2×C4⋊C4—42no—
156C2×C4⋊C4—42no—
157C2×C4⋊C4—42no—
158C2×C4⋊C4C22⋊C411yescharacteristic
159C2×C4⋊C4C22⋊C412yes—
160C2×C4⋊C4C22⋊C414yes—
161C2×C4⋊C4C22×C411yescharacteristic
162C2×C4⋊C4C22×C412yes—
163C2×C4⋊C4C22×C412yes—
164C2×C4⋊C4C2×D414yes—
165C2×C4⋊C4C2×D412yes—
166C2×C4⋊C4C2411yescharacteristic
167C2×C22⋊C4—22no—
168C2×C22⋊C4—21no—
169C2×C22⋊C4—22no—
170C2×C22⋊C4—22no—
171C2×C22⋊C4—24no—
172C2×C22⋊C4—22no—
173C2×C22⋊C4C22⋊C412yes—
174C2×C22⋊C4C22⋊C414yes—
175C2×C42—24no—
176C2×C42—21no—
177C2×C42—22no—
178C2×C42—21no—
179C2×C42—42no—
180C2×C42C22×C412yes—
181C2×C42C22×C411yescharacteristic
182C2×C42C22×C411yescharacteristic
183C2×C42C22×C411yescharacteristic
184C2×C42C2×D412yes—
185C4⋊C8—42no—
186C4⋊C8—42no—
187C22⋊C8—22no—
188C22⋊C8—24no—
189C22⋊C8—22no—
190C22⋊C8—44no—
191C22⋊C8—44no—
192#97—42no—
193C4×C8—42no—
194#95—24no—
195#95—24no—
196#95—21no—
197#95—22no—
198#95—22no—
199#95—24no—
200#95—24no—
201#95—22no—
202#95—24no—
203C24—21no—
204C24C2×C22⋊C412yes—
205#41—88no—
206#41—84no—
207#41—84no—
208C2×Q8—21no—
209C2×Q8—21no—
210C2×Q8—41no—
211C2×Q8—41no—
212C2×Q8—42no—
213C2×Q8—44no—
214C2×Q8—41no—
215C2×Q8—41no—
216C2×Q8—82no—
217C2×D4—24no—
218C2×D4—21no—
219C2×D4—21no—
220C2×D4—41no—
221C2×D4—41no—
222C2×D4—48no—
223C2×D4—42no—
224C2×D4—42no—
225C2×D4#9912yes—
226C2×D4#9911yescharacteristic
227C2×D4#9911yescharacteristic
228C22×C4—21no—
229C22×C4—22no—
230C22×C4—22no—
231C22×C4—24no—
232C22×C4—24no—
233C22×C4—22no—
234C22×C4—22no—
235C22×C4—21no—
236C22×C4—22no—
237C22×C4—21no—
238C22×C4—22no—
239C22×C4—42no—
240C22×C4—44no—
241C22×C4—44no—
242C22×C4—44no—
243C22×C4—48no—
244C22×C4—42no—
245C22×C4—42no—
246C22×C4—41no—
247C22×C4—41no—
248C22×C4—82no—
249C22×C4—84no—
250C22×C4C2×C22⋊C412yes—
251C22×C4C2×C22⋊C412yes—
252C22×C4C2×C22⋊C411yescharacteristic
253C22×C4C2×C22⋊C414yes—
254C22×C4C2×C22⋊C411yescharacteristic
255C22×C4C23×C411yescharacteristic
256C22×C4C22×D412yes—
257M16—164no—
258C2×C8—42no—
259C2×C8—42no—
260C2×C8—44no—
261C2×C8—164no—
262C4⋊C4—21no—
263C4⋊C4—21no—
264C4⋊C4—22no—
265C4⋊C4—22no—
266C4⋊C4—41no—
267C4⋊C4—42no—
268C4⋊C4—44no—
269C4⋊C4—44no—
270C4⋊C4—42no—
271C4⋊C4—41no—
272C4⋊C4—42no—
273C4⋊C4—44no—
274C4⋊C4—42no—
275C4⋊C4—42no—
276C22⋊C4—24no—
277C22⋊C4—24no—
278C22⋊C4—22no—
279C22⋊C4—22no—
280C22⋊C4—44no—
281C22⋊C4—42no—
282C22⋊C4—44no—
283C22⋊C4—44no—
284C22⋊C4—82no—
285C22⋊C4—84no—
286C42—21no—
287C42—22no—
288C42—22no—
