Finite Group Databasebeta

← #460469

Subgroups — #460469

Order 512.

1230 conjugacy classes of subgroups.

The subgroup lattice has 367 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2629C211yesmaximal, characteristic
2C2×#2083C212yesmaximal, complemented
3#14340C211yesmaximal, characteristic
4#11385C212yesmaximal, complemented
5#11324C212yesmaximal, complemented
6#11208C212yesmaximal, complemented
7#11096C214yesmaximal
8#10713C211yesmaximal, characteristic
9C22×#125C2211yescharacteristic
10C22×#117C2211yescharacteristic
11C22×#107C2214yes—
12C22×C22⋊C8C2212yes—
13#2629—28no—
14C2×#398—22no—
15C2×#396—22no—
16C2×#389C2212yes—
17C2×#384C2211yescharacteristic
18C2×#384C2211yescharacteristic
19C2×#382C2211yescharacteristic
20C2×#380C2212yes—
21C2×#380C2212yes—
22#2083—28no—
23C2×#340C2214yes—
24C2×#340C2214yes—
25C2×#340C2214yes—
26C2×#336C2212yes—
27C2×#336C2212yes—
28#1441—22no—
29#1439—22no—
30C4⋊#95C2212yes—
31C24×C4C2311yescharacteristic
32C23×C8C2314yes—
33C2×#125—24no—
34C2×#125—24no—
35C2×#125—42no—
36C2×#117—24no—
37C2×#117—24no—
38C2×#117—42no—
39C22×C4⋊C4—22no—
40C22×C4⋊C4—22no—
41C22×C4⋊C4—22no—
42C22×C4⋊C4—22no—
43C22×C4⋊C4D412yes—
44C22×C4⋊C4C2312yes—
45C22×C4⋊C4C2312yes—
46C22×C4⋊C4C2312yes—
47C22×C4⋊C4C2312yes—
48C22×C22⋊C4—22no—
49C22×C22⋊C4—22no—
50C22×C22⋊C4D412yes—
51C22×C42C2311yescharacteristic
52C22×C42C2311yescharacteristic
53C2×#107—28no—
54C2×#107—44no—
55C2×#107—44no—
56C2×C22⋊C8—42no—
57C2×C22⋊C8—42no—
58C2×C22⋊C8Q818yes—
59#398—216no—
60#396—216no—
61#389—24no—
62#389D418yes—
63#384—24no—
64#384—24no—
65#382—42no—
66#380—24no—
67#380—24no—
68#380—44no—
69C2×#95—22no—
70C2×#95—22no—
71C2×#95—22no—
72C2×#95—22no—
73C2×#95—24no—
74#340—28no—
75#340—44no—
76#340—44no—
77#336—24no—
78#336—24no—
79C25C2×D411yescharacteristic
80C23×C4—22no—
81C23×C4—22no—
82C23×C4—22no—
83C23×C4—22no—
84C23×C4—21no—
85C23×C4—22no—
86C23×C4—21no—
87C23×C4—22no—
88C23×C4—22no—
89C23×C4—22no—
90C23×C4—22no—
91C23×C4C2×D412yes—
92C23×C4C2×D412yes—
93C23×C4C2×D411yescharacteristic
94C23×C4C2×Q814yes—
95C23×C4C2411yescharacteristic
96C22×C8—24no—
97C22×C8—24no—
98C22×C8—24no—
99C22×C8—24no—
100C22×C8—24no—
101C22×C8—24no—
102C22×C8C2×Q818yes—
103#125—416no—
104#117—416no—
105C2×C4⋊C4—24no—
106C2×C4⋊C4—24no—
107C2×C4⋊C4—22no—
108C2×C4⋊C4—22no—
109C2×C4⋊C4—22no—
110C2×C4⋊C4—22no—
111C2×C4⋊C4—28no—
112C2×C4⋊C4—24no—
113C2×C4⋊C4—24no—
114C2×C4⋊C4—22no—
115C2×C4⋊C4—22no—
116C2×C4⋊C4—24no—
117C2×C4⋊C4—24no—
118C2×C4⋊C4—24no—
119C2×C4⋊C4—24no—
120C2×C4⋊C4—22no—
121C2×C4⋊C4—22no—
122C2×C4⋊C4—24no—
123C2×C4⋊C4—28no—
124C2×C4⋊C4—28no—
125C2×C4⋊C4—42no—
126C2×C4⋊C4—44no—
127C2×C4⋊C4—44no—
128C2×C4⋊C4—44no—
129C2×C4⋊C4—84no—
130C2×C4⋊C4—84no—
131C2×C4⋊C4—84no—
132C2×C4⋊C4C2×D414yes—
133C2×C4⋊C4C2×D414yes—
134C2×C22⋊C4—24no—
135C2×C22⋊C4—24no—
136C2×C22⋊C4—28no—
137C2×C22⋊C4—42no—
138C2×C22⋊C4—44no—
139C2×C22⋊C4—84no—
140C2×C22⋊C4—84no—
141C2×C42—22no—
142C2×C42—22no—
143C2×C42—22no—
144C2×C42—22no—
145C2×C42C2×D414yes—
146C2×C42#4114yes—
147#107—416no—
148C22⋊C8—48no—
149#95—28no—
150#95—28no—
151#95—44no—
152#95—44no—
153#95—84no—
154C24—21no—
155C24—21no—
156C24—21no—
157C24—21no—
158C24—21no—
159C24—21no—
160C24—24no—
161C24—24no—
162C24C4⋊Q812yes—
163C24C22×D411yescharacteristic
164C22×C4—21no—
165C22×C4—22no—
166C22×C4—22no—
167C22×C4—21no—
168C22×C4—22no—
169C22×C4—22no—
170C22×C4—21no—
171C22×C4—21no—
172C22×C4—21no—
173C22×C4—21no—
174C22×C4—24no—
175C22×C4—24no—
176C22×C4—24no—
177C22×C4—24no—
178C22×C4—24no—
179C22×C4—24no—
180C22×C4—28no—
181C22×C4—28no—
