Finite Group Databasebeta

← #103127

Subgroups — #103127

Order 512.

586 conjugacy classes of subgroups; only the 134 normal subgroups are shown.

normal subgroups only

This group has 586 conjugacy classes of subgroups — too many to draw. Only its 134 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#33312C211yesmaximal, characteristic
2#28689C211yesmaximal, characteristic
3#28440C211yesmaximal, characteristic
4#9898C211yesmaximal, characteristic
5#9870C211yesmaximal, characteristic
6#9694C211yesmaximal, characteristic
7#7015C211yesmaximal, characteristic
8C2×#500—21no—
9C2×#500—22no—
10C2×#494—21no—
11C2×#462—22no—
12#2868C2211yescharacteristic
13#2868C2211yescharacteristic
14C4×C4⋊Q8C2211yescharacteristic
15#2083—22no—
16#2016—22no—
17#1990—21no—
18#1956—21no—
19#1947—22no—
20#1845—22no—
21#1845—22no—
22#1842C2211yescharacteristic
23C4×C4⋊C8C2211yescharacteristic
24C4×C4⋊C8C2211yescharacteristic
25C42×C8C2211yescharacteristic
26C2×#332—22no—
27#1296—22no—
28#1279—22no—
29C22×Q16—24no—
30C2×C4⋊Q8—21no—
31C2×C4⋊Q8—21no—
32C2×C4⋊Q8—21no—
33C2×C4⋊Q8—21no—
34C2×C4⋊Q8—21no—
35C2×C4⋊Q8D411yescharacteristic
36C2×C4⋊Q8D412yes—
37C2×C4×Q8—21no—
38#500—48no—
39#500—42no—
40#500—42no—
41#494—42no—
42#494—42no—
43#462—44no—
44#462—44no—
45C2×#107—24no—
46C2×#107—28no—
47C2×C4⋊C8—21no—
48C2×C4⋊C8—21no—
49C2×C4⋊C8—21no—
50C2×C4⋊C8D412yes—
51C2×C4⋊C8D412yes—
52C2×#103—24no—
53C2×C4×C8—21no—
54C2×C4×C8—21no—
55C2×C4×C8—21no—
56C2×C4×C8—21no—
57C2×C4×C8D412yes—
58#391—21no—
59#389—21no—
60#389—21no—
61#384—21no—
62#384—21no—
63#384—21no—
64#384—21no—
65#384—21no—
66#384—21no—
67#384—21no—
68#384D411yescharacteristic
69#384D412yes—
70C4×C4⋊C4—21no—
71C4×C4⋊C4—21no—
72C43C2311yescharacteristic
73#340—24no—
74#336—21no—
75#336—21no—
76#332—44no—
77C22×Q8—21no—
78C22×Q8—22no—
79C22×Q8—41no—
80C2×Q16—48no—
81C2×Q16—48no—
82C2×Q16—84no—
83C22×C8—22no—
84C22×C8—22no—
85C22×C8—22no—
86C22×C8—41no—
87C4⋊Q8—21no—
88C4⋊Q8—21no—
89C4⋊Q8—21no—
90C4⋊Q8—44no—
91C4⋊Q8—44no—
92C4⋊Q8—42no—
93C4⋊Q8—41no—
94C4⋊Q8—42no—
95C4⋊Q8—41no—
96C4⋊Q8—41no—
97C4⋊Q8—41no—
98C4⋊Q8—41no—
99C4⋊Q8—42no—
100C4⋊Q8—41no—
101C4⋊Q8—41no—
102C4⋊Q8—42no—
103C4⋊Q8—41no—
104C4⋊Q8D812yes—
105C4×Q8—42no—
106C4×Q8—42no—
107C2×C4⋊C4—21no—
108C2×C4⋊C4—21no—
109C2×C4⋊C4—21no—
110C2×C4⋊C4—22no—
111C2×C4⋊C4—22no—
112C2×C4⋊C4—22no—
113C2×C4⋊C4—21no—
114C2×C4⋊C4—22no—
115C2×C4⋊C4—21no—
116C2×C4⋊C4—22no—
117C2×C4⋊C4—41no—
118C2×C4⋊C4—41no—
119C2×C4⋊C4—41no—
120C2×C4⋊C4—41no—
121C2×C4⋊C4—41no—
122C2×C4⋊C4—41no—
123C2×C4⋊C4—41no—
124C2×C4⋊C4—41no—
125C2×C42—41no—
126C2×C42C2×D411yescharacteristic
127C2×C42C2×D411yescharacteristic
128C2×C42C2×D411yescharacteristic
129C2×C42C2×D411yescharacteristic
130C2×C42C2×D411yescharacteristic
131C2×C42C2×D411yescharacteristic
132C2×C42#4111yescharacteristic
133#107—48no—
134#107—416no—
135C4⋊C8—22no—
136C4⋊C8—22no—
137C4⋊C8—42no—
138C4⋊C8—42no—
139C4⋊C8—42no—
140C4⋊C8D814yes—
141C4⋊C8Q1614yescomplemented
142#103—48no—
143C4×C8—24no—
144C4×C8—24no—
145C4×C8—22no—
146C4×C8—41no—
147C4×C8—41no—
148C4×C8—41no—
149C4×C8—41no—
150C4×C8Q1614yescomplemented
151#95—41no—
152#95—41no—
153C2×Q8—22no—
154C2×Q8—41no—
155C2×Q8—42no—
156C2×Q8—44no—
157C2×Q8—42no—
158C2×Q8—42no—
159C2×Q8—42no—
160C2×Q8—44no—
161C2×Q8—81no—
162C2×Q8—82no—
163C2×Q8—81no—
164C2×Q8—82no—
