Finite Group Databasebeta

← #416738

Subgroups — #416738

Order 512.

1170 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35167C211yesmaximal, characteristic
2#20764C212yesmaximal
3#13910C211yesmaximal, characteristic
4#11392C214yesmaximal
5#11210C214yesmaximal
6#10851C212yesmaximal
7C4×#343C211yesmaximal, characteristic
8C2×#544C2212yes—
9C2×C4×M16C2212yes—
10#2857—22no—
11#2659—22no—
12#2552—22no—
13#2551—22no—
14#2550—24no—
15#2316C2212yes—
16C2×#389—22no—
17#2285C2212yes—
18C2×#384C2212yes—
19C2×#384C2211yescharacteristic
20C2×C4×C4⋊C4C2211yescharacteristic
21C2×C4×C4⋊C4C2211yescharacteristic
22#1836—24no—
23#1834—22no—
24#1834—22no—
25C2×#343C2212yes—
26C2×#343C2212yes—
27#1736C2218yes—
28C2×#339C2218yes—
29#1444—22no—
30C4⋊#95C2212yes—
31C22×M16C2314yes—
32#544—28no—
33#544—44no—
34C2×C4⋊Q8—22no—
35C2×C4⋊Q8—24no—
36C2×#125—24no—
37C2×C22⋊Q8—24no—
38C2×#117C2314yes—
39C22×C4⋊C4—21no—
40C22×C4⋊C4—21no—
41C22×C4⋊C4—24no—
42C22×C4⋊C4—21no—
43C22×C4⋊C4C2312yes—
44C22×C4⋊C4C2311yescharacteristic
45C22×C4⋊C4C2311yescharacteristic
46C22×C42D412yes—
47C22×C42C2312yes—
48C22×C42C2311yescharacteristic
49C4×M16—24no—
50C4×M16Q818yescomplemented
51C2×#97—22no—
52C2×C4×C8—22no—
53#398—28no—
54#395—28no—
55#391—24no—
56#389—24no—
57#389—24no—
58#389—24no—
59#389—24no—
60#389—48no—
61#384—22no—
62#384—28no—
63#384—22no—
64#384—22no—
65#384D412yes—
66#384D412yes—
67#382D414yes—
68#380—24no—
69#380—22no—
70#380—22no—
71C4×C4⋊C4—22no—
72C4×C4⋊C4—22no—
73C4×C4⋊C4—22no—
74C4×C4⋊C4—22no—
75C4×C4⋊C4—22no—
76C4×C4⋊C4Q814yescomplemented
77C2×#95—21no—
78C2×#95—24no—
79C2×#95D412yes—
80#343—24no—
81#343—24no—
82#340—28no—
83#340—28no—
84#339—48no—
85C22×Q8—24no—
86C23×C4—22no—
87C23×C4—22no—
88C23×C4C2×D411yescharacteristic
89C23×C4C2×D412yes—
90C23×C4C2×D412yes—
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—
100C4⋊Q8—416no—
101C4⋊Q8—84no—
102#125—416no—
103C22⋊Q8—48no—
104C22⋊Q8—48no—
105#117—28no—
106#117—28no—
107C2×C4⋊C4—24no—
108C2×C4⋊C4—24no—
109C2×C4⋊C4—24no—
110C2×C4⋊C4—22no—
111C2×C4⋊C4—22no—
112C2×C4⋊C4—22no—
113C2×C4⋊C4—22no—
114C2×C4⋊C4—24no—
115C2×C4⋊C4—28no—
116C2×C4⋊C4—22no—
117C2×C4⋊C4—22no—
118C2×C4⋊C4—24no—
119C2×C4⋊C4—24no—
120C2×C4⋊C4—24no—
121C2×C4⋊C4—24no—
122C2×C4⋊C4—24no—
123C2×C4⋊C4—22no—
124C2×C4⋊C4—22no—
125C2×C4⋊C4—44no—
126C2×C4⋊C4—48no—
127C2×C4⋊C4—41no—
128C2×C4⋊C4—42no—
129C2×C4⋊C4—41no—
130C2×C4⋊C4—48no—
131C2×C4⋊C4—82no—
132C2×C4⋊C4—84no—
133C2×C4⋊C4C2×D414yes—
134C2×C4⋊C4C2×D414yes—
135C2×C4⋊C4#4114yes—
136C2×C4⋊C4#4114yes—
137C2×C22⋊C4—24no—
138C2×C42—24no—
139C2×C42—24no—
140C2×C42—22no—
141C2×C42—24no—
142C2×C42—22no—
143C2×C42—22no—
144C2×C42—44no—
145C2×C42C2×D412yes—
146C2×C42C2×D412yes—
147C2×C42C2×Q814yes—
148C2×C42C2×Q814yes—
149C2×C42C2×Q812yes—
150C2×C42#4112yes—
151#97—28no—
152C4×C8—28no—
153#95—28no—
154#95—24no—
155#95—24no—
156#95—24no—
157#95—24no—
158#95—48no—
159C24C22×D411yescharacteristic
160C2×Q8—48no—
161C2×Q8—84no—
162C2×Q8—84no—
163C22×C4—22no—
164C22×C4—22no—
165C22×C4—24no—
166C22×C4—24no—
167C22×C4—22no—
168C22×C4—22no—
169C22×C4—24no—
170C22×C4—22no—
171C22×C4—22no—
172C22×C4—24no—
173C22×C4—24no—
174C22×C4—41no—
175C22×C4—41no—
176C22×C4—42no—
