Finite Group Databasebeta

← #103131

Subgroups — #103131

Order 512.

524 conjugacy classes of subgroups; only the 130 normal subgroups are shown.

normal subgroups only

This group has 524 conjugacy classes of subgroups — too many to draw. Only its 130 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#33310C211yesmaximal, characteristic
2#29141C211yesmaximal, characteristic
3#28689C211yesmaximal, characteristic
4#9870C211yesmaximal, characteristic
5#9694C211yesmaximal, characteristic
6#9691C211yesmaximal, characteristic
7#7015C211yesmaximal, characteristic
8C2×#500—21no—
9C2×#498—22no—
10#2868C2211yescharacteristic
11#2868C2211yescharacteristic
12C4×C4⋊Q8C2211yescharacteristic
13#2083—22no—
14#2024—22no—
15#1990—21no—
16#1956—21no—
17#1947—22no—
18#1844—22no—
19#1844—22no—
20#1842C2211yescharacteristic
21#1834—21no—
22#1803—22no—
23C4×C4⋊C8C2211yescharacteristic
24C4×C4⋊C8C2211yescharacteristic
25C42×C8C2211yescharacteristic
26C2×#333—22no—
27C2×#332—22no—
28#1279—22no—
29C2×C4⋊Q8—21no—
30C2×C4⋊Q8—21no—
31C2×C4⋊Q8—21no—
32C2×C4⋊Q8—21no—
33C2×C4⋊Q8—21no—
34C2×C4⋊Q8D411yescharacteristic
35C2×C4⋊Q8D412yes—
36C2×C4×Q8—21no—
37#500—42no—
38#500—42no—
39#498—48no—
40C2×#107—24no—
41C2×#106—28no—
42C2×C4⋊C8—21no—
43C2×C4⋊C8—21no—
44C2×C4⋊C8—21no—
45C2×C4⋊C8D412yes—
46C2×C4⋊C8D412yes—
47C2×#103—24no—
48C2×C4×C8—21no—
49C2×C4×C8—21no—
50C2×C4×C8—21no—
51C2×C4×C8—21no—
52C2×C4×C8D412yes—
53#391—21no—
54#389—21no—
55#389—21no—
56#384—21no—
57#384—21no—
58#384—21no—
59#384—21no—
60#384—21no—
61#384—21no—
62#384—21no—
63#384D411yescharacteristic
64#384D412yes—
65C4×C4⋊C4—21no—
66C4×C4⋊C4—21no—
67C43C2311yescharacteristic
68#340—24no—
69#340—24no—
70#336—21no—
71#336—21no—
72#333—44no—
73#332—44no—
74C22×Q8—21no—
75C22×Q8—22no—
76C22×Q8—41no—
77C22×C8—22no—
78C22×C8—22no—
79C22×C8—22no—
80C22×C8—41no—
81C4⋊Q8—21no—
82C4⋊Q8—21no—
83C4⋊Q8—21no—
84C4⋊Q8—44no—
85C4⋊Q8—44no—
86C4⋊Q8—41no—
87C4⋊Q8—42no—
88C4⋊Q8—41no—
89C4⋊Q8—42no—
90C4⋊Q8—42no—
91C4⋊Q8—41no—
92C4⋊Q8—41no—
93C4⋊Q8—42no—
94C4⋊Q8—82no—
95C4⋊Q8SD1612yes—
96C4×Q8—42no—
97C4×Q8—42no—
98C2×C4⋊C4—21no—
99C2×C4⋊C4—21no—
100C2×C4⋊C4—21no—
101C2×C4⋊C4—22no—
102C2×C4⋊C4—22no—
103C2×C4⋊C4—22no—
104C2×C4⋊C4—22no—
105C2×C4⋊C4—21no—
106C2×C4⋊C4—22no—
107C2×C4⋊C4—21no—
108C2×C4⋊C4—41no—
109C2×C4⋊C4—41no—
110C2×C4⋊C4—41no—
111C2×C4⋊C4—41no—
112C2×C4⋊C4—41no—
113C2×C4⋊C4—41no—
114C2×C4⋊C4—41no—
115C2×C4⋊C4—41no—
116C2×C42—41no—
117C2×C42C2×D411yescharacteristic
118C2×C42C2×D411yescharacteristic
119C2×C42C2×D411yescharacteristic
120C2×C42C2×D411yescharacteristic
121C2×C42C2×D411yescharacteristic
122C2×C42C2×D411yescharacteristic
123C2×C42#4111yescharacteristic
124#107—48no—
125#106—416no—
126C4⋊C8—22no—
127C4⋊C8—22no—
128C4⋊C8—22no—
129C4⋊C8—42no—
130C4⋊C8—42no—
131C4⋊C8—42no—
132C4⋊C8SD1614yes—
133#103—84no—
134C4×C8—24no—
135C4×C8—24no—
136C4×C8—22no—
137C4×C8—41no—
138C4×C8—41no—
139C4×C8—41no—
140C4×C8—41no—
141C4×C8Q1614yes—
142#95—41no—
143#95—41no—
144C2×Q8—22no—
145C2×Q8—41no—
146C2×Q8—42no—
147C2×Q8—42no—
148C2×Q8—44no—
149C2×Q8—82no—
150C2×Q8—82no—
151C2×Q8—81no—
152C2×Q8—81no—
153C2×Q8—82no—
154C2×Q8—82no—
155C22×C4—41no—
156C22×C4—41no—
