Finite Group Databasebeta

← #207245

Subgroups — #207245

Order 512.

1050 conjugacy classes of subgroups.

The subgroup lattice has 365 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35167C211yesmaximal, characteristic
2#21951C211yesmaximal, characteristic
3#13804C211yesmaximal, characteristic
4#9911C214yesmaximal
5#9908C214yesmaximal
6#8026C212yesmaximal
7#8014C212yesmaximal
8C2×#544—21no—
9C2×#431C2212yes—
10#2857C2214yes—
11#2659—22no—
12#2552C414yescomplemented
13#2551C414yescomplemented
14#2550—24no—
15#2316—21no—
16#2285C2212yes—
17C2×#382C2211yescharacteristic
18C2×C4×C4⋊C4—21no—
19C2×C4×C4⋊C4C2211yescharacteristic
20C2×C4×C4⋊C4C2211yescharacteristic
21#1939—24no—
22#1797C2214yes—
23#1436—22no—
24C4⋊#95—21no—
25#1434C2212yes—
26C4×#95C2212yes—
27#1294C2214yes—
28#1294C2214yes—
29#1294C2218yes—
30#544—44no—
31#544—44no—
32C2×C4⋊Q8—24no—
33C2×C4⋊Q8C2×C414yes—
34C2×#125—24no—
35C2×C22⋊Q8—24no—
36C2×#117—22no—
37C22×C4⋊C4—21no—
38C22×C4⋊C4—21no—
39C22×C4⋊C4D411yescharacteristic
40C22×C4⋊C4D411yescharacteristic
41C22×C42—21no—
42C22×C42—22no—
43C22×C42C2312yes—
44C22×C42C2311yescharacteristic
45#431—24no—
46#431—24no—
47C2×C4⋊C8—22no—
48C2×C4⋊C8—24no—
49C2×C22⋊C8C2314yes—
50C2×C4×C8—22no—
51#398—28no—
52#395—28no—
53#391—24no—
54#389—24no—
55#389—24no—
56#389—24no—
57#389C2×C418yes—
58#384C2×C414yes—
59#384C2×C412yes—
60#384C2×C412yes—
61#382—22no—
62#382—22no—
63#382Q814yescomplemented
64#380—24no—
65#380—22no—
66#380C2314yes—
67C4×C4⋊C4—22no—
68C4×C4⋊C4—22no—
69C4×C4⋊C4—44no—
70C4×C4⋊C4C2×C414yes—
71C4×C4⋊C4C2×C414yes—
72C4×C4⋊C4D414yes—
73C4×C4⋊C4Q814yescomplemented
74C2×#95—21no—
75C2×#95—22no—
76C2×#95—22no—
77C2×#95—22no—
78C2×#95D412yes—
79C2×#95C2311yescharacteristic
80C2×#95C2312yes—
81C2×#95C2311yescharacteristic
82#336—24no—
83C22×Q8—24no—
84C23×C4—21no—
85C23×C4—22no—
86C23×C4—21no—
87C23×C4C2×D412yes—
88C23×C4C2×D411yescharacteristic
89C23×C4C2×D412yes—
90C23×C4C2×D411yescharacteristic
91C23×C4C2411yescharacteristic
92C22×C8—24no—
93C4⋊Q8—416no—
94C4⋊Q8—84no—
95#125—416no—
96C22⋊Q8—48no—
97C22⋊Q8—48no—
98#117—44no—
99#117—44no—
100C2×C4⋊C4—22no—
101C2×C4⋊C4—22no—
102C2×C4⋊C4—22no—
103C2×C4⋊C4—22no—
104C2×C4⋊C4—24no—
105C2×C4⋊C4—28no—
106C2×C4⋊C4—24no—
107C2×C4⋊C4—24no—
108C2×C4⋊C4—24no—
109C2×C4⋊C4—22no—
110C2×C4⋊C4—22no—
111C2×C4⋊C4—24no—
112C2×C4⋊C4—42no—
113C2×C4⋊C4—42no—
114C2×C4⋊C4—42no—
115C2×C4⋊C4—42no—
116C2×C4⋊C4C22⋊C414yes—
117C2×C4⋊C4C22⋊C414yes—
118C2×C4⋊C4SD1614yes—
