Finite Group Databasebeta

← #194766

Subgroups — #194766

Order 512.

1654 conjugacy classes of subgroups.

The subgroup lattice has 404 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2580C211yesmaximal, characteristic
2#34181C211yesmaximal, characteristic
3#20734C212yesmaximal, complemented
4C2×#1460C211yesmaximal, characteristic
5#9021C214yesmaximal, complemented
6#9013C214yesmaximal, complemented
7#8026C212yesmaximal, complemented
8C2×#518—21no—
9C22×C4×Q8C2211yescharacteristic
10C22×C4×D4C2211yescharacteristic
11C2×#431C2212yes—
12#2580—24no—
13#2580C418yescomplemented
14C2×#399C2212yes—
15C2×#393C2212yes—
16#2349—24no—
17D4×C4⋊C4—24no—
18#2316—21no—
19#2315C2214yes—
20C2×#390—21no—
21C2×#389—21no—
22C2×#385—22no—
23#2285C2212yes—
24C2×#382C2211yescharacteristic
25C2×#380—22no—
26#1824C2214yes—
27#1803—24no—
28#1787C2214yes—
29C4×#103—24no—
30#1460—24no—
31C2×#323C2214yes—
32#1294C2218yes—
33C23×Q8C2311yescharacteristic
34C23×D4D411yescharacteristic
35C24×C4D412yes—
36#518—44no—
37#518—44no—
38C2×C4×Q8—22no—
39C2×C4×Q8—22no—
40C2×C4×Q8—22no—
41C2×C4×Q8—24no—
42C2×C4×Q8C2×C414yescomplemented
43C2×C4×D4—28no—
44C2×C4×D4—22no—
45C2×C4×D4—22no—
46C2×C4×D4—24no—
47C2×C4×D4C2×C414yes—
48C2×C4×D4D414yes—
49C2×#117—22no—
50C22×C4⋊C4—21no—
51C22×C4⋊C4—21no—
52C22×C4⋊C4—24no—
53C22×C4⋊C4—22no—
54C22×C4⋊C4D411yescharacteristic
55C22×C4⋊C4C2311yescharacteristic
56C22×C22⋊C4—24no—
57C22×C22⋊C4—22no—
58C22×C22⋊C4—22no—
59C22×C22⋊C4C2312yes—
60C22×C42—21no—
61C22×C42C2311yescharacteristic
62#431—24no—
63#431—24no—
64C2×C4⋊C8—22no—
65C2×#103—28no—
66C2×C22⋊C8C2314yes—
67C2×C4×C8—22no—
68#399—24no—
69#399C2×C418yescomplemented
70#393—24no—
71#393C2×C418yescomplemented
72#390—28no—
73#389—28no—
74#389—28no—
75#385—216no—
76#385—216no—
77#384D414yes—
78#382—22no—
79#382—22no—
80#382C2×C414yescomplemented
81#380—24no—
82#380—24no—
83#380—24no—
84#380—22no—
85#380C2314yes—
86C4×C4⋊C4—22no—
87C4×C4⋊C4D414yes—
88C2×#95—24no—
89C2×#95—22no—
90C2×#95—21no—
91C2×#95C2312yes—
92#340—28no—
93#323—28no—
94C25C2×D412yes—
95C22×Q8—21no—
96C22×Q8—22no—
97C22×Q8—22no—
98C22×Q8—44no—
99C22×Q8C22×C412yes—
100C22×D4—22no—
101C22×D4—22no—
102C22×D4—22no—
103C22×D4—22no—
104C22×D4C22⋊C414yes—
105C22×D4SD1614yes—
106C22×D4Q1614yes—
107C23×C4—21no—
108C23×C4—22no—
109C23×C4—22no—
110C23×C4—24no—
111C23×C4—21no—
112C23×C4—22no—
113C23×C4—28no—
114C23×C4—84no—
115C23×C4C22⋊C418yes—
116C23×C4C2×D412yes—
117C23×C4C2×D411yescharacteristic
118C23×C4C2×D411yescharacteristic
119C23×C4C2411yescharacteristic
120C22×C8—24no—
121C4×Q8—28no—
122C4×Q8—28no—
123C4×Q8—44no—
124C4×Q8—44no—
125C4×D4—28no—
126C4×D4—28no—
127C4×D4—28no—
