Finite Group Databasebeta

← #213548

Subgroups — #213548

Order 512.

1358 conjugacy classes of subgroups.

The subgroup lattice has 403 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2324C211yesmaximal, characteristic
2#21742C212yesmaximal, complemented
3#9085C214yesmaximal, complemented
4#9083C214yesmaximal, complemented
5#8701C212yesmaximal, complemented
6#8300C212yesmaximal, complemented
7C22×#121C411yescharacteristic
8C22×C4×D4C2211yescharacteristic
9C22×#117C2211yescharacteristic
10C2×#407C2214yes—
11#2324—28no—
12C2×#390C412yes—
13C2×#390C411yescharacteristic
14C2×#389C411yescharacteristic
15C2×#388C412yes—
16C2×#386C2212yes—
17C2×#385C2212yes—
18C2×#380C411yescharacteristic
19C2×C4×C4⋊C4C2211yescharacteristic
20#1870—28no—
21#1817C2214yes—
22#1779C2214yes—
23C2×#324C2214yes—
24C2×#323C2214yes—
25#1280C2218yes—
26C23×D4C2×C412yes—
27C23×D4C2×C411yescharacteristic
28C24×C4C2×C411yescharacteristic
29C24×C4C2312yes—
30C22×M16—24no—
31C2×#121—24no—
32C2×#121—24no—
33C2×#121—24no—
34C2×C4×D4—44no—
35C2×C4×D4D418yes—
36C2×#117—24no—
37C2×#117—44no—
38C22×C4⋊C4C2×C411yescharacteristic
39C22×C4⋊C4C2×C411yescharacteristic
40C22×C4⋊C4C2×C411yescharacteristic
41C22×C4⋊C4C2×C412yes—
42C22×C4⋊C4C2×C411yescharacteristic
43C22×C4⋊C4C2311yescharacteristic
44C22×C22⋊C4C2×C412yes—
45C22×C22⋊C4C2×C411yescharacteristic
46C22×C22⋊C4C2×C412yes—
47C22×C22⋊C4C2×C412yes—
48C22×C22⋊C4C2×C412yes—
49C22×C22⋊C4C2×C412yes—
50C22×C22⋊C4C2311yescharacteristic
51C22×C42C2×C411yescharacteristic
52C22×C42C2×C412yes—
53C22×C42C2311yescharacteristic
54#407—48no—
55C2×C22⋊C8—24no—
56C2×C22⋊C8—24no—
57C2×C22⋊C8—24no—
58C2×C22⋊C8C2318yes—
59#390—24no—
60#390—28no—
61#389—24no—
62#388—28no—
63#386—24no—
64#386—24no—
65#385—24no—
66#385—24no—
67#380—24no—
68C4×C4⋊C4—24no—
69C2×#95C2×C412yes—
70C2×#95C2×C412yes—
71C2×#95C2312yes—
72#336—28no—
73#324—44no—
74#323—44no—
75C25C22×C412yes—
76C25C22×C411yescharacteristic
77C22×D4—24no—
78C22×D4—24no—
79C22×D4—22no—
80C22×D4—22no—
81C22×D4—24no—
82C22×D4—42no—
83C22×D4—42no—
84C22×D4—82no—
85C22×D4—82no—
86C22×D4C22⋊C414yes—
87C23×C4—22no—
88C23×C4—22no—
89C23×C4—28no—
90C23×C4—42no—
91C23×C4—42no—
92C23×C4—82no—
93C23×C4—82no—
94C23×C4M1614yescomplemented
95C23×C4C22×C411yescharacteristic
96C23×C4C22×C411yescharacteristic
97C23×C4C22×C411yescharacteristic
98C23×C4C22×C412yes—
99C23×C4C22×C411yescharacteristic
100C23×C4C22×C411yescharacteristic
101C23×C4C22×C412yes—
102C23×C4C22×C412yes—
103C23×C4C2×D414yes—
104C23×C4C2×D414yes—
105C23×C4C2411yescharacteristic
106C2×M16—48no—
107C2×M16—84no—
108C2×M16—84no—
109C22×C8—28no—
110#121—232no—
111C4×D4—48no—
112C4×D4—48no—
