Finite Group Databasebeta

← #6408868

Subgroups — #6408868

Order 512.

1394 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×#125C211yesmaximal, characteristic
2#23372C212yesmaximal, complemented
3#23369C212yesmaximal, complemented
4#23265C214yesmaximal, complemented
5#22629C214yesmaximal
6#22627C214yesmaximal
7#22379C218yesmaximal, complemented
8#21914C214yesmaximal
9#20778C212yesmaximal
10C22×#125C411yescharacteristic
11C4×#125C412yes—
12C4×#125C414yes—
13C4×#125C411yescharacteristic
14C4×#125C411yescharacteristic
15C4×#125C2211yescharacteristic
16C4×#125C2211yescharacteristic
17C4×#125C2214yes—
18C4×#125C2212yes—
19C2×#389C412yes—
20C2×#387C414yes—
21C2×#382C411yescharacteristic
22C2×C4×C4⋊C4C2214yes—
23C2×C4×C4⋊C4C2212yes—
24C2×C43C2211yescharacteristic
25#1940C2218yes—
26#1939C2218yes—
27#1917C2218yes—
28#1917C2218yes—
29#1917C2218yes—
30#1917C2218yes—
31#1917C2218yes—
32#1916C2218yes—
33#1916C2218yes—
34#1916C2218yes—
35#1839C2214yes—
36#1831C2214yes—
37#1817C2214yes—
38#1797C2212yes—
39#1797C2212yes—
40#1797C2214yes—
41#1797C2214yes—
42#1771C2218yes—
43#1770C2218yes—
44#1767C2214yes—
45#1749C2214yes—
46#1749C2214yes—
47C4×C22⋊C8C2214yes—
48#1745C2214yes—
49C2×#125—28no—
50C2×#125C2×C412yes—
51C2×#125C2×C414yes—
52C2×#125C2×C411yescharacteristic
53C2×#125C2×C411yescharacteristic
54C22×C4⋊C4C2×C414yes—
55C22×C4⋊C4C2×C412yes—
56C22×C4⋊C4C2×C412yes—
57C22×C4⋊C4C2×C414yes—
58C22×C42C2×C411yescharacteristic
59C22×C42C2×C412yes—
60C22×C42C2×C411yescharacteristic
61C22×C42C2311yescharacteristic
62C22×C42C2314yes—
63C22×C42C2312yes—
64C2×C4⋊C8C2314yes—
65C2×C4⋊C8C2314yes—
66C2×C4⋊C8C2318yes—
67C2×C4⋊C8C2318yes—
68C2×C22⋊C8C2314yes—
69C2×C22⋊C8C2314yes—
70C2×#97C2318yes—
71C2×#97C2318yes—
72C2×C4×C8C2318yes—
73#389C2×C414yes—
74#389C2×C414yes—
75#389C2×C412yes—
76#389C2×C412yes—
77#389C2×C412yes—
78#389C2×C412yes—
79#387C2×C414yes—
80#387C2×C414yes—
81#387C2×C414yes—
82#387C2×C414yes—
83#387C2×C414yes—
84#387C2×C414yes—
85#387C2×C414yes—
86#387C2×C414yes—
87#382C2×C414yes—
88#382C2×C412yes—
89#382C2×C411yescharacteristic
90#382C2×C411yescharacteristic
91C4×C4⋊C4C2×C414yes—
92C4×C4⋊C4C2×C414yes—
93C4×C4⋊C4C2×C414yes—
94C4×C4⋊C4C2×C414yes—
95C4×C4⋊C4C2×C414yes—
96C4×C4⋊C4C2×C412yes—
97C4×C4⋊C4C2×C412yes—
98C4×C4⋊C4C2312yes—
99C4×C4⋊C4C2314yes—
100C4×C4⋊C4C2314yes—
101C4×C4⋊C4C2312yes—
102C4×C4⋊C4C2314yes—
103C4×C4⋊C4C2314yes—
104C4×C4⋊C4C2314yes—
105C2×#95C2×C414yes—
106C2×#95C2×C414yes—
107C2×#95C2×C414yes—
108C43C2×C412yes—
109C43C2×C411yescharacteristic
110C43C2×C411yescharacteristic
111C43C2311yescharacteristic
112C43C2312yes—
113C43C2311yescharacteristic