182C22×C4—24no—
183C22×C4—24no—
184C22×C4—42no—
185C22×C4—42no—
186C22×C4—42no—
187C22×C4—42no—
188C22×C4—44no—
189C22×C4—44no—
190C22×C4—42no—
191C22×C4—42no—
192C22×C4—42no—
193C22×C4—42no—
194C22×C4—44no—
195C22×C4—44no—
196C22×C4—44no—
197C22×C4—44no—
198C22×C4—42no—
199C22×C4—42no—
200C22×C4—42no—
201C22×C4—42no—
202C22×C4—42no—
203C22×C4—42no—
204C22×C4—44no—
205C22×C4—44no—
206C22×C4—44no—
207C22×C4—44no—
208C22×C4#12014yes—
209C22×C4#12314yes—
210C22×C4C4⋊Q812yes—
211C22×C4C22×D412yes—
212C22×C4C22×Q812yes—
213C22×C4C2×#4112yes—
214C22×C4H(5,𝔽2)11yescharacteristic
215C22×C4H(5,𝔽2)12yes—
216C22×C4H(5,𝔽2)12yes—
217C22×C42−1+411yescharacteristic
218C22×C42−1+412yes—
219C2×C8—28no—
220C2×C8—28no—
221C2×C8—28no—
222C2×C8—44no—
223C2×C8—44no—
224C2×C8—44no—
225C2×C8—44no—
226C4⋊C4—24no—
227C4⋊C4—24no—
228C4⋊C4—28no—
229C4⋊C4—48no—
230C4⋊C4—416no—
231C4⋊C4—42no—
232C4⋊C4—42no—
233C4⋊C4—44no—
234C4⋊C4—44no—
235C4⋊C4—44no—
236C4⋊C4—48no—
237C4⋊C4—88no—
238C4⋊C4—88no—
239C4⋊C4—88no—
240C22⋊C4—48no—
241C22⋊C4—88no—
242C22⋊C4—88no—
243C42—28no—
244C42—28no—
245C23—28no—
246C23—22no—
247C23—22no—
248C23—22no—
249C23—41no—
250C23—44no—
251C23—41no—
252C23—41no—
253C23—41no—
254C23—44no—
255C23—42no—
256C23—42no—
257C23—44no—
258C23—42no—
259C23—42no—
260C23—42no—
261C23—42no—
262C23C2×#12011yescharacteristic
263C23C2×#12311yescharacteristic
264C23C2×C4⋊Q811yescharacteristic
265C23#53411yescharacteristic
266C23#53511yescharacteristic
267C23#53611yescharacteristic
268C23#53711yescharacteristic
269C23#57614yes—
270C23#57814yes—
271C2×C4—28no—
272C2×C4—22no—
273C2×C4—24no—
274C2×C4—24no—
275C2×C4—22no—
276C2×C4—24no—
277C2×C4—24no—
278C2×C4—22no—
279C2×C4—41no—
280C2×C4—44no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—44no—
285C2×C4—44no—
286C2×C4—42no—
287C2×C4—42no—
288C2×C4—48no—
289C2×C4—48no—
290C2×C4—44no—
291C2×C4—44no—
292C2×C4—44no—
293C2×C4—44no—
294C2×C4—44no—
295C2×C4—44no—
296C2×C4—84no—
297C2×C4—84no—
298C2×C4—84no—
299C2×C4—84no—
300C2×C4—82no—
301C2×C4—82no—
302C2×C4—84no—
303C2×C4—82no—
304C2×C4—82no—
305C2×C4—82no—
306C2×C4—82no—
307C2×C4—84no—
308C2×C4—84no—
309C2×C4—84no—
310C2×C4#53512yes—
311C2×C4#53712yes—
312C2×C4#54112yes—
313C2×C4#54212yes—
314C2×C4#54312yes—
315C2×C4#54412yes—
316C2×C4#54614yes—
317C2×C4D4×Q814yes—
318C2×C4#55414yes—
319C2×C4#56214yes—
320C8—48no—
321C22—44no—
322C22—44no—
323C22—44no—
324C22—42no—
325C22—82no—
326C22—82no—
327C22—81no—
328C22—81no—
329C22—82no—
330C22#240511yescharacteristic
331C22#243411yescharacteristic
332C22#250211yescharacteristic
333C22#261511yescharacteristic
334C22#262511yescharacteristic
335C22#265911yescharacteristic
336C22#281611yescharacteristic
337C22#300812yes—
338C22#301112yes—
339C22#309814yes—
340C22#316414yes—
341C22#320314yes—
342C22#322012yes—
343C22#322312yes—
344C22#324314yes—
345C22#325912yes—
346C22#326112yes—
347C4—44no—
348C4—44no—
349C4—42no—
350C4—84no—
351C4—84no—
352C4—162no—
353C4—162no—
354C4#269214yes—
355C4#270414yes—
356C2—82no—
357C2#1636511yescharacteristic
358C2#2401112yes—
359C2#2582112yes—
360C2#2795812yes—
361C2#2967011yescharacteristic
362C2#2967411yescharacteristic
363C2#2994812yes—
364C2#3030112yes—
365C2#3278312yes—
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×#117 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2083C2semidirect
4. #11385C2semidirect
5. #11324C2semidirect
6. #11208C2semidirect

Maximal subgroups

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

subgroupindex
C2×#26292
C2×#20832
#143402
#113852
#113242
#112082
#110962
#107132