Finite Group Databasebeta

← #20014

Subgroups — #20014

Order 256.

284 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 284 conjugacy classes of subgroups — too many to draw. Only its 103 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#3395C211yesmaximal, characteristic, complemented
2#3364C212yesmaximal
3#3342C211yesmaximal, characteristic
4#1707C211yesmaximal, characteristic, complemented
5#1680C212yesmaximal, complemented
6#1674C211yesmaximal, characteristic, complemented
7#1669C211yesmaximal, characteristic
8#1653C212yesmaximal, complemented
9#1652C212yesmaximal
10C8×Q16C211yesmaximal, characteristic, complemented
11C8×SD16C211yesmaximal, characteristic
12#557C2212yes—
13#556C2212yes—
14#500C2211yescharacteristic
15#498C2212yes—
16#494—21no—
17#494C2211yescharacteristic
18#488—22no—
19#487C2212yes—
20#485—22no—
21#479—22no—
22#475C2212yes—
23#474C2212yes—
24#467—21no—
25#464—22no—
26#462—22no—
27#462C2212yes—
28#462C2212yes—
29C8×Q8C2212yes—
30C8×Q8C2211yescharacteristic
31C4×Q16—21no—
32C4×Q16—22no—
33C4×Q16C2211yescharacteristic
34C4×SD16C2211yescharacteristic
35C8×D4C2211yescharacteristic
36#427—21no—
37#335C2211yescharacteristic
38#334C2211yescharacteristic
39#332C2212yes—
40#332C2212yes—
41Q8⋊C8C2211yescharacteristic
42Q8⋊C8C2212yes—
43D4⋊C8C2211yescharacteristic
44C82C2211yescharacteristic
45C2×Q16—22no—
46C2×Q16—22no—
47C2×Q16—21no—
48C2×Q16—21no—
49C2×Q16D411yescharacteristic
50C2×SD16D411yescharacteristic
51C22×C8—21no—
52C4⋊Q8—22no—
53C4⋊Q8C2312yes—
54C4⋊Q8C2312yes—
55#126—42no—
56#125—22no—
57#125—42no—
58#124—22no—
59C22⋊Q8—22no—
60C4×Q8—22no—
61C4×Q8—22no—
62C4×Q8—22no—
63C4×Q8C2312yes—
64C4×Q8C2311yescharacteristic
65C4×D4C2311yescharacteristic
66#117—22no—
67#107—21no—
68#107—21no—
69#107—22no—
70#107D411yescharacteristic
71#106—22no—
72#106—42no—
73#106D411yescharacteristic
74C4⋊C8—21no—
75C4⋊C8—22no—
76C4⋊C8C2311yescharacteristic
77C4⋊C8C2311yescharacteristic
78C4⋊C8C2312yes—
79#103—22no—
80#103—22no—
81#103—22no—
82#103—22no—
83#103—22no—
84#103D411yescharacteristic
85#103D412yes—
86#102—22no—
87#102D411yescharacteristic
88C22⋊C8—21no—
89C4×C8—22no—
90C4×C8—21no—
91C4×C8C2311yescharacteristic
92C4×C8C2311yescharacteristic
93C4×C8C2311yescharacteristic
94C2×Q8—22no—
95C2×Q8—22no—
96C2×Q8C2×D412yes—
97C2×Q8C2×D411yescharacteristic
98C2×D4C2×D411yescharacteristic
99C22×C4—21no—
100Q16—22no—
101Q16—44no—
102Q16—42no—
103Q16—81no—
104Q16—82no—
105SD16—22no—
106C2×C8—21no—
107C2×C8—22no—
108C2×C8—21no—
109C2×C8—21no—
110C2×C8—42no—
111C2×C8C2×D411yescharacteristic
112C2×C8C2×D411yescharacteristic
113C2×C8C2×D411yescharacteristic
114C2×C8C2×D411yescharacteristic
115C4⋊C4—21no—
116C4⋊C4—22no—
117C4⋊C4—22no—
118C4⋊C4—22no—
119C4⋊C4—22no—
120C4⋊C4—22no—
121C4⋊C4—42no—
122C4⋊C4—42no—
123C4⋊C4—42no—
124C4⋊C4—42no—
125C4⋊C4C2×D411yescharacteristic
126C4⋊C4C2×D412yes—
127C4⋊C4C2×D411yescharacteristic
128C22⋊C4—21no—
129C22⋊C4—42no—
130C22⋊C4—42no—
131C42—22no—
132C42—21no—
133C42—42no—
134C42—42no—
135C42C2411yescharacteristic
136C23—21no—
137Q8—21no—
138Q8—22no—
139Q8—22no—
140Q8—42no—
141Q8—42no—
142Q8—42no—
143Q8—42no—
144Q8#13512yes—
145D4—21no—
146D4—21no—
147C2×C4—21no—
148C2×C4—22no—
149C2×C4—22no—
150C2×C4—21no—
151C2×C4—21no—
152C2×C4—41no—
153C2×C4—41no—
154C2×C4—42no—
155C2×C4—42no—
156C2×C4C22×D411yescharacteristic
157C2×C4C22×D411yescharacteristic
158C2×C4H(5,𝔽2)11yescharacteristic
159C8—22no—
160C8—21no—
161C8—21no—
162C8—21no—
163C8—41no—
164C8—42no—
165C8—41no—
166C8#13512yes—
167C22—41no—
168C22—41no—
169C22—41no—
170C22D4211yescharacteristic
171C4—22no—
172C4—21no—
173C4—41no—
174C4—42no—
175C4—42no—
176C4—41no—
177C4—82no—
178C4—82no—
179C4C2×#13511yescharacteristic
180C4C2×#13511yescharacteristic
181C4#57711yescharacteristic
182C4#57811yescharacteristic
183C2—81no—
184C2#328411yescharacteristic
185C2#328511yescharacteristic
186C2#329311yescharacteristic
subgroupG/H
center, socleC22D42
commutator, Frattini, cosocleC42C24
prime radicalC2×D4C2×D4
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C42 ◁ G
LECSG ▷ C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #3395C2semidirect
4. #1707C2semidirect
5. #1680C2semidirect
6. #1674C2semidirect
8. #1653C2semidirect
10. C8×Q16C2semidirect

Maximal subgroups

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

subgroupindex
#33952
#33642
#33422
#17072
#16802
#16742
#16692
#16532
#16522
C8×Q162
C8×SD162