Finite Group Databasebeta

← #215147

Subgroups — #215147

Order 512.

1286 conjugacy classes of subgroups.

The subgroup lattice has 393 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2324C211yesmaximal, characteristic
2#21916C212yesmaximal, complemented
3#9653C214yesmaximal, complemented
4#9652C214yesmaximal, complemented
5#8302C212yesmaximal, complemented
6#8300C212yesmaximal, complemented
7C22×#121C411yescharacteristic
8C22×C4×D4C2211yescharacteristic
9C22×#117C2211yescharacteristic
10#2324—28no—
11C2×#390C411yescharacteristic
12C2×#390C2212yes—
13C2×#389C411yescharacteristic
14C2×#388C2212yes—
15C2×#386C412yes—
16C2×#385C412yes—
17C2×#380C411yescharacteristic
18C2×C4×C4⋊C4C2211yescharacteristic
19#1917—28no—
20#1817C2214yes—
21#1797C2214yes—
22#1749C2214yes—
23C2×#324C2214yes—
24C2×#323C2214yes—
25C2×#323C2218yes—
26C23×D4C2×C412yes—
27C23×D4C2×C411yescharacteristic
28C24×C4C2×C411yescharacteristic
29C24×C4C2×C412yes—
30C2×#121—24no—
31C2×#121—24no—
32C2×#121—24no—
33C2×C4×D4—24no—
34C2×C4×D4—44no—
35C2×#117—24no—
36C2×#117—44no—
37C22×C4⋊C4C2×C411yescharacteristic
38C22×C4⋊C4C2×C411yescharacteristic
39C22×C4⋊C4C2×C411yescharacteristic
40C22×C4⋊C4C2×C412yes—
41C22×C4⋊C4C2×C411yescharacteristic
42C22×C4⋊C4C2311yescharacteristic
43C22×C22⋊C4C2×C412yes—
44C22×C22⋊C4C2×C411yescharacteristic
45C22×C22⋊C4C2×C412yes—
46C22×C22⋊C4C2×C412yes—
47C22×C22⋊C4C2×C412yes—
48C22×C22⋊C4C2311yescharacteristic
49C22×C22⋊C4C2312yes—
50C22×C42C2×C411yescharacteristic
51C22×C42C2311yescharacteristic
52C22×C42C2312yes—
53C2×C4⋊C8—24no—
54C2×C22⋊C8C2318yes—
55C2×#97—24no—
56#390—24no—
57#390—24no—
58#390—24no—
59#389—24no—
60#388—24no—
61#388—24no—
62#386—28no—
63#385—28no—
64#380—24no—
65C4×C4⋊C4D418yes—
66C2×#95C2×C412yes—
67C2×#95C2×C412yes—
68C2×#95C2×C412yes—
69#336—24no—
70#336—28no—
71#336—24no—
72#324—44no—
73#323—44no—
74#323—48no—
75C25C22×C412yes—
76C25C22×C411yescharacteristic
77C22×D4—24no—
78C22×D4—24no—
79C22×D4—22no—
80C22×D4—24no—
81C22×D4—22no—
82C22×D4—42no—
83C22×D4—42no—
84C22×D4—82no—
85C22×D4—82no—
86C22×D4M1614yes—
87C23×C4—22no—
88C23×C4—22no—
89C23×C4—24no—
90C23×C4—42no—
91C23×C4—42no—
92C23×C4—44no—
93C23×C4—82no—
94C23×C4—82no—
95C23×C4M1614yes—
96C23×C4C22×C411yescharacteristic
97C23×C4C22×C411yescharacteristic
98C23×C4C22×C411yescharacteristic
99C23×C4C22×C411yescharacteristic
100C23×C4C22×C412yes—
101C23×C4C22×C412yes—
102C23×C4C22×C412yes—
103C23×C4C22×C411yescharacteristic
104C23×C4C2411yescharacteristic
105C22×C8—28no—
106#121—232no—
107C4×D4—416no—
