Finite Group Databasebeta

← #20308

Subgroups — #20308

Order 256.

313 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 313 conjugacy classes of subgroups — too many to draw. Only its 99 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#3366C212yesmaximal, complemented
2#3333C212yesmaximal, complemented
3#1709C211yesmaximal, characteristic, complemented
4#1698C212yesmaximal, complemented
5#1697C212yesmaximal, complemented
6#1632C212yesmaximal, complemented
7#1621C212yesmaximal, complemented
8#1591C211yesmaximal, characteristic, complemented
9#1589C211yesmaximal, characteristic, complemented
10#554C2212yes—
11#550C2212yes—
12#494C2211yescharacteristic
13#493C2211yescharacteristic
14#490—22no—
15#486C2212yes—
16#486C2212yes—
17#480—22no—
18#478—22no—
19#477C2212yes—
20#476C2212yes—
21#468—22no—
22#460C2212yes—
23#459—22no—
24#459C2212yes—
25#452—22no—
26#446C2212yes—
27#442—22no—
28C4×Q16C2211yescharacteristic
29C4×D8C2211yescharacteristic
30#436C2212yes—
31#417—22no—
32#335C2211yescharacteristic
33#335C2211yescharacteristic
34#333C2212yes—
35#331C2212yes—
36Q8⋊C8C2212yes—
37D4⋊C8C2212yes—
38C82C2211yescharacteristic
39C2×#41—22no—
40C2×Q16—41no—
41C2×Q16D411yescharacteristic
42C2×SD16—22no—
43C2×D8—22no—
44C2×D8—41no—
45C2×D8D411yescharacteristic
46C2×M16—22no—
47C4⋊Q8C2312yes—
48#127—22no—
49#127C2312yes—
50#125—22no—
51C22⋊Q8—22no—
52C22⋊Q8—42no—
53#121—22no—
54#121—42no—
55C4×Q8C2312yes—
56C4×D4—22no—
57C4×D4—22no—
58C4×D4C2312yes—
59C2×C4⋊C4—22no—
60#107—22no—
61#107D411yescharacteristic
62#107D411yescharacteristic
63#106—22no—
64C4⋊C8—22no—
65C4⋊C8C2312yes—
66C4⋊C8C2312yes—
67#103—22no—
68#103D412yes—
69#102—22no—
70#102—22no—
71#102—22no—
72#102—22no—
73#102—22no—
74#102—22no—
75#102—22no—
76#102D412yes—
77C22⋊C8—22no—
78#97—22no—
79C4×C8C2311yescharacteristic
80C4×C8C2311yescharacteristic
81C4×C8C2311yescharacteristic
82#41—44no—
83#41—82no—
84C2×Q8—22no—
85C2×Q8C2×D412yes—
86C2×D4—22no—
87C2×D4—22no—
88C2×D4—22no—
89C2×D4—42no—
90C2×D4—42no—
91C2×D4C2×D412yes—
92C22×C4—22no—
93C22×C4—42no—
94C22×C4—42no—
95Q16—41no—
96Q16—82no—
97SD16—82no—
98D8—41no—
99D8—44no—
100D8—82no—
101M16—82no—
102C2×C8—21no—
103C2×C8—22no—
104C2×C8—22no—
105C2×C8C2×D411yescharacteristic
106C2×C8C2×D411yescharacteristic
107C2×C8C2×D411yescharacteristic
108C2×C8C2×D411yescharacteristic
109C4⋊C4—22no—
110C4⋊C4—22no—
111C4⋊C4—22no—
112C4⋊C4—42no—
113C4⋊C4—42no—
114C4⋊C4C2×D412yes—
115C4⋊C4C2×D412yes—
116C22⋊C4—22no—
117C22⋊C4—42no—
118C22⋊C4—42no—
119C42—22no—
120C42C2411yescharacteristic
121C23—22no—
122C23—42no—
123Q8—22no—
124Q8—22no—
125Q8—82no—
126D4—22no—
127D4—22no—
128D4—42no—
129D4—42no—
130D4—42no—
131D4—42no—
132D4—82no—
133D4—84no—
134D4—162no—
135C2×C4—22no—
136C2×C4—22no—
137C2×C4—22no—
138C2×C4—42no—
139C2×C4—42no—
140C2×C4—42no—
141C2×C4—84no—
142C2×C4—162no—
143C2×C4C22×D411yescharacteristic
144C2×C4C22×D411yescharacteristic
145C2×C4H(5,𝔽2)11yescharacteristic
146C8—22no—
147C8—21no—
148C8—21no—
149C8—21no—
150C8—21no—
151C8—42no—
152C8—42no—
153C22—42no—
154C22—42no—
155C22—42no—
156C22—82no—
157C22—82no—
158C22—82no—
159C22D4211yescharacteristic
160C4—21no—
161C4—42no—
162C4—42no—
163C4—42no—
164C4—82no—
165C4#57511yescharacteristic
166C4#57511yescharacteristic
167C4#57611yescharacteristic
168C4#57811yescharacteristic
169C2—82no—
170C2—162no—
171C2#327311yescharacteristic
172C2#327711yescharacteristic
173C2#329211yescharacteristic
subgroupG/H
center, socleC22D42
commutator, Frattini, cosocleC42C24
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. #3366C2semidirect
2. #3333C2semidirect
3. #1709C2semidirect
4. #1698C2semidirect
5. #1697C2semidirect
6. #1632C2semidirect
7. #1621C2semidirect
8. #1591C2semidirect
9. #1589C2semidirect

Maximal subgroups

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

subgroupindex
#33662
#33332
#17092
#16982
#16972
#16322
#16212
#15912
#15892