Finite Group Databasebeta

← #10138

Subgroups — #10138

Order 256.

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

normal subgroups only

This group has 234 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#3206C211yesmaximal, characteristic
3C2×#501C211yesmaximal, characteristic, complemented
4#1556C211yesmaximal, characteristic, complemented
5#1530C211yesmaximal, characteristic
6#1470C211yesmaximal, characteristic, complemented
7#1302C211yesmaximal, characteristic, complemented
8#532C2211yescharacteristic
9C2×C4⋊Q8C2211yescharacteristic
10C2×#125C2211yescharacteristic
11#501—22no—
12#501—21no—
13#501—22no—
14#492—21no—
15#490—21no—
16#487—21no—
17#481—21no—
18#480—21no—
19#478—22no—
20#461—22no—
21#448—22no—
22#432C2211yescharacteristic
23#432C2211yescharacteristic
24#431C2211yescharacteristic
25C2×#107—21no—
26C2×#106—21no—
27C2×#97C2211yescharacteristic
28#332—22no—
29#329—22no—
30#324—22no—
31C2×#41—21no—
32C22×Q8—21no—
33C2×SD16—24no—
34C22×C8—21no—
35C4⋊Q8—21no—
36C4⋊Q8—21no—
37C4⋊Q8—21no—
38C4⋊Q8D412yes—
39C4⋊Q8D411yescharacteristic
40#127D411yescharacteristic
41#125—21no—
42#125—21no—
43#125—21no—
44#125—21no—
45#124—21no—
46#121—21no—
47#121—21no—
48C4×D4—21no—
49C4×D4—21no—
50C2×C4⋊C4—21no—
51C2×C4⋊C4—21no—
52C2×C4⋊C4—21no—
53C2×C4⋊C4—21no—
54C2×C4⋊C4D411yescharacteristic
55C2×C4⋊C4D411yescharacteristic
56C2×C42C2311yescharacteristic
57#107—24no—
58#107—22no—
59#106—24no—
60#106—22no—
61C4⋊C8—21no—
62C4⋊C8—21no—
63C4⋊C8D412yes—
64#103—24no—
65#102—22no—
66#102—22no—
67C22⋊C8—21no—
68C22⋊C8—21no—
69C22⋊C8D412yes—
70#97—21no—
71#97—21no—
72#97—21no—
73#97D412yes—
74C4×C8—21no—
75#41—82no—
76C2×Q8—22no—
77C2×Q8—21no—
78C2×Q8—41no—
79C2×Q8—82no—
80C2×D4—21no—
81C2×D4—21no—
82C2×D4—21no—
83C2×D4—41no—
84C22×C4—21no—
85C22×C4—21no—
86C22×C4—41no—
87C22×C4—41no—
88C22×C4C2×D411yescharacteristic
89C22×C4C2×D411yescharacteristic
90C22×C4#4111yescharacteristic
91SD16—84no—
92C2×C8—24no—
93C2×C8—22no—
94C2×C8—22no—
95C2×C8—22no—
96C2×C8—41no—
97C4⋊C4—21no—
98C4⋊C4—21no—
99C4⋊C4—21no—
100C4⋊C4—21no—
101C4⋊C4—22no—
102C4⋊C4—21no—
103C4⋊C4—21no—
104C4⋊C4—22no—
105C4⋊C4—41no—
106C4⋊C4—41no—
107C4⋊C4—41no—
108C4⋊C4—41no—
109C4⋊C4—41no—
110C4⋊C4—41no—
111C22⋊C4—41no—
112C22⋊C4—41no—
113C42C2×D411yescharacteristic
114C42C2×D411yescharacteristic
115C42C2×D411yescharacteristic
116C42C2×D411yescharacteristic
117C23—41no—
118C23#12111yescharacteristic
119Q8—44no—
120Q8—81no—
121Q8—82no—
122D4—42no—
123D4—42no—
124D4—81no—
125D4—82no—
126C2×C4—22no—
127C2×C4—22no—
128C2×C4—41no—
129C2×C4—41no—
130C2×C4—41no—
131C2×C4—41no—
132C2×C4—42no—
133C2×C4—42no—
134C2×C4—82no—
135C2×C4—82no—
136C2×C4—81no—
137C2×C4#12011yescharacteristic
138C2×C4#12011yescharacteristic
139C2×C4#12011yescharacteristic
140C2×C4#12111yescharacteristic
141C2×C4#12111yescharacteristic
142C2×C4#12711yescharacteristic
143C8—24no—
144C8—42no—
145C8—42no—
146C8—81no—
147C22—21no—
148C22—21no—
149C22—81no—
150C22—81no—
151C22—81no—
152C22#39211yescharacteristic
153C22#47112yes—
154C4—21no—
155C4—21no—
156C4—21no—
157C4—21no—
158C4—21no—
159C4—81no—
160C4—82no—
161C4—81no—
162C4—81no—
163C4#45612yes—
164C2—22no—
165C2—161no—
166C2#200511yescharacteristic
167C2#200811yescharacteristic
168C2#201911yescharacteristic
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. C2×#501C2semidirect
4. #1556C2semidirect
6. #1470C2semidirect
7. #1302C2semidirect

Maximal subgroups

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

subgroupindex
#32342
#32062
C2×#5012
#15562
#15302
#14702
#13022