Finite Group Databasebeta

← #215144

Subgroups — #215144

Order 512.

1596 conjugacy classes of subgroups.

The subgroup lattice has 410 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2321C211yesmaximal, characteristic
2#21906C212yesmaximal, complemented
3#9653C214yesmaximal, complemented
4#9652C214yesmaximal, complemented
5#8301C212yesmaximal, complemented
6#8298C212yesmaximal, complemented
7C22×#120C411yescharacteristic
8C22×C4×D4C2211yescharacteristic
9C23×C22⋊C4C2211yescharacteristic
10#2321—28no—
11C2×#390C2212yes—
12C2×#388C412yes—
13C2×#388C2212yes—
14C2×#386C412yes—
15C2×#386C411yescharacteristic
16C2×#385C411yescharacteristic
17C2×#379C411yescharacteristic
18C2×C4×C22⋊C4C2211yescharacteristic
19#1924—28no—
20#1797C2214yescomplemented
21#1779C2214yes—
22#1749C2214yescomplemented
23C2×#324C2214yescomplemented
24C2×#323C2218yes—
25C2×#323C2214yescomplemented
26C26C2×C411yescharacteristic
27C23×D4C2×C412yes—
28C23×D4C2×C411yescharacteristic
29C24×C4C2×C411yescharacteristic
30C24×C4C2×C411yescharacteristic
31C24×C4C2311yescharacteristic
32C2×#120—24no—
33C2×#120—24no—
34C2×#120—24no—
35C2×C4×D4—24no—
36C2×C4×D4—44no—
37C22×C4⋊C4C2×C412yes—
38C22×C4⋊C4C2311yescharacteristic
39C22×C22⋊C4—24no—
40C22×C22⋊C4—44no—
41C22×C22⋊C4C2×C412yes—
42C22×C22⋊C4C2×C411yescharacteristic
43C22×C22⋊C4C2×C412yes—
44C22×C22⋊C4C2×C411yescharacteristic
45C22×C22⋊C4C2×C412yes—
46C22×C22⋊C4C2×C412yes—
47C22×C22⋊C4C2×C412yes—
48C22×C22⋊C4C2×C411yescharacteristic
49C22×C22⋊C4C2312yes—
50C22×C22⋊C4C2311yescharacteristic
51C22×C42C2×C411yescharacteristic
52C22×C42C2312yes—
53C2×C4⋊C8—24no—
54C2×C22⋊C8—24no—
55C2×C22⋊C8—24no—
56C2×C22⋊C8C2318yes—
57C2×#97—24no—
58#390—24no—
59#390—24no—
60#388—24no—
61#388—24no—
62#388—28no—
63#386—24no—
64#386—28no—
65#385—24no—
66#379—24no—
67C4×C22⋊C4D418yes—
68C2×#95C2×C412yes—
69C2×#95C2×C412yes—
70C2×#95C2×C412yes—
71#336—24no—
72#336—24no—
73#324—44no—
74#323—48no—
75#323—44no—
76C25—22no—
77C25—22no—
78C25—42no—
79C25—42no—
80C25—82no—
81C25—82no—
82C25M1614yes—
83C25C22×C412yes—
84C25C22×C411yescharacteristic
85C25C22×C411yescharacteristic
86C22×D4—22no—
87C22×D4—22no—
88C22×D4—24no—
89C22×D4—24no—
90C22×D4—24no—
91C22×D4—42no—
92C22×D4—42no—
93C22×D4—82no—
94C22×D4—82no—
95C22×D4M1614yes—
96C23×C4—24no—
97C23×C4—22no—
98C23×C4—22no—
99C23×C4—42no—
100C23×C4—42no—
101C23×C4—82no—
102C23×C4—82no—
103C23×C4—82no—
104C23×C4C22×C411yescharacteristic
105C23×C4C22×C411yescharacteristic
106C23×C4C22×C411yescharacteristic
107C23×C4C22×C412yes—
108C23×C4C22×C412yes—
109C23×C4C22×C412yes—
110C23×C4C22×C411yescharacteristic
