Finite Group Databasebeta

← #10111

Subgroups — #10111

Order 256.

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

normal subgroups only

This group has 238 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#3229C211yesmaximal, characteristic, complemented
3#3141C211yesmaximal, characteristic
4#1997C211yesmaximal, characteristic, complemented
5#1526C211yesmaximal, characteristic, complemented
6#1484C211yesmaximal, characteristic
7#1299C211yesmaximal, characteristic, complemented
8#533C2211yescharacteristic
9#532C2211yescharacteristic
10C2×C4⋊Q8C2211yescharacteristic
11#498—21no—
12#489—22no—
13#487—21no—
14#487—21no—
15#486—21no—
16#483—22no—
17#477—21no—
18#475—22no—
19#474—21no—
20#468—22no—
21#461—21no—
22#460—21no—
23#431C2211yescharacteristic
24#423C2211yescharacteristic
25#417—22no—
26C4×M16C2211yescharacteristic
27C4×M16C2211yescharacteristic
28#343—22no—
29#330—22no—
30D4⋊C8—22no—
31C2×#41—21no—
32C22×Q8—21no—
33C2×SD16—22no—
34C2×M16—21no—
35C2×M16—21no—
36C2×M16D412yes—
37C4⋊Q8—21no—
38C4⋊Q8—21no—
39C4⋊Q8—21no—
40C4⋊Q8—21no—
41C4⋊Q8D411yescharacteristic
42C4⋊Q8D411yescharacteristic
43C4⋊Q8D411yescharacteristic
44#127D411yescharacteristic
45#125—21no—
46#124—21no—
47C22⋊Q8—21no—
48C22⋊Q8—21no—
49#121—21no—
50#121—21no—
51C4×Q8—21no—
52C4×Q8—21no—
53C4×D4—21no—
54C4×D4—21no—
55#117—21no—
56C2×C4⋊C4—21no—
57C2×C4⋊C4D412yes—
58C2×C42C2311yescharacteristic
59#107—22no—
60#106—24no—
61C4⋊C8—21no—
62C4⋊C8—21no—
63C4⋊C8D412yes—
64#103—24no—
65#103—22no—
66#103—24no—
67#102—24no—
68#102—22no—
69C22⋊C8—21no—
70#97—21no—
71#97D412yes—
72C4×C8—21no—
73C4×C8—21no—
74C4×C8—21no—
75#41—44no—
76C2×Q8—21no—
77C2×Q8—21no—
78C2×Q8—21no—
79C2×Q8—41no—
80C2×Q8—41no—
81C2×Q8—82no—
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—
92M16—22no—
93M16—42no—
94M16—41no—
95M16—42no—
96C2×C8—22no—
97C2×C8—22no—
98C2×C8—22no—
99C2×C8—41no—
100C4⋊C4—22no—
101C4⋊C4—21no—
102C4⋊C4—22no—
103C4⋊C4—21no—
104C4⋊C4—21no—
105C4⋊C4—21no—
106C4⋊C4—21no—
107C4⋊C4—21no—
108C4⋊C4—41no—
109C4⋊C4—41no—
110C4⋊C4—41no—
111C4⋊C4—41no—
112C22⋊C4—41no—
113C22⋊C4—41no—
114C22⋊C4—41no—
115C42—41no—
116C42C2×D411yescharacteristic
117C42C2×D411yescharacteristic
118C42C2×D411yescharacteristic
119C42C2×D411yescharacteristic
120C23—41no—
121C23#12111yescharacteristic
122Q8—42no—
123Q8—42no—
124Q8—41no—
125Q8—41no—
126Q8—82no—
127Q8—82no—
128D4—42no—
129D4—41no—
130D4—41no—
131D4—42no—
132D4—82no—
133C2×C4—22no—
134C2×C4—21no—
135C2×C4—21no—
136C2×C4—22no—
137C2×C4—41no—
138C2×C4—41no—
139C2×C4—41no—
140C2×C4—42no—
141C2×C4—82no—
142C2×C4—82no—
143C2×C4#12011yescharacteristic
144C2×C4#12011yescharacteristic
145C2×C4#12011yescharacteristic
146C2×C4#12111yescharacteristic
147C2×C4#12111yescharacteristic
148C2×C4#12711yescharacteristic
149C8—42no—
150C8—42no—
151C8—42no—
152C8—42no—
153C22—21no—
154C22—21no—
155C22—21no—
156C22—81no—
157C22—81no—
158C22—81no—
159C22#39211yescharacteristic
160C4—21no—
161C4—21no—
162C4—21no—
163C4—81no—
164C4—81no—
165C4—81no—
166C4—82no—
167C4#45412yes—
168C4#45511yescharacteristic
169C4#45611yescharacteristic
170C4#47112yes—
171C2—41no—
172C2—161no—
173C2#200311yescharacteristic
174C2#200711yescharacteristic
175C2#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
2. #3229C2semidirect
4. #1997C2semidirect
5. #1526C2semidirect
7. #1299C2semidirect

Maximal subgroups

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

subgroupindex
#32542
#32292
#31412
#19972
#15262
#14842
#12992