119C2×C4⋊C4Q1614yes—
120C2×C4⋊C4C22×C414yes—
121C2×C4⋊C4C22×C414yes—
122C2×C22⋊C4—24no—
123C2×C42—24no—
124C2×C42—24no—
125C2×C42—42no—
126C2×C42—42no—
127C2×C42—44no—
128C2×C42—44no—
129C2×C42—82no—
130C2×C42—82no—
131C2×C42C22×C414yes—
132C2×C42C22×C412yes—
133C2×C42C2×D414yes—
134C2×C42C2×D412yes—
135C2×C42C2×Q812yes—
136C2×C42C2×Q812yes—
137C2×C42#4112yes—
138C2×C42#4112yes—
139C4⋊C8—44no—
140C4⋊C8—84no—
141C22⋊C8—24no—
142C22⋊C8—24no—
143C4×C8—44no—
144#95—24no—
145#95—28no—
146#95—24no—
147#95—24no—
148#95—21no—
149#95—21no—
150#95—22no—
151#95—44no—
152#95—44no—
153#95—82no—
154#95C22⋊C418yes—
155#95C2×Q812yes—
156#95C2×Q812yes—
157#95C2×Q814yes—
158C24C22×D411yescharacteristic
159C2×Q8—48no—
160C2×Q8—84no—
161C2×Q8—84no—
162C22×C4—22no—
163C22×C4—22no—
164C22×C4—22no—
165C22×C4—22no—
166C22×C4—22no—
167C22×C4—24no—
168C22×C4—24no—
169C22×C4—24no—
170C22×C4—22no—
171C22×C4—42no—
172C22×C4—42no—
173C22×C4—41no—
174C22×C4—41no—
175C22×C4—41no—
176C22×C4—41no—
177C22×C4—42no—
178C22×C4—42no—
179C22×C4—41no—
180C22×C4—41no—
181C22×C4—41no—
182C22×C4—41no—
183C22×C4—42no—
184C22×C4—41no—
185C22×C4—42no—
186C22×C4—42no—
187C22×C4—41no—
188C22×C4—41no—
189C22×C4—42no—
190C22×C4—41no—
191C22×C4—41no—
192C22×C4—42no—
193C22×C4—41no—
194C22×C4—42no—
195C22×C4C2×C22⋊C414yes—
196C22×C4C2×C22⋊C412yes—
197C22×C4C2×C22⋊C414yes—
198C22×C4C2×C22⋊C412yes—
199C22×C4C22⋊Q812yes—
200C22×C4C22⋊Q812yes—
201C22×C4C22⋊Q812yes—
202C22×C4C22⋊Q812yes—
203C22×C4C2×SD1614yes—
204C22×C4C2×SD1612yes—
205C22×C4C2×Q1614yes—
206C22×C4C2×Q1612yes—
207C22×C4AGL(1,ℤ/8)12yes—
208C22×C4AGL(1,ℤ/8)12yes—
209C22×C4C23×C411yescharacteristic
210C22×C4C22×D411yescharacteristic
211C22×C4C22×Q811yescharacteristic
212C22×C4C22×Q811yescharacteristic
213C22×C4C2×#4111yescharacteristic
214C22×C4C2×#4111yescharacteristic
215C2×C8—44no—
216C2×C8—44no—
217C2×C8—44no—
218C4⋊C4—28no—
219C4⋊C4—44no—
220C4⋊C4—44no—
221C4⋊C4—48no—
222C4⋊C4—48no—
223C4⋊C4—416no—
224C4⋊C4—44no—
225C4⋊C4—44no—
226C4⋊C4—48no—
227C4⋊C4—82no—
228C4⋊C4—84no—
229C4⋊C4—82no—
230C4⋊C4—84no—
231C4⋊C4#103116yes—
232C22⋊C4—44no—
233C22⋊C4—44no—
234C42—44no—
235C42—44no—
236C42—48no—
237C42—48no—
238C42—88no—
239C42C4×Q814yes—
240C42C4×Q814yes—
241C42#12414yes—
242C42C4⋊Q814yes—
243C23—21no—
244C23—21no—
245C23—21no—
246C23—21no—
247C23C22×C22⋊C411yescharacteristic
248C23C2×C22⋊Q811yescharacteristic
249C23C2×C22⋊Q811yescharacteristic
250C23C22×SD1611yescharacteristic
251C23C22×Q1611yescharacteristic
