Finite Group Databasebeta

← #101845

Subgroups — #101845

Order 512.

550 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 550 conjugacy classes of subgroups — too many to draw. Only its 102 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#28438C211yesmaximal, characteristic
2#23175C211yesmaximal, characteristic
3#9903C212yesmaximal, complemented
4#7651C211yesmaximal, characteristic
5#7022C212yesmaximal, complemented
6C2×#492C412yes—
7C2×#461—21no—
8C2×#461—21no—
9C2×#435—22no—
10#2868C2211yescharacteristic
11C4×#127C2211yescharacteristic
12#2013—21no—
13#2013—21no—
14#1956C412yes—
15#1955—22no—
16#1870—21no—
17#1869—21no—
18#1831C2212yes—
19#1767C2212yes—
20C4×C4⋊C8C2211yescharacteristic
21#1296—22no—
22#1296—22no—
23#1270—21no—
24#1270—21no—
25C22×SD16—22no—
26C22×SD16—22no—
27C22×M16—21no—
28C23×C8—21no—
29C2×C4⋊Q8—21no—
30C2×C4⋊Q8—21no—
31C2×C4⋊Q8C2×C412yes—
32C2×#127C2×C411yescharacteristic
33C2×C4×D4—22no—
34C2×C4×D4D411yescharacteristic
35C2×C4×D4D411yescharacteristic
36#492—44no—
37#461—42no—
38#461—42no—
39#461—42no—
40#461—42no—
41#435—44no—
42#435—44no—
43C2×C4⋊C8—21no—
44C2×C4⋊C8—22no—
45C2×C4⋊C8D412yes—
46C2×C4⋊C8D412yes—
47C2×#103—22no—
48C2×#103—22no—
49C2×C22⋊C8—22no—
50C2×C22⋊C8—22no—
51C2×C22⋊C8—21no—
52C2×C22⋊C8—21no—
53C2×#97—22no—
54C2×C4×C8C2×C412yes—
55#390—22no—
56#390D411yescharacteristic
57#390D411yescharacteristic
58#384—22no—
59#384—22no—
60#384C2×C411yescharacteristic
61C43C2311yescharacteristic
62#336—22no—
63#336—21no—
64#336—22no—
65#336—22no—
66C22×Q8—22no—
67C22×D4—22no—
68C22×D4C22⋊C411yescharacteristic
69C22×D4C22⋊C411yescharacteristic
70C23×C4—21no—
71C23×C4—21no—
72C2×SD16—44no—
73C2×SD16—44no—
74C2×SD16—44no—
75C2×SD16—44no—
76C2×SD16—82no—
77C2×SD16—82no—
78C2×M16—44no—
79C2×M16—81no—
80C22×C8—22no—
81C22×C8—22no—
82C22×C8—22no—
83C22×C8—44no—
84C22×C8—42no—
85C22×C8—81no—
86C4⋊Q8—42no—
87C4⋊Q8—42no—
88C4⋊Q8—41no—
89C4⋊Q8—41no—
90C4⋊Q8—41no—
91C4⋊Q8—41no—
92C4⋊Q8—82no—
93#127—22no—
94#127—41no—
95C4×D4—41no—
96C4×D4—41no—
97C4×D4—41no—
98C4×D4—41no—
99C4×D4—42no—
100C4×D4—42no—
101C4×D4—42no—
102C4×D4—42no—
103C2×C4⋊C4—22no—
104C2×C4⋊C4—24no—
105C2×C4⋊C4—41no—
106C2×C4⋊C4—42no—
107C2×C4⋊C4—41no—
108C2×C4⋊C4C22⋊C411yescharacteristic
109C2×C4⋊C4C22⋊C411yescharacteristic
110C2×C22⋊C4—22no—
111C2×C22⋊C4—22no—
112C2×C22⋊C4—21no—
113C2×C22⋊C4—21no—
114C2×C42C22×C411yescharacteristic
115C2×C42C2×D411yescharacteristic
116C2×C42C2×D411yescharacteristic
117C2×C42C2×D412yes—
118C2×C42#4112yes—
119C4⋊C8—22no—
120C4⋊C8—22no—
121C4⋊C8—22no—
122C4⋊C8—22no—
123C4⋊C8—22no—
124C4⋊C8—42no—
125C4⋊C8—42no—
126C4⋊C8SD1614yescomplemented
127#103—44no—
128#103—44no—
129C22⋊C8—44no—
130C22⋊C8—44no—
131C22⋊C8—81no—
132C22⋊C8—81no—
133#97—42no—
134#97—42no—
135C4×C8—42no—
136C24—21no—
137C24—21no—
138C2×Q8—42no—
139C2×Q8—42no—
140C2×Q8—42no—
141C2×Q8—42no—
142C2×Q8—42no—
143C2×Q8—42no—
144C2×D4—21no—
145C2×D4—21no—
146C2×D4—21no—
147C2×D4—41no—
148C2×D4—41no—
149C2×D4—41no—
150C2×D4—41no—
151C2×D4—44no—
