Finite Group Databasebeta

← #481028

Subgroups — #481028

Order 512.

804 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#39746C211yesmaximal, characteristic
2#17429C212yesmaximal
3#11568C212yesmaximal, complemented
4#11564C214yesmaximal, complemented
5#11535C212yesmaximal, complemented
6#11532C211yesmaximal, characteristic, complemented
7#11481C212yesmaximal, complemented
8#11478C211yesmaximal, characteristic, complemented
9C2×#559C2211yescharacteristic
10C2×#557C411yescharacteristic
11#2974—22no—
12#2858C411yescharacteristic
13#2694—24no—
14#2658C412yes—
15#2548C411yescharacteristic
16#2547C2212yes—
17#2546C2211yescharacteristic
18#2545C412yes—
19#2454—22no—
20#2448—22no—
21#2326C2212yes—
22C4×C4⋊Q8C2211yescharacteristic
23#2295C411yescharacteristic
24#1469—22no—
25#1468—24no—
26#1467C2212yes—
27#1465C2214yes—
28#1461C2212yes—
29#1459C2214yes—
30#1325C2212yes—
31#1325C2211yescharacteristic
32#1325C2211yescharacteristic
33#1319C2212yes—
34#1319C2214yes—
35#1319C2212yes—
36#1319C2214yes—
37#559—42no—
38#559—42no—
39#557—42no—
40#557—42no—
41C2×C4⋊Q8C2×C411yescharacteristic
42C2×C4⋊Q8C2×C411yescharacteristic
43C2×C4⋊Q8C2×C412yes—
44C2×C4⋊Q8C2×C412yes—
45C2×#127C2×C411yescharacteristic
46C2×#125—22no—
47C2×#125—21no—
48C2×#124C2×C412yes—
49C2×#124C2×C412yes—
50C2×#124C2×C412yes—
51C2×#124C2×C412yes—
52C2×#124D412yes—
53C2×#123D414yes—
54C2×C4×Q8—21no—
55C2×C4×Q8C2×C411yescharacteristic
56C2×C4×Q8C2×C412yes—
57C2×C4×Q8C2×C411yescharacteristic
58C2×C4×Q8C2311yescharacteristic
59C2×C4×D4D412yes—
60C2×C4×D4C2312yes—
61#436—42no—
62#435—44no—
63C8×D4—42no—
64#432—22no—
65#431—22no—
66#407—24no—
67C2×C22⋊C8—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#391C2311yescharacteristic
77#391C2312yes—
78#390—22no—
79#390—22no—
80#390—22no—
81#390C2×C412yes—
82#389—21no—
83#389—22no—
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—22no—
97#324—22no—
98#324C2312yes—
99#324C2314yes—
100#324C2312yes—
101#323—24no—
102#323—24no—
103C22×Q8C22⋊C411yescharacteristic
104C22×Q8C22×C412yes—
105C22×Q8C22×C411yescharacteristic
106C22×Q8C22×C411yescharacteristic
107C22×D4C22⋊C412yes—
108C22×D4C22⋊C412yes—
109C22×D4C22×C412yes—
110C23×C4C2×D414yes—
111C2×M16—44no—
112C22×C8—44no—
113C4⋊Q8—22no—
114C4⋊Q8—41no—
115C4⋊Q8—44no—
116C4⋊Q8—41no—
117C4⋊Q8—41no—
118C4⋊Q8—82no—
119#127—41no—
120#127—41no—
121#125—48no—
122#125—42no—
123#125—42no—
124#124—24no—
125#124—44no—
126#124—42no—
127#124—42no—
128#124—42no—
129#124—82no—
130#124—82no—
131#123—44no—
132#123—44no—
133C4×Q8—22no—
134C4×Q8—22no—
135C4×Q8—41no—
136C4×Q8—41no—
137C4×Q8—44no—
138C4×Q8—41no—
139C4×Q8—41no—
140C4×Q8—42no—
141C4×Q8—42no—
142C4×D4—22no—
143C4×D4—24no—
144C4×D4—22no—
145C4×D4—44no—
146C2×C4⋊C4—21no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—21no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—21no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—21no—
154C2×C4⋊C4—24no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4C22⋊C412yes—
158C2×C4⋊C4C22⋊C411yescharacteristic
159C2×C4⋊C4C22×C411yescharacteristic
160C2×C4⋊C4C2×D412yes—
161C2×C4⋊C4C2411yescharacteristic
162C2×C22⋊C4—24no—
163C2×C22⋊C4—24no—
164C2×C22⋊C4—24no—
165C2×C22⋊C4—24no—
166C2×C22⋊C4C22⋊C414yes—
167C2×C22⋊C4C22⋊C414yes—
168C2×C22⋊C4C2×D414yes—
169C2×C42—21no—
170C2×C42—22no—
171C2×C42—21no—
172C2×C42—21no—
173C2×C42C22×C412yes—
174C2×C42C22×C411yescharacteristic
175C2×C42C22×C411yescharacteristic
176C2×C42C22×C411yescharacteristic
177C2×C42C22×C412yes—
178C2×C42C2×D412yes—
179C4⋊C8—42no—
180C4⋊C8—42no—
181C22⋊C8—22no—
182C22⋊C8—24no—
183C22⋊C8—22no—
