Finite Group Databasebeta

← #481029

Subgroups — #481029

Order 512.

786 conjugacy classes of subgroups.

The subgroup lattice has 369 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#39746C211yesmaximal, characteristic
2#17430C212yesmaximal, complemented
3#11567C212yesmaximal
4#11565C214yesmaximal, complemented
5#11534C212yesmaximal, complemented
6#11532C211yesmaximal, characteristic, complemented
7#11480C212yesmaximal, complemented
8#11478C211yesmaximal, characteristic, complemented
9C2×#559C411yescharacteristic
10C2×#557C2211yescharacteristic
11#2984—22no—
12#2858C411yescharacteristic
13#2694—24no—
14#2658C412yes—
15#2548C2211yescharacteristic
16#2547C412yes—
17#2546C411yescharacteristic
18#2545C2212yes—
19#2454—22no—
20#2448—22no—
21#2326C2212yes—
22C4×C4⋊Q8C2211yescharacteristic
23#2295C411yescharacteristic
24#1470—24no—
25#1470—22no—
26#1467C2212yes—
27#1467C2214yes—
28#1461C2214yes—
29#1461C2212yes—
30#1325C2212yes—
31#1325C2211yescharacteristic
32#1325C2211yescharacteristic
33#1319C2212yes—
34#1319C2212yes—
35#1318C2214yes—
36#1318C2214yes—
37#559—42no—
38#559—42no—
39#557—42no—
40#557—42no—
41C2×C4⋊Q8C2×C411yescharacteristic
42C2×C4⋊Q8C2×C412yes—
43C2×C4⋊Q8C2×C411yescharacteristic
44C2×C4⋊Q8C2×C412yes—
45C2×#127C2×C411yescharacteristic
46C2×#125D414yes—
47C2×#125D412yes—
48C2×#124—21no—
49C2×#124C2×C412yes—
50C2×#124C2×C412yes—
51C2×#124C2×C412yes—
52C2×#124C2×C412yes—
53C2×#123—22no—
54C2×C4×Q8C2×C411yescharacteristic
55C2×C4×Q8C2×C411yescharacteristic
56C2×C4×Q8D412yes—
57C2×C4×Q8C2311yescharacteristic
58C2×C4×Q8C2312yes—
59C2×C4×D4—21no—
60C2×C4×D4C2×C412yes—
61#446—42no—
62#446—44no—
63C8×Q8—42no—
64#432—22no—
65#432—24no—
66#431—22no—
67#431—24no—
68#399—24no—
69#399—24no—
70#398—22no—
71#397—24no—
72#396—24no—
73#394—22no—
74#391—21no—
75#391C2×C411yescharacteristic
76#391C2×C412yes—
77#391C2311yescharacteristic
78#390—22no—
79#390—22no—
80#390—22no—
81#390C2312yes—
82#389—22no—
83#389—21no—
84#389—22no—
85#388—24no—
86#388—24no—
87#387—22no—
88#385—24no—
89#384C2311yescharacteristic
90#380—24no—
91C4×C4⋊C4—22no—
92C4×C4⋊C4C2×C412yes—
93C4×C4⋊C4C2×C411yescharacteristic
94#376C2×C412yes—
95C43C2×C411yescharacteristic
96#324—24no—
97#324—22no—
98#324—24no—
99#324—22no—
100#324C2312yes—
101#324C2312yes—
102#324C2314yes—
103C22×Q8C22⋊C411yescharacteristic
104C22×Q8C22×C411yescharacteristic
105C22×Q8C22×C412yes—
106C22×Q8C22×C411yescharacteristic
107C22×D4—21no—
108C22×D4—21no—
109C22×D4C22×C412yes—
110C23×C4—22no—
111C4⋊Q8—22no—
112C4⋊Q8—44no—
113C4⋊Q8—41no—
114C4⋊Q8—41no—
115C4⋊Q8—41no—
116C4⋊Q8—82no—
117#127—41no—
118#127—41no—
119#125—42no—
120#125—44no—
121#125—42no—
122#125—44no—
123#124—24no—
124#124—42no—
125#124—44no—
126#124—42no—
127#124—42no—
128#124—82no—
129#124—82no—
130#123—48no—
131C4×Q8—22no—
132C4×Q8—22no—
133C4×Q8—22no—
134C4×Q8—22no—
135C4×Q8—24no—
136C4×Q8—41no—
137C4×Q8—44no—
138C4×Q8—41no—
139C4×Q8—41no—
140C4×Q8—41no—
141C4×D4—44no—
142C4×D4—42no—
143C4×D4—42no—
144C2×C4⋊C4—21no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—21no—
147C2×C4⋊C4—24no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4C22⋊C411yescharacteristic
151C2×C4⋊C4C22⋊C412yes—
152C2×C4⋊C4C22⋊C412yes—
153C2×C4⋊C4C22⋊C414yes—
154C2×C4⋊C4C22⋊C414yes—
155C2×C4⋊C4C22⋊C412yes—
156C2×C4⋊C4C22×C411yescharacteristic
157C2×C4⋊C4C2×D414yes—
158C2×C4⋊C4C2×D412yes—
159C2×C4⋊C4C2411yescharacteristic
160C2×C22⋊C4—22no—
161C2×C22⋊C4—22no—
162C2×C22⋊C4—22no—
163C2×C22⋊C4—24no—
164C2×C22⋊C4—24no—
165C2×C22⋊C4—24no—
166C2×C22⋊C4—24no—
167C2×C42—21no—
168C2×C42—21no—
169C2×C42—21no—
170C2×C42C22×C412yes—
171C2×C42C22×C411yescharacteristic
172C2×C42C22×C411yescharacteristic
