Finite Group Databasebeta

← #6498006

Subgroups — #6498006

Order 512.

1474 conjugacy classes of subgroups.

The subgroup lattice has 381 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36640C211yesmaximal, characteristic
2C2×#2859C211yesmaximal, characteristic
3#33299C212yesmaximal, complemented
4#33295C212yesmaximal, complemented
5#30235C214yesmaximal, complemented
6#29227C214yesmaximal, complemented
7#29221C214yesmaximal, complemented
8#27966C212yesmaximal
9#27965C212yesmaximal
10#27964C214yesmaximal
11#21838C214yesmaximal
12C4×#432C211yesmaximal, characteristic
13C22×#125C2212yes—
14C2×#432C2212yes—
15#2864—22no—
16#2859—22no—
17#2859—22no—
18#2859C2212yes—
19#2859C2211yescharacteristic
20#2859C2212yes—
21#2859C2211yescharacteristic
22#2859C2212yes—
23#2859C2212yes—
24#2859C2212yes—
25#2851—22no—
26#2851C2212yes—
27#2851C2212yes—
28C4×#125—22no—
29C4×#125C2212yes—
30C4×#125C2212yes—
31#2296—22no—
32C2×#389—22no—
33C2×#389—22no—
34#2285C2212yes—
35C2×#384C2211yescharacteristic
36C2×#384C2211yescharacteristic
37C2×#382C2212yes—
38C2×C43C2211yescharacteristic
39#2101C22116yes—
40#2100C2218yes—
41#2099C2218yes—
42#1992C2218yes—
43#1992C2218yes—
44#1992C2214yes—
45#1992C2214yes—
46#1991C2218yes—
47#1989C22116yes—
48#1836C2214yes—
49#1836C2214yes—
50#1835C2218yes—
51#1824C2218yes—
52#1823C2218yes—
53#1767C2212yes—
54C4×C4⋊C8C2212yes—
55#1749C2212yes—
56C4×C22⋊C8C2212yes—
57#1745C2212yes—
58C4×#97C2212yes—
59C2×#125—22no—
60C2×#125—22no—
61C2×#125—22no—
62C2×#125—216no—
63C2×#125C2312yes—
64C2×#125C2312yes—
65C2×#125C2314yes—
66C2×#125C2314yes—
67C2×#117—22no—
68C22×C4⋊C4—22no—
69C22×C4⋊C4—22no—
70C22×C4⋊C4—22no—
71C22×C4⋊C4—22no—
72C22×C4⋊C4C2312yes—
73C22×C4⋊C4C2312yes—
74C22×C42D412yes—
75C22×C42D412yes—
76C22×C42C2311yescharacteristic
77C22×C42C2311yescharacteristic
78C22×C42C2311yescharacteristic
79#432—28no—
80C2×C4⋊C8C2314yes—
81C2×C22⋊C8C2314yes—
82C2×#97C2314yes—
83C2×#97C2314yes—
84C2×C4×C8C2314yes—
85#389—22no—
86#389—22no—
87#389—24no—
88#389—24no—
89#389—24no—
90#389—24no—
91#389—24no—
92#389—24no—
93#389—24no—
94#389—24no—
95#389—22no—
96#389—22no—
97#389—22no—
98#389—22no—
99#389C2314yes—
100#389C2314yes—
101#387—24no—
102#387—24no—
103#387—24no—
104#387—24no—
105#385—24no—
106#385—24no—
107#384—22no—
108#384—22no—
109#384C2312yes—
110#384C2312yes—
111#384C2312yes—
112#384C2312yes—
113#384C2312yes—
114#384C2312yes—
115#382—22no—
116#382—24no—
117#382—22no—
118#382C2312yes—
119#382C2312yes—
120#382C2314yes—
121#382C2314yes—
122#380—22no—