152C2×D4—42no—
153C2×D4—42no—
154C2×D4—82no—
155C2×D4#10012yes—
156C22×C4—21no—
157C22×C4—21no—
158C22×C4—42no—
159C22×C4—42no—
160C22×C4—42no—
161C22×C4—42no—
162C22×C4—41no—
163C22×C4—41no—
164C22×C4—41no—
165C22×C4—41no—
166C22×C4—42no—
167C22×C4C2×C22⋊C411yescharacteristic
168C22×C4C4×D412yes—
169C22×C4#12011yescharacteristic
170C22×C4#12112yes—
171C22×C4#12311yescharacteristic
172SD16—88no—
173SD16—88no—
174M16—84no—
175C2×C8—24no—
176C2×C8—24no—
177C2×C8—42no—
178C2×C8—42no—
179C2×C8—42no—
180C2×C8—42no—
181C2×C8—42no—
182C2×C8—42no—
183C2×C8—42no—
184C2×C8—84no—
185C2×C8—82no—
186C2×C8—82no—
187C2×C8—82no—
188C4⋊C4—41no—
189C4⋊C4—41no—
190C4⋊C4—42no—
191C4⋊C4—42no—
192C4⋊C4—84no—
193C4⋊C4—81no—
194C4⋊C4—81no—
195C4⋊C4—84no—
196C4⋊C4—81no—
197C4⋊C4—81no—
198C22⋊C4—42no—
199C22⋊C4—42no—
200C22⋊C4—42no—
201C22⋊C4—42no—
202C22⋊C4—81no—
203C22⋊C4—81no—
204C42—21no—
205C42—21no—
206C42—21no—
207C42—21no—
208C42—41no—
209C42—42no—
210C42C2×SD1612yes—
211C42#13512yes—
212C42AGL(1,ℤ/8)12yes—
213C42#13712yes—
214C23—22no—
215C23—42no—
216C23—41no—
217C23—42no—
218C23—41no—
219C23—42no—
220C23—41no—
221C23—41no—
222C23—42no—
223C23—41no—
224C23#38611yescharacteristic
225Q8—82no—
226Q8—82no—
227Q8—82no—
228Q8—82no—
229D4—42no—
230D4—42no—
231D4—81no—
232D4—84no—
233D4—84no—
234D4—81no—
235C2×C4—21no—
236C2×C4—22no—
237C2×C4—22no—
238C2×C4—22no—
239C2×C4—22no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—22no—
243C2×C4—42no—
244C2×C4—42no—
245C2×C4—41no—
246C2×C4—41no—
247C2×C4—81no—
248C2×C4—81no—
249C2×C4—81no—
250C2×C4—81no—
251C2×C4—82no—
252C2×C4—82no—
253C2×C4—82no—
254C2×C4—82no—
255C2×C4—82no—
256C2×C4C2×#10011yescharacteristic
257C2×C4#41311yescharacteristic
258C2×C4C4×SD1612yes—
259C2×C4#44012yes—
260C2×C4#45012yes—
261C2×C4#45212yes—
262C2×C4#45311yescharacteristic
263C2×C4#45311yescharacteristic
264C2×C4#46112yes—
265C2×C4#46312yes—
266C8—44no—
267C8—44no—
268C8—82no—
269C8—162no—
270C22—44no—
271C22—42no—
272C22—82no—
273C22—81no—
274C22—81no—
275C22—82no—
276C22—82no—
277C22—81no—
278C22—81no—
279C22—81no—
280C22#187712yes—
281C22#188711yescharacteristic
282C22#190011yescharacteristic
283C22#190212yes—
284C22#191011yescharacteristic
285C4—22no—
286C4—22no—
287C4—22no—
288C4—22no—
289C4—21no—
290C4—21no—
291C4—41no—
292C4—81no—
293C4—81no—
294C4—162no—
295C4#162911yescharacteristic
296C4#163011yescharacteristic
297C4#163112yes—
298C2—82no—
299C2—161no—
300C2#906411yescharacteristic
301C2#924712yes—
302C2#928912yes—
303C2#942111yescharacteristic
304C2#945711yescharacteristic
subgroupG/H
center, socleC23#386
commutatorC2×C42C22×C4
Frattini, cosocleC43C23
prime radicalC2×#127C2×C4
LCSG ▷ C2×C42 ▷ C23 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ G
LECSG ▷ C43 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #9903C2semidirect
5. #7022C2semidirect
126. C4⋊C8SD16semidirect

Maximal subgroups

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

subgroupindex
#284382
#231752
#99032
#76512
#70222