165C22×C4—41no—
166C22×C4—41no—
167C22×C4—42no—
168C22×C4—42no—
169C22×C4#12011yescharacteristic
170C22×C4#12011yescharacteristic
171C22×C4#12011yescharacteristic
172C22×C4#12111yescharacteristic
173C22×C4#12111yescharacteristic
174C22×C4#12111yescharacteristic
175C22×C4#12711yescharacteristic
176Q16—816no—
177C2×C8—24no—
178C2×C8—24no—
179C2×C8—24no—
180C2×C8—24no—
181C2×C8—24no—
182C2×C8—42no—
183C2×C8—42no—
184C2×C8—42no—
185C2×C8—42no—
186C2×C8—81no—
187C2×C8—81no—
188C2×C8—81no—
189C4⋊C4—22no—
190C4⋊C4—22no—
191C4⋊C4—44no—
192C4⋊C4—44no—
193C4⋊C4—41no—
194C4⋊C4—41no—
195C4⋊C4—42no—
196C4⋊C4—42no—
197C4⋊C4—42no—
198C4⋊C4—42no—
199C4⋊C4—44no—
200C4⋊C4—42no—
201C4⋊C4—82no—
202C4⋊C4—82no—
203C4⋊C4—81no—
204C4⋊C4—81no—
205C4⋊C4—82no—
206C4⋊C4—82no—
207C4⋊C4—81no—
208C4⋊C4—81no—
209C4⋊C4—81no—
210C4⋊C4—81no—
211C4⋊C4—81no—
212C4⋊C4—81no—
213C4⋊C4—81no—
214C4⋊C4—81no—
215C42—22no—
216C42—22no—
217C42—22no—
218C42—81no—
219C42—81no—
220C42C2×D811yescharacteristic
221C42C2×D812yes—
222C42C2×Q1612yes—
223C42C2×Q1612yes—
224C42#13511yescharacteristic
225C42AGL(1,ℤ/8)11yescharacteristic
226C42AGL(1,ℤ/8)11yescharacteristic
227C42AGL(1,ℤ/8)12yes—
228C42#13712yes—
229C42#13712yes—
230C23#39211yescharacteristic
231Q8—44no—
232Q8—82no—
233Q8—84no—
234Q8—84no—
235Q8—162no—
236Q8—162no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—42no—
244C2×C4—81no—
245C2×C4—81no—
246C2×C4—82no—
247C2×C4—82no—
248C2×C4—82no—
249C2×C4—82no—
250C2×C4—81no—
251C2×C4—81no—
252C2×C4—82no—
253C2×C4—82no—
254C2×C4—81no—
255C2×C4#44711yescharacteristic
256C2×C4#44712yes—
257C2×C4#44711yescharacteristic
258C2×C4#44911yescharacteristic
259C2×C4#44911yescharacteristic
260C2×C4#45112yes—
261C2×C4#45112yes—
262C2×C4#45312yes—
263C2×C4#45612yes—
264C2×C4#45912yes—
265C2×C4#46212yes—
266C2×C4#46212yes—
267C2×C4#46611yescharacteristic
268C2×C4#46911yescharacteristic
269C2×C4#47312yes—
270C2×C4#49311yescharacteristic
271C2×C4#49411yescharacteristic
272C2×C4#49411yescharacteristic
273C2×C4#49611yescharacteristic
274C2×C4#49711yescharacteristic
275C2×C4#49711yescharacteristic
276C8—28no—
277C8—44no—
278C8—44no—
279C8—161no—
280C22#199911yescharacteristic
281C22#200211yescharacteristic
282C22#200511yescharacteristic
283C22#201111yescharacteristic
284C22#201611yescharacteristic
285C22#201611yescharacteristic
286C22#202011yescharacteristic
287C4—22no—
288C4—22no—
289C4—22no—
290C4—82no—
291C4—162no—
292C4—162no—
293C4—161no—
294C4#161912yes—
295C4#162112yes—
296C4#164812yes—
297C4#164912yes—
298C4#167412yes—
299C4#169412yes—
300C4#171311yescharacteristic
301C4#172211yescharacteristic
302C4#172311yescharacteristic
303C4#172311yescharacteristic
304C2#1004611yescharacteristic
305C2#1006711yescharacteristic
306C2#1015511yescharacteristic
307C2#1017411yescharacteristic
308C2#1026511yescharacteristic
309C2#1029311yescharacteristic
310C2#1030911yescharacteristic
subgroupG/H
center, socle, prime radicalC23#392
commutator, Frattini, cosocleC43C23
LCSG ▷ C43 ▷ C23 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ G
LECSG ▷ C43 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
141. C4⋊C8Q16semidirect
150. C4×C8Q16semidirect

Maximal subgroups

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

subgroupindex
#333122
#286892
#284402
#98982
#98702
#96942
#70152