Finite Group Databasebeta

← #327988

Subgroups — #327988

Order 512.

1083 conjugacy classes of subgroups.

The subgroup lattice has 393 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2551C211yesmaximal, characteristic
2#33149C211yesmaximal, characteristic
3#33148C211yesmaximal, characteristic
4#27972C211yesmaximal, characteristic
5#10575C214yesmaximal, complemented
6#10349C214yesmaximal, complemented
7#8724C211yesmaximal, characteristic
8#8707C211yesmaximal, characteristic
9#8706C211yesmaximal, characteristic
10C22×C4⋊Q8—21no—
11C22×C4⋊Q8—21no—
12#2857C2211yescharacteristic
13#2551—22no—
14#2551—22no—
15#2551—22no—
16#2551C2214yes—
17C2×#395—22no—
18C2×#389—21no—
19C2×#389—21no—
20C2×#384C2211yescharacteristic
21C2×C4×C4⋊C4C2211yescharacteristic
22C2×C4×C4⋊C4C2211yescharacteristic
23#2072—22no—
24#1987C2212yes—
25#1987C2212yes—
26#1956C2212yes—
27#1948C2212yes—
28#1947C2212yes—
29#1921—22no—
30#1920—22no—
31#1872—22no—
32#1853—21no—
33#1852—21no—
34#1834C2212yes—
35#1824—21no—
36#1823—21no—
37#1797C2211yescharacteristic
38#1797C2211yescharacteristic
39C4×C22⋊C8C2211yescharacteristic
40#1296C2214yes—
41#1279C2214yes—
42#1279C2214yes—
43C23×Q8—22no—
44C2×C4⋊Q8—24no—
45C2×C4⋊Q8—24no—
46C2×C4⋊Q8—24no—
47C2×C4⋊Q8—24no—
48C2×C4⋊Q8—48no—
49C2×C4⋊Q8—44no—
50C2×C4⋊Q8—44no—
51C2×C4⋊Q8C2311yescharacteristic
52C2×C22⋊Q8—21no—
53C22×C4⋊C4—22no—
54C22×C4⋊C4—22no—
55C22×C4⋊C4—21no—
56C22×C4⋊C4—21no—
57C22×C4⋊C4D411yescharacteristic
58C22×C4⋊C4D411yescharacteristic
59C22×C4⋊C4D411yescharacteristic
60C22×C4⋊C4D411yescharacteristic
61C22×C4⋊C4D412yes—
62C22×C42—21no—
63C22×C42—21no—
64C22×C42C2311yescharacteristic
65C2×#107—22no—
66C2×#106—22no—
67C2×C4⋊C8C2312yes—
68C2×C4⋊C8C2312yes—
69C2×#103—24no—
70C2×C22⋊C8—21no—
71C2×C22⋊C8—21no—
72C2×C22⋊C8—21no—
73C2×C4×C8C2312yes—
74#395—28no—
75#395—28no—
76#391—21no—
77#389—24no—
78#389—24no—
79#389—24no—
80#389—24no—
81#389—21no—
82#389—21no—
83#384—21no—
84#384—21no—
85#384D412yes—
86#384C2311yescharacteristic
87#384C2312yes—
88#380—21no—
89#380—21no—
90C4×C4⋊C4—21no—
91C4×C4⋊C4—21no—
92C4×C4⋊C4—21no—
93C4×C4⋊C4—21no—
94C4×C4⋊C4D412yes—
95C4×C4⋊C4D412yes—
96C4×C4⋊C4C2312yes—
97C4×C4⋊C4C2312yes—
98C2×#95—22no—
99C2×#95—21no—
100C2×#95—21no—
101#340—24no—
102#340—22no—
103#340—22no—
104#340—22no—
105#340—22no—
106#340—22no—
107#340—22no—
108#336—21no—
109#336—21no—
110#336—21no—
111C22×Q8—28no—
112C22×Q8—21no—
113C22×Q8—48no—
114C22×Q8—48no—
115C22×Q8—84no—
116C23×C4—21no—
117C23×C4—21no—
118C23×C4—21no—
119C23×C4—21no—
120C23×C4—22no—
121C23×C4C2×D411yescharacteristic
