Finite Group Databasebeta

← #192126

Subgroups — #192126

Order 512.

1390 conjugacy classes of subgroups.

The subgroup lattice has 371 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2316C211yesmaximal, characteristic
2#21940C212yesmaximal
3C22×#324C211yesmaximal, characteristic
4#14339C211yesmaximal, characteristic
5#8856C218yesmaximal, complemented
6#8023C212yesmaximal
7C22×#117—21no—
8C23×C4⋊C4C2211yescharacteristic
9C22×C22⋊C8C2212yes—
10#2316—24no—
11#2316—22no—
12#2316C2214yes—
13C2×#389—24no—
14C2×#381—21no—
15C2×#380—21no—
16C2×#380C2211yescharacteristic
17C2×#380C2212yes—
18C2×#380C2211yescharacteristic
19#1852—24no—
20#1852C2218yes—
21#1808—24no—
22C2×#324C418yescomplemented
23C2×#324C2214yes—
24#1441—22no—
25#1441C2214yes—
26C22⋊#95—21no—
27#1294C2218yes—
28C24×C4D411yescharacteristic
29C24×C4D411yescharacteristic
30C24×C4C2311yescharacteristic
31C23×C8—22no—
32C2×#117—24no—
33C2×#117—24no—
34C2×#117—44no—
35C2×#117—44no—
36C22×C4⋊C4—22no—
37C22×C4⋊C4—22no—
38C22×C4⋊C4—24no—
39C22×C4⋊C4—22no—
40C22×C4⋊C4—22no—
41C22×C4⋊C4—21no—
42C22×C4⋊C4C2×C414yes—
43C22×C4⋊C4D412yes—
44C22×C4⋊C4D414yes—
45C22×C4⋊C4Q814yes—
46C22×C4⋊C4C2312yes—
47C22×C4⋊C4C2312yes—
48C22×C22⋊C4—22no—
49C22×C22⋊C4—21no—
50C22×C22⋊C4—21no—
51C22×C42—22no—
52C2×#107—28no—
53C2×C22⋊C8—22no—
54C2×C22⋊C8—24no—
55C2×C22⋊C8C2×C418yes—
56C2×C22⋊C8C2314yes—
57#389—216no—
58#389—216no—
59#381—44no—
60#380—24no—
61#380—22no—
62#380—22no—
63#380—24no—
64#380—24no—
65#380—22no—
66#380—22no—
67#380D414yes—
68#380C2314yes—
69C2×#95—22no—
70C2×#95—22no—
71C2×#95—21no—
72C2×#95—24no—
73C2×#95—24no—
74C2×#95C2312yes—
75#340—28no—
76#324C2×C4116yescomplemented
77C25C2×D411yescharacteristic
78C23×C4—21no—
79C23×C4—21no—
80C23×C4—22no—
81C23×C4—22no—
82C23×C4—22no—
83C23×C4—21no—
84C23×C4—21no—
85C23×C4—22no—
86C23×C4—24no—
87C23×C4—22no—
88C23×C4—22no—
89C23×C4—82no—
90C23×C4—82no—
91C23×C4C4⋊C414yes—
92C23×C4C4⋊C414yes—
93C23×C4D814yes—
94C23×C4Q1614yes—
95C23×C4C22×C412yes—
96C23×C4C2×D412yes—
97C23×C4C2×D412yes—
98C23×C4C2×D411yescharacteristic
99C23×C4C2×D412yes—
100C23×C4C2×D412yes—
101C23×C4C2×Q812yes—
102C23×C4#4114yes—
103C23×C4C2411yescharacteristic
104C22×C8—24no—
105C22×C8—28no—
106C22×C8—84no—
107#117—48no—
108#117—416no—
109#117—48no—
110C2×C4⋊C4—28no—
111C2×C4⋊C4—28no—
112C2×C4⋊C4—24no—
113C2×C4⋊C4—28no—
114C2×C4⋊C4—28no—
115C2×C4⋊C4—24no—
