Finite Group Databasebeta

← #138343

Subgroups — #138343

Order 512.

686 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 686 conjugacy classes of subgroups — too many to draw. Only its 110 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#32931C211yesmaximal, characteristic
2C2×#1807C211yesmaximal, characteristic
3#12679C211yesmaximal, characteristic
4C2×#1348C212yesmaximal, complemented
5#7258C212yesmaximal, complemented
6C24×C8C2211yescharacteristic
7C2×#467C412yescomplemented
8C2×#467C2211yescharacteristic
9C22×#106—21no—
10#2848—21no—
11C2×#380—21no—
12C2×C22⋊C16—22no—
13C2×C22⋊C16C2212yes—
14#1892—22no—
15#1892—21no—
16#1808C412yescomplemented
17#1808C2211yescharacteristic
18#1807—24no—
19C2×#340C2212yes—
20#1348—24no—
21C24×C4D411yescharacteristic
22C22×Q16C2×C411yescharacteristic
23C23×C8—22no—
24C23×C8—22no—
25C23×C8—44no—
26C23×C8C2×C412yes—
27C23×C8D411yescharacteristic
28C23×C8D412yes—
29C23×C8C2311yescharacteristic
30C2×C22⋊Q8—22no—
31C2×C22⋊Q8—21no—
32C22×C4⋊C4—22no—
33C22×C22⋊C4—41no—
34C22×C16—22no—
35#467—22no—
36#467—22no—
37#467—44no—
38C2×#107C2×C411yescharacteristic
39C2×#107C2×C412yes—
40C2×#106—24no—
41C2×#106—44no—
42C2×#103—22no—
43C2×#103—21no—
44#385—41no—
45#385—42no—
46#380—24no—
47#380—22no—
48#380—21no—
49#380—42no—
50#379—41no—
51C2×#95—41no—
52C22⋊C16—22no—
53C22⋊C16—22no—
54C22⋊C16—44no—
55#340—22no—
56#340—22no—
57#340—21no—
58#340D414yes—
59C25D811yescharacteristic
60C22×Q8—21no—
61C23×C4—21no—
62C23×C4—21no—
63C23×C4—21no—
64C23×C4—21no—
65C23×C4—42no—
66C23×C4—42no—
67C23×C4—42no—
68C23×C4C22⋊C412yes—
69C23×C4D811yescharacteristic
70C23×C4SD1612yes—
71C23×C4C2×D411yescharacteristic
72C2×Q16—22no—
73C2×Q16—42no—
74C22×C8—21no—
75C22×C8—21no—
76C22×C8—24no—
77C22×C8—24no—
78C22×C8—24no—
79C22×C8—24no—
80C22×C8—24no—
81C22×C8—42no—
82C22×C8—42no—
83C22×C8—42no—
84C22×C8—48no—
85C22×C8—48no—
86C22×C8C22⋊C412yes—
87C22×C8C22×C411yescharacteristic
88C22×C8C2×D411yescharacteristic
89C22×C8C2×D412yes—
90C22×C8#4112yes—
91C22⋊Q8—44no—
92C22⋊Q8—44no—
93C22⋊Q8—41no—
94C22⋊Q8—41no—
95C22⋊Q8—41no—
96C22⋊Q8—41no—
97C2×C4⋊C4—21no—
98C2×C4⋊C4—22no—
99C2×C4⋊C4—24no—
100C2×C4⋊C4—44no—
101C2×C4⋊C4—44no—
102C2×C4⋊C4—42no—
103C2×C4⋊C4—41no—
104C2×C4⋊C4—42no—
105C2×C22⋊C4—44no—
106C2×C22⋊C4—42no—
107C2×C22⋊C4—42no—
108C2×C22⋊C4—41no—
109C2×C22⋊C4—41no—
110C2×C22⋊C4—84no—
111C2×C16—44no—
112C2×C16—42no—
113#107—22no—
114#107—42no—
115#106—48no—
116#103—44no—
117#103—42no—
118#95—44no—
119#95—42no—
120#95—41no—
121C24—24no—
122C24—21no—
123C24—21no—
124C24—22no—
125C24—22no—
126C24—42no—
127C24#10212yes—
128C24C2×D811yescharacteristic
129C2×Q8—41no—
130C2×Q8—42no—
131C2×Q8—41no—
132C2×Q8—42no—
133C22×C4—22no—
134C22×C4—22no—
135C22×C4—21no—
136C22×C4—21no—
137C22×C4—22no—
138C22×C4—22no—
139C22×C4—22no—
140C22×C4—22no—
141C22×C4—22no—
142C22×C4—22no—
143C22×C4—22no—
144C22×C4—22no—
145C22×C4—42no—
146C22×C4—42no—
147C22×C4—42no—
148C22×C4—42no—
149C22×C4—42no—
150C22×C4—42no—
151C22×C4—42no—
152C22×C4—44no—
153C22×C4—44no—
154C22×C4—41no—