113#117—416no—
114C2×C4⋊C4—22no—
115C2×C4⋊C4—22no—
116C2×C4⋊C4—22no—
117C2×C4⋊C4—22no—
118C2×C4⋊C4—22no—
119C2×C4⋊C4—24no—
120C2×C4⋊C4—24no—
121C2×C4⋊C4—42no—
122C2×C4⋊C4—42no—
123C2×C4⋊C4—42no—
124C2×C4⋊C4—82no—
125C2×C4⋊C4C22⋊C414yes—
126C2×C4⋊C4M1614yescomplemented
127C2×C22⋊C4—24no—
128C2×C22⋊C4—24no—
129C2×C22⋊C4—24no—
130C2×C22⋊C4—22no—
131C2×C22⋊C4—22no—
132C2×C22⋊C4—24no—
133C2×C22⋊C4—24no—
134C2×C22⋊C4—24no—
135C2×C22⋊C4—24no—
136C2×C22⋊C4—42no—
137C2×C22⋊C4—44no—
138C2×C22⋊C4—82no—
139C2×C22⋊C4—82no—
140C2×C22⋊C4—82no—
141C2×C22⋊C4C22⋊C418yes—
142C2×C42—24no—
143C2×C42—22no—
144C2×C42—24no—
145C2×C42—42no—
146C2×C42—82no—
147C2×C42C2×D414yes—
148C22⋊C8—28no—
149C22⋊C8—28no—
150C22⋊C8—48no—
151C22⋊C8—84no—
152C22⋊C8—84no—
153#95—24no—
154#95—24no—
155#95—22no—
156#95—22no—
157C24—21no—
158C24—22no—
159C24—22no—
160C24—21no—
161C24—22no—
162C24—24no—
163C24—22no—
164C24—24no—
165C24—42no—
166C24—42no—
167C24—42no—
168C24—42no—
169C24—41no—
170C24—41no—
171C24—41no—
172C24—41no—
173C24C2×C22⋊C414yes—
174C24C2×M1612yes—
175C24C23×C411yescharacteristic
176C2×D4—216no—
177C2×D4—28no—
178C2×D4—44no—
179C2×D4—44no—
180C2×D4—48no—
181C2×D4—48no—
182C2×D4—84no—
183C2×D4—84no—
184C2×D4—88no—
185C2×D4—88no—
186C22×C4—22no—
187C22×C4—22no—
188C22×C4—22no—
189C22×C4—21no—
190C22×C4—22no—
191C22×C4—22no—
192C22×C4—21no—
193C22×C4—22no—
194C22×C4—22no—
195C22×C4—24no—
196C22×C4—24no—
197C22×C4—24no—
198C22×C4—28no—
199C22×C4—28no—
200C22×C4—21no—
201C22×C4—22no—
202C22×C4—21no—
203C22×C4—22no—
204C22×C4—22no—
205C22×C4—24no—
206C22×C4—41no—
207C22×C4—44no—
208C22×C4—44no—
209C22×C4—48no—
210C22×C4—41no—
211C22×C4—41no—
212C22×C4—41no—
213C22×C4—44no—
214C22×C4—44no—
215C22×C4—42no—
216C22×C4—42no—
217C22×C4—44no—
218C22×C4—44no—
219C22×C4—48no—
220C22×C4—42no—
221C22×C4—42no—
222C22×C4—42no—
223C22×C4—41no—
224C22×C4—42no—
225C22×C4—41no—
226C22×C4—41no—
227C22×C4—41no—
228C22×C4—816no—
229C22×C4C2×C22⋊C414yes—
230C22×C4C2×C22⋊C412yes—
231C22×C4C2×C22⋊C412yes—
232C22×C4C2×M1612yes—
233C22×C4C2×M1612yes—
234C22×C4C2×M1612yes—
235C22×C4C2×M1614yes—
236C22×C4C22×D412yes—
237C22×C4H(5,𝔽2)14yes—
238M16—816no—
239C2×C8—48no—
240C2×C8—48no—
241C2×C8—48no—
242C4⋊C4—28no—
243C4⋊C4—44no—
244C4⋊C4—44no—
245C4⋊C4—44no—
246C4⋊C4—44no—
247C4⋊C4—84no—
248C22⋊C4—216no—
249C22⋊C4—48no—
250C22⋊C4—44no—
251C22⋊C4—44no—
252C22⋊C4—84no—
253C22⋊C4—84no—
254C22⋊C4—84no—
255C42—28no—
256C42—44no—
257C42—84no—
258C23—21no—
259C23—21no—
260C23—21no—
261C23—21no—
262C23—24no—
263C23—24no—
264C23—22no—
265C23—24no—
266C23—24no—
267C23—24no—
268C23—24no—
269C23—24no—
270C23—24no—
271C23—42no—
272C23—44no—
273C23—44no—
