Finite Group Databasebeta

← #126366

Subgroups — #126366

Order 512.

280 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 280 conjugacy classes of subgroups — too many to draw. Only its 62 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#32529C211yesmaximal, characteristic
2#13526C211yesmaximal, characteristic
3#13506C211yesmaximal, characteristic, complemented
4#10651C211yesmaximal, characteristic, complemented
5#7121C211yesmaximal, characteristic, complemented
6#7118C211yesmaximal, characteristic, complemented
7#7112C211yesmaximal, characteristic
8#2513—21no—
9C2×#367—21no—
10C2×#366—21no—
11C2×#357—21no—
12#2092—21no—
13#2065—21no—
14#2054—21no—
15#1970C2211yescharacteristic
16#1949—21no—
17#1948—21no—
18#1942C2211yescharacteristic
19#1939C2211yescharacteristic
20#1925—21no—
21#1845C2211yescharacteristic
22#1380—21no—
23#1374C2211yescharacteristic
24#1374C2211yescharacteristic
25#1296—21no—
26#1277C2211yescharacteristic
27C22×D8D411yescharacteristic
28C22×C16—21no—
29C22×C16—21no—
30C2×#107—21no—
31C2×#107—21no—
32C2×#107D411yescharacteristic
33C2×#107D411yescharacteristic
34C2×#107D411yescharacteristic
35C2×#106—21no—
36C2×#106—41no—
37C2×C4⋊C8—21no—
38C2×C4⋊C8—21no—
39C2×C4⋊C8D411yescharacteristic
40C2×#102—21no—
41C2×#102—21no—
42C2×#102—21no—
43C2×C4×C8C2311yescharacteristic
44#397—41no—
45#397—41no—
46#390—21no—
47#390—21no—
48#390—21no—
49#389—21no—
50#388—41no—
51#388—41no—
52#384—21no—
53#382—21no—
54C4×C4⋊C4—21no—
55C4×C4⋊C4D411yescharacteristic
56#367—42no—
57#366—42no—
58#357—42no—
59#340—21no—
60#340—21no—
61#340—41no—
62#336—21no—
63C22×D4—21no—
64C2×D8—21no—
65C2×D8—21no—
66C2×D8—41no—
67C2×D8—41no—
68C22×C8—21no—
69C22×C8—41no—
70C22×C8C2×D411yescharacteristic
71C22×C8C2×D411yescharacteristic
72C2×C4⋊C4—21no—
73C2×C4⋊C4—21no—
74C2×C4⋊C4—21no—
75C2×C4⋊C4—41no—
76C2×C4⋊C4—41no—
77C2×C4⋊C4—41no—
78C2×C4⋊C4—41no—
79C2×C4⋊C4D811yescharacteristic
80C2×C4⋊C4D811yescharacteristic
81C2×C22⋊C4—41no—
82C2×C22⋊C4—41no—
83C2×C22⋊C4—41no—
84C2×C22⋊C4—41no—
85C2×C22⋊C4—41no—
86C2×C22⋊C4—41no—
87C2×C42—21no—
88C2×C42—41no—
89C2×C42C2×D411yescharacteristic
90C2×C16—22no—
91C2×C16—22no—
92C2×C16—41no—
93C2×C16—41no—
94C2×C16—41no—
95C2×C16—41no—
96#107—21no—
97#107—21no—
98#107—41no—
99#107—41no—
100#107—41no—
101#107—41no—
102#107—41no—
103#107—41no—
104#106—82no—
105#106—81no—
106C4⋊C8—22no—
107C4⋊C8—41no—
108C4⋊C8—41no—
109C4⋊C8—41no—
110C4⋊C8—41no—
111#102—41no—
112#102—41no—
113#102—81no—
114#102—81no—
115C4×C8—41no—
116#95—21no—
117#95—41no—
118#95—41no—
119#95—41no—
120#95—41no—
121C24—41no—
122C2×D4—41no—
123C2×D4—41no—
124C2×D4—41no—
125C2×D4—41no—
126C2×D4—41no—
127C2×D4—41no—
128C22×C4—21no—
129C22×C4—21no—
130C22×C4—41no—
131C22×C4—41no—
132C22×C4—41no—
133C22×C4—81no—
134C22×C4#12011yescharacteristic
135C22×C4C2×D811yescharacteristic
136C22×C4AGL(1,ℤ/8)11yescharacteristic
