Finite Group Databasebeta

← #10209

Subgroups — #10209

Order 256.

237 conjugacy classes of subgroups; only the 47 normal subgroups are shown.

normal subgroups only

This group has 237 conjugacy classes of subgroups — too many to draw. Only its 47 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#3216C211yesmaximal, characteristic, complemented
3#3214C211yesmaximal, characteristic
4#1515C211yesmaximal, characteristic, complemented
5#1514C211yesmaximal, characteristic, complemented
6#1514C211yesmaximal, characteristic, complemented
7#1308C211yesmaximal, characteristic, complemented
8#533C2211yescharacteristic
9#532C2211yescharacteristic
10#532C2211yescharacteristic
11#501—21no—
12#497—21no—
13#496—21no—
14#490—21no—
15#489—21no—
16#489—21no—
17#482—22no—
18#476—21no—
19#474—21no—
20#462—21no—
21#461—21no—
22#460—21no—
23#459—21no—
24#452—22no—
25#449—22no—
26#432C2211yescharacteristic
27#432C2211yescharacteristic
28#432C2211yescharacteristic
29#383C2211yescharacteristic
30#328—22no—
31#327—22no—
32#327—22no—
33C2×#41—21no—
34C2×#41—21no—
35C2×Q16—22no—
36C2×SD16—22no—
37C2×SD16—22no—
38C2×D8—22no—
39C4⋊Q8D411yescharacteristic
40C4⋊Q8D411yescharacteristic
41C4⋊Q8D411yescharacteristic
42C4⋊Q8D411yescharacteristic
43#127D411yescharacteristic
44#127D411yescharacteristic
45#125—21no—
46#124—21no—
47#124—21no—
48C22⋊Q8—21no—
49C22⋊Q8—21no—
50#121—21no—
51#121—21no—
52#121—21no—
53#121—21no—
54C4×Q8—21no—
55C4×Q8—21no—
56C4×D4—21no—
57C4×D4—21no—
58C4×D4—21no—
59C4×D4—21no—
60#117—21no—
61C2×C42C2311yescharacteristic
62#107—22no—
63#106—22no—
64C4⋊C8—21no—
65C4⋊C8—21no—
66C4⋊C8—21no—
67#103—22no—
68#103—22no—
69#102—22no—
70#102—22no—
71#102—22no—
72#102—22no—
73C22⋊C8D412yes—
74C22⋊C8D412yes—
75C22⋊C8D412yes—
76#97—21no—
77#97—21no—
78#97—21no—
79#95—21no—
80#95—21no—
81#95—21no—
82#41—82no—
83#41—82no—
84C2×Q8—21no—
85C2×Q8—21no—
86C2×Q8—21no—
87C2×Q8—21no—
88C2×D4—21no—
89C2×D4—21no—
90C2×D4—21no—
91C2×D4—21no—
92C2×D4—21no—
93C2×D4—21no—
94C2×D4—41no—
95C2×D4—41no—
96C22×C4—41no—
97C22×C4—41no—
98C22×C4—41no—
99C22×C4C2×D411yescharacteristic
100C22×C4C2×D411yescharacteristic
101C22×C4C2×D411yescharacteristic
102Q16—82no—
103SD16—82no—
104SD16—82no—
105D8—82no—
106C2×C8—22no—
107C2×C8—22no—
108C2×C8—22no—
109C4⋊C4—21no—
110C4⋊C4—21no—
111C4⋊C4—21no—
112C4⋊C4—21no—
113C4⋊C4—21no—
114C4⋊C4—21no—
115C4⋊C4—21no—
116C4⋊C4—21no—
117C4⋊C4—41no—
118C4⋊C4—41no—
119C22⋊C4—41no—
120C22⋊C4—41no—
121C22⋊C4—41no—
122C22⋊C4—41no—
123C22⋊C4—41no—
124C42—41no—
125C42C2×D411yescharacteristic
126C42C2×D411yescharacteristic
127C42C2×D411yescharacteristic
128C42#4111yescharacteristic
129C23—41no—
130C23—41no—
131C23#12711yescharacteristic
132Q8—41no—
133Q8—42no—
134Q8—41no—
135Q8—42no—
136Q8—81no—
137D4—41no—
138D4—42no—
139D4—41no—
140D4—42no—
141D4—42no—
142D4—81no—
143D4—81no—
144D4—82no—
145D4—82no—
146C2×C4—41no—
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—81no—
156C2×C4—81no—
157C2×C4—81no—
158C2×C4#12011yescharacteristic
159C2×C4#12011yescharacteristic
160C2×C4#12011yescharacteristic
161C2×C4#12111yescharacteristic
162C2×C4#12111yescharacteristic
163C2×C4#12111yescharacteristic
164C8—42no—
165C8—42no—
166C8—42no—
167C22—21no—
168C22—21no—
169C22—21no—
170C22—81no—
171C22—81no—
172C22—81no—
173C22—81no—
174C22—81no—
175C22—81no—
176C22#39211yescharacteristic
177C4—21no—
178C4—21no—
179C4—21no—
180C4—21no—
181C4—21no—
182C4—21no—
183C4—81no—
184C4—81no—
185C4—81no—
186C4—81no—
187C4—161no—
188C2—41no—
189C2—161no—
190C2—161no—
191C2#200811yescharacteristic
192C2#200811yescharacteristic
193C2#200911yescharacteristic
subgroupG/H
center, socleC22#392
commutator, Frattini, cosocleC2×C42C23
prime radical#3214C2
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. #3216C2semidirect
4. #1515C2semidirect
5. #1514C2semidirect
6. #1514C2semidirect
7. #1308C2semidirect

Maximal subgroups

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

subgroupindex
#32352
#32162
#32142
#15152
#15142
#13082