128C4×D4—28no—
129C4×D4—432no—
130#117—44no—
131#117—44no—
132C2×C4⋊C4—22no—
133C2×C4⋊C4—22no—
134C2×C4⋊C4—22no—
135C2×C4⋊C4—22no—
136C2×C4⋊C4—24no—
137C2×C4⋊C4—24no—
138C2×C4⋊C4—24no—
139C2×C4⋊C4—24no—
140C2×C4⋊C4—21no—
141C2×C4⋊C4—28no—
142C2×C4⋊C4—28no—
143C2×C4⋊C4—41no—
144C2×C4⋊C4—41no—
145C2×C4⋊C4—42no—
146C2×C4⋊C4—42no—
147C2×C4⋊C4—42no—
148C2×C4⋊C4—88no—
149C2×C4⋊C4—84no—
150C2×C4⋊C4C22⋊C414yes—
151C2×C4⋊C4C22×C412yes—
152C2×C4⋊C4C2×D418yes—
153C2×C22⋊C4—24no—
154C2×C22⋊C4—24no—
155C2×C22⋊C4—28no—
156C2×C22⋊C4—28no—
157C2×C22⋊C4—24no—
158C2×C22⋊C4—28no—
159C2×C22⋊C4—48no—
160C2×C22⋊C4—48no—
161C2×C22⋊C4—48no—
162C2×C22⋊C4—84no—
163C2×C22⋊C4C22×C414yes—
164C2×C22⋊C4C2×D414yes—
165C2×C42—24no—
166C2×C42—24no—
167C2×C42—42no—
168C2×C42—42no—
169C2×C42C22×C412yes—
170C2×C42C2×D412yes—
171C2×C42C2×D412yes—
172C2×C42C2×D412yes—
173C2×C42#4114yes—
174C4⋊C8—44no—
175#103—416no—
176C22⋊C8—24no—
177C22⋊C8—24no—
178C4×C8—44no—
179#95—28no—
180#95—24no—
181#95—28no—
182#95—24no—
183#95—22no—
184#95—24no—
185#95C22×C414yes—
186C24—22no—
187C24—22no—
188C24—22no—
189C24—22no—
190C24—44no—
191C24—44no—
192C24C2×C22⋊C414yes—
193C24C2×SD1614yes—
194C24C2×Q1614yes—
195C24C22×D411yescharacteristic
196C2×Q8—28no—
197C2×Q8—24no—
198C2×Q8—44no—
199C2×Q8—44no—
200C2×Q8—44no—
201C2×Q8—48no—
202C2×Q8—42no—
203C2×D4—28no—
204C2×D4—28no—
205C2×D4—28no—
206C2×D4—28no—
207C2×D4—28no—
208C2×D4—28no—
209C2×D4#103116yescomplemented
210C22×C4—24no—
211C22×C4—22no—
212C22×C4—22no—
213C22×C4—22no—
214C22×C4—28no—
215C22×C4—28no—
216C22×C4—24no—
217C22×C4—22no—
218C22×C4—28no—
219C22×C4—42no—
220C22×C4—42no—
221C22×C4—48no—
222C22×C4—42no—
223C22×C4—41no—
224C22×C4—41no—
225C22×C4—44no—
226C22×C4—44no—
227C22×C4—42no—
228C22×C4—44no—
229C22×C4—42no—
230C22×C4—41no—
231C22×C4—41no—
232C22×C4—42no—
233C22×C4—42no—
234C22×C4—416no—
235C22×C4—84no—
236C22×C4—88no—
237C22×C4—88no—
238C22×C4—88no—
239C22×C4—88no—
240C22×C4C2×C22⋊C414yes—
241C22×C4C2×C22⋊C412yes—
242C22×C4C2×C22⋊C412yes—
243C22×C4#12011yescharacteristic
244C22×C4#12011yescharacteristic
245C22×C4#12012yes—
246C22×C4#12312yes—
247C22×C4#12312yes—
248C22×C4C2×SD1612yes—
249C22×C4C2×Q1612yes—
250C22×C4AGL(1,ℤ/8)12yes—
251C22×C4#13712yes—
252C22×C4C23×C411yescharacteristic
253C22×C4C22×D412yes—
254C22×C4C22×D411yescharacteristic
255C22×C4C2×#4112yes—
256C2×C8—48no—
257C2×C8—44no—
258C4⋊C4—28no—
259C4⋊C4—24no—
260C4⋊C4—48no—
261C4⋊C4—48no—
262C4⋊C4—42no—
263C4⋊C4—44no—
264C4⋊C4—44no—
265C4⋊C4—48no—
266C4⋊C4—48no—
267C4⋊C4—88no—
268C4⋊C4—88no—
269C22⋊C4—28no—
270C22⋊C4—28no—
271C22⋊C4—416no—
272C22⋊C4—416no—
273C22⋊C4—44no—
274C22⋊C4—44no—
275C22⋊C4—88no—
276C42—48no—
