Finite Group Databasebeta

← #102386

Subgroups — #102386

Order 512.

409 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 409 conjugacy classes of subgroups — too many to draw. Only its 114 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#33226C211yesmaximal, characteristic
2#8807C211yesmaximal, characteristic, complemented
3#8260C211yesmaximal, characteristic, complemented
4#7660C212yesmaximal, complemented
5#7659C212yesmaximal
6C2×#488—21no—
7C2×#488—21no—
8#2859C2211yescharacteristic
9#2858C2211yescharacteristic
10#1958—21no—
11#1958—21no—
12#1957C411yescharacteristic
13#1957C411yescharacteristic
14#1956C411yescharacteristic
15#1956C411yescharacteristic
16#1831C2212yes—
17C4×C4⋊C8C2212yes—
18C42×C8C2211yescharacteristic
19C2×#335—21no—
20C2×#334—21no—
21C2×#322—21no—
22#1273—21no—
23#1271—24no—
24#1270—24no—
25C2×C4⋊Q8C2×C411yescharacteristic
26C2×#127C2×C411yescharacteristic
27C2×#125—21no—
28C2×#124—21no—
29C2×#124—21no—
30#488—42no—
31#488—42no—
32#488—42no—
33#488—42no—
34C2×C4⋊C8—22no—
35C2×C4⋊C8—22no—
36C2×#103—21no—
37C2×#103—21no—
38C2×#103—21no—
39C2×#103—21no—
40C2×#102—21no—
41C2×#102—21no—
42C2×#102—21no—
43C2×#102—21no—
44C2×C4×C8—22no—
45C2×C4×C8C2×C411yescharacteristic
46C2×C4×C8C2×C411yescharacteristic
47C2×C4×C8D411yescharacteristic
48C2×C4×C8D411yescharacteristic
49C2×C4×C8D411yescharacteristic
50C2×C4×C8D411yescharacteristic
51#390—22no—
52#390—22no—
53#389—22no—
54#389—22no—
55#384C2×C411yescharacteristic
56#384C2×C411yescharacteristic
57#382—21no—
58C43C2311yescharacteristic
59#336—22no—
60#336—22no—
61#336D414yes—
62#335—42no—
63#334—42no—
64#322—42no—
65C22×Q8—21no—
66C22×D4—21no—
67C22×D4—22no—
68C22×C8—24no—
69C22×C8—42no—
70C22×C8C22⋊C411yescharacteristic
71C22×C8C22⋊C411yescharacteristic
72C22×C8C22⋊C411yescharacteristic
73C22×C8C22⋊C411yescharacteristic
74C4⋊Q8—41no—
75C4⋊Q8—41no—
76#127—41no—
77#127—41no—
78#125—41no—
79#125—41no—
80#125—41no—
81#125—41no—
82#124—41no—
83#124—41no—
84#124—41no—
85#124—41no—
86#124—42no—
87#124—41no—
88#124—41no—
89C2×C4⋊C4—21no—
90C2×C4⋊C4—21no—
91C2×C4⋊C4—22no—
92C2×C4⋊C4—22no—
93C2×C4⋊C4—22no—
94C2×C4⋊C4—41no—
95C2×C4⋊C4—41no—
96C2×C22⋊C4—42no—
97C2×C22⋊C4—41no—
98C2×C22⋊C4—41no—
99C2×C42C22×C411yescharacteristic
100C2×C42C2×D412yes—
101C2×C42C2×D411yescharacteristic
102C2×C42C2×D411yescharacteristic
103C2×C42#4112yes—
104C4⋊C8—44no—
105C4⋊C8—42no—
106C4⋊C8—42no—
107#103—42no—
108#103—42no—
109#103—42no—
110#103—42no—
111#102—42no—
112#102—42no—
113#102—42no—
114#102—42no—
115C4×C8—22no—
116C4×C8—21no—
117C4×C8—21no—
118C4×C8—41no—
119C4×C8—41no—
120C4×C8—42no—
121C4×C8—42no—
122C4×C8D812yes—
123C4×C8D812yes—
124C4×C8SD1612yes—
125C4×C8SD1612yes—
126#95—42no—
127C24—41no—
128C2×Q8—41no—
129C2×Q8—41no—
130C2×Q8—41no—
131C2×Q8—41no—
132C2×Q8—41no—
133C2×Q8—41no—
134C2×D4—41no—
135C2×D4—41no—