122C23×C4C2×D411yescharacteristic
123C23×C4C2×D411yescharacteristic
124C22×C8—22no—
125C22×C8—22no—
126C22×C8—22no—
127C4⋊Q8—24no—
128C4⋊Q8—416no—
129C4⋊Q8—48no—
130C4⋊Q8—48no—
131C4⋊Q8—416no—
132C4⋊Q8—48no—
133C4⋊Q8—48no—
134C22⋊Q8—44no—
135C2×C4⋊C4—21no—
136C2×C4⋊C4—21no—
137C2×C4⋊C4—21no—
138C2×C4⋊C4—21no—
139C2×C4⋊C4—22no—
140C2×C4⋊C4—22no—
141C2×C4⋊C4—24no—
142C2×C4⋊C4—24no—
143C2×C4⋊C4—28no—
144C2×C4⋊C4—28no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—21no—
150C2×C4⋊C4—21no—
151C2×C4⋊C4—21no—
152C2×C4⋊C4—21no—
153C2×C4⋊C4—21no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—48no—
156C2×C4⋊C4—48no—
157C2×C4⋊C4—44no—
158C2×C4⋊C4—42no—
159C2×C4⋊C4—42no—
160C2×C4⋊C4—41no—
161C2×C4⋊C4—82no—
162C2×C4⋊C4SD1614yes—
163C2×C4⋊C4Q1614yes—
164C2×C4⋊C4C2×D412yes—
165C2×C4⋊C4C2×D412yes—
166C2×C4⋊C4C2×D412yes—
167C2×C4⋊C4C2×D412yes—
168C2×C4⋊C4C2×D414yes—
169C2×C22⋊C4—41no—
170C2×C42—24no—
171C2×C42—22no—
172C2×C42—22no—
173C2×C42—22no—
174C2×C42—22no—
175C2×C42—44no—
176C2×C42—42no—
177C2×C42—42no—
178C2×C42C2×D411yescharacteristic
179C2×C42C2×D411yescharacteristic
180C2×C42C2×D411yescharacteristic
181C2×C42C2411yescharacteristic
182#107—44no—
183#106—44no—
184C4⋊C8—24no—
185C4⋊C8—24no—
186#103—84no—
187C22⋊C8—42no—
188C22⋊C8—42no—
189C22⋊C8—42no—
190C4×C8—24no—
191#95—28no—
192#95—22no—
193#95—22no—
194#95—22no—
195#95—22no—
196#95—41no—
197#95—41no—
198C24#12011yescharacteristic
199C2×Q8—416no—
200C2×Q8—416no—
201C2×Q8—42no—
202C2×Q8—42no—
203C2×Q8—42no—
204C2×Q8—88no—
205C2×Q8—832no—
206C22×C4—22no—
207C22×C4—22no—
208C22×C4—22no—
209C22×C4—22no—
210C22×C4—24no—
211C22×C4—24no—
212C22×C4—22no—
213C22×C4—22no—
214C22×C4—22no—
215C22×C4—22no—
216C22×C4—24no—
217C22×C4—24no—
218C22×C4—22no—
219C22×C4—22no—
220C22×C4—24no—
221C22×C4—41no—
222C22×C4—42no—
223C22×C4—42no—
224C22×C4—42no—
225C22×C4—42no—
226C22×C4—42no—
227C22×C4—42no—
228C22×C4—42no—
229C22×C4—42no—
230C22×C4—44no—
231C22×C4—44no—
232C22×C4—44no—
233C22×C4—41no—
234C22×C4—41no—
235C22×C4#12011yescharacteristic
236C22×C4#12011yescharacteristic
237C22×C4#12011yescharacteristic
238C22×C4C2×SD1612yes—
239C22×C4C2×Q1612yes—
240C22×C4AGL(1,ℤ/8)12yes—
241C22×C4#13712yes—
242C22×C4C22×D411yescharacteristic
243C22×C4C22×D411yescharacteristic
244C22×C4C22×D411yescharacteristic
245C2×C8—42no—
246C2×C8—42no—
247C2×C8—42no—
248C2×C8—42no—
249C2×C8—42no—
250C2×C8—42no—
251C2×C8—42no—
252C2×C8—42no—
253C2×C8—42no—
254C4⋊C4—24no—
255C4⋊C4—24no—
256C4⋊C4—22no—
257C4⋊C4—22no—
258C4⋊C4—22no—
259C4⋊C4—22no—
260C4⋊C4—24no—
261C4⋊C4—24no—
262C4⋊C4—44no—
263C4⋊C4—44no—
264C4⋊C4—416no—
