Finite Group Databasebeta

← #327989

Subgroups — #327989

Order 512.

899 conjugacy classes of subgroups.

The subgroup lattice has 380 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2552C211yesmaximal, characteristic
2#33149C211yesmaximal, characteristic
3#33149C211yesmaximal, characteristic
4#27972C211yesmaximal, characteristic
5#10581C214yesmaximal, complemented
6#10355C214yesmaximal, complemented
7#8724C211yesmaximal, characteristic
8#8707C211yesmaximal, characteristic
9#8707C211yesmaximal, characteristic
10C22×#125—21no—
11#2857C2211yescharacteristic
12#2552—22no—
13#2552—22no—
14#2552—22no—
15#2552C2214yes—
16C2×#398—22no—
17C2×#389—21no—
18C2×#389—21no—
19C2×#384—21no—
20C2×#384C2211yescharacteristic
21C2×C4×C4⋊C4C2211yescharacteristic
22C2×C4×C4⋊C4C2211yescharacteristic
23#2072—22no—
24#1987C2212yes—
25#1987C2212yes—
26#1956C2212yes—
27#1947C2212yes—
28#1947C2212yes—
29#1921—22no—
30#1921—22no—
31#1872—22no—
32#1852—21no—
33#1852—21no—
34#1834C2212yes—
35#1824—21no—
36#1824—21no—
37#1797C2211yescharacteristic
38#1797C2211yescharacteristic
39C4×C22⋊C8C2211yescharacteristic
40#1296C2214yes—
41#1279C2214yes—
42#1279C2214yes—
43C2×C4⋊Q8C2311yescharacteristic
44C2×#125—24no—
45C2×#125—24no—
46C2×#125—48no—
47C2×C22⋊Q8—21no—
48C22×C4⋊C4—22no—
49C22×C4⋊C4—22no—
50C22×C4⋊C4—21no—
51C22×C4⋊C4—21no—
52C22×C4⋊C4—22no—
53C22×C4⋊C4D411yescharacteristic
54C22×C4⋊C4D411yescharacteristic
55C22×C4⋊C4D411yescharacteristic
56C22×C4⋊C4D411yescharacteristic
57C22×C4⋊C4D412yes—
58C22×C42—21no—
59C22×C42—21no—
60C22×C42C2311yescharacteristic
61C2×#107—22no—
62C2×#107—22no—
63C2×C4⋊C8C2312yes—
64C2×C4⋊C8C2312yes—
65C2×#103—24no—
66C2×C22⋊C8—21no—
67C2×C22⋊C8—21no—
68C2×C22⋊C8—21no—
69C2×C4×C8C2312yes—
70#398—28no—
71#398—28no—
72#391—21no—
73#389—24no—
74#389—24no—
75#389—24no—
76#389—24no—
77#389—21no—
78#389—21no—
79#384—24no—
80#384—24no—
81#384—21no—
82#384—21no—
83#384D412yes—
84#384C2311yescharacteristic
85#384C2312yes—
86#380—21no—
87#380—21no—
88C4×C4⋊C4—21no—
89C4×C4⋊C4—21no—
90C4×C4⋊C4—21no—
91C4×C4⋊C4—21no—
92C4×C4⋊C4D412yes—
93C4×C4⋊C4D412yes—
94C4×C4⋊C4C2312yes—
95C4×C4⋊C4C2312yes—
96C2×#95—22no—
97C2×#95—21no—
98C2×#95—21no—
99#340—24no—
100#340—22no—
101#340—22no—
102#340—22no—
103#340—22no—
104#340—22no—
105#340—22no—
106#336—21no—
107#336—21no—
108#336—21no—
109C22×Q8—21no—
110C23×C4—21no—
111C23×C4—21no—
112C23×C4—22no—
113C23×C4—21no—
114C23×C4—21no—
115C23×C4C2×D411yescharacteristic
