Finite Group Databasebeta

← #328011

Subgroups — #328011

Order 512.

1101 conjugacy classes of subgroups.

The subgroup lattice has 400 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2551C211yesmaximal, characteristic
2#33146C211yesmaximal, characteristic
3#33109C211yesmaximal, characteristic
4#27950C211yesmaximal, characteristic
5#10573C214yesmaximal, complemented
6#10350C214yesmaximal, complemented
7#8724C211yesmaximal, characteristic
8#8707C211yesmaximal, characteristic
9#8706C211yesmaximal, characteristic
10C22×C4⋊Q8—21no—
11C22×C4⋊Q8—21no—
12C2×#484—21no—
13#2857C2211yescharacteristic
14#2551—22no—
15#2551—22no—
16#2551—22no—
17#2551C2214yes—
18C2×#395—22no—
19C2×#389—21no—
20C2×#389—21no—
21C2×#384C2211yescharacteristic
22C2×C4×C4⋊C4C2211yescharacteristic
23C2×C4×C4⋊C4C2211yescharacteristic
24#2083—22no—
25#1990C2212yes—
26#1988C2212yes—
27#1981C2212yes—
28#1947C2212yes—
29#1947C2212yes—
30#1928—22no—
31#1918—22no—
32#1872—21no—
33#1852—22no—
34#1852—21no—
35#1845C2212yes—
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
65#484—44no—
66C2×#107—22no—
67C2×#107—22no—
68C2×#107—24no—
69C2×C4⋊C8C2312yes—
70C2×C4⋊C8C2312yes—
71C2×#103—22no—
72C2×#103—22no—
73C2×C22⋊C8—21no—
74C2×C22⋊C8—21no—
75C2×C22⋊C8—21no—
76C2×C4×C8C2312yes—
77#395—28no—
78#395—28no—
79#391—21no—
80#389—24no—
81#389—24no—
82#389—24no—
83#389—24no—
84#389—21no—
85#389—21no—
86#384—21no—
87#384—21no—
88#384D412yes—
89#384C2311yescharacteristic
90#384C2312yes—
91#380—21no—
92#380—21no—
93C4×C4⋊C4—21no—
94C4×C4⋊C4—21no—
95C4×C4⋊C4—21no—
96C4×C4⋊C4—21no—
97C4×C4⋊C4D412yes—
98C4×C4⋊C4D412yes—
99C4×C4⋊C4C2312yes—
100C4×C4⋊C4C2312yes—
101C2×#95—22no—
102C2×#95—21no—
103C2×#95—21no—
104#340—22no—
105#340—22no—
106#340—24no—
107#340—22no—
108#340—22no—
109#336—21no—
110#336—21no—
111#336—21no—
112C22×Q8—28no—
113C22×Q8—21no—
114C22×Q8—48no—
115C22×Q8—48no—
116C22×Q8—84no—
117C23×C4—21no—
118C23×C4—21no—
119C23×C4—21no—
120C23×C4—21no—
121C23×C4—22no—
122C23×C4C2×D411yescharacteristic
123C23×C4C2×D411yescharacteristic
124C23×C4C2×D411yescharacteristic
125C22×C8—22no—
126C22×C8—22no—
127C22×C8—22no—
128C4⋊Q8—24no—
129C4⋊Q8—416no—
130C4⋊Q8—48no—
131C4⋊Q8—48no—
132C4⋊Q8—416no—
133C4⋊Q8—48no—
134C4⋊Q8—48no—
135C22⋊Q8—42no—
136C22⋊Q8—42no—
137C2×C4⋊C4—21no—
138C2×C4⋊C4—21no—
139C2×C4⋊C4—21no—
140C2×C4⋊C4—21no—
141C2×C4⋊C4—22no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4—24no—
144C2×C4⋊C4—24no—
145C2×C4⋊C4—28no—
146C2×C4⋊C4—28no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—21no—
152C2×C4⋊C4—21no—
153C2×C4⋊C4—21no—
154C2×C4⋊C4—21no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—48no—
158C2×C4⋊C4—48no—
159C2×C4⋊C4—44no—
160C2×C4⋊C4—42no—
161C2×C4⋊C4—42no—
162C2×C4⋊C4—41no—
163C2×C4⋊C4—82no—
164C2×C4⋊C4D814yes—
165C2×C4⋊C4Q1614yes—
166C2×C4⋊C4C2×D412yes—
167C2×C4⋊C4C2×D412yes—
168C2×C4⋊C4C2×D412yes—
169C2×C4⋊C4C2×D412yes—
170C2×C4⋊C4C2×D414yes—
171C2×C22⋊C4—41no—
172C2×C42—24no—
173C2×C42—22no—
174C2×C42—22no—
175C2×C42—22no—
176C2×C42—22no—
177C2×C42—44no—
178C2×C42—42no—
179C2×C42—42no—
180C2×C42C2×D411yescharacteristic
181C2×C42C2×D411yescharacteristic
182C2×C42C2×D411yescharacteristic
183C2×C42C2411yescharacteristic
184#107—44no—
185#107—44no—
186#107—84no—
187C4⋊C8—24no—
188C4⋊C8—24no—
189#103—44no—
190#103—44no—
191C22⋊C8—24no—
192C22⋊C8—42no—
193C22⋊C8—42no—
194C4×C8—24no—
195#95—28no—
196#95—22no—
197#95—22no—
198#95—22no—
199#95—22no—
200#95—41no—
201#95—41no—
202C24#12011yescharacteristic
203C2×Q8—416no—
204C2×Q8—416no—
205C2×Q8—42no—
206C2×Q8—41no—
207C2×Q8—41no—
208C2×Q8—42no—
209C2×Q8—88no—
210C2×Q8—832no—
