Finite Group Databasebeta

← #501423

Subgroups — #501423

Order 512.

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

normal subgroups only

This group has 539 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
1C2×#2237C211yesmaximal, characteristic
2C2×#2159C211yesmaximal, characteristic
3#27931C211yesmaximal, characteristic
4#22614C211yesmaximal, characteristic
5#13507C212yesmaximal, complemented
6#13475C212yesmaximal, complemented
7#13178C212yesmaximal, complemented
8#13132C212yesmaximal, complemented
9#13062C211yesmaximal, characteristic
10#12913C211yesmaximal, characteristic
11#12867C211yesmaximal, characteristic
12C22×M32C2211yescharacteristic
13C2×#483—21no—
14C2×#467C2211yescharacteristic
15C2×#427C2211yescharacteristic
16C22×#107C2211yescharacteristic
17#2503—21no—
18#2347—21no—
19#2237—24no—
20#2159—24no—
21C2×#367C2212yes—
22C2×#367C2211yescharacteristic
23C2×#366C2211yescharacteristic
24C2×#366C2212yes—
25C2×C4⋊C16C2212yes—
26C2×#358C2212yes—
27C2×#358C2211yescharacteristic
28C2×#358C2211yescharacteristic
29C2×C22⋊C16C2211yescharacteristic
30#2097—22no—
31#1969C2212yes—
32#1942C2212yes—
33#1919—22no—
34#1846C2212yes—
35#1825—21no—
36#1808C2211yescharacteristic
37C4×#107C2212yes—
38C8×C22⋊C4C2211yescharacteristic
39#1380C2212yes—
40#1380C2212yes—
41#1380C2212yes—
42#1374C2212yes—
43C22×Q16C2311yescharacteristic
44C23×C8C2311yescharacteristic
45C2×C22⋊Q8—21no—
46C2×#117—21no—
47C22×C4⋊C4—21no—
48C2×M32—22no—
49C2×M32—24no—
50C22×C16C2311yescharacteristic
51C22×C16C2311yescharacteristic
52C22×C16C2312yes—
53#483—44no—
54#467—22no—
55#467—22no—
56#427—22no—
57#427D414yes—
58C2×#107—22no—
59C2×#107—22no—
60C2×#107—21no—
61C2×#107C2312yes—
62C2×#107C2311yescharacteristic
63C2×#107C2312yes—
64C2×#107C2311yescharacteristic
65C2×#107C2311yescharacteristic
66C2×#106—21no—
67C2×#106—22no—
68C2×#103—22no—
69C2×#103—21no—
70C2×C22⋊C8D411yescharacteristic
71C2×C4×C8C2312yes—
72#400—41no—
73#400—41no—
74#391—22no—
75#389—22no—
76#388—42no—
77#383—41no—
78#382—22no—
79#381—42no—
80#380—21no—
81C4×C4⋊C4—22no—
82C4×C4⋊C4—41no—
83C4×C22⋊C4D411yescharacteristic
84#367—24no—
85#367—22no—
86#366—24no—
87#366—22no—
88C4⋊C16—24no—
89#358—24no—
90#358—22no—
91#358—22no—
92C22⋊C16—22no—
93#340—21no—
94#340—22no—
95#340—22no—
96#336D412yes—
97C22×Q8—21no—
98C23×C4—41no—
99C23×C4C2×D411yescharacteristic
100C2×Q16—21no—
101C2×Q16—22no—
102C2×Q16—21no—
103C2×Q16—22no—
104C22×C8—22no—
105C22×C8—22no—
106C22×C8C2×D412yes—
107C22×C8C2×D411yescharacteristic
108C22×C8C2×D412yes—
109C22×C8#4112yes—
110C22×C8C2411yescharacteristic
111C22⋊Q8—42no—
112C22⋊Q8—42no—
113#117—24no—
114#117—42no—
115C2×C4⋊C4—21no—
116C2×C4⋊C4—21no—
117C2×C4⋊C4—22no—
118C2×C4⋊C4—21no—
119C2×C4⋊C4—22no—
120C2×C4⋊C4—42no—
121C2×C4⋊C4—42no—
122C2×C4⋊C4—41no—
123C2×C4⋊C4—41no—
124C2×C4⋊C4—42no—
125C2×C22⋊C4—41no—
126C2×C22⋊C4—41no—
127C2×C22⋊C4—41no—
128C2×C22⋊C4D812yes—
129C2×C42—41no—
130C2×C42—42no—
131C2×C42C2×D412yes—
132M32—28no—
133C2×C16—22no—
134C2×C16—22no—
135C2×C16—21no—
136C2×C16—21no—
137C2×C16—22no—
138C2×C16—22no—
139C2×C16#4114yes—
140#107—21no—
141#107—21no—
142#107—22no—
143#107—21no—
144#107—22no—
145#107—22no—