114#336C2314yes—
115#336C2314yes—
116#336C2318yes—
117#336C2318yes—
118#336C2318yes—
119#336C2318yes—
120#336C2318yes—
121#336C2318yes—
122#336C2318yes—
123C23×C4C22×C411yescharacteristic
124C23×C4C22×C411yescharacteristic
125C23×C4C22×C414yes—
126C23×C4C22×C412yes—
127C23×C4C22×C414yes—
128C23×C4C22×C412yes—
129C23×C4C2411yescharacteristic
130C22×C8C2418yes—
131C22×C8C2418yes—
132#125—216no—
133#125—28no—
134#125—24no—
135#125M1618yes—
136C2×C4⋊C4—28no—
137C2×C4⋊C4—216no—
138C2×C4⋊C4—44no—
139C2×C4⋊C4—48no—
140C2×C4⋊C4C22×C414yes—
141C2×C4⋊C4C22×C414yes—
142C2×C4⋊C4C22×C414yes—
143C2×C4⋊C4C22×C412yes—
144C2×C4⋊C4C22×C414yes—
145C2×C4⋊C4C22×C414yes—
146C2×C4⋊C4C22×C412yes—
147C2×C4⋊C4C22×C412yes—
148C2×C4⋊C4C22×C414yes—
149C2×C4⋊C4C22×C414yes—
150C2×C4⋊C4C22×C414yes—
151C2×C4⋊C4C22×C414yes—
152C2×C4⋊C4C22×C414yes—
153C2×C4⋊C4C22×C412yes—
154C2×C42—22no—
155C2×C42—22no—
156C2×C42—24no—
157C2×C42—42no—
158C2×C42—48no—
159C2×C42—44no—
160C2×C42—44no—
161C2×C42C22×C411yescharacteristic
162C2×C42C22×C412yes—
163C2×C42C22×C411yescharacteristic
164C2×C42C22×C411yescharacteristic
165C2×C42C22×C412yes—
166C2×C42C22×C412yes—
167C2×C42C22×C412yes—
168C2×C42C22×C412yes—
169C2×C42C22×C412yes—
170C2×C42C22×C412yes—
171C2×C42C22×C412yes—
172C2×C42C22×C411yescharacteristic
173C2×C42C22×C411yescharacteristic
174C2×C42C22×C414yes—
175C2×C42C22×C414yes—
176C2×C42C22×C411yescharacteristic
177C2×C42C2411yescharacteristic
178C2×C42C2411yescharacteristic
179C2×C42C2412yes—
180C2×C42C2412yes—
181C2×C42C2414yes—
182C2×C42C2414yes—
183C4⋊C8—48no—
184C4⋊C8—48no—
185C4⋊C8—44no—
186C4⋊C8—44no—
187C22⋊C8—28no—
188C22⋊C8—28no—
189#97—48no—
190#97—48no—
191C4×C8—48no—
192#95C22×C414yes—
193#95C22×C414yes—
194#95C22×C414yes—
195#95C22×C414yes—
196#95C22×C414yes—
197#95C22×C414yes—
198#95C22×C414yes—
199#95C22×C414yes—
200#95C22×C414yes—
201#95C22×C414yes—
202#95C22×C414yes—
203#95C22×C414yes—
204C24C23×C411yescharacteristic
205C22×C4—22no—
206C22×C4—22no—
207C22×C4—22no—
208C22×C4—24no—
209C22×C4—22no—
210C22×C4—24no—
211C22×C4—24no—
212C22×C4—24no—
213C22×C4—28no—
214C22×C4—28no—
215C22×C4—24no—
216C22×C4—28no—
217C22×C4—44no—
218C22×C4—42no—
219C22×C4—42no—
220C22×C4—42no—
221C22×C4—44no—
222C22×C4—44no—
223C22×C4C23×C411yescharacteristic
224C22×C4C23×C411yescharacteristic
225C22×C4C23×C414yes—
226C22×C4C23×C414yes—
227C22×C4C23×C412yes—
228C22×C4C23×C412yes—
229C22×C4C23×C412yes—
230C22×C4C23×C414yes—
231C22×C4C23×C411yescharacteristic
232C22×C4C23×C412yes—
233C22×C4C23×C414yes—
234C22×C4C23×C411yescharacteristic
235C22×C4C23×C411yescharacteristic
236C22×C42−1+412yes—
237C22×C42−1+412yes—
