Finite Group Databasebeta

← #323280

Subgroups — #323280

Order 512.

720 conjugacy classes of subgroups.

The subgroup lattice has 351 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33310C211yesmaximal, characteristic
2#29218C211yesmaximal, characteristic
3#29208C211yesmaximal, characteristic
4#29141C211yesmaximal, characteristic
5#28511C211yesmaximal, characteristic
6C4×#498C211yesmaximal, characteristic
7#10684C212yesmaximal
8#10680C212yesmaximal
9#9819C212yesmaximal
10#8814C212yesmaximal
11C8×C4⋊C8C211yesmaximal, characteristic
12C2×#498—22no—
13C2×#498C2211yescharacteristic
14C2×#477—22no—
15#2868C2211yescharacteristic
16#2868C2211yescharacteristic
17C4×C4⋊Q8C2211yescharacteristic
18C4×C4⋊Q8C2211yescharacteristic
19#2100—22no—
20#2095—22no—
21#2057—22no—
22#2024—22no—
23#1990C2212yes—
24#1990C2212yes—
25#1990C2211yescharacteristic
26#1990C2211yescharacteristic
27#1987—22no—
28#1956C2212yes—
29#1956C2212yes—
30#1947—22no—
31#1941—22no—
32#1845C2212yes—
33#1845C2212yes—
34#1844—22no—
35#1844C2212yes—
36#1844C2212yes—
37#1844C2211yescharacteristic
38#1834C2212yes—
39#1834C2212yes—
40#1803—22no—
41C4×#106—22no—
42C4×C4⋊C8C2211yescharacteristic
43C4×C4⋊C8C2211yescharacteristic
44C42×C8C2211yescharacteristic
45C2×C82C2212yes—
46#1273C2212yes—
47C2×C4⋊Q8—21no—
48C2×C4⋊Q8—21no—
49C2×C4⋊Q8—21no—
50C2×C4⋊Q8—21no—
51C2×C4⋊Q8—21no—
52C2×C4⋊Q8—21no—
53C2×C4⋊Q8C2311yescharacteristic
54C2×C4⋊Q8C2311yescharacteristic
55C2×C4×Q8—21no—
56C2×C4×Q8—21no—
57#498—22no—
58#498—48no—
59#498Q814yescomplemented
60#477—48no—
61C2×#107—24no—
62C2×#107—24no—
63C2×#106—22no—
64C2×#106—24no—
65C2×#106—24no—
66C2×#106—22no—
67C2×C4⋊C8D412yes—
68C2×C4⋊C8D412yes—
69C2×#103—24no—
70C2×#103—24no—
71C2×C4×C8D412yes—
72C2×C4×C8D412yes—
73C2×C4×C8C2312yes—
74C2×C4×C8C2311yescharacteristic
75C2×C4×C8C2312yes—
76C2×C4×C8C2311yescharacteristic
77#391—21no—
78#391—21no—
79#389—21no—
80#389—21no—
81#389—21no—
82#389—21no—
83#384—21no—
84#384—21no—
85#384—21no—
86#384—21no—
87#384—21no—
88#384—21no—
89#384C2311yescharacteristic
90#384C2311yescharacteristic
91#384C2312yes—
92#384C2312yes—
93C4×C4⋊C4—21no—
94C4×C4⋊C4—21no—
95C4×C4⋊C4—21no—
96C4×C4⋊C4—21no—
97C43C2311yescharacteristic
98#340—24no—
99#340—24no—
100#340—24no—
101#340—24no—
102#340—22no—
103#340—22no—
104#336D412yes—
105#336D412yes—
106C82—24no—
107C22×Q8—21no—
108C22×Q8—21no—
109C22×Q8—41no—
110C22×Q8—41no—
111C22×C8C2×D414yes—