265C4⋊C4—416no—
266C4⋊C4—42no—
267C4⋊C4—42no—
268C4⋊C4—42no—
269C4⋊C4—44no—
270C4⋊C4—44no—
271C4⋊C4—42no—
272C4⋊C4—42no—
273C4⋊C4—42no—
274C4⋊C4—42no—
275C4⋊C4—82no—
276C4⋊C4—82no—
277C4⋊C4C2×SD1614yes—
278C4⋊C4C2×Q1614yes—
279C4⋊C4AGL(1,ℤ/8)14yes—
280C4⋊C4#13714yes—
281C22⋊C4—82no—
282C42—22no—
283C42—22no—
284C42—22no—
285C42—44no—
286C42—42no—
287C42—42no—
288C42—44no—
289C42—44no—
290C422−1+414yes—
291C23—21no—
292C23—21no—
293C23—21no—
294C23—21no—
295C23#45012yes—
296C23#45112yes—
297C23#45412yes—
298C23C2×#12011yescharacteristic
299Q8—84no—
300Q8—832no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—24no—
308C2×C4—24no—
309C2×C4—24no—
310C2×C4—24no—
311C2×C4—21no—
312C2×C4—24no—
313C2×C4—24no—
314C2×C4—24no—
315C2×C4—24no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—44no—
319C2×C4—48no—
320C2×C4—48no—
321C2×C4—48no—
322C2×C4—48no—
323C2×C4—42no—
324C2×C4—42no—
325C2×C4—42no—
326C2×C4—42no—
327C2×C4—42no—
328C2×C4—42no—
329C2×C4—42no—
330C2×C4—42no—
331C2×C4—44no—
332C2×C4—44no—
333C2×C4—44no—
334C2×C4—88no—
335C2×C4—81no—
336C2×C4—81no—
337C2×C4—81no—
338C2×C4—81no—
339C2×C4—81no—
340C2×C4—81no—
341C2×C4#45012yes—
342C2×C4#45112yes—
343C2×C4#53612yes—
344C2×C4#53612yes—
345C2×C4#53612yes—
346C2×C4C22×SD1611yescharacteristic
347C2×C4C22×Q1611yescharacteristic
348C2×C4C2×AGL(1,ℤ/8)12yes—
349C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
350C2×C4C2×#13712yes—
351C2×C4C2×#13711yescharacteristic
352C8—82no—
353C8—82no—
354C8—82no—
355C22—22no—
356C22—22no—
357C22—22no—
358C22—22no—
359C22—22no—
360C22—22no—
361C22#163714yes—
362C22#261411yescharacteristic
363C22C2×#45011yescharacteristic
364C22C2×#45111yescharacteristic
365C22#300311yescharacteristic
366C22#300411yescharacteristic
367C22C2×#45411yescharacteristic
368C22#301911yescharacteristic
369C4—22no—
370C4—22no—
371C4—22no—
372C4—44no—
373C4—44no—
374C4—44no—
375C4—44no—
376C4—41no—
377C4—84no—
378C4—161no—
379C4—161no—
380C4#322912yes—
381C4#323012yes—
382C4#323812yes—
383C4#323912yes—
384C2—24no—
385C2C2×#163711yescharacteristic
386C2#1853411yescharacteristic
387C2#1874711yescharacteristic
388C2#1876911yescharacteristic
389C2#2964111yescharacteristic
390C2#2964211yescharacteristic
391C2#2965111yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC2×C42C24
prime radicalC24#120
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
5. #10575C2semidirect
6. #10349C2semidirect

Maximal subgroups

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

subgroupindex
C2×#25512
#331492
#331482
#279722
#105752
#103492
#87242
#87072
#87062