184C22⋊C8—44no—
185C22⋊C8—44no—
186#97—42no—
187C4×C8—42no—
188#95—22no—
189#95—22no—
190#95—22no—
191#95—21no—
192#95—22no—
193#95—22no—
194#95—24no—
195C24C2×C22⋊C414yes—
196C2×Q8—21no—
197C2×Q8—21no—
198C2×Q8—21no—
199C2×Q8—21no—
200C2×Q8—22no—
201C2×Q8—21no—
202C2×Q8—22no—
203C2×Q8—21no—
204C2×Q8—41no—
205C2×Q8—41no—
206C2×Q8—42no—
207C2×Q8—42no—
208C2×Q8—41no—
209C2×Q8—41no—
210C2×Q8—42no—
211C2×D4—22no—
212C2×D4—22no—
213C2×D4—22no—
214C2×D4—22no—
215C2×D4—44no—
216C2×D4—42no—
217C2×D4—42no—
218C2×D4—42no—
219C2×D4—82no—
220C22×C4—21no—
221C22×C4—24no—
222C22×C4—24no—
223C22×C4—21no—
224C22×C4—22no—
225C22×C4—21no—
226C22×C4—22no—
227C22×C4—21no—
228C22×C4—84no—
229C22×C4C2×C22⋊C412yes—
230C22×C4C2×C22⋊C411yescharacteristic
231C22×C4C2×C22⋊C412yes—
232C22×C4C2×C22⋊C411yescharacteristic
233C22×C4C2×C22⋊C412yes—
234C22×C4C23×C411yescharacteristic
235C22×C4C22×D412yes—
236M16—164no—
237C2×C8—42no—
238C2×C8—42no—
239C2×C8—44no—
240C2×C8—164no—
241C4⋊C4—24no—
242C4⋊C4—42no—
243C4⋊C4—41no—
244C4⋊C4—44no—
245C4⋊C4—41no—
246C4⋊C4—42no—
247C4⋊C4—42no—
248C4⋊C4—44no—
249C4⋊C4—42no—
250C4⋊C4—41no—
251C4⋊C4—41no—
252C4⋊C4—44no—
253C4⋊C4—44no—
254C4⋊C4—84no—
255C4⋊C4—82no—
256C4⋊C4—82no—
257C4⋊C4#9912yes—
258C4⋊C4#9912yes—
259C22⋊C4—24no—
260C22⋊C4—24no—
261C22⋊C4—44no—
262C22⋊C4—44no—
263C22⋊C4—48no—
264C22⋊C4—84no—
265C22⋊C4—84no—
266C22⋊C4—84no—
267C42—21no—
268C42—22no—
269C42—22no—
270C42—24no—
271C42—21no—
272C42—21no—
273C42—41no—
274C42—42no—
275C42—42no—
276C42—44no—
277C42—41no—
278C42—41no—
279C42—81no—
280C42#13112yes—
281C23—24no—
282C23—24no—
283C23—24no—
284C23—44no—
285C23—44no—
286C23C22×C22⋊C411yescharacteristic
287Q8—42no—
288Q8—42no—
289Q8—81no—
290Q8—81no—
291Q8—82no—
292Q8—81no—
293Q8—82no—
294Q8—81no—
295D4—44no—
296D4—82no—
297D4—82no—
298D4—82no—
299D4—162no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—22no—
303C2×C4—22no—
304C2×C4—22no—
305C2×C4—22no—
306C2×C4—22no—
307C2×C4—22no—
308C2×C4—21no—
309C2×C4—22no—
310C2×C4—42no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—42no—
315C2×C4—42no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—42no—
319C2×C4—42no—
320C2×C4—42no—
321C2×C4—42no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—42no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—42no—
328C2×C4—41no—
329C2×C4—42no—
330C2×C4—84no—
331C2×C4—84no—
332C2×C4C2×#9911yescharacteristic
333C2×C4C2×#9914yes—
334C2×C4C2×#9911yescharacteristic
335C2×C4C2×#13111yescharacteristic
336C2×C4#56811yescharacteristic
337C8—82no—
338C8—84no—
339C8—82no—
340C22—21no—
341C22—21no—
342C22—44no—
343C22—44no—
344C22—44no—
345C22—84no—
346C22—84no—
347C22#288011yescharacteristic
348C22C22×#9911yescharacteristic
349C22#288811yescharacteristic
350C4—21no—
351C4—21no—
352C4—24no—
353C4—21no—
354C4—42no—
355C4—42no—
356C4—42no—
357C4—41no—
358C4—42no—
359C4—81no—
360C4—82no—
361C4—82no—
362C4—81no—
363C4—81no—
364C4#213512yes—
365C2—21no—
366C2—21no—
367C2—84no—
368C2#1741911yescharacteristic
369C2#3336111yescharacteristic
370C2C2×#213511yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC22×C4C23×C4
Frattini, cosocleC2×C4⋊C4C24
prime radicalC2×#127C2×C4
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C4⋊C4 ◁ C2×#559 ◁ G
LECSG ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #11568C2semidirect
4. #11564C2semidirect
5. #11535C2semidirect
6. #11532C2semidirect
7. #11481C2semidirect
8. #11478C2semidirect

Maximal subgroups

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

subgroupindex
#397462
#174292
#115682
#115642
#115352
#115322
#114812
#114782