112C22×C8C2×D414yes—
113C22×C8C2×D412yes—
114C22×C8C2×D412yes—
115C4⋊Q8—21no—
116C4⋊Q8—21no—
117C4⋊Q8—21no—
118C4⋊Q8—21no—
119C4⋊Q8—21no—
120C4⋊Q8—21no—
121C4⋊Q8—44no—
122C4⋊Q8—44no—
123C4⋊Q8—44no—
124C4⋊Q8—44no—
125C4⋊Q8—82no—
126C4⋊Q8—82no—
127C4⋊Q8C2×Q812yes—
128C4⋊Q8C2×Q812yes—
129C4×Q8—44no—
130C4×Q8—44no—
131C2×C4⋊C4—21no—
132C2×C4⋊C4—21no—
133C2×C4⋊C4—21no—
134C2×C4⋊C4—21no—
135C2×C4⋊C4—21no—
136C2×C4⋊C4—21no—
137C2×C4⋊C4—22no—
138C2×C4⋊C4—22no—
139C2×C4⋊C4—22no—
140C2×C4⋊C4—22no—
141C2×C4⋊C4—22no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4—21no—
144C2×C4⋊C4—21no—
145C2×C4⋊C4—21no—
146C2×C4⋊C4—21no—
147C2×C4⋊C4—41no—
148C2×C4⋊C4—41no—
149C2×C4⋊C4—41no—
150C2×C4⋊C4—41no—
151C2×C4⋊C4—41no—
152C2×C4⋊C4—41no—
153C2×C4⋊C4—41no—
154C2×C4⋊C4—41no—
155C2×C42—41no—
156C2×C42—41no—
157C2×C42C2×D411yescharacteristic
158C2×C42C2×D411yescharacteristic
159C2×C42C2×D411yescharacteristic
160C2×C42C2×D411yescharacteristic
161C2×C42C2×D411yescharacteristic
162C2×C42C2×D411yescharacteristic
163C2×C42C2411yescharacteristic
164#107—48no—
165#107—84no—
166#106—28no—
167#106—48no—
168#106—44no—
169#106—48no—
170C4⋊C8—24no—
171C4⋊C8SD1618yes—
172#103—48no—
173#103—84no—
174C4×C8—24no—
175C4×C8—22no—
176C4×C8—22no—
177C4×C8—21no—
178C4×C8—21no—
179C4×C8—22no—
180C4×C8—22no—
181C4×C8Q1618yes—
182C4×C8C2×Q812yes—
183C4×C8#4112yes—
184#95—41no—
185#95—41no—
186#95—41no—
187#95—41no—
188C2×Q8—22no—
189C2×Q8—22no—
190C2×Q8—41no—
191C2×Q8—41no—
192C2×Q8—42no—
193C2×Q8—42no—
194C2×Q8—42no—
195C2×Q8—42no—
196C2×Q8—82no—
197C2×Q8—82no—
198C2×Q8—82no—
199C2×Q8—82no—
200C2×Q8—82no—
201C2×Q8—82no—
202C22×C4—41no—
203C22×C4—41no—
204C22×C4—41no—
205C22×C4—41no—
206C22×C4—42no—
207C22×C4—42no—
208C22×C4#12711yescharacteristic
209C22×C4#12711yescharacteristic
210C22×C4#12711yescharacteristic
211C22×C4#12711yescharacteristic
212C22×C4C22×D411yescharacteristic
213C22×C4C22×D411yescharacteristic
214C22×C4C22×D411yescharacteristic
215C2×C8—22no—
216C2×C8—24no—
217C2×C8—22no—
218C2×C8—24no—
219C2×C8—24no—
220C2×C8—24no—
221C2×C8—24no—
222C2×C8C22⋊Q814yes—
223C2×C8C2×SD1618yes—
224C2×C8C2×Q1614yes—
225C2×C8C2×Q1614yes—
226C2×C8AGL(1,ℤ/8)14yes—
227C4⋊C4—22no—
228C4⋊C4—22no—
229C4⋊C4—22no—
230C4⋊C4—22no—
231C4⋊C4—41no—
232C4⋊C4—41no—
233C4⋊C4—41no—
234C4⋊C4—41no—
235C4⋊C4—44no—
236C4⋊C4—42no—
237C4⋊C4—42no—
238C4⋊C4—44no—
239C4⋊C4—44no—
240C4⋊C4—44no—
241C4⋊C4—42no—
242C4⋊C4—42no—