173C2×C42C22×C411yescharacteristic
174C2×C42C22×C412yes—
175C2×C42C2×D412yes—
176C2×C42C2×D414yes—
177C4⋊C8—42no—
178C4⋊C8—44no—
179C4⋊C8—44no—
180C4⋊C8—42no—
181C22⋊C8—22no—
182C22⋊C8—24no—
183C22⋊C8—22no—
184#97—42no—
185#97—44no—
186C4×C8—44no—
187C4×C8—42no—
188#95—22no—
189#95—22no—
190#95—22no—
191#95—21no—
192#95—22no—
193#95—24no—
194#95—22no—
195C24—22no—
196C2×Q8—22no—
197C2×Q8—21no—
198C2×Q8—22no—
199C2×Q8—21no—
200C2×Q8—21no—
201C2×Q8—21no—
202C2×Q8—42no—
203C2×Q8—41no—
204C2×Q8—41no—
205C2×Q8—42no—
206C2×Q8—42no—
207C2×Q8—41no—
208C2×Q8—41no—
209C2×Q8#9912yes—
210C2×Q8#9912yes—
211C2×D4—22no—
212C2×D4—22no—
213C2×D4—42no—
214C2×D4—42no—
215C2×D4—44no—
216C2×D4—42no—
217C2×D4—41no—
218C2×D4—41no—
219C2×D4—82no—
220C22×C4—21no—
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—22no—
224C22×C4—21no—
225C22×C4—44no—
226C22×C4—84no—
227C22×C4C2×C22⋊C414yes—
228C22×C4C2×C22⋊C412yes—
229C22×C4C2×C22⋊C411yescharacteristic
230C22×C4C2×C22⋊C411yescharacteristic
231C22×C4C2×C22⋊C412yes—
232C22×C4C2×C22⋊C412yes—
233C22×C4C23×C411yescharacteristic
234C22×C4C22×D412yes—
235C2×C8—42no—
236C2×C8—44no—
237C2×C8—42no—
238C4⋊C4—21no—
239C4⋊C4—21no—
240C4⋊C4—24no—
241C4⋊C4—22no—
242C4⋊C4—24no—
243C4⋊C4—24no—
244C4⋊C4—22no—
245C4⋊C4—44no—
246C4⋊C4—42no—
247C4⋊C4—44no—
248C4⋊C4—42no—
249C4⋊C4—41no—
250C4⋊C4—44no—
251C4⋊C4—41no—
252C4⋊C4—42no—
253C4⋊C4—42no—
254C4⋊C4—84no—
255C4⋊C4—82no—
256C4⋊C4—82no—
257C22⋊C4—48no—
258C22⋊C4—44no—
259C22⋊C4—44no—
260C22⋊C4—44no—
261C22⋊C4—84no—
262C22⋊C4—84no—
263C22⋊C4—84no—
264C42—21no—
265C42—24no—
266C42—22no—
267C42—22no—
268C42—24no—
269C42—24no—
270C42—21no—
271C42—21no—
272C42—41no—
273C42—42no—
274C42—42no—
275C42—41no—
276C42—41no—
277C42—81no—
278C42#13112yes—
279C23—44no—
280C23—44no—
281C23—42no—
282C23—42no—
283C23—42no—
284C23C22×C22⋊C411yescharacteristic
285Q8—24no—
286Q8—42no—
287Q8—82no—
288Q8—81no—
289Q8—82no—
290Q8—81no—
291Q8—81no—
292Q8—81no—
293D4—82no—
294D4—82no—
295D4—82no—
296D4—82no—
297D4—162no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—22no—
301C2×C4—22no—
302C2×C4—24no—
303C2×C4—22no—
304C2×C4—22no—
305C2×C4—22no—
306C2×C4—21no—
307C2×C4—24no—
308C2×C4—22no—
309C2×C4—22no—
310C2×C4—24no—
311C2×C4—42no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—42no—
316C2×C4—42no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—42no—
320C2×C4—42no—
321C2×C4—42no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—42no—
327C2×C4—41no—
328C2×C4—164no—
329C2×C4C2×#9911yescharacteristic
330C2×C4C2×#9914yes—
331C2×C4C2×#9911yescharacteristic
332C2×C4C2×#13111yescharacteristic
333C2×C4#56811yescharacteristic
334C8—82no—
335C8—84no—
336C8—82no—
337C22—21no—
338C22—21no—
339C22—84no—
340C22—84no—
341C22—82no—
342C22—82no—
343C22—82no—
344C22#288011yescharacteristic
345C22C22×#9911yescharacteristic
346C22#288811yescharacteristic
347C4—21no—
348C4—21no—
349C4—21no—
350C4—24no—
351C4—42no—
352C4—44no—
353C4—42no—
354C4—42no—
355C4—42no—
356C4—41no—
357C4—81no—
358C4—82no—
359C4—81no—
360C4—81no—
361C4#213312yes—
362C2—21no—
363C2—21no—
364C2—162no—
365C2#1741911yescharacteristic
366C2#3336211yescharacteristic
367C2C2×#213311yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC22×C4C23×C4
Frattini, cosocleC2×C4⋊C4C24
prime radicalC2×#127C2×C4
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×Q8 ◁ C2×#557 ◁ G
LECSG ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #17430C2semidirect
4. #11565C2semidirect
5. #11534C2semidirect
6. #11532C2semidirect
7. #11480C2semidirect
8. #11478C2semidirect

Maximal subgroups

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

subgroupindex
#397462
#174302
#115672
#115652
#115342
#115322
#114802
#114782