252C23C2×AGL(1,ℤ/8)11yescharacteristic
253C23C2×AGL(1,ℤ/8)11yescharacteristic
254Q8—168no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—24no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—22no—
261C2×C4—22no—
262C2×C4—24no—
263C2×C4—24no—
264C2×C4—24no—
265C2×C4—24no—
266C2×C4—24no—
267C2×C4—44no—
268C2×C4—42no—
269C2×C4—42no—
270C2×C4—44no—
271C2×C4—41no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—42no—
276C2×C4—42no—
277C2×C4—42no—
278C2×C4—42no—
279C2×C4—42no—
280C2×C4—42no—
281C2×C4—42no—
282C2×C4—44no—
283C2×C4—44no—
284C2×C4—42no—
285C2×C4—44no—
286C2×C4—44no—
287C2×C4—44no—
288C2×C4—44no—
289C2×C4—44no—
290C2×C4—44no—
291C2×C4—42no—
292C2×C4—44no—
293C2×C4—82no—
294C2×C4—82no—
295C2×C4—82no—
296C2×C4—81no—
297C2×C4—82no—
298C2×C4—81no—
299C2×C4—81no—
300C2×C4—82no—
301C2×C4—81no—
302C2×C4—82no—
303C2×C4#39114yes—
304C2×C4#39114yes—
305C2×C4C2×#10318yes—
306C2×C4C2×#10314yes—
307C2×C4#41814yes—
308C2×C4#47614yes—
309C2×C4#47614yes—
310C2×C4#47714yes—
311C2×C4#47714yes—
312C2×C4C2×C4×Q811yescharacteristic
313C2×C4C2×C4×Q811yescharacteristic
314C2×C4C2×#12411yescharacteristic
315C2×C4C2×C4⋊Q811yescharacteristic
316C8—84no—
317C22—21no—
318C22—21no—
319C22—21no—
320C22—21no—
321C22—21no—
322C22—21no—
323C22—21no—
324C22—21no—
325C22—21no—
326C22—21no—
327C22—21no—
328C22—21no—
329C22C2×#39111yescharacteristic
330C22#288311yescharacteristic
331C22#288411yescharacteristic
332C22C22×#10311yescharacteristic
333C22C2×#41811yescharacteristic
334C22#290111yescharacteristic
335C22#290111yescharacteristic
336C22C2×#47611yescharacteristic
337C22C2×#47611yescharacteristic
338C22C2×#47711yescharacteristic
339C22C2×#47711yescharacteristic
340C4—24no—
341C4—21no—
342C4—21no—
343C4—21no—
344C4—21no—
345C4—28no—
346C4—44no—
347C4—44no—
348C4—82no—
349C4—84no—
350C4—84no—
351C4—84no—
352C4#198718yes—
353C2—21no—
354C2—21no—
355C2—21no—
356C2—21no—
357C2#1761411yescharacteristic
358C2#1761911yescharacteristic
359C2#1762611yescharacteristic
360C2#1763111yescharacteristic
361C2C2×#198711yescharacteristic
362C2#2133011yescharacteristic
363C2#2133411yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C22 ▷ 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
12. #2552C4semidirect
13. #2551C4semidirect
63. #382Q8semidirect
73. C4×C4⋊C4Q8semidirect

Maximal subgroups

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

subgroupindex
#351672
#219512
#138042
#99112
#99082
#80262
#80142