Finite Group Databasebeta

← #192166

Subgroups — #192166

Order 512.

1232 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34182C211yesmaximal, characteristic
2#21981C212yesmaximal
3C2×#1461C211yesmaximal, characteristic
4#13926C211yesmaximal, characteristic
5#8857C218yesmaximal
6#8014C212yesmaximal
7C2×#519—21no—
8C22×C4×Q8C2211yescharacteristic
9C2×#431C2212yes—
10#2351—24no—
11Q8×C4⋊C4—22no—
12Q8×C4⋊C4—22no—
13#2316—21no—
14#2316C2214yes—
15C2×#391—21no—
16#2285—22no—
17#2285C2212yes—
18C2×#384C2211yescharacteristic
19C2×C4×C4⋊C4C2211yescharacteristic
20#1948—24no—
21#1853C2218yes—
22#1844—22no—
23#1834—22no—
24#1461C418yescomplemented
25C2×#324C2214yes—
26#1441C2214yes—
27C4⋊#95—22no—
28C4⋊#95—21no—
29#1294C2218yes—
30C23×Q8D411yescharacteristic
31#519—44no—
32#519—44no—
33C2×C4×Q8—24no—
34C2×C4×Q8—24no—
35C2×C4×Q8—22no—
36C2×C4×Q8—22no—
37C2×C4×Q8—24no—
38C2×C4×Q8C2×C414yes—
39C2×C4×Q8Q814yes—
40C2×#117—22no—
41C2×#117—24no—
42C22×C4⋊C4—21no—
43C22×C4⋊C4—21no—
44C22×C4⋊C4—21no—
45C22×C4⋊C4D411yescharacteristic
46C22×C4⋊C4C2312yes—
47C22×C4⋊C4C2312yes—
48C22×C42—21no—
49C22×C42D412yes—
50C22×C42C2311yescharacteristic
51#431—24no—
52#431C2×C418yes—
53C2×#106—28no—
54C2×C4⋊C8—22no—
55C2×C22⋊C8C2314yes—
56C2×C4×C8—22no—
57#391—44no—
58#389—24no—
59#389—24no—
60#389—28no—
61#389—28no—
62#384—22no—
63#384—22no—
64#384—22no—
65#384—22no—
66#384D412yes—
67#384D412yes—
68#382—24no—
69#382—22no—
70#380—24no—
71#380—22no—
72#380C2314yes—
73C4×C4⋊C4—24no—
74C4×C4⋊C4—24no—
75C4×C4⋊C4—22no—
76C4×C4⋊C4—24no—
77C4×C4⋊C4D414yes—
78C2×#95—24no—
79C2×#95—24no—
80C2×#95C2312yes—
81#340—28no—
82#324C2×C4116yes—
83C22×Q8—22no—
84C22×Q8—22no—
85C22×Q8—22no—
86C22×Q8—22no—
87C22×Q8C4⋊C414yes—
88C22×Q8SD1614yes—
89C22×Q8SD1614yes—
90C23×C4—22no—
91C23×C4—22no—
92C23×C4C2×D411yescharacteristic
93C23×C4C2×D412yes—
94C23×C4C2×D412yes—
95C23×C4C2×D411yescharacteristic
96C23×C4C2411yescharacteristic
97C22×C8—24no—
98C4×Q8—28no—
99C4×Q8—28no—
100C4×Q8—28no—
101C4×Q8—28no—
102C4×Q8—416no—
103C4×Q8—416no—
104#117—48no—
105#117—48no—
106#117—44no—
107#117—44no—
108C2×C4⋊C4—22no—
109C2×C4⋊C4—22no—
110C2×C4⋊C4—24no—
111C2×C4⋊C4—24no—
112C2×C4⋊C4—24no—
113C2×C4⋊C4—24no—
114C2×C4⋊C4—24no—
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—24no—
122C2×C4⋊C4—42no—
123C2×C4⋊C4—42no—
124C2×C4⋊C4—44no—
125C2×C4⋊C4—41no—
126C2×C4⋊C4—42no—
127C2×C4⋊C4—41no—
128C2×C4⋊C4—42no—
129C2×C4⋊C4—42no—
130C2×C4⋊C4—82no—
131C2×C4⋊C4C4⋊C414yes—
132C2×C4⋊C4C22×C414yes—
133C2×C4⋊C4C2×D414yes—
134C2×C4⋊C4C2×D414yes—
135C2×C4⋊C4C2×Q814yes—
136C2×C22⋊C4—24no—
137C2×C42—22no—
138C2×C42—22no—
139C2×C42—24no—
140C2×C42—28no—
141C2×C42—24no—
142C2×C42—42no—
143C2×C42—42no—
144C2×C42—42no—
145C2×C42C4⋊C418yes—
146C2×C42C22×C412yes—
147C2×C42C2×D412yes—
148C2×C42C2×D412yes—
149C2×C42C2×Q812yes—
150C2×C42#4114yes—
151#106—88no—
152C4⋊C8—28no—
153C22⋊C8—24no—
154C22⋊C8C22×C418yes—
155C4×C8—28no—
156#95—28no—
157#95—24no—
158#95—24no—
159#95—24no—
160#95—48no—
161#95—84no—
162C24C22×D411yescharacteristic
163C2×Q8—28no—
164C2×Q8—28no—
165C2×Q8—28no—
166C2×Q8—28no—
167C2×Q8—28no—
168C2×Q8—28no—
169C2×Q8#106116yescomplemented
170C22×C4—22no—
171C22×C4—22no—
172C22×C4—22no—
173C22×C4—22no—
174C22×C4—22no—
175C22×C4—22no—
176C22×C4—22no—
177C22×C4—22no—
178C22×C4—24no—
179C22×C4—24no—
180C22×C4—24no—
181C22×C4—41no—
182C22×C4—41no—
183C22×C4—42no—
184C22×C4—42no—
185C22×C4—42no—
186C22×C4—42no—
187C22×C4—42no—
188C22×C4—42no—