116C2×C4⋊C4—24no—
117C2×C4⋊C4—24no—
118C2×C4⋊C4—28no—
119C2×C4⋊C4—24no—
120C2×C4⋊C4—28no—
121C2×C4⋊C4—24no—
122C2×C4⋊C4—48no—
123C2×C4⋊C4—42no—
124C2×C4⋊C4—82no—
125C2×C4⋊C4—84no—
126C2×C4⋊C4—84no—
127C2×C4⋊C4—82no—
128C2×C4⋊C4C4⋊C418yes—
129C2×C4⋊C4C22×C414yes—
130C2×C4⋊C4C2×D418yes—
131C2×C4⋊C4C2×Q814yes—
132C2×C22⋊C4—24no—
133C2×C22⋊C4—24no—
134C2×C22⋊C4—44no—
135C2×C22⋊C4—44no—
136C2×C22⋊C4—41no—
137C2×C22⋊C4—41no—
138C2×C22⋊C4—42no—
139C2×C22⋊C4—42no—
140C2×C22⋊C4—42no—
141C2×C22⋊C4—42no—
142C2×C22⋊C4—42no—
143C2×C42—28no—
144C2×C42—48no—
145#107—88no—
146C22⋊C8—28no—
147C22⋊C8—24no—
148C22⋊C8C22×C418yes—
149#95—28no—
150#95—24no—
151#95—24no—
152#95—24no—
153#95—42no—
154#95—48no—
155#95—84no—
156C24—42no—
157C24—42no—
158C24C2×C4⋊C412yes—
159C24#12011yescharacteristic
160C24#12011yescharacteristic
161C24#12312yes—
162C24C2×D812yes—
163C24C2×Q1612yes—
164C24AGL(1,ℤ/8)12yes—
165C24#13712yes—
166C24C22×D411yescharacteristic
167C22×C4—22no—
168C22×C4—22no—
169C22×C4—22no—
170C22×C4—22no—
171C22×C4—24no—
172C22×C4—24no—
173C22×C4—28no—
174C22×C4—24no—
175C22×C4—24no—
176C22×C4—24no—
177C22×C4—24no—
178C22×C4—28no—
179C22×C4—28no—
180C22×C4—22no—
181C22×C4—24no—
182C22×C4—22no—
183C22×C4—24no—
184C22×C4—28no—
185C22×C4—44no—
186C22×C4—42no—
187C22×C4—42no—
188C22×C4—44no—
189C22×C4—44no—
190C22×C4—44no—
191C22×C4—48no—
192C22×C4—48no—
193C22×C4—48no—
194C22×C4—44no—
195C22×C4—44no—
196C22×C4—44no—
197C22×C4—44no—
198C22×C4—42no—
199C22×C4—42no—
200C22×C4—42no—
201C22×C4—42no—
202C22×C4—42no—
203C22×C4—42no—
204C22×C4—82no—
205C22×C4—82no—
206C22×C4—84no—
207C22×C4—84no—
208C22×C4—82no—
209C22×C4—84no—
210C22×C4#107116yescomplemented
211C22×C4C2×C4⋊C414yes—
212C22×C4C2×C4⋊C414yes—
213C22×C4C2×C4⋊C412yes—
214C22×C4C4×D414yes—
215C22×C4C4×D414yes—
216C22×C4#12012yes—
217C22×C4C22⋊Q814yes—
218C22×C4C22⋊Q814yes—
219C22×C4#12312yes—
220C22×C4C2×D814yes—
221C22×C4C2×Q1614yes—
222C22×C4C23×C411yescharacteristic
223C22×C4C22×D412yes—
224C22×C4C22×Q811yescharacteristic
225C22×C4C2×#4112yes—
226C2×C8—28no—
227C2×C8—48no—
228C2×C8—88no—
229C4⋊C4—28no—
230C4⋊C4—28no—
231C4⋊C4—216no—
232C4⋊C4—48no—
233C4⋊C4—416no—
234C4⋊C4—48no—
235C4⋊C4—84no—
236C4⋊C4—88no—
237C4⋊C4—84no—
238C22⋊C4—44no—
239C22⋊C4—44no—
240C22⋊C4—48no—
241C22⋊C4—82no—
242C22⋊C4—82no—
243C22⋊C4—84no—
244C42—416no—
245C23—22no—
246C23—22no—
247C23—22no—
248C23—22no—
249C23—22no—
