Finite Group Databasebeta

← #511425

Subgroups — #511425

Order 512.

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

normal subgroups only

This group has 388 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#29270C212yesmaximal, complemented
2#13544C212yesmaximal, complemented
3#13519C211yesmaximal, characteristic, complemented
4#13489C212yesmaximal, complemented
5#13444C211yesmaximal, characteristic
6#13266C211yesmaximal, characteristic, complemented
7#13151C212yesmaximal, complemented
8#13021C212yesmaximal
9#12942C212yesmaximal, complemented
10#3414C2211yescharacteristic
11#2960C2211yescharacteristic
12#2957C2212yes—
13#2161C2211yescharacteristic
14#2150C2211yescharacteristic
15#2148C2212yes—
16C2×#359C2212yes—
17#2138C2212yes—
18C2×C4×C16C2211yescharacteristic
19#2098—22no—
20#2031—22no—
21#1976C2212yes—
22#1974C2212yes—
23#1953C2212yes—
24#1945C2212yes—
25#1868—22no—
26#1861—22no—
27#1833C2212yes—
28#1389C2212yes—
29#1385C2211yescharacteristic
30#1384C2211yescharacteristic
31#1382C2212yes—
32#1381C2212yes—
33#1379C2212yes—
34#1379C2212yes—
35C2×#135C2312yes—
36#568—22no—
37#533D411yescharacteristic
38#510—22no—
39#509—22no—
40#508—41no—
41C2×M32C2311yescharacteristic
42C2×M32C2311yescharacteristic
43C22×C16C2312yes—
44#446—41no—
45C8×Q8—41no—
46#444—42no—
47#443—42no—
48#430—22no—
49C2×#108C2312yes—
50C2×#108C2312yes—
51#427C2312yes—
52#424—22no—
53#424D412yes—
54#423D411yescharacteristic
55#421—22no—
56#421—22no—
57C2×#104—22no—
58#410—22no—
59#408—22no—
60#405C2312yes—
61C2×C4×C8C2311yescharacteristic
62#368—41no—
63C4⋊C16—41no—
64#362—22no—
65#361—22no—
66#359—42no—
67#358—22no—
68#357—22no—
69C4×C16—41no—
70#344—22no—
71#344—22no—
72#339—22no—
73H(3,ℤ/4)—22no—
74C2×#41—22no—
75#135—22no—
76#135—22no—
77#135—22no—
78#135—22no—
79C2×Q16#4112yes—
80C2×SD16—22no—
81C2×D8#4112yes—
82#131—44no—
83#131—82no—
84C2×M16—22no—
85C2×M16—22no—
86C2×M16—82no—
87C2×M16C2×D412yes—
88C22×C8C2×D411yescharacteristic
89C22×C8C2411yescharacteristic
90C4⋊Q8—21no—
91#125—21no—
92C22⋊Q8—41no—
93C4×Q8—41no—
94#117—22no—
95#117C2×D412yes—
96C2×C42C2×D411yescharacteristic
97Q32—41no—
98SD32—42no—
99D16—41no—
100M32—22no—
101M32—22no—
102C2×C16—22no—
103C2×C16—22no—
104C2×C16—21no—
105C2×C16—21no—
106C2×C16—22no—
107#108—22no—
108#108—22no—
109#108—22no—
110#108—22no—
111#108—82no—
112#107—22no—
113#106—22no—
114C4⋊C8—41no—
115C4⋊C8—42no—
116C4⋊C8—82no—
117#104—44no—
118#104—42no—
119#104—42no—
120#104—82no—
121#99—44no—
122C22⋊C8—82no—
123#97—22no—
124C4×C8—21no—
125C4×C8—21no—
126C4×C8—22no—
127#41—42no—
128#41—42no—
129#41—42no—
130#41—42no—
131C2×Q8—22no—
132C2×Q8—41no—
133C2×D4—22no—
134C2×D4—42no—
135C22×C4—21no—
136C22×C4—42no—
137C22×C4C22×D411yescharacteristic
138Q16—22no—
139Q16—22no—
140SD16—42no—
141SD16—42no—
142D8—22no—
143D8—22no—
144M16—22no—
145M16—22no—
146M16—42no—
147M16—42no—
148M16—42no—
149M16—42no—
150M16—84no—
151C2×C8—21no—
152C2×C8—22no—
153C2×C8—22no—
154C2×C8—42no—
155C2×C8—42no—
156C2×C8—84no—
157C2×C8C2×#4112yes—
158C2×C8H(5,𝔽2)11yescharacteristic
159C2×C8H(5,𝔽2)11yescharacteristic
160C2×C82−1+412yes—
161C4⋊C4—22no—
162C4⋊C4—42no—
163C4⋊C4—41no—
164C4⋊C4—42no—
165C22⋊C4—22no—
166C22⋊C4—42no—
167C42—21no—
168C42—22no—
169C42—42no—
170C42#13511yescharacteristic
171C42#13511yescharacteristic
172C16—42no—
173C16—41no—
174C16—41no—
175C23—42no—
176C23#57511yescharacteristic
177Q8—42no—
178Q8—42no—
179Q8—82no—
180D4—42no—
181D4—42no—
182D4—82no—
183D4—82no—
184C2×C4—21no—
185C2×C4—21no—
186C2×C4—42no—
187C2×C4—42no—
188C2×C4—42no—
189C2×C4—82no—
190C2×C4—82no—
191C2×C4—82no—
192C2×C4#53411yescharacteristic
193C2×C4#54812yes—
194C2×C4C2×#13511yescharacteristic
195C2×C4#57612yes—
196C8—41no—
197C8—42no—
198C8—82no—
199C8—82no—
200C8#53912yes—
201C8#55111yescharacteristic
202C8#56311yescharacteristic
203C22—41no—
204C22—82no—
205C22—162no—
206C22#319111yescharacteristic
207C22#337012yes—
208C4—22no—
209C4—41no—
210C4—82no—
211C4—82no—
212C4—162no—
213C4#272811yescharacteristic
214C4#319911yescharacteristic
215C4#336112yes—
216C2—21no—
217C2—22no—
218C2—162no—
219C2#3254511yescharacteristic
subgroupG/H
centerC4#3199
commutator, Frattini, cosocleC22×C8C24
socleC2#32545
prime radical#3414C22
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C4 ◁ C22×C4 ◁ #2960 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #29270C2semidirect
2. #13544C2semidirect
3. #13519C2semidirect
4. #13489C2semidirect
6. #13266C2semidirect
7. #13151C2semidirect
9. #12942C2semidirect

Maximal subgroups

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

subgroupindex
#292702
#135442
#135192
#134892
#134442
#132662
#131512
#130212
#129422