111C23×C4C2×D414yes—
112C23×C4C2411yescharacteristic
113C22×C8—28no—
114#120—232no—
115C4×D4—416no—
116C2×C4⋊C4—22no—
117C2×C4⋊C4—24no—
118C2×C4⋊C4—24no—
119C2×C4⋊C4—82no—
120C2×C22⋊C4—24no—
121C2×C22⋊C4—22no—
122C2×C22⋊C4—22no—
123C2×C22⋊C4—24no—
124C2×C22⋊C4—24no—
125C2×C22⋊C4—22no—
126C2×C22⋊C4—24no—
127C2×C22⋊C4—22no—
128C2×C22⋊C4—22no—
129C2×C22⋊C4—22no—
130C2×C22⋊C4—28no—
131C2×C22⋊C4—24no—
132C2×C22⋊C4—24no—
133C2×C22⋊C4—24no—
134C2×C22⋊C4—24no—
135C2×C22⋊C4—44no—
136C2×C22⋊C4—42no—
137C2×C22⋊C4—42no—
138C2×C22⋊C4—416no—
139C2×C22⋊C4—82no—
140C2×C22⋊C4—82no—
141C2×C22⋊C4—82no—
142C2×C22⋊C4C22⋊C414yes—
143C2×C22⋊C4C22⋊C414yes—
144C2×C42—22no—
145C2×C42—42no—
146C2×C42—82no—
147C2×C42C2×D414yes—
148C2×C42C2×D414yes—
149C4⋊C8—84no—
150C22⋊C8—28no—
151C22⋊C8—28no—
152C22⋊C8—84no—
153C22⋊C8—84no—
154#97—84no—
155#95—24no—
156#95—24no—
157#95C22⋊C418yes—
158C24—22no—
159C24—22no—
160C24—22no—
161C24—21no—
162C24—22no—
163C24—21no—
164C24—24no—
165C24—22no—
166C24—22no—
167C24—24no—
168C24—28no—
169C24—42no—
170C24—42no—
171C24—48no—
172C24—44no—
173C24—44no—
174C24—48no—
175C24—42no—
176C24—42no—
177C24—42no—
178C24—42no—
179C24—42no—
180C24—42no—
181C24—84no—
182C24—84no—
183C24—88no—
184C24—88no—
185C24—88no—
186C24—88no—
187C24—816no—
188C24C2×C22⋊C412yes—
189C24C2×M1614yes—
190C24C2×M1612yes—
191C24C2×M1612yes—
192C24C2×M1612yes—
193C24C23×C411yescharacteristic
194C2×D4—28no—
195C2×D4—216no—
196C2×D4—48no—
197C2×D4—44no—
198C2×D4—44no—
199C2×D4—48no—
200C2×D4—84no—
201C2×D4—84no—
202C2×D4—88no—
203C2×D4—88no—
204C22×C4—21no—
205C22×C4—22no—
206C22×C4—22no—
207C22×C4—22no—
208C22×C4—22no—
209C22×C4—21no—
210C22×C4—22no—
211C22×C4—24no—
212C22×C4—28no—
213C22×C4—22no—
214C22×C4—21no—
215C22×C4—21no—
216C22×C4—22no—
217C22×C4—24no—
218C22×C4—22no—
219C22×C4—42no—
220C22×C4—42no—
221C22×C4—42no—
222C22×C4—42no—
223C22×C4—44no—
224C22×C4—44no—
225C22×C4—42no—
226C22×C4—42no—
227C22×C4—44no—
228C22×C4—44no—
229C22×C4—42no—
230C22×C4—42no—
231C22×C4—42no—
232C22×C4—42no—
233C22×C4—84no—
234C22×C4—84no—
235C22×C4—88no—
236C22×C4—88no—
237C22×C4—88no—
238C22×C4C2×C22⋊C414yes—
239C22×C4C2×C22⋊C414yes—
240C22×C4C2×C22⋊C412yes—
241C22×C4C2×M1612yes—
242C22×C4C22×D412yes—
243C22×C4H(5,𝔽2)14yes—
244C2×C8—48no—
245C2×C8—48no—
246C2×C8—48no—
247C4⋊C4—44no—
248C4⋊C4—84no—
249C22⋊C4—216no—
250C22⋊C4—28no—
251C22⋊C4—44no—
252C22⋊C4—48no—
253C22⋊C4—44no—
254C22⋊C4—44no—
255C22⋊C4—44no—
256C22⋊C4—44no—
257C22⋊C4—84no—
258C22⋊C4—84no—
259C22⋊C4—84no—
260C42—44no—
261C42—44no—
262C42—44no—
263C23—22no—
264C23—22no—
265C23—22no—
266C23—22no—
267C23—24no—
268C23—24no—