123#380C2314yes—
124C4×C4⋊C4—22no—
125C4×C4⋊C4—24no—
126C4×C4⋊C4—24no—
127C4×C4⋊C4C2314yes—
128C2×#95—24no—
129C43D412yes—
130C43D412yes—
131C43C2311yescharacteristic
132C43C2311yescharacteristic
133C43C2311yescharacteristic
134C43C2311yescharacteristic
135#340C23116yes—
136#340C23116yes—
137#340C23116yes—
138#340C23116yes—
139#336C2314yes—
140#336C2314yes—
141C23×C4—22no—
142C23×C4—22no—
143C23×C4C2×D412yes—
144C23×C4C2×D412yes—
145C23×C4C2×D411yescharacteristic
146C23×C4C2×D411yescharacteristic
147C23×C4C2411yescharacteristic
148C22×C8C2418yes—
149#125—216no—
150#125—216no—
151#125—28no—
152#125—28no—
153#125—48no—
154#125—48no—
155#125—48no—
156#117—48no—
157C2×C4⋊C4—28no—
158C2×C4⋊C4—28no—
159C2×C4⋊C4—24no—
160C2×C4⋊C4—24no—
161C2×C4⋊C4—24no—
162C2×C4⋊C4—24no—
163C2×C4⋊C4—24no—
164C2×C4⋊C4—24no—
165C2×C4⋊C4—24no—
166C2×C4⋊C4—24no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—24no—
174C2×C4⋊C4—24no—
175C2×C4⋊C4—24no—
176C2×C4⋊C4—24no—
177C2×C4⋊C4—48no—
178C2×C4⋊C4—48no—
179C2×C4⋊C4—48no—
180C2×C4⋊C4—48no—
181C2×C4⋊C4C2414yes—
182C2×C4⋊C4C2414yes—
183C2×C4⋊C4C2414yes—
184C2×C4⋊C4C2414yes—
185C2×C22⋊C4—24no—
186C2×C42—24no—
187C2×C42—24no—
188C2×C42—24no—
189C2×C42—44no—
190C2×C42—44no—
191C2×C42C2×D412yes—
192C2×C42C2×D412yes—
193C2×C42C2×D412yes—
194C2×C42C2×D412yes—
195C2×C42C2×D412yes—
196C2×C42C2×D412yes—
197C2×C42C2×D412yes—
198C2×C42C2×D412yes—
199C2×C42C2×D412yes—
200C2×C42C2×D412yes—
201C2×C42C2×D411yescharacteristic
202C2×C42C2×D411yescharacteristic
203C2×C42#4118yes—
204C2×C42C2411yescharacteristic
205C2×C42C2411yescharacteristic
206C2×C42C2411yescharacteristic
207C2×C42C2411yescharacteristic
208C2×C42C2411yescharacteristic
209C2×C42C2411yescharacteristic
210C4⋊C8—28no—
211C22⋊C8—28no—
212#97—28no—
213#97—28no—
214C4×C8—28no—
215#95—28no—
216#95—28no—
217#95—24no—
218#95—24no—
219#95—24no—
220#95—24no—
221#95—24no—
222#95—24no—
223C24C22×D411yescharacteristic
224C22×C4—24no—
225C22×C4—24no—
226C22×C4—24no—
227C22×C4—24no—
228C22×C4—22no—
229C22×C4—22no—
230C22×C4—24no—
231C22×C4—24no—
232C22×C4—24no—
233C22×C4—24no—
234C22×C4—24no—
235C22×C4—24no—
236C22×C4—42no—
237C22×C4—42no—
238C22×C4—48no—
239C22×C4—44no—
240C22×C4—44no—
241C22×C4—48no—
242C22×C4#12414yes—
243C22×C4#12414yes—
244C22×C4C22×D412yes—
245C22×C4C22×D411yescharacteristic
246C22×C4C22×D411yescharacteristic
247C22×C4C22×D411yescharacteristic
248C22×C4C22×D411yescharacteristic
249C22×C4C22×D411yescharacteristic
250C22×C4C22×D412yes—
251C22×C4C22×D412yes—
252C22×C4C22×D412yes—