136C2×D4—41no—
137C2×D4—41no—
138C2×D4—41no—
139C2×D4—41no—
140C2×D4—44no—
141C2×D4—82no—
142C2×D4—82no—
143C22×C4—41no—
144C22×C4—41no—
145C22×C4—41no—
146C22×C4C2×C22⋊C411yescharacteristic
147C22×C4C4×D412yes—
148C22×C4#12112yes—
149C22×C4#12411yescharacteristic
150C22×C4#12711yescharacteristic
151C2×C8—21no—
152C2×C8—21no—
153C2×C8—21no—
154C2×C8—21no—
155C2×C8—21no—
156C2×C8—21no—
157C2×C8—21no—
158C2×C8—21no—
159C2×C8—44no—
160C2×C8—44no—
161C2×C8—44no—
162C2×C8—84no—
163C2×C8—82no—
164C2×C8#10212yes—
165C2×C8#10212yes—
166C2×C8#10212yes—
167C2×C8#10212yes—
168C4⋊C4—41no—
169C4⋊C4—41no—
170C4⋊C4—41no—
171C4⋊C4—41no—
172C4⋊C4—44no—
173C4⋊C4—82no—
174C4⋊C4—82no—
175C4⋊C4—81no—
176C4⋊C4—81no—
177C4⋊C4—81no—
178C4⋊C4—81no—
179C22⋊C4—82no—
180C22⋊C4—82no—
181C22⋊C4—81no—
182C22⋊C4—81no—
183C22⋊C4—81no—
184C22⋊C4—81no—
185C42—21no—
186C42—21no—
187C42—24no—
188C42—42no—
189C42C2×D811yescharacteristic
190C42C2×D811yescharacteristic
191C42C2×SD1611yescharacteristic
192C42C2×SD1611yescharacteristic
193C42#13512yes—
194C42AGL(1,ℤ/8)11yescharacteristic
195C42#13711yescharacteristic
196C23—81no—
197C23—81no—
198C23—82no—
199C23—82no—
200C23—81no—
201C23#39011yescharacteristic
202Q8—81no—
203Q8—81no—
204Q8—81no—
205Q8—81no—
206D4—81no—
207D4—81no—
208D4—81no—
209D4—81no—
210D4—164no—
211C2×C4—22no—
212C2×C4—22no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—22no—
216C2×C4—22no—
217C2×C4—81no—
218C2×C4—81no—
219C2×C4—81no—
220C2×C4—81no—
221C2×C4—82no—
222C2×C4—82no—
223C2×C4—81no—
224C2×C4#41311yescharacteristic
225C2×C4#41311yescharacteristic
226C2×C4C2×#10211yescharacteristic
227C2×C4C2×#10211yescharacteristic
228C2×C4#41611yescharacteristic
229C2×C4#41711yescharacteristic
230C2×C4#44414yes—
231C2×C4#46312yes—
232C2×C4#46412yes—
233C2×C4#48611yescharacteristic
234C2×C4#48611yescharacteristic
235C2×C4#49211yescharacteristic
236C2×C4#49311yescharacteristic
237C8—22no—
238C8—22no—
239C8—22no—
240C8—22no—
241C8—84no—
242C8—162no—
243C22—162no—
244C22—161no—
245C22—162no—
246C22—161no—
247C22—161no—
248C22#195711yescharacteristic
249C22#196711yescharacteristic
250C22#197512yes—
251C22#197612yes—
252C22#198311yescharacteristic
253C4—22no—
254C4—22no—
255C4—21no—
256C4—21no—
257C4—24no—
258C4—21no—
259C4—21no—
260C4—161no—
261C4—161no—
262C4—161no—
263C4#154011yescharacteristic
264C4#154011yescharacteristic
265C4#154711yescharacteristic
266C4#154711yescharacteristic
267C2—321no—
268C2#969811yescharacteristic
269C2#975412yes—
270C2#975611yescharacteristic
271C2#979012yes—
272C2#986211yescharacteristic
subgroupG/H
center, socleC23#390
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
2. #8807C2semidirect
3. #8260C2semidirect
4. #7660C2semidirect

Maximal subgroups

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

subgroupindex
#332262
#88072
#82602
#76602
#76592