269C23—24no—
270C23—24no—
271C23—24no—
272C23—22no—
273C23—24no—
274C23—28no—
275C23—44no—
276C23—44no—
277C23—44no—
278C23—44no—
279C23—48no—
280C23—44no—
281C23—44no—
282C23—416no—
283C23—44no—
284C23—44no—
285C23—44no—
286C23—44no—
287C23—42no—
288C23—41no—
289C23—41no—
290C23—42no—
291C23—42no—
292C23—42no—
293C23—42no—
294C23—42no—
295C23—42no—
296C23—42no—
297C23—42no—
298C23—44no—
299C23—816no—
300C23—82no—
301C23—82no—
302C23—84no—
303C23—84no—
304C23—84no—
305C23—84no—
306C23—84no—
307C23—832no—
308C23—832no—
309C23—88no—
310C23—88no—
311C23—816no—
312C23#40714yes—
313C23C22×C22⋊C411yescharacteristic
314C23#51812yes—
315C23C22×M1611yescharacteristic
316C23C22×M1611yescharacteristic
317C23#56812yes—
318D4—416no—
319D4—816no—
320C2×C4—22no—
321C2×C4—22no—
322C2×C4—22no—
323C2×C4—24no—
324C2×C4—22no—
325C2×C4—24no—
326C2×C4—24no—
327C2×C4—24no—
328C2×C4—24no—
329C2×C4—24no—
330C2×C4—24no—
331C2×C4—24no—
332C2×C4—44no—
333C2×C4—44no—
334C2×C4—44no—
335C2×C4—44no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—42no—
339C2×C4—41no—
340C2×C4—42no—
341C2×C4—41no—
342C2×C4—42no—
343C2×C4—42no—
344C2×C4—42no—
345C2×C4—42no—
346C2×C4—88no—
347C2×C4—84no—
348C2×C4—88no—
349C2×C4—88no—
350C2×C4—82no—
351C2×C4—82no—
352C2×C4—84no—
353C2×C4#53514yes—
354C8—88no—
355C22—22no—
356C22—22no—
357C22—22no—
358C22—22no—
359C22—24no—
360C22—24no—
361C22—28no—
362C22—48no—
363C22—48no—
364C22—42no—
365C22—42no—
366C22—44no—
367C22—42no—
368C22—44no—
369C22—44no—
370C22—44no—
371C22—44no—
372C22—84no—
373C22—88no—
374C22—84no—
375C22—88no—
376C22—832no—
377C22—81no—
378C22—82no—
379C22—81no—
380C22—82no—
381C22—82no—
382C22—82no—
383C22—82no—
384C22#230911yescharacteristic
385C22C2×#40711yescharacteristic
386C22#287911yescharacteristic
387C22#288312yes—
388C22#288812yes—
389C22#297312yes—
390C22#297312yes—
391C4—24no—
392C4—42no—
393C4—42no—
394C4—44no—
395C4—44no—
396C4—81no—
397C4—81no—
398C4—82no—
399C2—22no—
400C2—22no—
401C2—44no—
402C2—82no—
403C2—84no—
404C2#1736712yes—
405C2#1740512yes—
406C2#2179011yescharacteristic
407C2#2179511yescharacteristic
408C2#2180111yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC24C23×C4
Frattini, cosocleC23×C4C24
prime radicalC22×#120C4
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21906C2semidirect
3. #9653C2semidirect
4. #9652C2semidirect
5. #8301C2semidirect
6. #8298C2semidirect
20. #1797C22semidirect
22. #1749C22semidirect
23. C2×#324C22semidirect
25. C2×#323C22semidirect

Maximal subgroups

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

subgroupindex
C2×#23212
#219062
#96532
#96522
#83012
#82982