Finite Group Databasebeta

← #350969

Subgroups — #350969

Order 512.

576 conjugacy classes of subgroups.

The subgroup lattice has 302 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33290C211yesmaximal, characteristic
2#33069C211yesmaximal, characteristic
3#33027C211yesmaximal, characteristic
4C4×#446C211yesmaximal, characteristic
5#9897C211yesmaximal, characteristic
6#9693C211yesmaximal, characteristic
7#9576C212yesmaximal
8#9574C212yesmaximal
9#9226C212yesmaximal
10#9224C212yesmaximal
11#8785C211yesmaximal, characteristic
12C2×#477C2211yescharacteristic
13C2×#462C2211yescharacteristic
14C2×#446C2212yes—
15#2862C2211yescharacteristic
16#2859C2211yescharacteristic
17C42×Q8C2211yescharacteristic
18#1991—22no—
19#1990C2212yes—
20#1990C2212yes—
21#1987C2211yescharacteristic
22#1981C2211yescharacteristic
23#1947C2212yes—
24#1928—22no—
25#1921—22no—
26#1878—22no—
27#1863—22no—
28#1835—21no—
29#1835—21no—
30#1831C2211yescharacteristic
31#1831C2211yescharacteristic
32#1803C2212yes—
33#1771—22no—
34#1770C2212yes—
35C4×C4⋊C8—21no—
36C4×C4⋊C8C2211yescharacteristic
37C4×#97—21no—
38C42×C8C2211yescharacteristic
39C2×#333C2212yes—
40C2×#332C2212yes—
41#1296C2212yes—
42#1296C2212yes—
43#1277C2212yes—
44#1277C2212yes—
45C22×Q16—21no—
46C2×C4⋊Q8C2311yescharacteristic
47C2×C4⋊Q8C2311yescharacteristic
48C2×#125—21no—
49C2×C4×Q8D412yes—
50C2×C4×Q8C2311yescharacteristic
51#477—24no—
52#462—24no—
53#446—24no—
54#446—24no—
55C2×#107—21no—
56C2×#107—22no—
57C2×C4⋊C8—22no—
58C2×C4⋊C8C2311yescharacteristic
59C2×C4⋊C8C2312yes—
60C2×C4⋊C8C2311yescharacteristic
61C2×C4⋊C8C2312yes—
62C2×#103—21no—
63C2×#103—22no—
64C2×#103—22no—
65C2×#103—21no—
66C2×#97—22no—
67C2×#97—22no—
68C2×C4×C8—21no—
69C2×C4×C8D411yescharacteristic
70C2×C4×C8D412yes—
71C2×C4×C8D411yescharacteristic
72C2×C4×C8C2312yes—
73#391—22no—
74#391—22no—
75#389—22no—
76#389—22no—
77#384C2312yes—
78#382—21no—
79#382—21no—
80#382—21no—
81C4×C4⋊C4—21no—
82C4×C4⋊C4—22no—
83C4×C4⋊C4D412yes—
84C4×C4⋊C4C2311yescharacteristic
85C43—21no—
86C43C2311yescharacteristic
87#340—22no—
88#340—22no—
89#340—22no—
90#340—22no—
91#340—22no—
92#340—22no—
93#340—22no—
94#336—22no—
95#336—22no—
96#336—22no—
97#336—21no—
98#333—24no—
99#332—24no—
100C22×Q8—21no—
101C22×Q8—21no—
102C22×Q8C2×D411yescharacteristic
103C2×Q16—44no—
104C2×Q16—42no—
105C22×C8—22no—
106C22×C8—21no—
107C22×C8—22no—
108C22×C8—21no—
109C22×C8C2×D412yes—
110C22×C8C2×D412yes—
111C4⋊Q8—22no—
112C4⋊Q8—22no—
113C4⋊Q8—22no—
114C4⋊Q8—22no—
115#125—44no—
116C4×Q8—22no—
117C4×Q8—42no—
118C4×Q8—42no—
119C4×Q8#4114yes—
120C2×C4⋊C4—21no—
121C2×C4⋊C4—22no—
122C2×C4⋊C4—22no—
123C2×C4⋊C4—21no—
124C2×C4⋊C4—21no—
125C2×C4⋊C4—21no—
126C2×C4⋊C4—22no—
127C2×C4⋊C4—21no—
128C2×C4⋊C4—22no—
129C2×C4⋊C4—42no—
130C2×C4⋊C4C2×D412yes—
131C2×C4⋊C4C2×D411yescharacteristic
132C2×C42—21no—
133C2×C42—22no—
134C2×C42—22no—
135C2×C42—21no—
136C2×C42C2×D412yes—
137C2×C42C2×D411yescharacteristic
138C2×C42C2×D411yescharacteristic
139C2×C42#4112yes—
140C2×C42C2411yescharacteristic