137D8—42no—
138D8—81no—
139C2×C8—21no—
140C2×C8—21no—
141C2×C8—21no—
142C2×C8—21no—
143C2×C8—42no—
144C2×C8—41no—
145C2×C8—41no—
146C2×C8—41no—
147C2×C8—81no—
148C2×C8—81no—
149C2×C8#13511yescharacteristic
150C2×C8#13511yescharacteristic
151C2×C8AGL(1,ℤ/8)11yescharacteristic
152C2×C8#13711yescharacteristic
153C4⋊C4—21no—
154C4⋊C4—21no—
155C4⋊C4—41no—
156C4⋊C4—41no—
157C4⋊C4—41no—
158C4⋊C4—41no—
159C4⋊C4—41no—
160C4⋊C4—41no—
161C4⋊C4—41no—
162C4⋊C4—81no—
163C4⋊C4—81no—
164C4⋊C4—81no—
165C4⋊C4—81no—
166C4⋊C4—161no—
167C4⋊C4—161no—
168C22⋊C4—82no—
169C22⋊C4—81no—
170C22⋊C4—81no—
171C22⋊C4—161no—
172C22⋊C4—161no—
173C22⋊C4—161no—
174C22⋊C4—161no—
175C42—21no—
176C42—21no—
177C42—41no—
178C42—41no—
179C42—161no—
180C16—42no—
181C16—42no—
182C23—81no—
183C23—81no—
184C23—81no—
185C23—81no—
186C23—81no—
187C23—81no—
188C23—81no—
189C23#44711yescharacteristic
190D4—81no—
191D4—81no—
192D4—81no—
193D4—81no—
194C2×C4—21no—
195C2×C4—21no—
196C2×C4—21no—
197C2×C4—21no—
198C2×C4—21no—
199C2×C4—41no—
200C2×C4—41no—
201C2×C4—41no—
202C2×C4—41no—
203C2×C4—41no—
204C2×C4—41no—
205C2×C4—81no—
206C2×C4—81no—
207C2×C4—81no—
208C2×C4—81no—
209C2×C4—81no—
210C2×C4—81no—
211C2×C4—81no—
212C2×C4—81no—
213C2×C4—81no—
214C2×C4—161no—
215C2×C4—161no—
216C2×C4—161no—
217C2×C4#44911yescharacteristic
218C2×C4#45211yescharacteristic
219C2×C4#45311yescharacteristic
220C2×C4#45611yescharacteristic
221C2×C4#45611yescharacteristic
222C2×C4#45811yescharacteristic
223C2×C4#50811yescharacteristic
224C2×C4#50911yescharacteristic
225C8—21no—
226C8—21no—
227C8—21no—
228C8—21no—
229C8—82no—
230C8—81no—
231C22—161no—
232C22—161no—
233C22—161no—
234C22—161no—
235C22—161no—
236C22—161no—
237C22—161no—
238C22#161011yescharacteristic
239C22#162111yescharacteristic
240C22#164711yescharacteristic
241C22#218611yescharacteristic
242C22#219111yescharacteristic
243C22#219211yescharacteristic
244C22#219411yescharacteristic
245C4—21no—
246C4—21no—
247C4—41no—
248C4—81no—
249C4—81no—
250C4—161no—
251C4—161no—
252C4—161no—
253C4—321no—
254C4#160911yescharacteristic
255C4#161811yescharacteristic
256C4#165211yescharacteristic
257C4#166011yescharacteristic
258C2—321no—
259C2#779311yescharacteristic
260C2#1229011yescharacteristic
261C2#1229411yescharacteristic
262C2#1232911yescharacteristic
263C2#1233811yescharacteristic
264C2#1239511yescharacteristic
265C2#1240011yescharacteristic
subgroupG/H
center, socleC23#447
commutator, Frattini, cosocleC2×C4×C8C23
prime radicalC22×D8D4
LCSG ▷ C2×C4×C8 ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1939 ◁ G
LECSG ▷ C2×C4×C8 ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #13506C2semidirect
4. #10651C2semidirect
5. #7121C2semidirect
6. #7118C2semidirect

Maximal subgroups

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

subgroupindex
#325292
#135262
#135062
#106512
#71212
#71182
#71122