Finite Group Databasebeta

← #330449

Subgroups — #330449

Order 512.

512 conjugacy classes of subgroups; only the 154 normal subgroups are shown.

normal subgroups only

This group has 512 conjugacy classes of subgroups — too many to draw. Only its 154 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33290C211yesmaximal, characteristic
2#29999C211yesmaximal, characteristic
3#29830C211yesmaximal, characteristic
4#21341C211yesmaximal, characteristic
5#10594C212yesmaximal
6#10593C212yesmaximal
7#10483C212yesmaximal
8#10478C212yesmaximal
9#9902C211yesmaximal, characteristic
10#9701C211yesmaximal, characteristic
11#8806C211yesmaximal, characteristic
12C2×#477C2211yescharacteristic
13C2×#462C2211yescharacteristic
14#2862C2211yescharacteristic
15#2859C2211yescharacteristic
16#2295C2211yescharacteristic
17#2087—22no—
18#2068—22no—
19#2068—22no—
20#2056—22no—
21#1991—22no—
22#1990C2212yes—
23#1990C2212yes—
24#1987C2211yescharacteristic
25#1984C2212yes—
26#1981C2211yescharacteristic
27#1947C2212yes—
28#1939C2212yes—
29#1916—22no—
30#1842C2211yescharacteristic
31#1842C2211yescharacteristic
32#1835—21no—
33#1835—21no—
34#1803C2212yes—
35#1768—21no—
36#1767C2211yescharacteristic
37#1746—21no—
38C42×C8C2211yescharacteristic
39C2×#335C2212yes—
40C2×#335C2212yes—
41C2×#333C2212yes—
42C2×#332C2212yes—
43#1296C2212yes—
44#1296C2212yes—
45C22×Q16—21no—
46C2×C4⋊Q8C2311yescharacteristic
47C2×C4⋊Q8C2311yescharacteristic
48C2×#125—21no—
49C2×C4×Q8C2311yescharacteristic
50#477—42no—
51#462—42no—
52C2×#107—21no—
53C2×#107—22no—
54C2×C4⋊C8—22no—
55C2×C4⋊C8—22no—
56C2×C4⋊C8—22no—
57C2×C4⋊C8C2311yescharacteristic
58C2×C4⋊C8C2312yes—
59C2×C4⋊C8C2312yes—
60C2×C4⋊C8C2311yescharacteristic
61C2×#103—21no—
62C2×#103—21no—
63C2×#103—22no—
64C2×#103—22no—
65C2×#97—21no—
66C2×C4×C8D411yescharacteristic
67C2×C4×C8D412yes—
68C2×C4×C8D411yescharacteristic
69C2×C4×C8C2312yes—
70#391—22no—
71#391—22no—
72#391D412yes—
73#389—22no—
74#389—22no—
75#389D412yes—
76#387—22no—
77#384C2312yes—
78#382—21no—
79#382—21no—
80#382—21no—
81C4×C4⋊C4—21no—
82C4×C4⋊C4C2311yescharacteristic
83#376—21no—
84C43C2311yescharacteristic
85#340—22no—
86#340—22no—
87#340—22no—
88#340—22no—
89#340—22no—
90#340—22no—
91#340—22no—
92#336—22no—
93#336—22no—
94#336—22no—
95#336—21no—
96#335—42no—
97#335—42no—
98#333—42no—
99#332—42no—
100C22×Q8—21no—
101C22×Q8—21no—
102C22×Q8C2×D411yescharacteristic
103C2×Q16—42no—
104C2×Q16—82no—
105C22×C8—21no—
106C22×C8—22no—
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—22no—
118C2×C4⋊C4—22no—
119C2×C4⋊C4—21no—
120C2×C4⋊C4—22no—
121C2×C4⋊C4—21no—
122C2×C4⋊C4—21no—
123C2×C4⋊C4—21no—
124C2×C4⋊C4—22no—
125C2×C4⋊C4—21no—
126C2×C4⋊C4—22no—
127C2×C4⋊C4—42no—
128C2×C4⋊C4C2×D411yescharacteristic
129C2×C4⋊C4C2×D412yes—
130C2×C42—21no—
131C2×C42—21no—
132C2×C42C2×D411yescharacteristic
133C2×C42C2×D412yes—
134C2×C42C2×D411yescharacteristic
135C2×C42#4112yes—
136C2×C42C2411yescharacteristic