177C22×C4—42no—
178C22×C4—42no—
179C22×C4—42no—
180C22×C4—42no—
181C22×C4—42no—
182C22×C4—42no—
183C22×C4—42no—
184C22×C4—44no—
185C22×C4—44no—
186C22×C4—42no—
187C22×C4—42no—
188C22×C4—42no—
189C22×C4#12011yescharacteristic
190C22×C4#12012yes—
191C22×C4#12011yescharacteristic
192C22×C4C22⋊Q812yes—
193C22×C4C22⋊Q812yes—
194C22×C4#12414yes—
195C22×C4#12414yes—
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—216no—
214C4⋊C4—28no—
215C4⋊C4—28no—
216C4⋊C4—28no—
217C4⋊C4—28no—
218C4⋊C4—48no—
219C4⋊C4—416no—
220C4⋊C4—44no—
221C4⋊C4—44no—
222C4⋊C4—48no—
223C4⋊C4—82no—
224C4⋊C4—82no—
225C4⋊C4—82no—
226C4⋊C4—84no—
227C4⋊C4—84no—
228C4⋊C4—84no—
229C4⋊C4—816no—
230C4⋊C4—82no—
231C22⋊C4—44no—
232C22⋊C4—44no—
233C42—28no—
234C42—48no—
235C42—44no—
236C42—44no—
237C42—82no—
238C42C22⋊Q814yes—
239C42C22⋊Q814yes—
240C42C4⋊Q814yes—
241C42C4⋊Q814yes—
242C23—21no—
243C23—21no—
244C23—21no—
245C23—21no—
246C23C2×#12011yescharacteristic
247C23C2×C22⋊Q811yescharacteristic
248C23C2×#12411yescharacteristic
249C23C2×#12411yescharacteristic
250C23C2×C4⋊Q811yescharacteristic
251C23#53511yescharacteristic
252C23#53611yescharacteristic
253Q8—168no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—22no—
257C2×C4—22no—
258C2×C4—24no—
259C2×C4—24no—
260C2×C4—28no—
261C2×C4—24no—
262C2×C4—24no—
263C2×C4—42no—
264C2×C4—42no—
265C2×C4—44no—
266C2×C4—44no—
267C2×C4—44no—
268C2×C4—44no—
269C2×C4—44no—
270C2×C4—44no—
271C2×C4—42no—
272C2×C4—42no—
273C2×C4—42no—
274C2×C4—42no—
275C2×C4—44no—
276C2×C4—44no—
277C2×C4—44no—
278C2×C4—44no—
279C2×C4—44no—
280C2×C4—44no—
281C2×C4—44no—
282C2×C4—44no—
283C2×C4—82no—
284C2×C4—82no—
285C2×C4—84no—
286C2×C4—84no—
287C2×C4—82no—
288C2×C4—82no—
289C2×C4—82no—
290C2×C4#39512yes—
291C2×C4#39514yes—
292C2×C4#39512yes—
293C2×C4#45312yes—
294C2×C4#45612yes—
295C2×C4C2×C22⋊Q812yes—
296C2×C4C2×C4⋊Q812yes—
297C2×C4#53314yes—
298C2×C4#54414yes—
299C2×C4#54614yes—
300C2×C4#54814yes—
301C2×C4#55412yes—
302C2×C4#55512yes—
303C2×C4#55612yes—
304C2×C4#55712yes—
305C8—48no—
306C22—21no—
307C22—21no—
308C22—21no—
309C22—21no—
310C22—21no—
311C22—21no—
312C22—21no—
313C22—21no—
314C22—21no—
315C22—21no—
316C22—21no—
317C22—21no—
318C22C2×#39511yescharacteristic
319C22#243511yescharacteristic
320C22#243611yescharacteristic
321C22#256311yescharacteristic
322C22#256811yescharacteristic
323C22#266011yescharacteristic
324C22#286211yescharacteristic
325C22C2×#45311yescharacteristic
326C22C2×#45611yescharacteristic
327C22#302111yescharacteristic
328C22#302211yescharacteristic
329C4—22no—
330C4—24no—
331C4—48no—
332C4—44no—
333C4—44no—
334C4—84no—
335C4—84no—
336C4—162no—
337C4#206014yes—
338C4#243914yes—
339C4#255014yes—
340C2—21no—
341C2—21no—
342C2—21no—
343C2—21no—
344C2#1523011yescharacteristic
345C2C2×#206011yescharacteristic
346C2#2356211yescharacteristic
347C2#2587011yescharacteristic
348C2#2587111yescharacteristic
349C2#2591811yescharacteristic
350C2#2591911yescharacteristic
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
50. C4×M16Q8semidirect
76. C4×C4⋊C4Q8semidirect

Maximal subgroups

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

subgroupindex
#351672
#207642
#139102
#113922
#112102
#108512
C4×#3432