Finite Group Databasebeta

← #10095

Subgroups — #10095

Order 256.

240 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 240 conjugacy classes of subgroups — too many to draw. Only its 51 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#3234C211yesmaximal, characteristic
2#3141C211yesmaximal, characteristic
3#2642C211yesmaximal, characteristic, complemented
4#1556C211yesmaximal, characteristic, complemented
5#1528C211yesmaximal, characteristic, complemented
6#1505C211yesmaximal, characteristic
7#1300C211yesmaximal, characteristic, complemented
8#532C2211yescharacteristic
9C2×C4⋊Q8C2211yescharacteristic
10C2×#125C2211yescharacteristic
11#501—21no—
12#490—21no—
13#489—22no—
14#487—21no—
15#486—21no—
16#481—21no—
17#480—21no—
18#478—22no—
19#432C2211yescharacteristic
20#432C2211yescharacteristic
21#417—22no—
22C4×M16C2211yescharacteristic
23#400—22no—
24#400—22no—
25#391—21no—
26#391—21no—
27#376C2211yescharacteristic
28#332—22no—
29#330—22no—
30#327—22no—
31C2×#41—21no—
32C22×Q8—21no—
33C2×M16—21no—
34C4⋊Q8—21no—
35C4⋊Q8—21no—
36C4⋊Q8—21no—
37C4⋊Q8—21no—
38C4⋊Q8D411yescharacteristic
39#127D411yescharacteristic
40#125—21no—
41#125—21no—
42#125—21no—
43#125—21no—
44#124—21no—
45#121—21no—
46#121—21no—
47C4×D4—21no—
48C4×D4—21no—
49C2×C4⋊C4—21no—
50C2×C4⋊C4—21no—
51C2×C4⋊C4—21no—
52C2×C4⋊C4D411yescharacteristic
53C2×C4⋊C4D411yescharacteristic
54C2×C4⋊C4D412yes—
55C2×C42—21no—
56C2×C42C2311yescharacteristic
57#107—22no—
58#106—22no—
59C4⋊C8—21no—
60C4⋊C8D412yes—
61#103—24no—
62#102—24no—
63#102—22no—
64#102—22no—
65C22⋊C8—21no—
66C22⋊C8—21no—
67#97—21no—
68#97—21no—
69#97D412yes—
70C4×C8—21no—
71#95—24no—
72#95—24no—
73#95—21no—
74#95D412yes—
75#41—44no—
76C2×Q8—21no—
77C2×Q8—44no—
78C2×Q8—41no—
79C2×Q8—41no—
80C2×D4—21no—
81C2×D4—21no—
82C2×D4—21no—
83C2×D4—41no—
84C22×C4—22no—
85C22×C4—21no—
86C22×C4—21no—
87C22×C4—22no—
88C22×C4—41no—
89C22×C4C2×D411yescharacteristic
90C22×C4C2×D411yescharacteristic
91C22×C4C2×D411yescharacteristic
92M16—42no—
93C2×C8—22no—
94C2×C8—22no—
95C2×C8—41no—
96C4⋊C4—22no—
97C4⋊C4—21no—
98C4⋊C4—21no—
99C4⋊C4—21no—
100C4⋊C4—21no—
101C4⋊C4—21no—
102C4⋊C4—22no—
103C4⋊C4—21no—
104C4⋊C4—41no—
105C4⋊C4—41no—
106C4⋊C4—41no—
107C4⋊C4—41no—
108C4⋊C4—41no—
109C4⋊C4—41no—
110C22⋊C4—41no—
111C22⋊C4—41no—
112C42—42no—
113C42C2×D411yescharacteristic
114C42C2×D411yescharacteristic
115C42C2×D411yescharacteristic
116C42#4111yescharacteristic
117C23—41no—
118C23#12011yescharacteristic
119Q8—42no—
120Q8—82no—
121Q8—82no—
122D4—42no—
123D4—42no—
124D4—81no—
125D4—82no—
126C2×C4—22no—
127C2×C4—22no—
128C2×C4—41no—
129C2×C4—42no—
130C2×C4—42no—
131C2×C4—42no—
132C2×C4—44no—
133C2×C4—41no—
134C2×C4—41no—
135C2×C4—41no—
136C2×C4—42no—
137C2×C4—42no—
138C2×C4—42no—
139C2×C4—82no—
140C2×C4#12011yescharacteristic
141C2×C4#12011yescharacteristic
142C2×C4#12111yescharacteristic
143C2×C4#12111yescharacteristic
144C2×C4#12111yescharacteristic
145C2×C4#12711yescharacteristic
146C8—42no—
147C8—42no—
148C8—81no—
149C22—21no—
150C22—21no—
151C22—81no—
152C22—81no—
153C22—81no—
154C22#39211yescharacteristic
155C22#45412yes—
156C4—21no—
157C4—21no—
158C4—21no—
159C4—21no—
160C4—21no—
161C4—82no—
162C4—81no—
163C4—82no—
164C4—81no—
165C4—81no—
166C4#45412yes—
167C2—22no—
168C2—161no—
169C2#200311yescharacteristic
170C2#200311yescharacteristic
171C2#200911yescharacteristic
subgroupG/H
center, socleC22#392
commutator, Frattini, cosocleC2×C42C23
prime radical#532C22
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #2642C2semidirect
4. #1556C2semidirect
5. #1528C2semidirect
7. #1300C2semidirect

Maximal subgroups

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

subgroupindex
#32342
#31412
#26422
#15562
#15282
#15052
#13002