Finite Group Databasebeta

← #6482283

Subgroups — #6482283

Order 512.

1720 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35167C211yesmaximal, characteristic
2C2×#2365C211yesmaximal, characteristic
3#33148C214yesmaximal
4#29968C214yesmaximal
5#26510C218yesmaximal, complemented
6#26444C218yesmaximal, complemented
7#21951C211yesmaximal, characteristic
8#21833C212yesmaximal
9#21359C212yesmaximal
10C2×#544—21no—
11C22×C4⋊Q8C2211yescharacteristic
12C22×C4×Q8—21no—
13C2×#431C2212yes—
14#2857C2214yes—
15#2659—22no—
16#2552C2214yes—
17#2551C2214yes—
18#2550—24no—
19#2365—24no—
20#2365C2218yes—
21#2316—21no—
22C2×#391—21no—
23C2×#389C2212yes—
24C2×#387—22no—
25#2285C2212yes—
26C2×C4×C4⋊C4C2211yescharacteristic
27C2×C4×C4⋊C4C2212yes—
28C2×C4×C4⋊C4C2211yescharacteristic
29#2100C2218yes—
30#2088C22116yes—
31#2058C2218yes—
32#1990C2214yes—
33#1990C2214yes—
34#1988C2218yes—
35#1987C2218yes—
36#1948C2214yes—
37#1948C2214yes—
38#1948C2218yes—
39#1939C2218yes—
40#1920C22116yes—
41#1853C2214yes—
42#1853C2214yes—
43#1823C2218yes—
44#1797C2214yes—
45C4×#106C2218yes—
46C23×Q8—21no—
47#544—48no—
48C2×C4⋊Q8—24no—
49C2×C4⋊Q8—22no—
50C2×C4⋊Q8—24no—
51C2×C4⋊Q8Q818yescomplemented
52C2×C4⋊Q8C2314yes—
53C2×C4⋊Q8C2314yes—
54C2×#125—24no—
55C2×C22⋊Q8—24no—
56C2×C4×Q8—28no—
57C2×C4×Q8—48no—
58C2×#117—22no—
59C22×C4⋊C4—21no—
60C22×C4⋊C4—22no—
61C22×C4⋊C4D411yescharacteristic
62C22×C4⋊C4D411yescharacteristic
63C22×C4⋊C4C2311yescharacteristic
64C22×C4⋊C4C2312yes—
65C22×C4⋊C4C2311yescharacteristic
66C22×C42—22no—
67C22×C42—21no—
68C22×C42C2312yes—
69C22×C42C2311yescharacteristic
70#431—28no—
71C2×#106C2318yes—
72C2×#106C2318yes—
73C2×C4⋊C8C2314yes—
74C2×C4⋊C8C2318yes—
75C2×C22⋊C8C2314yes—
76C2×C4×C8C2314yes—
77#398—28no—
78#395—28no—
79#391—28no—
80#391—24no—
81#389—24no—
82#389—24no—
83#389—24no—
84#389—24no—
85#389C2318yes—
86#389C2318yes—
87#387—216no—
88#384C2314yes—
89#384C2312yes—
90#384C2312yes—
91#382—22no—
92#380—24no—
93#380—22no—
94#380C2314yes—
95C4×C4⋊C4—22no—
96C4×C4⋊C4—24no—
97C4×C4⋊C4—22no—
98C4×C4⋊C4D414yes—
99C4×C4⋊C4C2318yes—
100C4×C4⋊C4C2314yes—
101C4×C4⋊C4C2314yes—
102C4×C4⋊C4C2314yes—
103C2×#95—21no—
104C2×#95—22no—
105C2×#95—22no—
106C2×#95D412yes—
107#340C23116yes—
108#340C23116yes—
109#340C2318yes—
110#340C2318yes—
111#336C2318yes—
112C22×Q8—28no—
113C22×Q8—24no—
114C22×Q8—24no—
115C22×Q8—44no—