146#107—21no—
147#107—22no—
148#107—42no—
149#107C2×D412yes—
150#107C2×D412yes—
151#106—24no—
152#106—44no—
153#103—44no—
154#103—42no—
155C22⋊C8—22no—
156C4×C8—22no—
157C4×C8—22no—
158#95—42no—
159#95—41no—
160#95—42no—
161#95—42no—
162#95—42no—
163#95—42no—
164#95—42no—
165#95D811yescharacteristic
166#95D811yescharacteristic
167C24C2×D811yescharacteristic
168C2×Q8—41no—
169C2×Q8—42no—
170C2×Q8—41no—
171C2×Q8—42no—
172C22×C4—22no—
173C22×C4—22no—
174C22×C4—41no—
175C22×C4—41no—
176C22×C4—41no—
177C22×C4—41no—
178C22×C4—42no—
179C22×C4—42no—
180C22×C4—84no—
181C22×C4—82no—
182C22×C4#12112yes—
183C22×C4C2×D812yes—
184C22×C4C2×D812yes—
185C22×C4C2×D811yescharacteristic
186C22×C4AGL(1,ℤ/8)12yes—
187C22×C4C22×D411yescharacteristic
188Q16—41no—
189Q16—41no—
190Q16—42no—
191C2×C8—22no—
192C2×C8—22no—
193C2×C8—22no—
194C2×C8—24no—
195C2×C8—22no—
196C2×C8—22no—
197C2×C8—22no—
198C2×C8#12112yes—
199C2×C8C22×D411yescharacteristic
200C2×C8C2×#4112yes—
201C2×C8C2×#4111yescharacteristic
202C2×C82−1+412yes—
203C4⋊C4—22no—
204C4⋊C4—22no—
205C4⋊C4—41no—
206C4⋊C4—41no—
207C4⋊C4—41no—
208C4⋊C4—42no—
209C4⋊C4—42no—
210C4⋊C4—42no—
211C4⋊C4—41no—
212C4⋊C4—42no—
213C4⋊C4—84no—
214C4⋊C4—82no—
215C4⋊C4—82no—
216C22⋊C4—42no—
217C22⋊C4—44no—
218C22⋊C4—82no—
219C22⋊C4—161no—
220C42—22no—
221C42—22no—
222C42—44no—
223C42—162no—
224C16—24no—
225C16—41no—
226C16—41no—
227C23—22no—
228C23—22no—
229C23—21no—
230C23—21no—
231C23#45912yes—
232C23C22×D811yescharacteristic
233Q8—82no—
234Q8—81no—
235Q8—81no—
236C2×C4—22no—
237C2×C4—24no—
238C2×C4—22no—
239C2×C4—22no—
240C2×C4—22no—
241C2×C4—22no—
242C2×C4—22no—
243C2×C4—22no—
244C2×C4—22no—
245C2×C4—22no—
246C2×C4—42no—
247C2×C4—42no—
248C2×C4—82no—
249C2×C4—81no—
250C2×C4—81no—
251C2×C4—82no—
252C2×C4—82no—
253C2×C4—82no—
254C2×C4—84no—
255C2×C4—82no—
256C2×C4—81no—
257C2×C4—81no—
258C2×C4—81no—
259C2×C4—81no—
260C2×C4—82no—
261C2×C4—164no—
262C2×C4#45912yes—
263C2×C4C2×#12111yescharacteristic
264C2×C4#54712yes—
265C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
266C2×C4#57812yes—
267C8—22no—
268C8—42no—
269C8#52511yescharacteristic
270C8#53611yescharacteristic
271C8#55212yes—
272C22—22no—
273C22—22no—
274C22—22no—
275C22—42no—
276C22—42no—
277C22C2×#45911yescharacteristic
278C22#333412yes—
279C22#341412yes—
280C22#341611yescharacteristic
281C22#341711yescharacteristic
282C4—22no—
283C4—42no—
284C4—42no—
285C4—82no—
286C4—82no—
287C4—161no—
288C4—162no—
289C4—161no—
290C4—162no—
291C4#241311yescharacteristic
292C4#304411yescharacteristic
293C4#334312yes—
294C2—42no—
295C2#2457611yescharacteristic
296C2#3358411yescharacteristic
297C2#3359011yescharacteristic
298C2#3391612yes—
299C2#3391812yes—
subgroupG/H
center, socleC23C22×D8
commutator, Frattini, cosocleC22×C8C24
prime radicalC24C2×D8
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ C8×C22⋊C4 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
5. #13507C2semidirect
6. #13475C2semidirect
7. #13178C2semidirect
8. #13132C2semidirect

Maximal subgroups

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

subgroupindex
C2×#22372
C2×#21592
#279312
#226142
#135072
#134752
#131782
#131322
#130622
#129132
#128672