157C22×C4—42no—
158C22×C4—42no—
159C22×C4#12011yescharacteristic
160C22×C4#12011yescharacteristic
161C22×C4#12011yescharacteristic
162C22×C4#12111yescharacteristic
163C22×C4#12111yescharacteristic
164C22×C4#12111yescharacteristic
165C22×C4#12711yescharacteristic
166C2×C8—24no—
167C2×C8—24no—
168C2×C8—24no—
169C2×C8—24no—
170C2×C8—42no—
171C2×C8—42no—
172C2×C8—42no—
173C2×C8—42no—
174C2×C8—42no—
175C2×C8—81no—
176C2×C8—81no—
177C2×C8—81no—
178C4⋊C4—22no—
179C4⋊C4—22no—
180C4⋊C4—44no—
181C4⋊C4—44no—
182C4⋊C4—41no—
183C4⋊C4—41no—
184C4⋊C4—42no—
185C4⋊C4—42no—
186C4⋊C4—42no—
187C4⋊C4—44no—
188C4⋊C4—42no—
189C4⋊C4—44no—
190C4⋊C4—42no—
191C4⋊C4—82no—
192C4⋊C4—82no—
193C4⋊C4—82no—
194C4⋊C4—82no—
195C4⋊C4—81no—
196C4⋊C4—81no—
197C4⋊C4—82no—
198C4⋊C4—81no—
199C4⋊C4—81no—
200C4⋊C4—81no—
201C4⋊C4—81no—
202C42—22no—
203C42—22no—
204C42—22no—
205C42—81no—
206C42—81no—
207C42C2×SD1611yescharacteristic
208C42C2×SD1612yes—
209C42C2×Q1612yes—
210C42#13511yescharacteristic
211C42#13512yes—
212C42AGL(1,ℤ/8)11yescharacteristic
213C42AGL(1,ℤ/8)12yes—
214C42#13711yescharacteristic
215C42#13712yes—
216C42#13712yes—
217C23#39211yescharacteristic
218Q8—44no—
219Q8—82no—
220Q8—164no—
221Q8—162no—
222Q8—162no—
223C2×C4—21no—
224C2×C4—21no—
225C2×C4—21no—
226C2×C4—21no—
227C2×C4—21no—
228C2×C4—21no—
229C2×C4—42no—
230C2×C4—81no—
231C2×C4—81no—
232C2×C4—82no—
233C2×C4—82no—
234C2×C4—82no—
235C2×C4—81no—
236C2×C4—81no—
237C2×C4—82no—
238C2×C4—82no—
239C2×C4—81no—
240C2×C4—82no—
241C2×C4#44811yescharacteristic
242C2×C4#44812yes—
243C2×C4#44912yes—
244C2×C4#44911yescharacteristic
245C2×C4#45011yescharacteristic
246C2×C4#45112yes—
247C2×C4#45211yescharacteristic
248C2×C4#45312yes—
249C2×C4#45612yes—
250C2×C4#46112yes—
251C2×C4#46212yes—
252C2×C4#46312yes—
253C2×C4#46511yescharacteristic
254C2×C4#46811yescharacteristic
255C2×C4#47112yes—
256C2×C4#49211yescharacteristic
257C2×C4#49411yescharacteristic
258C2×C4#49511yescharacteristic
259C2×C4#49611yescharacteristic
260C2×C4#49711yescharacteristic
261C2×C4#49711yescharacteristic
262C8—28no—
263C8—44no—
264C8—82no—
265C8—161no—
266C22#200011yescharacteristic
267C22#200211yescharacteristic
268C22#200511yescharacteristic
269C22#201211yescharacteristic
270C22#201311yescharacteristic
271C22#201611yescharacteristic
272C22#201911yescharacteristic
273C4—22no—
274C4—22no—
275C4—22no—
276C4—82no—
277C4—162no—
278C4—162no—
279C4—161no—
280C4#162212yes—
281C4#162412yes—
282C4#165112yes—
283C4#165312yes—
284C4#167212yes—
285C4#169212yes—
286C4#170711yescharacteristic
287C4#171411yescharacteristic
288C4#171811yescharacteristic
289C4#172311yescharacteristic
290C2#1004811yescharacteristic
291C2#1007511yescharacteristic
292C2#1015311yescharacteristic
293C2#1016611yescharacteristic
294C2#1027311yescharacteristic
295C2#1029311yescharacteristic
296C2#1031411yescharacteristic
subgroupG/H
center, socle, prime radicalC23#392
commutator, Frattini, cosocleC43C23
LCSG ▷ C43 ▷ C23 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ G
LECSG ▷ C43 ▷ C23 ▷ C1

Maximal subgroups

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

subgroupindex
#333102
#291412
#286892
#98702
#96942
#96912
#70152