277C42—48no—
278C42C4×D414yes—
279C42C4×D414yes—
280C42#12114yes—
281C42#12114yes—
282C23—22no—
283C23—22no—
284C23—24no—
285C23—24no—
286C23—24no—
287C23—24no—
288C23—24no—
289C23—24no—
290C23—44no—
291C23—44no—
292C23—48no—
293C23—48no—
294C23—44no—
295C23—44no—
296C23—82no—
297C23—84no—
298C23—84no—
299C23—82no—
300C23C2×#10318yes—
301C23C22×C22⋊C411yescharacteristic
302C23C2×#12011yescharacteristic
303C23C2×#12311yescharacteristic
304C23C22×SD1611yescharacteristic
305C23C22×Q1611yescharacteristic
306C23C2×AGL(1,ℤ/8)11yescharacteristic
307C23C2×#13711yescharacteristic
308Q8—28no—
309Q8—48no—
310Q8—44no—
311D4—232no—
312C2×C4—22no—
313C2×C4—22no—
314C2×C4—24no—
315C2×C4—24no—
316C2×C4—24no—
317C2×C4—42no—
318C2×C4—42no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—42no—
322C2×C4—416no—
323C2×C4—42no—
324C2×C4—44no—
325C2×C4—44no—
326C2×C4—44no—
327C2×C4—42no—
328C2×C4—44no—
329C2×C4—44no—
330C2×C4—42no—
331C2×C4—44no—
332C2×C4—42no—
333C2×C4—44no—
334C2×C4—48no—
335C2×C4—48no—
336C2×C4—48no—
337C2×C4—42no—
338C2×C4—88no—
339C2×C4—82no—
340C2×C4—84no—
341C2×C4—84no—
342C2×C4—82no—
343C2×C4—82no—
344C2×C4—1616no—
345C2×C4#38612yes—
346C2×C4#38612yes—
347C2×C4#38614yes—
348C2×C4C2×#10314yes—
349C2×C4#41714yes—
350C2×C4#45014yes—
351C2×C4#45114yes—
352C2×C4#48314yes—
353C2×C4#48414yes—
354C2×C4C2×C4×D412yes—
355C2×C4C2×#12112yes—
356C8—84no—
357C22—22no—
358C22—22no—
359C22—22no—
360C22—22no—
361C22—21no—
362C22—21no—
363C22—21no—
364C22—21no—
365C22—28no—
366C22—48no—
367C22—48no—
368C22—48no—
369C22—44no—
370C22—44no—
371C22—84no—
372C22—84no—
373C22—84no—
374C22C2×#38611yescharacteristic
375C22C22×#10311yescharacteristic
376C22C2×#41711yescharacteristic
377C22C2×#45011yescharacteristic
378C22C2×#45111yescharacteristic
379C22#301912yes—
380C22#302512yes—
381C22C2×#48311yescharacteristic
382C22C2×#48411yescharacteristic
383C4—22no—
384C4—22no—
385C4—22no—
386C4—22no—
387C4—44no—
388C4—44no—
389C4—84no—
390C4—88no—
391C4—82no—
392C4#187218yes—
393C2—21no—
394C2—21no—
395C2—21no—
396C2—21no—
397C2—48no—
398C2#1821112yes—
399C2#1821312yes—
400C2C2×#187211yescharacteristic
401C2#2095311yescharacteristic
402C2#2095911yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×D4D4
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
3. #20734C2semidirect
5. #9021C2semidirect
6. #9013C2semidirect
7. #8026C2semidirect
13. #2580C4semidirect
42. C2×C4×Q8C2×C4semidirect
69. #399C2×C4semidirect
71. #393C2×C4semidirect
80. #382C2×C4semidirect
209. C2×D4#103semidirect

Maximal subgroups

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

subgroupindex
C2×#25802
#341812
#207342
C2×#14602
#90212
#90132
#80262