238C22×C4C2511yescharacteristic
239C2×C8—28no—
240C2×C8—28no—
241C2×C8—28no—
242C2×C8—28no—
243C2×C8—28no—
244C2×C8—28no—
245C4⋊C4—28no—
246C4⋊C4—28no—
247C4⋊C4—28no—
248C4⋊C4—28no—
249C4⋊C4—24no—
250C4⋊C4—28no—
251C4⋊C4—42no—
252C4⋊C4—42no—
253C4⋊C4—44no—
254C4⋊C4—44no—
255C4⋊C4—44no—
256C4⋊C4—44no—
257C4⋊C4C2×M16116yes—
258C4⋊C4C2×M1618yes—
259C42—22no—
260C42—22no—
261C42—24no—
262C42—22no—
263C42—24no—
264C42—21no—
265C42—21no—
266C42—21no—
267C42—21no—
268C42—28no—
269C42—24no—
270C42—28no—
271C42—24no—
272C42—42no—
273C42—44no—
274C42—42no—
275C42—44no—
276C42—41no—
277C42—41no—
278C42—42no—
279C42—42no—
280C42—42no—
281C42—42no—
282C42C2×M1614yes—
283C23—22no—
284C23—21no—
285C23—21no—
286C23#51912yes—
287C23#56814yes—
288C23C24×C411yescharacteristic
289C2×C4—28no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—24no—
293C2×C4—22no—
294C2×C4—22no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—22no—
298C2×C4—24no—
299C2×C4—24no—
300C2×C4—24no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—22no—
304C2×C4—24no—
305C2×C4—22no—
306C2×C4—22no—
307C2×C4—24no—
308C2×C4—24no—
309C2×C4—22no—
310C2×C4—24no—
311C2×C4—22no—
312C2×C4—22no—
313C2×C4—22no—
314C2×C4—22no—
315C2×C4—22no—
316C2×C4—22no—
317C2×C4—22no—
318C2×C4—24no—
319C2×C4—24no—
320C2×C4—24no—
321C2×C4—24no—
322C2×C4—24no—
323C2×C4—48no—
324C2×C4—416no—
325C2×C4—48no—
326C2×C4—44no—
327C2×C4—88no—
328C2×C4#51912yes—
329C2×C4C22×M1612yes—
330C2×C4C22×M1614yes—
331C2×C4C22×M1618yes—
332C2×C4#56812yes—
333C2×C4C2×2−1+411yescharacteristic
334C2×C4C2×2−1+411yescharacteristic
335C2×C4#58514yes—
336C8—48no—
337C8—48no—
338C22—22no—
339C22—22no—
340C22—24no—
341C22—24no—
342C22#298714yes—
343C22C2×#51911yescharacteristic
344C22#342812yes—
345C22C23×M1611yescharacteristic
346C22C2×#56812yes—
347C22C2×#56811yescharacteristic
348C4—22no—
349C4—22no—
350C4—22no—
351C4—24no—
352C4—28no—
353C4—41no—
354C4—41no—
355C4—42no—
356C4—44no—
357C4—42no—
358C4—42no—
359C4—44no—
360C4—44no—
361C4#355312yes—
362C4#355312yes—
363C2—24no—
364C2#3437211yescharacteristic
365C2C2×#298711yescharacteristic
366C2#6024811yescharacteristic
367C2#6028312yes—
368C2#6028912yes—
subgroupG/H
center, Frattini, cosocleC22×C4C25
commutator, socleC23C24×C4
prime radicalC24C23×C4
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C22×C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #23372C2semidirect
3. #23369C2semidirect
4. #23265C2semidirect
7. #22379C2semidirect

Maximal subgroups

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

subgroupindex
C2×C4×#1252
#233722
#233692
#232652
#226292
#226272
#223792
#219142
#207782