243C4⋊C4—44no—
244C4⋊C4—42no—
245C4⋊C4—44no—
246C4⋊C4—42no—
247C4⋊C4—82no—
248C4⋊C4—82no—
249C4⋊C4—82no—
250C4⋊C4—82no—
251C4⋊C4—82no—
252C4⋊C4—82no—
253C4⋊C4—82no—
254C4⋊C4—82no—
255C42—22no—
256C42—22no—
257C42—22no—
258C42—82no—
259C42—82no—
260C42C2×SD1614yes—
261C42C2×Q1614yes—
262C42#13514yes—
263C42C22×Q811yescharacteristic
264C42C2×#4111yescharacteristic
265C422−1+411yescharacteristic
266C422−1+411yescharacteristic
267C23C2×#12711yescharacteristic
268Q8—44no—
269Q8—44no—
270Q8—82no—
271Q8—82no—
272Q8—164no—
273Q8—164no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—42no—
281C2×C4—42no—
282C2×C4—81no—
283C2×C4—81no—
284C2×C4—81no—
285C2×C4—81no—
286C2×C4—81no—
287C2×C4—82no—
288C2×C4—81no—
289C2×C4—82no—
290C2×C4—82no—
291C2×C4—82no—
292C2×C4—81no—
293C2×C4—81no—
294C2×C4—82no—
295C2×C4—82no—
296C2×C4—81no—
297C2×C4—81no—
298C2×C4#49212yes—
299C2×C4#49212yes—
300C2×C4#49412yes—
301C2×C4#49412yes—
302C2×C4#49512yes—
303C2×C4#49512yes—
304C2×C4C2×C22⋊Q811yescharacteristic
305C2×C4#53611yescharacteristic
306C2×C4#54711yescharacteristic
307C2×C4#54711yescharacteristic
308C2×C4D4×Q811yescharacteristic
309C2×C4D4×Q811yescharacteristic
310C2×C4C22×SD1612yes—
311C2×C4C22×Q1612yes—
312C2×C4C2×#13512yes—
313C2×C4C2×AGL(1,ℤ/8)12yes—
314C2×C4#57712yes—
315C2×C4#57712yes—
316C8—28no—
317C8—24no—
318C8—44no—
319C8#47718yescomplemented
320C22#254211yescharacteristic
321C22C2×#49211yescharacteristic
322C22C2×#49411yescharacteristic
323C22C2×#49511yescharacteristic
324C22#315111yescharacteristic
325C22#315411yescharacteristic
326C22#315411yescharacteristic
327C4—22no—
328C4—22no—
329C4—22no—
330C4—82no—
331C4—82no—
332C4—161no—
333C4—162no—
334C4—161no—
335C4—162no—
336C4#170714yes—
337C4C2×#47712yes—
338C4#323012yes—
339C4#333512yes—
340C4#334012yes—
341C4Q8×SD1612yes—
342C4#338112yes—
343C2C2×#170711yescharacteristic
344C2#1962111yescharacteristic
345C2#1962511yescharacteristic
346C2#1966311yescharacteristic
347C2#2851111yescharacteristic
348C2#2852911yescharacteristic
349C2#2853011yescharacteristic
subgroupG/H
center, socle, prime radicalC23C2×#127
commutator, Frattini, cosocleC2×C42C24
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
59. #498Q8semidirect
319. C8#477semidirect

Maximal subgroups

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

subgroupindex
#333102
#292182
#292082
#291412
#285112
C4×#4982
#106842
#106802
#98192
#88142
C8×C4⋊C82