108#117—416no—
109C2×C4⋊C4—22no—
110C2×C4⋊C4—22no—
111C2×C4⋊C4—22no—
112C2×C4⋊C4—22no—
113C2×C4⋊C4—22no—
114C2×C4⋊C4—24no—
115C2×C4⋊C4—24no—
116C2×C4⋊C4—42no—
117C2×C4⋊C4—42no—
118C2×C4⋊C4—42no—
119C2×C4⋊C4—82no—
120C2×C4⋊C4C22⋊C414yes—
121C2×C4⋊C4C22⋊C414yes—
122C2×C22⋊C4—24no—
123C2×C22⋊C4—24no—
124C2×C22⋊C4—24no—
125C2×C22⋊C4—22no—
126C2×C22⋊C4—22no—
127C2×C22⋊C4—22no—
128C2×C22⋊C4—22no—
129C2×C22⋊C4—24no—
130C2×C22⋊C4—28no—
131C2×C22⋊C4—24no—
132C2×C22⋊C4—24no—
133C2×C22⋊C4—24no—
134C2×C22⋊C4—42no—
135C2×C22⋊C4—82no—
136C2×C22⋊C4—82no—
137C2×C22⋊C4—82no—
138C2×C42—24no—
139C2×C42—22no—
140C2×C42—42no—
141C2×C42—82no—
142C2×C42C2×D414yes—
143C2×C42C2×D414yes—
144C2×C42C2×D414yes—
145C4⋊C8—84no—
146C22⋊C8—28no—
147C22⋊C8—28no—
148#97—84no—
149#95—24no—
150#95—24no—
151#95C22⋊C418yes—
152C24—22no—
153C24—21no—
154C24—22no—
155C24—21no—
156C24—22no—
157C24—24no—
158C24—22no—
159C24—24no—
160C24—41no—
161C24—42no—
162C24—42no—
163C24—42no—
164C24—41no—
165C24—41no—
166C24—42no—
167C24—41no—
168C24C2×M1614yes—
169C24C2×M1612yes—
170C24C23×C411yescharacteristic
171C2×D4—216no—
172C2×D4—28no—
173C2×D4—48no—
174C2×D4—44no—
175C2×D4—44no—
176C2×D4—48no—
177C2×D4—84no—
178C2×D4—84no—
179C2×D4—88no—
180C2×D4—88no—
181C22×C4—22no—
182C22×C4—22no—
183C22×C4—21no—
184C22×C4—22no—
185C22×C4—22no—
186C22×C4—21no—
187C22×C4—22no—
188C22×C4—22no—
189C22×C4—22no—
190C22×C4—24no—
191C22×C4—28no—
192C22×C4—21no—
193C22×C4—22no—
194C22×C4—21no—
195C22×C4—22no—
196C22×C4—22no—
197C22×C4—24no—
198C22×C4—41no—
199C22×C4—42no—
200C22×C4—42no—
201C22×C4—48no—
202C22×C4—41no—
203C22×C4—41no—
204C22×C4—41no—
205C22×C4—42no—
206C22×C4—42no—
207C22×C4—44no—
208C22×C4—44no—
209C22×C4—48no—
210C22×C4—48no—
211C22×C4—41no—
212C22×C4—41no—
213C22×C4—42no—
214C22×C4—41no—
215C22×C4—42no—
216C22×C4—41no—
217C22×C4—84no—
218C22×C4—84no—
219C22×C4—816no—
220C22×C4C2×C22⋊C414yes—
221C22×C4C2×C22⋊C414yes—
222C22×C4C2×C22⋊C412yes—
223C22×C4C2×C22⋊C412yes—
224C22×C4C2×M1612yes—
225C22×C4C2×M1612yes—
226C22×C4C2×M1612yes—
227C22×C4C22×D412yes—
228C22×C4H(5,𝔽2)14yes—
229C2×C8—48no—
230C2×C8—48no—
231C2×C8—48no—
232C4⋊C4—28no—
233C4⋊C4—44no—
234C4⋊C4—44no—
235C4⋊C4—44no—
236C4⋊C4—44no—
237C4⋊C4—84no—
238C22⋊C4—216no—
239C22⋊C4—44no—
240C22⋊C4—44no—
241C22⋊C4—44no—
242C22⋊C4—44no—
243C22⋊C4—84no—
244C22⋊C4—84no—
245C22⋊C4—84no—
246C42—28no—
247C42—44no—
248C42—44no—
249C42—44no—
250C23—21no—
251C23—21no—
252C23—21no—
253C23—21no—
254C23—24no—
255C23—22no—
256C23—24no—
257C23—28no—
258C23—42no—
259C23—44no—
260C23—44no—