137#107—42no—
138#107—82no—
139C4⋊C8—22no—
140C4⋊C8—22no—
141C4⋊C8—22no—
142C4⋊C8—22no—
143C4⋊C8—22no—
144C4⋊C8—22no—
145C4⋊C8—44no—
146C4⋊C8—82no—
147C4⋊C8—82no—
148#103—42no—
149#103—42no—
150#103—82no—
151#103—82no—
152#97—42no—
153C4×C8—22no—
154C4×C8—22no—
155C4×C8—22no—
156C4×C8—42no—
157C4×C8Q1614yescomplemented
158#95—22no—
159#95—22no—
160#95—22no—
161#95—21no—
162#95—42no—
163#95—42no—
164#95—42no—
165C2×Q8—21no—
166C2×Q8—21no—
167C2×Q8—22no—
168C2×Q8—22no—
169C2×Q8—41no—
170C2×Q8—41no—
171C2×Q8—42no—
172C2×Q8—42no—
173C2×Q8—42no—
174C2×Q8—82no—
175C22×C4—22no—
176C22×C4—21no—
177C22×C4—21no—
178C22×C4—41no—
179C22×C4—41no—
180C22×C4—42no—
181C22×C4#12112yes—
182C22×C4#12112yes—
183C22×C4C22×D411yescharacteristic
184C22×C4C22×D411yescharacteristic
185C22×C4C2×#4111yescharacteristic
186Q16—84no—
187C2×C8—22no—
188C2×C8—22no—
189C2×C8—22no—
190C2×C8—22no—
191C2×C8—22no—
192C2×C8—41no—
193C2×C8—41no—
194C2×C8—42no—
195C2×C8—42no—
196C2×C8—42no—
197C2×C8—42no—
198C2×C8—42no—
199C2×C8—42no—
200C2×C8—42no—
201C2×C8—42no—
202C2×C8C2×Q1614yes—
203C4⋊C4—21no—
204C4⋊C4—21no—
205C4⋊C4—22no—
206C4⋊C4—22no—
207C4⋊C4—44no—
208C4⋊C4—42no—
209C4⋊C4—41no—
210C4⋊C4—41no—
211C4⋊C4—42no—
212C4⋊C4—42no—
213C4⋊C4—42no—
214C4⋊C4—44no—
215C4⋊C4—42no—
216C4⋊C4—42no—
217C4⋊C4—82no—
218C4⋊C4—84no—
219C42—22no—
220C42—22no—
221C42—22no—
222C42—42no—
223C42—42no—
224C42—42no—
225C42C2×Q1612yes—
226C42#13712yes—
227C42H(5,𝔽2)12yes—
228C422−1+412yes—
229C23C2×#12111yescharacteristic
230Q8—41no—
231Q8—41no—
232Q8—42no—
233Q8—81no—
234Q8—81no—
235Q8—82no—
236Q8—162no—
237C2×C4—22no—
238C2×C4—22no—
239C2×C4—22no—
240C2×C4—22no—
241C2×C4—22no—
242C2×C4—22no—
243C2×C4—41no—
244C2×C4—41no—
245C2×C4—42no—
246C2×C4—42no—
247C2×C4—42no—
248C2×C4—42no—
249C2×C4—42no—
250C2×C4—81no—
251C2×C4—81no—
252C2×C4—82no—
253C2×C4—82no—
254C2×C4—82no—
255C2×C4—82no—
256C2×C4—82no—
257C2×C4#46214yes—
258C2×C4#53511yescharacteristic
259C2×C4#53611yescharacteristic
260C2×C4#53712yes—
261C2×C4#54212yes—
262C2×C4C22×Q1611yescharacteristic
263C2×C4C2×#13711yescharacteristic
264C2×C4#57512yes—
265C2×C4#57612yes—
266C2×C4#57812yes—
267C8—24no—
268C8—42no—
269C8—81no—
270C8—82no—
271C8—81no—
272C8—82no—
273C22#262011yescharacteristic
274C22C2×#46211yescharacteristic
275C22#304011yescharacteristic
276C22#306112yes—
277C22#306712yes—
278C4—24no—
279C4—22no—
280C4—42no—
281C4—81no—
282C4—81no—
283C4—82no—
284C4—161no—
285C4—161no—
286C4—162no—
287C4#322011yescharacteristic
288C4#322311yescharacteristic
289C4#324312yes—
290C4#324912yes—
291C4#325212yes—
292C2#1921112yes—
293C2#1936712yes—
294C2#2983011yescharacteristic
295C2#2984911yescharacteristic
296C2#2986011yescharacteristic
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
157. C4×C8Q16semidirect

Maximal subgroups

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

subgroupindex
#332902
#299992
#298302
#213412
#105942
#105932
#104832
#104782
#99022
#97012
#88062