Finite Group Databasebeta

← #351619

Subgroups — #351619

Order 512.

545 conjugacy classes of subgroups; only the 150 normal subgroups are shown.

normal subgroups only

This group has 545 conjugacy classes of subgroups — too many to draw. Only its 150 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#29942C212yesmaximal
2#29878C211yesmaximal, characteristic
3#20548C211yesmaximal, characteristic
4#9730C212yesmaximal, complemented
5#9616C212yesmaximal, complemented
6#9576C212yesmaximal, complemented
7#9575C212yesmaximal, complemented
8#9135C212yesmaximal
9#8599C211yesmaximal, characteristic
10C2×#500C2211yescharacteristic
11C2×#431C2211yescharacteristic
12C2×#423C2211yescharacteristic
13#2857C2212yes—
14#2283C2211yescharacteristic
15#2101—22no—
16#2087—22no—
17#2085—22no—
18#2077—22no—
19#2076—22no—
20#2075—22no—
21#1990C2212yes—
22#1987C2212yes—
23#1987C2212yes—
24#1948C2212yes—
25#1947C2212yes—
26#1834C2211yescharacteristic
27#1797C2212yes—
28#1770C2212yes—
29C8×C4⋊C4C2212yes—
30#1768—22no—
31#1753—22no—
32C2×#335C2212yes—
33C2×#333C2212yes—
34C2×#333C2212yes—
35#1296C2212yes—
36#1296C2212yes—
37#1277C2211yescharacteristic
38#1277C2211yescharacteristic
39C22×M16—21no—
40C2×C4⋊Q8C2312yes—
41C2×C22⋊Q8—22no—
42C22×C42C2311yescharacteristic
43#500—24no—
44#431—24no—
45C2×#107—22no—
46C2×#107—22no—
47C2×#106—22no—
48#423—24no—
49C2×C4⋊C8—21no—
50C2×C4⋊C8D412yes—
51C2×C4⋊C8C2312yes—
52C2×C4⋊C8C2312yes—
53C2×C4⋊C8C2312yes—
54C2×C22⋊C8—22no—
55C2×C22⋊C8D412yes—
56C2×#97—22no—
57C2×C4×C8—22no—
58C2×C4×C8C2311yescharacteristic
59C2×C4×C8C2311yescharacteristic
60#391—22no—
61#389—22no—
62#389—22no—
63#384C2312yes—
64#383—21no—
65#382D411yescharacteristic
66#382D411yescharacteristic
67#381—21no—
68#380—22no—
69#380—22no—
70#380D412yes—
71C4×C4⋊C4C2312yes—
72C4×C22⋊C4—21no—
73#376—21no—
74#340—22no—
75#340—24no—
76#340—22no—
77#340—22no—
78#340—22no—
79#340—22no—
80#340—22no—
81#340—22no—
82#336—22no—
83#336—22no—
84#336—22no—
85#335—24no—
86#333—24no—
87#333—24no—
88C22×Q8—22no—
89C23×C4C2×D411yescharacteristic
90C23×C4C2×D411yescharacteristic
91C23×C4#4111yescharacteristic
92C2×M16—24no—
93C2×M16—82no—
94C22×C8—22no—
95C22×C8—22no—
96C22×C8—22no—
97C22×C8C2×D412yes—
98C22×C8C2×D412yes—
99C4⋊Q8—22no—
100C4⋊Q8—22no—
101C4⋊Q8—22no—
102C4⋊Q8—22no—
103C22⋊Q8—84no—
104C2×C4⋊C4—22no—
105C2×C4⋊C4—22no—
106C2×C4⋊C4—22no—
107C2×C4⋊C4—22no—
108C2×C4⋊C4—22no—
109C2×C4⋊C4—42no—
110C2×C4⋊C4C2×D412yes—
111C2×C4⋊C4C2×D412yes—
112C2×C22⋊C4—22no—
113C2×C22⋊C4—42no—
114C2×C42—22no—
115C2×C42—22no—
116C2×C42—22no—
117C2×C42C2×D411yescharacteristic
118C2×C42C2×D411yescharacteristic
119C2×C42#4111yescharacteristic
120C2×C42C2411yescharacteristic
121#107—44no—
122#107—82no—
123#106—82no—
124C4⋊C8—24no—
125C4⋊C8—24no—
126C4⋊C8—22no—
127C4⋊C8—22no—
128C4⋊C8—22no—
129C4⋊C8—22no—
130C4⋊C8—22no—
131C4⋊C8#4114yes—
132C22⋊C8—24no—
133C22⋊C8—82no—
134#97—82no—
135C4×C8—22no—
136C4×C8—82no—
137C4×C8#4114yes—