116C23×C4C2×D411yescharacteristic
117C23×C4C2×D411yescharacteristic
118C22×C8—22no—
119C22×C8—22no—
120C22×C8—22no—
121C4⋊Q8—24no—
122#125—416no—
123#125—48no—
124#125—48no—
125C22⋊Q8—44no—
126C2×C4⋊C4—21no—
127C2×C4⋊C4—21no—
128C2×C4⋊C4—22no—
129C2×C4⋊C4—24no—
130C2×C4⋊C4—22no—
131C2×C4⋊C4—22no—
132C2×C4⋊C4—28no—
133C2×C4⋊C4—28no—
134C2×C4⋊C4—22no—
135C2×C4⋊C4—22no—
136C2×C4⋊C4—21no—
137C2×C4⋊C4—21no—
138C2×C4⋊C4—22no—
139C2×C4⋊C4—22no—
140C2×C4⋊C4—21no—
141C2×C4⋊C4—21no—
142C2×C4⋊C4—21no—
143C2×C4⋊C4—21no—
144C2×C4⋊C4—21no—
145C2×C4⋊C4—28no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—42no—
148C2×C4⋊C4—48no—
149C2×C4⋊C4—48no—
150C2×C4⋊C4—44no—
151C2×C4⋊C4—42no—
152C2×C4⋊C4—42no—
153C2×C4⋊C4—41no—
154C2×C4⋊C4—82no—
155C2×C4⋊C4—84no—
156C2×C4⋊C4Q1614yes—
157C2×C4⋊C4C2×D412yes—
158C2×C4⋊C4C2×D412yes—
159C2×C4⋊C4C2×D412yes—
160C2×C4⋊C4C2×D412yes—
161C2×C4⋊C4C2×D414yes—
162C2×C22⋊C4—41no—
163C2×C42—24no—
164C2×C42—22no—
165C2×C42—22no—
166C2×C42—22no—
167C2×C42—22no—
168C2×C42—44no—
169C2×C42—44no—
170C2×C42C2×D411yescharacteristic
171C2×C42C2×D411yescharacteristic
172C2×C42C2×D411yescharacteristic
173C2×C42C2411yescharacteristic
174#107—44no—
175#107—44no—
176C4⋊C8—24no—
177C4⋊C8—24no—
178#103—84no—
179C22⋊C8—42no—
180C22⋊C8—42no—
181C22⋊C8—42no—
182C4×C8—24no—
183#95—28no—
184#95—22no—
185#95—22no—
186#95—22no—
187#95—22no—
188#95—41no—
189#95—41no—
190C24#12011yescharacteristic
191C2×Q8—42no—
192C2×Q8—42no—
193C2×Q8—42no—
194C22×C4—22no—
195C22×C4—22no—
196C22×C4—22no—
197C22×C4—22no—
198C22×C4—24no—
199C22×C4—22no—
200C22×C4—22no—
201C22×C4—24no—
202C22×C4—22no—
203C22×C4—22no—
204C22×C4—22no—
205C22×C4—22no—
206C22×C4—24no—
207C22×C4—41no—
208C22×C4—42no—
209C22×C4—42no—
210C22×C4—42no—
211C22×C4—42no—
212C22×C4—42no—
213C22×C4—42no—
214C22×C4—44no—
215C22×C4—42no—
216C22×C4—44no—
217C22×C4—42no—
218C22×C4—42no—
219C22×C4—44no—
220C22×C4—42no—
221C22×C4—41no—
222C22×C4—41no—
223C22×C4#12011yescharacteristic
224C22×C4#12011yescharacteristic
225C22×C4#12011yescharacteristic
226C22×C4C2×Q1612yes—
227C22×C4#13512yes—
228C22×C4#13712yes—
229C22×C4#13712yes—
230C22×C4C22×D411yescharacteristic
231C22×C4C22×D411yescharacteristic
232C22×C4C22×D411yescharacteristic
233C2×C8—42no—
234C2×C8—42no—
235C2×C8—42no—
236C2×C8—42no—
237C2×C8—42no—
238C2×C8—42no—
239C2×C8—42no—
240C2×C8—42no—
241C2×C8—42no—
242C4⋊C4—24no—
243C4⋊C4—22no—
244C4⋊C4—22no—
245C4⋊C4—24no—
246C4⋊C4—22no—
247C4⋊C4—22no—
248C4⋊C4—22no—
249C4⋊C4—22no—