250C23—22no—
251C23—42no—
252C23—42no—
253C23—42no—
254C23—42no—
255C23—44no—
256C23—44no—
257C23—44no—
258C23—44no—
259C23#38512yes—
260C23#38512yes—
261C23C2×#10714yes—
262C23#42814yes—
263C23#44712yes—
264C23#44712yes—
265C23#45112yes—
266C23#45112yes—
267C23#48014yes—
268C23#48414yes—
269C23C22×C4⋊C411yescharacteristic
270C23C2×#12011yescharacteristic
271C23C2×#12311yescharacteristic
272C23C22×D811yescharacteristic
273C23C22×Q1611yescharacteristic
274C23C2×AGL(1,ℤ/8)11yescharacteristic
275C23C2×#13711yescharacteristic
276C2×C4—24no—
277C2×C4—24no—
278C2×C4—216no—
279C2×C4—216no—
280C2×C4—24no—
281C2×C4—24no—
282C2×C4—24no—
283C2×C4—24no—
284C2×C4—24no—
285C2×C4—24no—
286C2×C4—24no—
287C2×C4—42no—
288C2×C4—44no—
289C2×C4—42no—
290C2×C4—44no—
291C2×C4—44no—
292C2×C4—44no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—42no—
296C2×C4—48no—
297C2×C4—44no—
298C2×C4—44no—
299C2×C4—48no—
300C2×C4—48no—
301C2×C4—48no—
302C2×C4—82no—
303C2×C4—84no—
304C2×C4—816no—
305C2×C4—82no—
306C2×C4—84no—
307C2×C4—84no—
308C2×C4—82no—
309C2×C4—84no—
310C2×C4—84no—
311C2×C4—82no—
312C2×C4—82no—
313C2×C4—82no—
314C2×C4—82no—
315C2×C4#38514yes—
316C2×C4C2×#10718yes—
317C2×C4C2×C4×D412yes—
318C2×C4C2×C22⋊Q812yes—
319C8—48no—
320C22—22no—
321C22—24no—
322C22—22no—
323C22—24no—
324C22—22no—
325C22—22no—
326C22—24no—
327C22—24no—
328C22—21no—
329C22—21no—
330C22—22no—
331C22—22no—
332C22—21no—
333C22—22no—
334C22—21no—
335C22—22no—
336C22—44no—
337C22—44no—
338C22—44no—
339C22—44no—
340C22#185214yes—
341C22#185214yes—
342C22C2×#38511yescharacteristic
343C22C22×#10711yescharacteristic
344C22C2×#42811yescharacteristic
345C22C2×#44711yescharacteristic
346C22C2×#45111yescharacteristic
347C22#302012yes—
348C22#302612yes—
349C22C2×#48011yescharacteristic
350C22C2×#48411yescharacteristic
351C4—28no—
352C4—22no—
353C4—22no—
354C4—24no—
355C4—44no—
356C4—88no—
357C4—162no—
358C4—162no—
359C2—21no—
360C2—21no—
361C2—24no—
362C2—24no—
363C2—21no—
364C2—21no—
365C2#1829212yes—
366C2#1829512yes—
367C2C2×#185211yescharacteristic
368C2#2085711yescharacteristic
369C2#2088911yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
5. #8856C2semidirect
22. C2×#324C4semidirect
76. #324C2×C4semidirect
210. C22×C4#107semidirect

Maximal subgroups

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

subgroupindex
C2×#23162
#219402
C22×#3242
#143392
#88562
#80232