189C22×C4—42no—
190C22×C4—42no—
191C22×C4—44no—
192C22×C4—44no—
193C22×C4—42no—
194C22×C4—42no—
195C22×C4—42no—
196C22×C4C2×C4⋊C412yes—
197C22×C4C2×C4⋊C414yes—
198C22×C4C2×C4⋊C414yes—
199C22×C4C2×C4⋊C412yes—
200C22×C4#12011yescharacteristic
201C22×C4#12012yes—
202C22×C4#12011yescharacteristic
203C22×C4#12312yes—
204C22×C4#12312yes—
205C22×C4C2×SD1614yes—
206C22×C4C2×SD1614yes—
207C22×C4C2×SD1612yes—
208C22×C4C2×SD1612yes—
209C22×C4AGL(1,ℤ/8)12yes—
210C22×C4#13712yes—
211C22×C4C23×C411yescharacteristic
212C22×C4C22×D412yes—
213C22×C4C22×Q811yescharacteristic
214C22×C4C2×#4112yes—
215C2×C8—28no—
216C2×C8—44no—
217C2×C8—44no—
218C4⋊C4—28no—
219C4⋊C4—28no—
220C4⋊C4—28no—
221C4⋊C4—44no—
222C4⋊C4—44no—
223C4⋊C4—48no—
224C4⋊C4—48no—
225C4⋊C4—48no—
226C4⋊C4—48no—
227C4⋊C4—44no—
228C4⋊C4—44no—
229C4⋊C4—44no—
230C4⋊C4—44no—
231C4⋊C4—82no—
232C4⋊C4—84no—
233C4⋊C4—84no—
234C4⋊C4—84no—
235C4⋊C4—82no—
236C22⋊C4—44no—
237C22⋊C4—44no—
238C42—28no—
239C42—28no—
240C42—416no—
241C42—48no—
242C42—84no—
243C42C4×D414yes—
244C42C4×D414yes—
245C42C22⋊Q814yes—
246C42C22⋊Q814yes—
247C23—21no—
248C23—21no—
249C23—21no—
250C23—21no—
251C23C22×C4⋊C411yescharacteristic
252C23C2×#12011yescharacteristic
253C23C2×#12311yescharacteristic
254C23C22×SD1611yescharacteristic
255C23C22×SD1611yescharacteristic
256C23C2×AGL(1,ℤ/8)11yescharacteristic
257C23C2×#13711yescharacteristic
258Q8—232no—
259C2×C4—24no—
260C2×C4—24no—
261C2×C4—24no—
262C2×C4—24no—
263C2×C4—24no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—22no—
267C2×C4—22no—
268C2×C4—24no—
269C2×C4—24no—
270C2×C4—24no—
271C2×C4—24no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—42no—
275C2×C4—42no—
276C2×C4—42no—
277C2×C4—44no—
278C2×C4—44no—
279C2×C4—44no—
280C2×C4—44no—
281C2×C4—44no—
282C2×C4—44no—
283C2×C4—44no—
284C2×C4—44no—
285C2×C4—44no—
286C2×C4—44no—
287C2×C4—82no—
288C2×C4—82no—
289C2×C4—82no—
290C2×C4—82no—
291C2×C4—82no—
292C2×C4—84no—
293C2×C4—84no—
294C2×C4—82no—
295C2×C4—82no—
296C2×C4—82no—
297C2×C4#38512yes—
298C2×C4#38514yes—
299C2×C4#38512yes—
300C2×C4C2×#10618yes—
301C2×C4C2×#10614yes—
302C2×C4#42814yes—
303C2×C4#44814yes—
304C2×C4#45014yes—
305C2×C4#45412yes—
306C2×C4#45612yes—
307C2×C4#48114yes—
308C2×C4#48314yes—
309C2×C4C2×C4×D412yes—
310C2×C4C2×C22⋊Q812yes—
311C8—48no—
312C22—21no—
313C22—21no—
314C22—21no—
315C22—21no—
316C22—21no—
317C22—21no—
318C22—21no—
319C22—21no—
320C22—21no—
321C22—21no—
322C22—21no—
323C22—21no—
324C22C2×#38511yescharacteristic
325C22C22×#10611yescharacteristic
326C22C2×#42811yescharacteristic
327C22C2×#45011yescharacteristic
328C22C2×#44811yescharacteristic
329C22C2×#45411yescharacteristic
330C22C2×#45611yescharacteristic
331C22#302711yescharacteristic
332C22#302811yescharacteristic
333C22C2×#48311yescharacteristic
334C22C2×#48111yescharacteristic
335C4—28no—
336C4—22no—
337C4—22no—
338C4—22no—
339C4—44no—
340C4—84no—
341C4—84no—
342C4—84no—
343C4—162no—
344C4#185318yes—
345C4#185714yes—
346C2—21no—
347C2—21no—
348C2—21no—
349C2—21no—
350C2#1840211yescharacteristic
351C2#1840311yescharacteristic
352C2#1840911yescharacteristic
353C2#1841011yescharacteristic
354C2C2×#185311yescharacteristic
355C2C2×#185711yescharacteristic
356C2#2089011yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
24. #1461C4semidirect
169. C2×Q8#106semidirect

Maximal subgroups

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

subgroupindex
#341822
#219812
C2×#14612
#139262
#88572
#80142