Finite Group Databasebeta

← #10114

Subgroups — #10114

Order 256.

244 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 244 conjugacy classes of subgroups — too many to draw. Only its 53 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#3254C211yesmaximal, characteristic
2#3206C211yesmaximal, characteristic
3#3139C211yesmaximal, characteristic, complemented
4#1997C211yesmaximal, characteristic, complemented
5#1525C211yesmaximal, characteristic
6#1485C211yesmaximal, characteristic, complemented
7#1299C211yesmaximal, characteristic, complemented
8#533C2211yescharacteristic
9#532C2211yescharacteristic
10C2×C4⋊Q8C2211yescharacteristic
11#492—21no—
12#491—22no—
13#487—21no—
14#487—21no—
15#487—21no—
16#477—21no—
17#474—21no—
18#468—22no—
19#461—21no—
20#461—22no—
21#460—21no—
22#448—22no—
23#431C2211yescharacteristic
24#423C2211yescharacteristic
25#419—22no—
26C4×M16C2211yescharacteristic
27C4×M16C2211yescharacteristic
28#343—22no—
29#329—22no—
30Q8⋊C8—22no—
31C2×#41—21no—
32C22×Q8D412yes—
33C2×SD16—24no—
34C2×SD16—22no—
35C2×M16—21no—
36C2×M16—21no—
37C2×M16D412yes—
38C4⋊Q8—21no—
39C4⋊Q8—21no—
40C4⋊Q8—21no—
41C4⋊Q8—21no—
42C4⋊Q8D411yescharacteristic
43C4⋊Q8D411yescharacteristic
44C4⋊Q8D411yescharacteristic
45#127D411yescharacteristic
46#125—21no—
47#124—21no—
48C22⋊Q8—21no—
49C22⋊Q8—21no—
50#121—21no—
51#121—21no—
52C4×Q8—21no—
53C4×Q8—21no—
54C4×D4—21no—
55C4×D4—21no—
56#117—21no—
57C2×C4⋊C4—21no—
58C2×C4⋊C4—21no—
59C2×C42C2311yescharacteristic
60#107—22no—
61C4⋊C8—21no—
62C4⋊C8—21no—
63C4⋊C8D412yes—
64#103—24no—
65#103—24no—
66#103—24no—
67#103—22no—
68#102—22no—
69C22⋊C8—21no—
70#97—21no—
71#97D412yes—
72C4×C8—21no—
73C4×C8—21no—
74C4×C8—21no—
75#41—82no—
76C2×Q8—22no—
77C2×Q8—21no—
78C2×Q8—22no—
79C2×Q8—21no—
80C2×Q8—21no—
81C2×Q8—44no—
82C2×D4—21no—
83C2×D4—21no—
84C2×D4—21no—
85C2×D4—41no—
86C22×C4—22no—
87C22×C4—41no—
88C22×C4C2×D411yescharacteristic
89C22×C4C2×D411yescharacteristic
90C22×C4#4111yescharacteristic
91SD16—82no—
92SD16—84no—
93M16—22no—
94M16—42no—
95M16—41no—
96M16—42no—
97C2×C8—22no—
98C2×C8—22no—
99C2×C8—22no—
100C2×C8—41no—
101C4⋊C4—21no—
102C4⋊C4—21no—
103C4⋊C4—21no—
104C4⋊C4—21no—
105C4⋊C4—21no—
106C4⋊C4—21no—
107C4⋊C4—41no—
108C4⋊C4—41no—
109C4⋊C4—41no—
110C4⋊C4—41no—
111C4⋊C4—41no—
112C4⋊C4—41no—
113C22⋊C4—41no—
114C22⋊C4—41no—
115C22⋊C4—41no—
116C42—41no—
117C42C2×D411yescharacteristic
118C42C2×D411yescharacteristic
119C42C2×D411yescharacteristic
120C42C2×D411yescharacteristic
121C23—41no—
122C23#12111yescharacteristic
123Q8—44no—
124Q8—41no—
125Q8—44no—
126Q8—41no—
127Q8—42no—
128Q8—81no—
129D4—42no—
130D4—41no—
131D4—41no—
132D4—81no—
133D4—82no—
134C2×C4—21no—
135C2×C4—22no—
136C2×C4—21no—
137C2×C4—22no—
138C2×C4—41no—
139C2×C4—41no—
140C2×C4—41no—
141C2×C4—42no—
142C2×C4—82no—
143C2×C4—82no—
144C2×C4#12011yescharacteristic
145C2×C4#12011yescharacteristic
146C2×C4#12011yescharacteristic
147C2×C4#12111yescharacteristic
148C2×C4#12111yescharacteristic
149C2×C4#12711yescharacteristic
150C8—42no—
151C8—42no—
152C8—42no—
153C8—42no—
154C22—21no—
155C22—21no—
156C22—21no—
157C22—81no—
158C22—81no—
159C22—81no—
160C22#39211yescharacteristic
161C4—21no—
162C4—21no—
163C4—21no—
164C4—82no—
165C4—81no—
166C4—81no—
167C4—81no—
168C4#45512yes—
169C4#45511yescharacteristic
170C4#45611yescharacteristic
171C4#47112yes—
172C2—41no—
173C2—161no—
174C2#200411yescharacteristic
175C2#200711yescharacteristic
176C2#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. #3139C2semidirect
4. #1997C2semidirect
6. #1485C2semidirect
7. #1299C2semidirect

Maximal subgroups

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

subgroupindex
#32542
#32062
#31392
#19972
#15252
#14852
#12992