Finite Group Databasebeta

← #416101

Subgroups — #416101

Order 512.

1150 conjugacy classes of subgroups.

The subgroup lattice has 354 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35181C211yesmaximal, characteristic
2#21170C212yesmaximal
3C4×#384C211yesmaximal, characteristic
4#11340C214yesmaximal
5#11217C214yesmaximal
6#10847C211yesmaximal, characteristic
7#10839C212yesmaximal
8C2×#428C2214yes—
9C2×#423C2212yes—
10#2851—22no—
11#2646—22no—
12#2571—22no—
13#2570—24no—
14#2552—22no—
15#2317C2214yes—
16#2285C2212yes—
17C2×#384C2212yes—
18C2×C4×C4⋊C4—22no—
19C2×C4×C4⋊C4C2211yescharacteristic
20C2×C4×C4⋊C4C2211yescharacteristic
21C2×C43C2211yescharacteristic
22#1949—24no—
23#1948—22no—
24#1947—22no—
25C2×#343C2214yes—
26#1736C2214yes—
27C2×#339C2214yes—
28C2×#339C2214yes—
29C4⋊#95—22no—
30C4⋊#95C2212yes—
31C22×M16C2314yes—
32C2×#125—22no—
33C2×#117C2314yes—
34C22×C4⋊C4—24no—
35C22×C4⋊C4C2312yes—
36C22×C4⋊C4C2312yes—
37C22×C42—21no—
38C22×C42—21no—
39C22×C42—21no—
40C22×C42D412yes—
41C22×C42C2311yescharacteristic
42C22×C42C2311yescharacteristic
43C22×C42C2311yescharacteristic
44#428—28no—
45#428—44no—
46C2×#107—24no—
47C2×#106—24no—
48#423—24no—
49#423Q818yes—
50C2×C4⋊C8—22no—
51C2×C4⋊C8—22no—
52#400—28no—
53#398—28no—
54#389—24no—
55#389—24no—
56#389—24no—
57#387—28no—
58#385—28no—
59#384—24no—
60#384—24no—
61#384—24no—
62#384D412yes—
63#384D412yes—
64#382—22no—
65#382—24no—
66#382D414yes—
67#380—22no—
68#380—22no—
69C4×C4⋊C4—22no—
70C4×C4⋊C4—22no—
71C4×C4⋊C4—22no—
72C4×C4⋊C4—22no—
73C4×C4⋊C4—22no—
74C4×C4⋊C4—22no—
75C4×C4⋊C4—48no—
76C2×#95—21no—
77C2×#95—24no—
78C2×#95D412yes—
79C43—22no—
80C43Q814yes—
81#343—44no—
82#340—24no—
83#340—24no—
84#339—28no—
85#339—44no—
86C23×C4—21no—
87C23×C4—21no—
88C23×C4—22no—
89C23×C4C2×D411yescharacteristic
90C23×C4C2×D414yes—
91C23×C4C2×D411yescharacteristic
92C23×C4C2411yescharacteristic
93C2×M16—24no—
94C2×M16—24no—
95C2×M16—24no—
96C2×M16—24no—
97C2×M16—24no—
98C2×M16C2×Q818yes—
99C22×C8—24no—
100#125—84no—
101#117—24no—
102#117—24no—
103#117—24no—
104#117—24no—
105C2×C4⋊C4—22no—
106C2×C4⋊C4—22no—
107C2×C4⋊C4—22no—
108C2×C4⋊C4—22no—
109C2×C4⋊C4—216no—
110C2×C4⋊C4—22no—
111C2×C4⋊C4—22no—
112C2×C4⋊C4—22no—
113C2×C4⋊C4—22no—
114C2×C4⋊C4—22no—
115C2×C4⋊C4—22no—
116C2×C4⋊C4—22no—
117C2×C4⋊C4—22no—
118C2×C4⋊C4—22no—
119C2×C4⋊C4—28no—
120C2×C4⋊C4—28no—
121C2×C4⋊C4—22no—
122C2×C4⋊C4—22no—
123C2×C4⋊C4—48no—
124C2×C4⋊C4—48no—
125C2×C4⋊C4C2×D414yes—
126C2×C4⋊C4C2×D414yes—
127C2×C4⋊C4#4114yes—
128C2×C22⋊C4—24no—
129C2×C42—24no—
130C2×C42—24no—
131C2×C42—24no—
132C2×C42—21no—
133C2×C42—22no—
134C2×C42—22no—
135C2×C42—22no—
136C2×C42—22no—
137C2×C42—24no—
138C2×C42—24no—
139C2×C42—21no—
140C2×C42—22no—
141C2×C42—44no—
142C2×C42—44no—
143C2×C42—44no—
144C2×C42—41no—
145C2×C42—41no—
146C2×C42C2×D412yes—
147C2×C42C2×D412yes—
148C2×C42C2×Q812yes—
149C2×C42C2×Q812yes—
150C2×C42C2×Q814yes—
151C2×C42C2×Q812yes—
152C2×C42#4114yes—
153C2×C42#4112yes—
154#107—48no—
155#106—48no—
156C4⋊C8—28no—
157C4⋊C8—28no—
158#95—28no—
159#95—28no—
160#95—24no—
161#95—24no—
162#95—24no—
163#95—48no—
164#95—42no—
165C24C22×D411yescharacteristic
166C22×C4—22no—
167C22×C4—22no—
168C22×C4—24no—
169C22×C4—22no—
170C22×C4—24no—
171C22×C4—22no—
172C22×C4—22no—
173C22×C4—22no—
174C22×C4—22no—
175C22×C4—22no—
176C22×C4—24no—
177C22×C4—24no—
178C22×C4—24no—
179C22×C4—24no—
180C22×C4—24no—
181C22×C4—24no—
