Finite Group Databasebeta

← #136123

Subgroups — #136123

Order 512.

240 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 240 conjugacy classes of subgroups — too many to draw. Only its 78 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#17824C211yesmaximal, characteristic
2#12030C211yesmaximal, characteristic, complemented
3#12028C211yesmaximal, characteristic, complemented
4#11463C211yesmaximal, characteristic
5#8607C211yesmaximal, characteristic
6#7295C211yesmaximal, characteristic, complemented
7#7293C211yesmaximal, characteristic, complemented
8#2989—21no—
9#2922C2211yescharacteristic
10#2163C2211yescharacteristic
11#2163C2211yescharacteristic
12#1574—21no—
13#1573—21no—
14#1572—21no—
15#1570—21no—
16#1569—21no—
17#1568C2211yescharacteristic
18#1566C2211yescharacteristic
19#1564C2211yescharacteristic
20C2×#334C2211yescharacteristic
21#1477—21no—
22#1368—21no—
23#1368—21no—
24#1368—21no—
25#1368—21no—
26#1323C412yes—
27#1315C412yes—
28#1283—21no—
29C4×#41D411yescharacteristic
30#446—41no—
31#436—41no—
32#435—41no—
33#435—41no—
34#433—21no—
35#433—21no—
36#432—21no—
37#432—21no—
38#431—21no—
39#431—21no—
40#431—21no—
41#431—21no—
42#424C2×C411yescharacteristic
43#424Q811yescharacteristic
44#423—21no—
45#423—21no—
46C2×C4⋊C8—21no—
47C2×C4⋊C8—21no—
48#408C2×C411yescharacteristic
49#408Q811yescharacteristic
50C4×M16D411yescharacteristic
51C2×C4×C8C2311yescharacteristic
52C4⋊C16—21no—
53C4⋊C16—21no—
54C22⋊C16C2×C412yes—
55C22⋊C16C2×C412yes—
56#346—21no—
57#346—21no—
58#335—21no—
59#335—21no—
60#335—21no—
61#335—21no—
62#334—21no—
63#334—21no—
64#334—21no—
65#334—21no—
66#334—21no—
67#334—21no—
68#334D411yescharacteristic
69#334D411yescharacteristic
70#334D411yescharacteristic
71#334D411yescharacteristic
72Q8⋊C8—41no—
73D4⋊C8—41no—
74C2×#41C4⋊C411yescharacteristic
75C2×M16—41no—
76C2×M16—41no—
77C2×M16C4⋊C411yescharacteristic
78C2×M16C4⋊C411yescharacteristic
79C22×C8—41no—
80C22×C8—41no—
81C22×C8C22×C411yescharacteristic
82C22×C8C2×Q811yescharacteristic
83C4×Q8SD1611yescharacteristic
84C4×Q8SD1611yescharacteristic
85C4×D4—21no—
86C4×D4—21no—
87C4×D4SD1611yescharacteristic
88C4×D4SD1611yescharacteristic
89#117—21no—
90#117C4⋊C411yescharacteristic
91C2×C42—21no—
92C2×C42C2×D411yescharacteristic
93C2×C16—22no—
94C2×C16—22no—
95C4⋊C8—21no—
96C4⋊C8—21no—
97C4⋊C8—21no—
98C4⋊C8—21no—
99C4⋊C8—21no—
100C4⋊C8—21no—
101C4⋊C8—21no—
102C4⋊C8—21no—
103C4⋊C8—41no—
104C4⋊C8—41no—
105C4⋊C8—41no—
106C4⋊C8—41no—
107C4⋊C8—41no—
108C4⋊C8—41no—
109C22⋊C8—21no—
110C22⋊C8—21no—
111C22⋊C8—21no—
112C22⋊C8—21no—
113C22⋊C8—41no—
114C22⋊C8—41no—
115C22⋊C8—41no—
116C22⋊C8—41no—
117#97—21no—
118#97—41no—
119#97—41no—
120C4×C8—21no—
121C4×C8—41no—
122C4×C8—41no—
123C4×C8C2×D411yescharacteristic
124C4×C8C2×D411yescharacteristic
125C4×C8#4111yescharacteristic
126C4×C8#4111yescharacteristic
127#41—41no—
128#41—41no—
129C2×Q8#10611yescharacteristic
130C2×D4—21no—
131C2×D4#10611yescharacteristic
132C22×C4—21no—
133C22×C4—21no—
134C22×C4—21no—
135C22×C4C2×C4⋊C411yescharacteristic
136C22×C4AGL(1,ℤ/8)11yescharacteristic
137C22×C4#13711yescharacteristic
138M16—41no—
139M16—41no—
140M16—161no—
141M16—161no—
142C2×C8—21no—
143C2×C8—21no—
144C2×C8—41no—
145C2×C8—41no—
146C2×C8—41no—
147C2×C8—41no—
148C2×C8—41no—
149C2×C8—41no—
150C2×C8—81no—
151C2×C8—81no—
152C2×C8—161no—
153C2×C8—161no—
154C2×C8C4×D411yescharacteristic
155C2×C8C4×D411yescharacteristic
156C2×C8C22⋊Q811yescharacteristic
157C2×C8C22⋊Q811yescharacteristic
158C4⋊C4—21no—
159C4⋊C4—21no—
160C4⋊C4#10611yescharacteristic
161C4⋊C4#10611yescharacteristic
162C22⋊C4—21no—
163C22⋊C4—21no—
164C22⋊C4—21no—
165C42—21no—
166C42—21no—
167C42—21no—
168C42#12011yescharacteristic
169C42#12311yescharacteristic
170C42C2×SD1611yescharacteristic
171C42C2×SD1611yescharacteristic
172C16—42no—
173C16—42no—
174C23—21no—
175C23#42811yescharacteristic
176Q8—41no—
177D4—41no—
178D4—81no—
179C2×C4—21no—
180C2×C4—21no—
181C2×C4—21no—
182C2×C4—21no—
183C2×C4—21no—
184C2×C4—41no—
185C2×C4—41no—
186C2×C4—41no—
187C2×C4—41no—
188C2×C4—41no—
189C2×C4—41no—
190C2×C4—81no—
191C2×C4#38511yescharacteristic
192C2×C4C2×#10611yescharacteristic
193C2×C4#44811yescharacteristic
194C2×C4#45011yescharacteristic
195C2×C4#48111yescharacteristic
196C2×C4#48311yescharacteristic
197C8—21no—
198C8—21no—
199C8—21no—
200C8—21no—
201C8—41no—
202C8—41no—
203C8—81no—
204C8—81no—
205C8—81no—
206C8—81no—
207C8—161no—
208C8—161no—
209C22—21no—
210C22—21no—
211C22—21no—
212C22—41no—
213C22—41no—
214C22—41no—
215C22#185311yescharacteristic
216C4—21no—
217C4—21no—
218C4—21no—
219C4—21no—
220C4—21no—
221C4—41no—
222C4—41no—
223C4—41no—
224C4#185811yescharacteristic
225C4#186811yescharacteristic
226C2—41no—
227C2—81no—
228C2#886211yescharacteristic
229C2#1293311yescharacteristic
230C2#1293411yescharacteristic
subgroupG/H
center, socleC22#1853
commutatorC22×C8C22×C4
Frattini, cosocleC2×C4×C8C23
prime radicalC2×#41C4⋊C4
LCSG ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2922 ◁ G
LECSG ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #12030C2semidirect
3. #12028C2semidirect
6. #7295C2semidirect
7. #7293C2semidirect

Maximal subgroups

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

subgroupindex
#178242
#120302
#120282
#114632
#86072
#72952
#72932