253C22×C4C2×#4114yes—
254C22×C4C2511yescharacteristic
255C2×C8—28no—
256C2×C8—28no—
257C2×C8—28no—
258C4⋊C4—28no—
259C4⋊C4—28no—
260C4⋊C4—28no—
261C4⋊C4—28no—
262C4⋊C4—48no—
263C4⋊C4—48no—
264C4⋊C4—48no—
265C4⋊C4—44no—
266C4⋊C4—44no—
267C4⋊C4—44no—
268C4⋊C4—44no—
269C4⋊C4—48no—
270C4⋊C4—44no—
271C4⋊C4—44no—
272C4⋊C4—48no—
273C4⋊C4—48no—
274C4⋊C4—48no—
275C4⋊C4—48no—
276C4⋊C4—48no—
277C4⋊C4—48no—
278C4⋊C4—48no—
279C4⋊C4—48no—
280C22⋊C4—48no—
281C42—24no—
282C42—24no—
283C42—24no—
284C42—24no—
285C42—22no—
286C42—22no—
287C42—42no—
288C42—42no—
289C42—42no—
290C42—42no—
291C42—42no—
292C42—42no—
293C42—48no—
294C42#12414yes—
295C42#12414yes—
296C42C2×#4114yes—
297C42C2×#4114yes—
298C422−1+414yes—
299C422−1+414yes—
300C23—21no—
301C23—21no—
302C23—22no—
303C23C2×#12412yes—
304C23#57514yes—
305C23C23×D411yescharacteristic
306C2×C4—216no—
307C2×C4—28no—
308C2×C4—22no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—22no—
312C2×C4—22no—
313C2×C4—22no—
314C2×C4—22no—
315C2×C4—22no—
316C2×C4—22no—
317C2×C4—22no—
318C2×C4—22no—
319C2×C4—22no—
320C2×C4—28no—
321C2×C4—22no—
322C2×C4—22no—
323C2×C4—22no—
324C2×C4—44no—
325C2×C4—48no—
326C2×C4—44no—
327C2×C4—44no—
328C2×C4—44no—
329C2×C4—44no—
330C2×C4—44no—
331C2×C4—44no—
332C2×C4—44no—
333C2×C4—44no—
334C2×C4—48no—
335C2×C4—48no—
336C2×C4—88no—
337C2×C4—88no—
338C2×C4C2×#12412yes—
339C2×C4C2×#12412yes—
340C2×C4C2×#12412yes—
341C2×C4C2×#12412yes—
342C2×C4C2×#12412yes—
343C2×C4#53614yes—
344C2×C4#53614yes—
345C2×C4#57514yes—
346C2×C4C22×#4112yes—
347C2×C4C2×2−1+412yes—
348C2×C4C2×2−1+412yes—
349C8—48no—
350C22—24no—
351C22—24no—
352C22—22no—
353C22—22no—
354C22#314214yes—
355C22C22×#12411yescharacteristic
356C22C2×#53611yescharacteristic
357C22C2×#53611yescharacteristic
358C22C2×#57512yes—
359C22C2×#57512yes—
360C4—24no—
361C4—24no—
362C4—24no—
363C4—22no—
364C4—22no—
365C4—22no—
366C4—22no—
367C4—22no—
368C4—42no—
369C4—84no—
370C4—84no—
371C4—84no—
372C4—84no—
373C4#350914yes—
374C2—24no—
375C2#3532011yescharacteristic
376C2C2×#314211yescharacteristic
377C2#6061811yescharacteristic
378C2#6085412yes—
379C2#6086112yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C43 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #33299C2semidirect
4. #33295C2semidirect
5. #30235C2semidirect
6. #29227C2semidirect
7. #29221C2semidirect

Maximal subgroups

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

subgroupindex
#366402
C2×#28592
#332992
#332952
#302352
#292272
#292212
#279662
#279652
#279642
#218382
C4×#4322