211C22×C4—22no—
212C22×C4—22no—
213C22×C4—22no—
214C22×C4—22no—
215C22×C4—24no—
216C22×C4—24no—
217C22×C4—22no—
218C22×C4—22no—
219C22×C4—22no—
220C22×C4—22no—
221C22×C4—24no—
222C22×C4—24no—
223C22×C4—22no—
224C22×C4—22no—
225C22×C4—24no—
226C22×C4—41no—
227C22×C4—42no—
228C22×C4—42no—
229C22×C4—42no—
230C22×C4—42no—
231C22×C4—42no—
232C22×C4—42no—
233C22×C4—42no—
234C22×C4—42no—
235C22×C4—44no—
236C22×C4—44no—
237C22×C4—44no—
238C22×C4—41no—
239C22×C4—41no—
240C22×C4#12011yescharacteristic
241C22×C4#12011yescharacteristic
242C22×C4#12011yescharacteristic
243C22×C4C2×D812yes—
244C22×C4C2×Q1612yes—
245C22×C4AGL(1,ℤ/8)12yes—
246C22×C4#13712yes—
247C22×C4C22×D411yescharacteristic
248C22×C4C22×D411yescharacteristic
249C22×C4C22×D411yescharacteristic
250C2×C8—24no—
251C2×C8—42no—
252C2×C8—42no—
253C2×C8—42no—
254C2×C8—42no—
255C2×C8—42no—
256C2×C8—42no—
257C2×C8—42no—
258C2×C8—42no—
259C4⋊C4—24no—
260C4⋊C4—24no—
261C4⋊C4—22no—
262C4⋊C4—22no—
263C4⋊C4—22no—
264C4⋊C4—22no—
265C4⋊C4—24no—
266C4⋊C4—24no—
267C4⋊C4—44no—
268C4⋊C4—44no—
269C4⋊C4—416no—
270C4⋊C4—416no—
271C4⋊C4—42no—
272C4⋊C4—42no—
273C4⋊C4—42no—
274C4⋊C4—44no—
275C4⋊C4—44no—
276C4⋊C4—42no—
277C4⋊C4—42no—
278C4⋊C4—41no—
279C4⋊C4—42no—
280C4⋊C4—41no—
281C4⋊C4—82no—
282C4⋊C4—82no—
283C4⋊C4C2×D814yes—
284C4⋊C4C2×Q1614yes—
285C4⋊C4AGL(1,ℤ/8)14yes—
286C4⋊C4#13714yes—
287C22⋊C4—82no—
288C42—22no—
289C42—22no—
290C42—22no—
291C42—44no—
292C42—42no—
293C42—42no—
294C42—44no—
295C42—44no—
296C422−1+414yes—
297C23—21no—
298C23—21no—
299C23—21no—
300C23—21no—
301C23#44712yes—
302C23#45112yes—
303C23#45612yes—
304C23C2×#12011yescharacteristic
305Q8—832no—
306Q8—82no—
307Q8—82no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—24no—
315C2×C4—24no—
316C2×C4—24no—
317C2×C4—24no—
318C2×C4—21no—
319C2×C4—24no—
320C2×C4—24no—
321C2×C4—24no—
322C2×C4—24no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—44no—
326C2×C4—48no—
327C2×C4—48no—
328C2×C4—48no—
329C2×C4—48no—
330C2×C4—42no—
331C2×C4—42no—
332C2×C4—42no—
333C2×C4—42no—
334C2×C4—42no—
335C2×C4—42no—
336C2×C4—42no—
337C2×C4—42no—
338C2×C4—44no—
339C2×C4—44no—
340C2×C4—44no—
341C2×C4—88no—
342C2×C4—81no—
343C2×C4—81no—
344C2×C4—81no—
345C2×C4—81no—
346C2×C4—81no—
347C2×C4—81no—
348C2×C4#44712yes—
349C2×C4#45112yes—
350C2×C4#53612yes—
351C2×C4#53612yes—
352C2×C4#53612yes—
353C2×C4C22×D811yescharacteristic
354C2×C4C22×Q1611yescharacteristic
355C2×C4C2×AGL(1,ℤ/8)12yes—
356C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
357C2×C4C2×#13712yes—
358C2×C4C2×#13711yescharacteristic
359C8—44no—
360C8—82no—
361C8—82no—
362C22—22no—
363C22—22no—
364C22—22no—
365C22—22no—
366C22—22no—
367C22—22no—
368C22#164914yes—
369C22#261411yescharacteristic
370C22C2×#44711yescharacteristic
371C22C2×#45111yescharacteristic
372C22#300311yescharacteristic
373C22#300411yescharacteristic
374C22C2×#45611yescharacteristic
375C22#301911yescharacteristic
376C4—22no—
377C4—22no—
378C4—22no—
379C4—44no—
380C4—44no—
381C4—44no—
382C4—44no—
383C4—41no—
384C4—84no—
385C4—161no—
386C4—161no—
387C4#322612yes—
388C4#323012yes—
389C4#323812yes—
390C4#323912yes—
391C2—24no—
392C2C2×#164911yescharacteristic
393C2#1854011yescharacteristic
394C2#1874611yescharacteristic
395C2#1876911yescharacteristic
396C2#2963811yescharacteristic
397C2#2964211yescharacteristic
398C2#2965311yescharacteristic
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. #10573C2semidirect
6. #10350C2semidirect

Maximal subgroups

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

subgroupindex
C2×#25512
#331462
#331092
#279502
#105732
#103502
#87242
#87072
#87062