155C22×C4—42no—
156C22×C4—41no—
157C22×C4—44no—
158C22×C4—84no—
159C22×C4—84no—
160C22×C4—82no—
161C22×C4—82no—
162C22×C4#10212yes—
163C22×C4C2×C22⋊C411yescharacteristic
164C22×C4#12011yescharacteristic
165C22×C4#12311yescharacteristic
166C22×C4C2×SD1611yescharacteristic
167C22×C4AGL(1,ℤ/8)11yescharacteristic
168C22×C4#13711yescharacteristic
169Q16—82no—
170C2×C8—24no—
171C2×C8—24no—
172C2×C8—22no—
173C2×C8—22no—
174C2×C8—22no—
175C2×C8—22no—
176C2×C8—42no—
177C2×C8—42no—
178C2×C8—48no—
179C2×C8—48no—
180C2×C8#9914yes—
181C2×C8C4×D412yes—
182C2×C8#12112yes—
183C4⋊C4—44no—
184C4⋊C4—44no—
185C4⋊C4—41no—
186C4⋊C4—42no—
187C4⋊C4—41no—
188C4⋊C4—42no—
189C4⋊C4—84no—
190C4⋊C4—84no—
191C4⋊C4—81no—
192C4⋊C4—81no—
193C22⋊C4—88no—
194C22⋊C4—84no—
195C22⋊C4—84no—
196C22⋊C4—81no—
197C22⋊C4—81no—
198C22⋊C4—82no—
199C16—82no—
200C23—22no—
201C23—22no—
202C23—24no—
203C23—24no—
204C23—22no—
205C23—22no—
206C23—22no—
207C23—22no—
208C23—22no—
209C23—22no—
210C23—48no—
211C23—48no—
212C23—42no—
213C23—42no—
214C23—42no—
215C23—42no—
216C23#36014yes—
217C23C2×#10211yescharacteristic
218C23#44711yescharacteristic
219C23#48011yescharacteristic
220C23#50812yes—
221C23#51012yes—
222Q8—81no—
223Q8—81no—
224Q8—82no—
225C2×C4—22no—
226C2×C4—21no—
227C2×C4—21no—
228C2×C4—21no—
229C2×C4—21no—
230C2×C4—22no—
231C2×C4—22no—
232C2×C4—22no—
233C2×C4—22no—
234C2×C4—22no—
235C2×C4—44no—
236C2×C4—44no—
237C2×C4—42no—
238C2×C4—42no—
239C2×C4—42no—
240C2×C4—42no—
241C2×C4—42no—
242C2×C4—42no—
243C2×C4—84no—
244C2×C4—84no—
245C2×C4—81no—
246C2×C4—81no—
247C2×C4—82no—
248C2×C4—82no—
249C2×C4—82no—
250C2×C4—84no—
251C2×C4—82no—
252C2×C4—82no—
253C2×C4—164no—
254C2×C4—164no—
255C2×C4#38611yescharacteristic
256C2×C4C2×#9912yes—
257C2×C4#41711yescharacteristic
258C2×C4#42112yes—
259C2×C4#44811yescharacteristic
260C2×C4#48111yescharacteristic
261C8—22no—
262C8—24no—
263C8—22no—
264C22—24no—
265C22—24no—
266C22—21no—
267C22—21no—
268C22—21no—
269C22—21no—
270C22—24no—
271C22—24no—
272C22—24no—
273C22—48no—
274C22—48no—
275C22—42no—
276C22—42no—
277C22—42no—
278C22—42no—
279C22#151912yes—
280C22#187411yescharacteristic
281C22#213811yescharacteristic
282C22#214111yescharacteristic
283C22C2×#36012yes—
284C22#218912yes—
285C22#223712yes—
286C4—21no—
287C4—21no—
288C4—42no—
289C4—164no—
290C4—162no—
291C4—161no—
292C4—161no—
293C4#151312yes—
294C4#189711yescharacteristic
295C4#189811yescharacteristic
296C2—22no—
297C2—22no—
298C2—44no—
299C2#915511yescharacteristic
300C2#1188512yes—
301C2#1188912yes—
302C2#1299411yescharacteristic
303C2#1300111yescharacteristic
subgroupG/H
center, socleC23C2×#102
commutatorC22×C8C22×C4
Frattini, cosocleC23×C8C23
prime radicalC25D8
LCSG ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ C24×C8 ◁ G
LECSG ▷ C23×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. C2×#1348C2semidirect
5. #7258C2semidirect
7. C2×#467C4semidirect
16. #1808C4semidirect

Maximal subgroups

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

subgroupindex
#329312
C2×#18072
#126792
C2×#13482
#72582