274C23—44no—
275C23—44no—
276C23—42no—
277C23—42no—
278C23—42no—
279C23—41no—
280C23—41no—
281C23—41no—
282C23—41no—
283C23—41no—
284C23—41no—
285C23—44no—
286C23—42no—
287C23—44no—
288C23—84no—
289C23—84no—
290C23—84no—
291C23—84no—
292C23—82no—
293C23—82no—
294C23—82no—
295C23—82no—
296C23C22×C22⋊C411yescharacteristic
297C23#51812yes—
298C23C22×M1611yescharacteristic
299C23C22×M1611yescharacteristic
300C23#56812yes—
301D4—416no—
302D4—816no—
303C2×C4—22no—
304C2×C4—22no—
305C2×C4—22no—
306C2×C4—22no—
307C2×C4—24no—
308C2×C4—24no—
309C2×C4—22no—
310C2×C4—24no—
311C2×C4—22no—
312C2×C4—24no—
313C2×C4—24no—
314C2×C4—24no—
315C2×C4—22no—
316C2×C4—22no—
317C2×C4—24no—
318C2×C4—44no—
319C2×C4—48no—
320C2×C4—416no—
321C2×C4—48no—
322C2×C4—44no—
323C2×C4—44no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—42no—
327C2×C4—42no—
328C2×C4—42no—
329C2×C4—44no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—42no—
333C2×C4—42no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—42no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—88no—
340C2×C4—84no—
341C2×C4—84no—
342C2×C4—82no—
343C2×C4—82no—
344C2×C4#40714yes—
345C2×C4#53414yes—
346C8—88no—
347C22—22no—
348C22—21no—
349C22—21no—
350C22—21no—
351C22—21no—
352C22—22no—
353C22—22no—
354C22—22no—
355C22—44no—
356C22—42no—
357C22—44no—
358C22—44no—
359C22—42no—
360C22—42no—
361C22—44no—
362C22—44no—
363C22—44no—
364C22—44no—
365C22—84no—
366C22—88no—
367C22—88no—
368C22—84no—
369C22—81no—
370C22—81no—
371C22—81no—
372C22—81no—
373C22#230811yescharacteristic
374C22C2×#40711yescharacteristic
375C22#287911yescharacteristic
376C22#288311yescharacteristic
377C22#288311yescharacteristic
378C22#288811yescharacteristic
379C22#288911yescharacteristic
380C22#297312yes—
381C22#297512yes—
382C4—24no—
383C4—24no—
384C4—44no—
385C4—42no—
386C4—44no—
387C4—42no—
388C4—82no—
389C4—81no—
390C4—81no—
391C2—22no—
392C2—22no—
393C2—84no—
394C2—82no—
395C2#1736911yescharacteristic
396C2#1737011yescharacteristic
397C2#1740711yescharacteristic
398C2#1740811yescharacteristic
399C2#2174211yescharacteristic
400C2#2175011yescharacteristic
401C2#2175711yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC24C23×C4
Frattini, cosocleC23×C4C24
prime radicalC22×#121C4
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×D4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21742C2semidirect
3. #9085C2semidirect
4. #9083C2semidirect
5. #8701C2semidirect
6. #8300C2semidirect
94. C23×C4M16semidirect
126. C2×C4⋊C4M16semidirect

Maximal subgroups

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

subgroupindex
C2×#23242
#217422
#90852
#90832
#87012
#83002