116C22×Q8—44no—
117C23×C4—22no—
118C23×C4—21no—
119C23×C4—21no—
120C23×C4C2×D412yes—
121C23×C4C2×D411yescharacteristic
122C23×C4C2×D412yes—
123C23×C4C2×D411yescharacteristic
124C23×C4C2411yescharacteristic
125C22×C8C2418yes—
126C4⋊Q8—216no—
127C4⋊Q8—28no—
128C4⋊Q8—48no—
129C4⋊Q8—416no—
130C4⋊Q8C2×Q8116yes—
131#125—416no—
132C22⋊Q8—416no—
133C4×Q8—432no—
134#117—48no—
135C2×C4⋊C4—28no—
136C2×C4⋊C4—24no—
137C2×C4⋊C4—24no—
138C2×C4⋊C4—22no—
139C2×C4⋊C4—21no—
140C2×C4⋊C4—24no—
141C2×C4⋊C4—24no—
142C2×C4⋊C4—28no—
143C2×C4⋊C4—21no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—24no—
147C2×C4⋊C4—28no—
148C2×C4⋊C4—24no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—24no—
152C2×C4⋊C4—42no—
153C2×C4⋊C4—48no—
154C2×C4⋊C4SD1618yes—
155C2×C4⋊C4C2×D414yes—
156C2×C4⋊C4C2×D414yes—
157C2×C4⋊C4C2×Q818yes—
158C2×C4⋊C4C2412yes—
159C2×C4⋊C4C2414yes—
160C2×C4⋊C4C2414yes—
161C2×C4⋊C4C2414yes—
162C2×C4⋊C4C2412yes—
163C2×C22⋊C4—24no—
164C2×C42—24no—
165C2×C42—28no—
166C2×C42—24no—
167C2×C42—44no—
168C2×C42—48no—
169C2×C42—44no—
170C2×C42C2×D414yes—
171C2×C42C2×D412yes—
172C2×C42C2×Q814yes—
173C2×C42#4114yes—
174C2×C42C2414yes—
175C2×C42C2412yes—
176#106—216no—
177#106—48no—
178C4⋊C8—28no—
179C4⋊C8—48no—
180C22⋊C8—28no—
181C4×C8—28no—
182#95—24no—
183#95—28no—
184#95—28no—
185#95—24no—
186#95—24no—
187#95—28no—
188#95C2×D418yes—
189C24C22×D411yescharacteristic
190C2×Q8—216no—
191C2×Q8—48no—
192C2×Q8—48no—
193C2×Q8—48no—
194C2×Q8—432no—
195C2×Q8—84no—
196C2×Q8—84no—
197C22×C4—22no—
198C22×C4—22no—
199C22×C4—24no—
200C22×C4—28no—
201C22×C4—22no—
202C22×C4—24no—
203C22×C4—24no—
204C22×C4—22no—
205C22×C4—24no—
206C22×C4—24no—
207C22×C4—42no—
208C22×C4—41no—
209C22×C4—41no—
210C22×C4—42no—
211C22×C4—42no—
212C22×C4—42no—
213C22×C4—42no—
214C22×C4—42no—
215C22×C4—44no—
216C22×C4—44no—
217C22×C4—42no—
218C22×C4—42no—
219C22×C4—42no—
220C22×C4C22⋊Q814yes—
221C22×C4C22⋊Q814yes—
222C22×C4C2×SD1618yes—
223C22×C4C2×SD1614yes—
224C22×C4AGL(1,ℤ/8)14yes—
225C22×C4C22×D414yes—
226C22×C4C22×D412yes—
227C22×C4C22×D411yescharacteristic
228C22×C4C22×D414yes—
229C22×C4C22×D412yes—
230C22×C4C22×Q812yes—
231C22×C4C2×#4112yes—
232C22×C4C2511yescharacteristic
233C2×C8—28no—
234C2×C8—28no—
235C2×C8—28no—
236C4⋊C4—24no—
237C4⋊C4—216no—
238C4⋊C4—28no—
239C4⋊C4—28no—
240C4⋊C4—24no—
241C4⋊C4—22no—
242C4⋊C4—28no—
243C4⋊C4—24no—