141#107—44no—
142#107—42no—
143C4⋊C8—28no—
144C4⋊C8—22no—
145C4⋊C8—22no—
146C4⋊C8—22no—
147C4⋊C8—22no—
148C4⋊C8—22no—
149C4⋊C8#4114yes—
150#103—44no—
151#103—44no—
152#103—42no—
153#103—42no—
154#97—28no—
155#97—44no—
156C4×C8—22no—
157C4×C8—22no—
158C4×C8—22no—
159C4×C8—42no—
160C4×C8—42no—
161C4×C8Q1614yescomplemented
162#95—22no—
163#95—21no—
164#95—42no—
165#95—42no—
166#95—42no—
167C2×Q8—21no—
168C2×Q8—22no—
169C2×Q8—22no—
170C2×Q8—41no—
171C2×Q8—42no—
172C2×Q8—41no—
173C2×Q8—42no—
174C2×Q8—42no—
175C2×Q8—82no—
176C2×Q8#12412yes—
177C22×C4—21no—
178C22×C4—21no—
179C22×C4—22no—
180C22×C4—41no—
181C22×C4—41no—
182C22×C4—42no—
183C22×C4#12112yes—
184C22×C4#12112yes—
185C22×C4C22×D411yescharacteristic
186C22×C4C22×D411yescharacteristic
187C22×C4C2×#4111yescharacteristic
188Q16—84no—
189C2×C8—22no—
190C2×C8—22no—
191C2×C8—22no—
192C2×C8—24no—
193C2×C8—22no—
194C2×C8—22no—
195C2×C8—42no—
196C2×C8—42no—
197C2×C8—41no—
198C2×C8—41no—
199C2×C8—42no—
200C2×C8—42no—
201C2×C8—42no—
202C2×C8—42no—
203C2×C8—42no—
204C2×C8C2×Q1614yes—
205C4⋊C4—24no—
206C4⋊C4—21no—
207C4⋊C4—22no—
208C4⋊C4—22no—
209C4⋊C4—44no—
210C4⋊C4—42no—
211C4⋊C4—42no—
212C4⋊C4—41no—
213C4⋊C4—42no—
214C4⋊C4—41no—
215C4⋊C4—42no—
216C4⋊C4—42no—
217C4⋊C4—44no—
218C4⋊C4—82no—
219C4⋊C4—84no—
220C4⋊C4#12412yes—
221C42—24no—
222C42—22no—
223C42—22no—
224C42—22no—
225C42—42no—
226C42—44no—
227C42—44no—
228C42—42no—
229C42C2×Q1612yes—
230C42#13712yes—
231C42C2×#4112yes—
232C42C2×#4112yes—
233C23C2×#12111yescharacteristic
234Q8—22no—
235Q8—22no—
236Q8—42no—
237Q8—81no—
238Q8—81no—
239Q8—82no—
240Q8—162no—
241C2×C4—22no—
242C2×C4—22no—
243C2×C4—22no—
244C2×C4—22no—
245C2×C4—22no—
246C2×C4—22no—
247C2×C4—22no—
248C2×C4—22no—
249C2×C4—42no—
250C2×C4—42no—
251C2×C4—42no—
252C2×C4—42no—
253C2×C4—42no—
254C2×C4—81no—
255C2×C4—81no—
256C2×C4—82no—
257C2×C4—82no—
258C2×C4—82no—
259C2×C4—82no—
260C2×C4—82no—
261C2×C4#46214yes—
262C2×C4#52512yes—
263C2×C4C2×#12411yescharacteristic
264C2×C4#52912yes—
265C2×C4#53211yescharacteristic
266C2×C4C22×Q1611yescharacteristic
267C2×C4C2×#13711yescharacteristic
268C2×C4#57512yes—
269C2×C4#57512yes—
270C2×C4#57512yes—
271C8—24no—
272C8—42no—
273C8—44no—
274C8—81no—
275C8—81no—
276C8—82no—
277C22#285011yescharacteristic
278C22C2×#46211yescharacteristic
279C22#304011yescharacteristic
280C22#305412yes—
281C22#306712yes—
282C4—22no—
283C4—24no—
284C4—42no—
285C4—42no—
286C4—42no—
287C4—82no—
288C4—161no—
289C4—161no—
290C4—162no—
291C4#310412yes—
292C4#311612yes—
293C4#314011yescharacteristic
294C4#314211yescharacteristic
295C4#318312yes—
296C2#1915012yes—
297C2#1930312yes—
298C2#3302711yescharacteristic
299C2#3305011yescharacteristic
300C2#3306211yescharacteristic
subgroupG/H
center, socle, prime radicalC23C2×#121
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
161. C4×C8Q16semidirect

Maximal subgroups

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

subgroupindex
#332902
#330692
#330272
C4×#4462
#98972
#96932
#95762
#95742
#92262
#92242
#87852