Finite Group Databasebeta

← #10042

Subgroups — #10042

Order 256.

249 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 249 conjugacy classes of subgroups — too many to draw. Only its 49 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#3235C211yesmaximal, characteristic, complemented
2#3110C211yesmaximal, characteristic, complemented
3#3100C211yesmaximal, characteristic
4#1536C211yesmaximal, characteristic, complemented
5#1497C211yesmaximal, characteristic, complemented
6#1466C211yesmaximal, characteristic, complemented
7#1287C211yesmaximal, characteristic, complemented
8#533C2211yescharacteristic
9#532C2211yescharacteristic
10C2×C4×D4C2211yescharacteristic
11#501—21no—
12#490—21no—
13#484—21no—
14#482—22no—
15#480—21no—
16#476—21no—
17#474—21no—
18#474—21no—
19#474—21no—
20#461—21no—
21#459—21no—
22#450—21no—
23#447—21no—
24#440—22no—
25C4×D8—22no—
26#432C2211yescharacteristic
27#432C2211yescharacteristic
28#432C2211yescharacteristic
29#431C2211yescharacteristic
30#329—22no—
31Q8⋊C8—22no—
32#323—22no—
33C2×#41—21no—
34C22×D4D411yescharacteristic
35C23×C4—21no—
36C2×SD16—22no—
37C2×D8—22no—
38C4⋊Q8D411yescharacteristic
39C4⋊Q8D411yescharacteristic
40C4⋊Q8D411yescharacteristic
41#127D411yescharacteristic
42#125—21no—
43#124—21no—
44C22⋊Q8—21no—
45C22⋊Q8—21no—
46#121—21no—
47#121—21no—
48C4×Q8—21no—
49C4×Q8—21no—
50C4×D4—21no—
51C4×D4—21no—
52C4×D4—21no—
53C4×D4—21no—
54C4×D4—21no—
55C4×D4—21no—
56#117—21no—
57C2×C4⋊C4D411yescharacteristic
58C2×C22⋊C4—21no—
59C2×C42C2311yescharacteristic
60#107—22no—
61#107—22no—
62#107—22no—
63#106—22no—
64C4⋊C8—21no—
65C4⋊C8—21no—
66C4⋊C8—21no—
67C4⋊C8—21no—
68#103—22no—
69#102—22no—
70#102—22no—
71#102—22no—
72#102—22no—
73#102—22no—
74C22⋊C8—21no—
75C22⋊C8—21no—
76C22⋊C8—21no—
77C22⋊C8D412yes—
78#97—21no—
79#97—21no—
80#97D412yes—
81C4×C8D412yes—
82C24—21no—
83#41—82no—
84C2×Q8—21no—
85C2×Q8—21no—
86C2×Q8—21no—
87C2×D4—21no—
88C2×D4—21no—
89C2×D4—21no—
90C2×D4—21no—
91C2×D4—21no—
92C2×D4—42no—
93C2×D4—41no—
94C22×C4—21no—
95C22×C4—41no—
96C22×C4—41no—
97C22×C4—41no—
98C22×C4—82no—
99C22×C4C2×D411yescharacteristic
100C22×C4C2×D411yescharacteristic
101C22×C4#4111yescharacteristic
102SD16—44no—
103D8—44no—
104C2×C8—22no—
105C2×C8—22no—
106C2×C8—22no—
107C2×C8—41no—
108C4⋊C4—21no—
109C4⋊C4—21no—
110C4⋊C4—21no—
111C4⋊C4—21no—
112C4⋊C4—21no—
113C4⋊C4—21no—
114C4⋊C4—21no—
115C4⋊C4—21no—
116C4⋊C4—41no—
117C4⋊C4—41no—
118C22⋊C4—41no—
119C22⋊C4—41no—
120C22⋊C4—41no—
121C22⋊C4—41no—
122C22⋊C4—41no—
123C42—41no—
124C42C2×D411yescharacteristic
125C42C2×D411yescharacteristic
126C42C2×D411yescharacteristic
127C42C2×D411yescharacteristic
128C23—22no—
129C23—41no—
130C23—41no—
131C23—41no—
132C23—81no—
133C23—81no—
134C23#12111yescharacteristic
135Q8—42no—
136Q8—42no—
137Q8—81no—
138D4—22no—
139D4—22no—
140D4—41no—
141D4—41no—
142D4—42no—
143D4—42no—
144D4—81no—
145D4—82no—
146C2×C4—22no—
147C2×C4—41no—
148C2×C4—41no—
149C2×C4—41no—
150C2×C4—41no—
151C2×C4—41no—
152C2×C4—41no—
153C2×C4—82no—
154C2×C4—82no—
155C2×C4—82no—
156C2×C4—81no—
157C2×C4#12011yescharacteristic
158C2×C4#12011yescharacteristic
159C2×C4#12011yescharacteristic
160C2×C4#12111yescharacteristic
161C2×C4#12111yescharacteristic
162C2×C4#12711yescharacteristic
163C8—42no—
164C8—42no—
165C8—42no—
166C8—81no—
167C22—21no—
168C22—21no—
169C22—21no—
170C22—42no—
171C22—82no—
172C22—82no—
173C22—81no—
174C22—81no—
175C22—81no—
176C22—81no—
177C22—81no—
178C22#39211yescharacteristic
179C4—21no—
180C4—21no—
181C4—21no—
182C4—21no—
183C4—21no—
184C4—81no—
185C4—81no—
186C4—81no—
187C4—81no—
188C4#47212yes—
189C2—41no—
190C2—82no—
191C2—161no—
192C2#200911yescharacteristic
193C2#201011yescharacteristic
194C2#201811yescharacteristic
subgroupG/H
center, socleC22#392
commutator, Frattini, cosocleC2×C42C23
prime radical#3100C2
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
1. #3235C2semidirect
2. #3110C2semidirect
4. #1536C2semidirect
5. #1497C2semidirect
6. #1466C2semidirect
7. #1287C2semidirect

Maximal subgroups

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

subgroupindex
#32352
#31102
#31002
#15362
#14972
#14662
#12872