244C4⋊C4—28no—
245C4⋊C4—42no—
246C4⋊C4—48no—
247C4⋊C4—416no—
248C4⋊C4—416no—
249C4⋊C4—44no—
250C4⋊C4—44no—
251C4⋊C4—82no—
252C4⋊C4—84no—
253C4⋊C4—82no—
254C4⋊C4—84no—
255C4⋊C4C2×SD16116yes—
256C4⋊C4C22×Q818yes—
257C4⋊C42−1+414yes—
258C22⋊C4—48no—
259C42—28no—
260C42—44no—
261C42—416no—
262C42—48no—
263C42—48no—
264C42C22⋊Q818yes—
265C42C22×Q814yes—
266C42C2×#4114yes—
267C23—21no—
268C23—21no—
269C23—21no—
270C23—21no—
271C23C2×C22⋊Q812yes—
272C23C22×SD1612yes—
273C23C2×AGL(1,ℤ/8)12yes—
274C23C23×D411yescharacteristic
275Q8—432no—
276Q8—168no—
277C2×C4—22no—
278C2×C4—22no—
279C2×C4—24no—
280C2×C4—24no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—24no—
284C2×C4—24no—
285C2×C4—24no—
286C2×C4—28no—
287C2×C4—24no—
288C2×C4—24no—
289C2×C4—24no—
290C2×C4—44no—
291C2×C4—48no—
292C2×C4—48no—
293C2×C4—42no—
294C2×C4—42no—
295C2×C4—44no—
296C2×C4—44no—
297C2×C4—416no—
298C2×C4—44no—
299C2×C4—44no—
300C2×C4—42no—
301C2×C4—42no—
302C2×C4—44no—
303C2×C4—44no—
304C2×C4—42no—
305C2×C4—44no—
306C2×C4—44no—
307C2×C4—84no—
308C2×C4—84no—
309C2×C4#476116yes—
310C2×C4C2×C22⋊Q812yes—
311C2×C4C2×C22⋊Q814yes—
312C2×C4C2×C22⋊Q814yes—
313C2×C4C22×SD1618yes—
314C2×C4C22×SD1614yes—
315C2×C4C2×AGL(1,ℤ/8)14yes—
316C2×C4C23×Q811yescharacteristic
317C2×C4C22×#4111yescharacteristic
318C2×C4C2×2−1+412yes—
319C2×C4#58512yes—
320C8—48no—
321C22—22no—
322C22—22no—
323C22—22no—
324C22—22no—
325C22—22no—
326C22—22no—
327C22C2×#47614yes—
328C22C22×C22⋊Q811yescharacteristic
329C22#348511yescharacteristic
330C22#348611yescharacteristic
331C22C23×SD1611yescharacteristic
332C22C22×AGL(1,ℤ/8)11yescharacteristic
333C22#358611yescharacteristic
334C22#358611yescharacteristic
335C4—24no—
336C4—22no—
337C4—28no—
338C4—44no—
339C4—48no—
340C4—44no—
341C4—84no—
342C4—84no—
343C4—82no—
344C4C2×#47618yes—
345C4#346112yes—
346C4#350212yes—
347C2—24no—
348C2#3515411yescharacteristic
349C2C22×#47611yescharacteristic
350C2#6049111yescharacteristic
351C2#6169511yescharacteristic
352C2#6172211yescharacteristic
353C2#6181811yescharacteristic
354C2#6182811yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
5. #26510C2semidirect
6. #26444C2semidirect
51. C2×C4⋊Q8Q8semidirect

Maximal subgroups

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

subgroupindex
#351672
C2×#23652
#331482
#299682
#265102
#264442
#219512
#218332
#213592