250C4⋊C4—44no—
251C4⋊C4—42no—
252C4⋊C4—44no—
253C4⋊C4—416no—
254C4⋊C4—416no—
255C4⋊C4—42no—
256C4⋊C4—42no—
257C4⋊C4—44no—
258C4⋊C4—42no—
259C4⋊C4—42no—
260C4⋊C4—44no—
261C4⋊C4—42no—
262C4⋊C4—42no—
263C4⋊C4—42no—
264C4⋊C4—42no—
265C4⋊C4—82no—
266C4⋊C4—82no—
267C4⋊C4—88no—
268C4⋊C4C2×Q1614yes—
269C4⋊C4AGL(1,ℤ/8)14yes—
270C22⋊C4—82no—
271C42—22no—
272C42—22no—
273C42—22no—
274C42—44no—
275C42—44no—
276C42—44no—
277C42—44no—
278C422−1+414yes—
279C23—21no—
280C23—21no—
281C23—21no—
282C23—21no—
283C23#45112yes—
284C23#45212yes—
285C23#45412yes—
286C23C2×#12011yescharacteristic
287Q8—84no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—24no—
295C2×C4—24no—
296C2×C4—24no—
297C2×C4—24no—
298C2×C4—21no—
299C2×C4—24no—
300C2×C4—24no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—44no—
304C2×C4—48no—
305C2×C4—48no—
306C2×C4—42no—
307C2×C4—42no—
308C2×C4—42no—
309C2×C4—42no—
310C2×C4—42no—
311C2×C4—44no—
312C2×C4—42no—
313C2×C4—42no—
314C2×C4—44no—
315C2×C4—42no—
316C2×C4—42no—
317C2×C4—44no—
318C2×C4—42no—
319C2×C4—84no—
320C2×C4—88no—
321C2×C4—84no—
322C2×C4—81no—
323C2×C4—81no—
324C2×C4—81no—
325C2×C4—81no—
326C2×C4—81no—
327C2×C4—81no—
328C2×C4#45112yes—
329C2×C4#45212yes—
330C2×C4#53612yes—
331C2×C4#53612yes—
332C2×C4#53612yes—
333C2×C4C22×Q1611yescharacteristic
334C2×C4C2×#13511yescharacteristic
335C2×C4C2×AGL(1,ℤ/8)12yes—
336C2×C4C2×#13711yescharacteristic
337C2×C4C2×#13711yescharacteristic
338C2×C4#57512yes—
339C8—82no—
340C8—82no—
341C8—82no—
342C22—22no—
343C22—22no—
344C22—22no—
345C22—22no—
346C22—22no—
347C22—22no—
348C22#163614yes—
349C22#261411yescharacteristic
350C22C2×#45111yescharacteristic
351C22C2×#45211yescharacteristic
352C22#300311yescharacteristic
353C22#300511yescharacteristic
354C22C2×#45411yescharacteristic
355C22#301911yescharacteristic
356C4—22no—
357C4—22no—
358C4—22no—
359C4—44no—
360C4—44no—
361C4—41no—
362C4—82no—
363C4—84no—
364C4—82no—
365C4—161no—
366C4—161no—
367C4#323012yes—
368C4#323212yes—
369C4#323612yes—
370C4#323912yes—
371C2—24no—
372C2C2×#163611yescharacteristic
373C2#1853311yescharacteristic
374C2#1874911yescharacteristic
375C2#1877111yescharacteristic
376C2#2964211yescharacteristic
377C2#2964311yescharacteristic
378C2#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. #10581C2semidirect
6. #10355C2semidirect

Maximal subgroups

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

subgroupindex
C2×#25522
#331492
#279722
#105812
#103552
#87242
#87072