261C23—42no—
262C23—42no—
263C23—44no—
264C23—42no—
265C23—44no—
266C23—41no—
267C23—42no—
268C23—41no—
269C23—42no—
270C23—42no—
271C23—41no—
272C23—42no—
273C23—41no—
274C23—41no—
275C23—42no—
276C23—42no—
277C23—41no—
278C23—42no—
279C23—44no—
280C23—84no—
281C23—82no—
282C23—84no—
283C23—82no—
284C23—84no—
285C23—82no—
286C23—84no—
287C23—82no—
288C23C22×C22⋊C411yescharacteristic
289C23#51812yes—
290C23C22×M1611yescharacteristic
291C23C22×M1611yescharacteristic
292C23#56812yes—
293D4—416no—
294D4—816no—
295C2×C4—22no—
296C2×C4—22no—
297C2×C4—22no—
298C2×C4—22no—
299C2×C4—24no—
300C2×C4—24no—
301C2×C4—22no—
302C2×C4—22no—
303C2×C4—24no—
304C2×C4—24no—
305C2×C4—24no—
306C2×C4—24no—
307C2×C4—24no—
308C2×C4—24no—
309C2×C4—22no—
310C2×C4—22no—
311C2×C4—24no—
312C2×C4—44no—
313C2×C4—416no—
314C2×C4—44no—
315C2×C4—44no—
316C2×C4—44no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—42no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—42no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—42no—
327C2×C4—41no—
328C2×C4—42no—
329C2×C4—84no—
330C2×C4—816no—
331C2×C4—82no—
332C2×C4—82no—
333C2×C4—84no—
334C2×C4#40714yes—
335C2×C4#53514yes—
336C8—88no—
337C22—22no—
338C22—22no—
339C22—22no—
340C22—22no—
341C22—21no—
342C22—21no—
343C22—21no—
344C22—21no—
345C22—42no—
346C22—44no—
347C22—42no—
348C22—44no—
349C22—44no—
350C22—42no—
351C22—84no—
352C22—88no—
353C22—84no—
354C22—88no—
355C22—81no—
356C22—82no—
357C22—81no—
358C22—82no—
359C22—81no—
360C22—82no—
361C22—82no—
362C22—81no—
363C22#230911yescharacteristic
364C22C2×#40711yescharacteristic
365C22#287911yescharacteristic
366C22#288311yescharacteristic
367C22#288411yescharacteristic
368C22#288811yescharacteristic
369C22#288811yescharacteristic
370C22#297312yes—
371C22#297312yes—
372C4—24no—
373C4—24no—
374C4—44no—
375C4—42no—
376C4—42no—
377C4—44no—
378C4—81no—
379C4—81no—
380C4—82no—
381C2—22no—
382C2—22no—
383C2—82no—
384C2—84no—
385C2#1736911yescharacteristic
386C2#1737211yescharacteristic
387C2#1740811yescharacteristic
388C2#1740911yescharacteristic
389C2#2179011yescharacteristic
390C2#2179611yescharacteristic
391C2#2180111yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC24C23×C4
Frattini, cosocleC23×C4C24
prime radicalC22×#121C4
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21916C2semidirect
3. #9653C2semidirect
4. #9652C2semidirect
5. #8302C2semidirect
6. #8300C2semidirect

Maximal subgroups

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

subgroupindex
C2×#23242
#219162
#96532
#96522
#83022
#83002