289C42—21no—
290C42—21no—
291C42—22no—
292C42—22no—
293C42—44no—
294C42—41no—
295C42—42no—
296C42—81no—
297C42#13112yes—
298C23—22no—
299C23—22no—
300C23—24no—
301C23—42no—
302C23—42no—
303C23—42no—
304C23—41no—
305C23—41no—
306C23—41no—
307C23C2×#9912yes—
308C23C2×#9912yes—
309C23C22×C22⋊C411yescharacteristic
310Q8—81no—
311Q8—81no—
312Q8—82no—
313Q8—82no—
314Q8—162no—
315D4—24no—
316D4—22no—
317D4—81no—
318D4—81no—
319D4—84no—
320D4—84no—
321D4#35514yes—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—22no—
325C2×C4—22no—
326C2×C4—22no—
327C2×C4—22no—
328C2×C4—21no—
329C2×C4—22no—
330C2×C4—22no—
331C2×C4—22no—
332C2×C4—24no—
333C2×C4—22no—
334C2×C4—22no—
335C2×C4—42no—
336C2×C4—42no—
337C2×C4—42no—
338C2×C4—42no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—42no—
343C2×C4—44no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—42no—
347C2×C4—42no—
348C2×C4—42no—
349C2×C4—44no—
350C2×C4—42no—
351C2×C4—41no—
352C2×C4—42no—
353C2×C4—41no—
354C2×C4—82no—
355C2×C4—84no—
356C2×C4—84no—
357C2×C4—88no—
358C2×C4—84no—
359C2×C4—84no—
360C2×C4—82no—
361C2×C4C2×#9911yescharacteristic
362C2×C4C2×#9911yescharacteristic
363C2×C4C2×#13111yescharacteristic
364C2×C4#56811yescharacteristic
365C8—82no—
366C8—84no—
367C8—82no—
368C22—22no—
369C22—21no—
370C22—22no—
371C22—21no—
372C22—22no—
373C22—22no—
374C22—22no—
375C22—44no—
376C22—44no—
377C22—82no—
378C22—82no—
379C22—81no—
380C22—81no—
381C22—81no—
382C22C2×#35514yes—
383C22#288011yescharacteristic
384C22C22×#9911yescharacteristic
385C22#288911yescharacteristic
386C4—21no—
387C4—21no—
388C4—21no—
389C4—42no—
390C4—42no—
391C4—42no—
392C4—42no—
393C4—44no—
394C4—41no—
395C4—81no—
396C4—82no—
397C4—81no—
398C4—82no—
399C4C2×#35512yes—
400C2—21no—
401C2—21no—
402C2—24no—
403C2—42no—
404C2—161no—
405C2#1741611yescharacteristic
406C2#3336111yescharacteristic
407C2C22×#35511yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC22×C4C23×C4
Frattini, cosocleC2×C4⋊C4C24
prime radicalC2×#547C4
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×D4 ◁ C2×#554 ◁ G
LECSG ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #17426C2semidirect
3. #11573C2semidirect
4. #11553C2semidirect
5. #11530C2semidirect
6. #11511C2semidirect
7. #11476C2semidirect
8. C2×#1324C2semidirect

Maximal subgroups

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

subgroupindex
#361642
#174262
#115732
#115532
#115302
#115112
#114762
C2×#13242