182C22×C4—42no—
183C22×C4—42no—
184C22×C4—42no—
185C22×C4—44no—
186C22×C4—44no—
187C22×C4—44no—
188C22×C4—44no—
189C22×C4—42no—
190C22×C4—42no—
191C22×C4#12012yes—
192C22×C4#12011yescharacteristic
193C22×C4#12011yescharacteristic
194C22×C4C22⋊Q812yes—
195C22×C4C22⋊Q812yes—
196C22×C4C4⋊Q812yes—
197C22×C4C4⋊Q812yes—
198C22×C4C22×D412yes—
199C22×C4C22×Q812yes—
200C22×C4C22×Q811yescharacteristic
201C22×C4C2×#4111yescharacteristic
202C22×C4C2×#4112yes—
203C22×C4C2×#4112yes—
204C22×C4H(5,𝔽2)12yes—
205C22×C42−1+412yes—
206M16—28no—
207M16—28no—
208M16—44no—
209M16—44no—
210C2×C8—28no—
211C2×C8—44no—
212C2×C8—44no—
213C4⋊C4—24no—
214C4⋊C4—24no—
215C4⋊C4—24no—
216C4⋊C4—24no—
217C4⋊C4—24no—
218C4⋊C4—24no—
219C4⋊C4—432no—
220C4⋊C4—42no—
221C4⋊C4—42no—
222C4⋊C4—42no—
223C4⋊C4—42no—
224C4⋊C4—42no—
225C4⋊C4—42no—
226C4⋊C4—44no—
227C4⋊C4—44no—
228C4⋊C4—82no—
229C4⋊C4—82no—
230C4⋊C4—88no—
231C4⋊C4—88no—
232C22⋊C4—44no—
233C22⋊C4—44no—
234C42—216no—
235C42—24no—
236C42—22no—
237C42—24no—
238C42—48no—
239C42—41no—
240C42—41no—
241C42—44no—
242C42—44no—
243C42—44no—
244C42—82no—
245C42—82no—
246C42—82no—
247C42C22⋊Q814yes—
248C42C22⋊Q814yes—
249C42#12514yescomplemented
250C42C4⋊Q814yes—
251C42C4⋊Q814yes—
252C23—21no—
253C23—21no—
254C23—22no—
255C23C2×#12011yescharacteristic
256C23C2×C22⋊Q811yescharacteristic
257C23#52512yes—
258C23C2×C4⋊Q811yescharacteristic
259C23#53712yes—
260C2×C4—24no—
261C2×C4—24no—
262C2×C4—28no—
263C2×C4—28no—
264C2×C4—24no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—22no—
268C2×C4—22no—
269C2×C4—24no—
270C2×C4—24no—
271C2×C4—24no—
272C2×C4—42no—
273C2×C4—42no—
274C2×C4—42no—
275C2×C4—44no—
276C2×C4—44no—
277C2×C4—42no—
278C2×C4—48no—
279C2×C4—48no—
280C2×C4—44no—
281C2×C4—44no—
282C2×C4—44no—
283C2×C4—44no—
284C2×C4—44no—
285C2×C4—44no—
286C2×C4—44no—
287C2×C4—44no—
288C2×C4—44no—
289C2×C4—44no—
290C2×C4—84no—
291C2×C4—82no—
292C2×C4—82no—
293C2×C4—84no—
294C2×C4#39512yes—
295C2×C4#39514yes—
296C2×C4#39512yes—
297C2×C4#45312yes—
298C2×C4#45612yes—
299C2×C4C2×C22⋊Q812yes—
300C2×C4C2×#12511yescharacteristic
301C2×C4#52912yes—
302C2×C4C2×C4⋊Q812yes—
303C2×C4#53311yescharacteristic
304C2×C4#54011yescharacteristic
305C2×C4#54111yescharacteristic
306C2×C4#54412yes—
307C2×C4#54612yes—
308C2×C4#54712yes—
309C2×C4#54812yes—
310C2×C4#55012yes—
311C2×C4#55412yes—
312C2×C4#55412yes—
313C2×C4#55712yes—
314C2×C4#55712yes—
315C8—48no—
316C22—21no—
317C22—21no—
318C22—21no—
319C22—21no—
320C22—22no—
321C22—22no—
322C22—22no—
323C22—22no—
324C22C2×#39511yescharacteristic
325C22#240712yes—
326C22#258412yes—
327C22#262911yescharacteristic
328C22#285111yescharacteristic
329C22C2×#45311yescharacteristic
330C22C2×#45611yescharacteristic
331C22#301812yes—
332C4—28no—
333C4—22no—
334C4—24no—
335C4—44no—
336C4—84no—
337C4—84no—
338C4—84no—
339C4—84no—
340C4#206014yes—
341C4#245312yes—
342C4#245412yes—
343C4#255012yes—
344C4#255212yes—
345C2—21no—
346C2—21no—
347C2—22no—
348C2#1521811yescharacteristic
349C2C2×#206011yescharacteristic
350C2#2356011yescharacteristic
351C2#2417612yes—
352C2#2417912yes—
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC23×C4C24
prime radicalC24C22×D4
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
249. C42#125semidirect

Maximal subgroups

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

subgroupindex
#351812
#211702
C4×#3842
#113402
#112172
#108472
#108392