Finite Group Databasebeta

← #9377

Subgroups — #9377

Order 256.

217 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 217 conjugacy classes of subgroups — too many to draw. Only its 61 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#3239C211yesmaximal, characteristic
2#2992C211yesmaximal, characteristic
3#1981C211yesmaximal, characteristic, complemented
4#1556C211yesmaximal, characteristic, complemented
5#1509C211yesmaximal, characteristic
6#1290C211yesmaximal, characteristic, complemented
7#1285C211yesmaximal, characteristic, complemented
8C2×C4⋊Q8C2211yescharacteristic
9#529C2211yescharacteristic
10#501C412yescomplemented
11#489C412yescomplemented
12#484—21no—
13#481—21no—
14#475—21no—
15#474—21no—
16#446—21no—
17#435—21no—
18#435—21no—
19C8×D4—21no—
20#433—21no—
21#432C2211yescharacteristic
22#432C2211yescharacteristic
23#432C2211yescharacteristic
24#424—21no—
25C2×C4⋊C8C2211yescharacteristic
26C2×#103—21no—
27C2×#103—21no—
28C4×C4⋊C4C2211yescharacteristic
29#332—21no—
30#332—21no—
31#328—21no—
32#327—21no—
33C22×Q8—21no—
34C2×M16—21no—
35C22×C8—21no—
36C22×C8D411yescharacteristic
37C22×C8D411yescharacteristic
38C4⋊Q8—21no—
39C4⋊Q8—21no—
40C4⋊Q8—21no—
41C4⋊Q8C2×C412yes—
42#126—21no—
43#125C2×C411yescharacteristic
44#124C2×C411yescharacteristic
45#123—21no—
46C22⋊Q8D411yescharacteristic
47#121D411yescharacteristic
48C4×Q8D411yescharacteristic
49C4×D4—21no—
50C4×D4D411yescharacteristic
51#117—21no—
52C2×C4⋊C4—21no—
53C2×C4⋊C4—21no—
54C2×C4⋊C4—21no—
55C2×C42—21no—
56C2×C42C2311yescharacteristic
57#107—22no—
58#106—22no—
59C4⋊C8—21no—
60C4⋊C8—21no—
61C4⋊C8—21no—
62C4⋊C8—21no—
63C4⋊C8—21no—
64C4⋊C8—21no—
65C4⋊C8D411yescharacteristic
66C4⋊C8D411yescharacteristic
67#103—22no—
68#103—42no—
69#103—42no—
70#102—22no—
71C22⋊C8—21no—
72C22⋊C8—21no—
73C22⋊C8—21no—
74C22⋊C8—21no—
75C22⋊C8—21no—
76#97—21no—
77#97—21no—
78#97—21no—
79#97C2×C412yes—
80C4×C8—21no—
81#95—21no—
82C2×Q8—22no—
83C2×Q8—41no—
84C2×Q8—41no—
85C2×Q8—41no—
86C2×Q8—41no—
87C2×Q8—41no—
88C2×Q8C22⋊C411yescharacteristic
89C2×D4—21no—
90C2×D4C22⋊C411yescharacteristic
91C22×C4—21no—
92C22×C4—41no—
93C22×C4—41no—
94C22×C4C2×D411yescharacteristic
95C22×C4C2×D411yescharacteristic
96C22×C4#4111yescharacteristic
97M16—81no—
98C2×C8—22no—
99C2×C8—21no—
100C2×C8—21no—
101C2×C8—21no—
102C2×C8—21no—
103C2×C8—41no—
104C2×C8—41no—
105C2×C8—41no—
106C2×C8—81no—
107C4⋊C4—21no—
108C4⋊C4—21no—
109C4⋊C4—21no—
110C4⋊C4—21no—
111C4⋊C4—22no—
112C4⋊C4—22no—
113C4⋊C4—22no—
114C4⋊C4—41no—
115C4⋊C4—41no—
116C4⋊C4—41no—
117C4⋊C4C22⋊C411yescharacteristic
118C4⋊C4C22⋊C411yescharacteristic
119C22⋊C4—21no—
120C22⋊C4—21no—
121C22⋊C4—21no—
122C22⋊C4—21no—
123C22⋊C4—21no—
124C42—21no—
125C42—81no—
126C42C22×C411yescharacteristic
127C42C2×D411yescharacteristic
128C42C2×D411yescharacteristic
129C42#4111yescharacteristic
130C23—21no—
131C23#12111yescharacteristic
132Q8—41no—
133Q8—42no—
134Q8—42no—
135Q8—81no—
136Q8—81no—
137D4—41no—
138D4—41no—
139D4—41no—
140C2×C4—21no—
141C2×C4—21no—
142C2×C4—21no—
143C2×C4—21no—
144C2×C4—21no—
145C2×C4—21no—
146C2×C4—21no—
147C2×C4—41no—
148C2×C4—41no—
149C2×C4—41no—
150C2×C4—42no—
151C2×C4—42no—
152C2×C4—81no—
153C2×C4—81no—
154C2×C4—81no—
155C2×C4—81no—
156C2×C4C2×C22⋊C411yescharacteristic
157C2×C4C4×D411yescharacteristic
158C2×C4C4×D411yescharacteristic
159C2×C4#12011yescharacteristic
160C2×C4#12111yescharacteristic
161C2×C4#12311yescharacteristic
162C8—42no—
163C8—41no—
164C8—41no—
165C8—41no—
166C8—41no—
167C8—81no—
168C22—21no—
169C22—21no—
170C22—41no—
171C22—41no—
172C22—41no—
173C22#38611yescharacteristic
174C22#47111yescharacteristic
175C22#47311yescharacteristic
176C4—21no—
177C4—21no—
178C4—21no—
179C4—21no—
180C4—21no—
181C4—41no—
182C4—41no—
183C4—41no—
184C4—82no—
185C4—82no—
186C4#45311yescharacteristic
187C4#45411yescharacteristic
188C2—21no—
189C2—21no—
190C2—81no—
191C2#188311yescharacteristic
192C2#190811yescharacteristic
193C2#190911yescharacteristic
subgroupG/H
center, socleC22#386
commutatorC42C22×C4
Frattini, cosocleC2×C42C23
prime radical#121D4
LCSG ▷ 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. #1981C2semidirect
4. #1556C2semidirect
6. #1290C2semidirect
7. #1285C2semidirect
10. #501C4semidirect
11. #489C4semidirect

Maximal subgroups

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

subgroupindex
#32392
#29922
#19812
#15562
#15092
#12902
#12852