138#95—22no—
139#95—21no—
140#95—21no—
141#95—21no—
142#95—21no—
143#95—22no—
144#95—42no—
145#95—42no—
146C24#12111yescharacteristic
147C2×Q8—42no—
148C2×Q8—42no—
149C2×Q8—82no—
150C2×Q8—82no—
151C22×C4—21no—
152C22×C4—21no—
153C22×C4—22no—
154C22×C4—21no—
155C22×C4—22no—
156C22×C4—22no—
157C22×C4—22no—
158C22×C4—21no—
159C22×C4—22no—
160C22×C4—22no—
161C22×C4—22no—
162C22×C4—42no—
163C22×C4—42no—
164C22×C4—42no—
165C22×C4#12111yescharacteristic
166C22×C4#12111yescharacteristic
167C22×C4#12111yescharacteristic
168C22×C4C22×D411yescharacteristic
169C22×C4C22×D411yescharacteristic
170C22×C4C2×#4111yescharacteristic
171M16—84no—
172C2×C8—22no—
173C2×C8—22no—
174C2×C8—22no—
175C2×C8—24no—
176C2×C8—22no—
177C2×C8—22no—
178C2×C8—42no—
179C2×C8—42no—
180C2×C8—42no—
181C2×C8—42no—
182C2×C8—42no—
183C2×C8—42no—
184C2×C8—42no—
185C2×C8—42no—
186C2×C8#12314yes—
187C4⋊C4—22no—
188C4⋊C4—24no—
189C4⋊C4—22no—
190C4⋊C4—42no—
191C4⋊C4—42no—
192C4⋊C4—42no—
193C4⋊C4—42no—
194C4⋊C4—42no—
195C4⋊C4—42no—
196C4⋊C4—42no—
197C4⋊C4—42no—
198C4⋊C4—82no—
199C4⋊C4—84no—
200C22⋊C4—82no—
201C22⋊C4—84no—
202C42—21no—
203C42—21no—
204C42—22no—
205C42—22no—
206C42—42no—
207C42—42no—
208C42C2×#4112yes—
209C42C2×#4112yes—
210C23—21no—
211C23—22no—
212C23—21no—
213C23—21no—
214C23—22no—
215C23C2×#12111yescharacteristic
216Q8—162no—
217Q8—162no—
218C2×C4—22no—
219C2×C4—21no—
220C2×C4—21no—
221C2×C4—21no—
222C2×C4—22no—
223C2×C4—21no—
224C2×C4—21no—
225C2×C4—22no—
226C2×C4—42no—
227C2×C4—41no—
228C2×C4—41no—
229C2×C4—42no—
230C2×C4—42no—
231C2×C4—42no—
232C2×C4—42no—
233C2×C4—42no—
234C2×C4—42no—
235C2×C4—42no—
236C2×C4—42no—
237C2×C4—82no—
238C2×C4—82no—
239C2×C4—82no—
240C2×C4—82no—
241C2×C4—82no—
242C2×C4—82no—
243C2×C4—82no—
244C2×C4C2×#12311yescharacteristic
245C2×C4#52511yescharacteristic
246C2×C4#52912yes—
247C2×C4#53212yes—
248C2×C4#57611yescharacteristic
249C2×C4#57712yes—
250C2×C4#57812yes—
251C2×C4#57812yes—
252C2×C4#57811yescharacteristic
253C8—24no—
254C8—42no—
255C8—44no—
256C8—82no—
257C8—82no—
258C22—42no—
259C22—41no—
260C22—42no—
261C22—41no—
262C22—41no—
263C22#285411yescharacteristic
264C22#304612yes—
265C22#305911yescharacteristic
266C22#306111yescharacteristic
267C22#306812yes—
268C4—22no—
269C4—41no—
270C4—41no—
271C4—41no—
272C4—82no—
273C4—82no—
274C4—162no—
275C4—162no—
276C4#309712yes—
277C4#312311yescharacteristic
278C4#312711yescharacteristic
279C4#317512yes—
280C4#317712yes—
281C2—81no—
282C2#1954712yes—
283C2#1955112yes—
284C2#3311711yescharacteristic
285C2#3312511yescharacteristic
286C2#3313911yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutator, Frattini, cosocleC2×C42C24
prime radicalC24#121
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
4. #9730C2semidirect
5. #9616C2semidirect
6. #9576C2semidirect
7. #9575C2semidirect

Maximal subgroups

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

subgroupindex
#299422
#298782
#205482
#97302
#96162
#95762
#95752
#91352
#85992