Finite Group Databasebeta

← #8998

Subgroups — #8998

Order 256.

207 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 207 conjugacy classes of subgroups — too many to draw. Only its 59 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#3172C211yesmaximal, characteristic
2#2995C211yesmaximal, characteristic
3#1995C211yesmaximal, characteristic, complemented
4#1500C211yesmaximal, characteristic
5#1497C211yesmaximal, characteristic, complemented
6#1300C211yesmaximal, characteristic, complemented
7#1299C211yesmaximal, characteristic, complemented
8#533C2211yescharacteristic
9#529C2211yescharacteristic
10#501C411yescharacteristic, complemented
11#497C411yescharacteristic, complemented
12#491C411yescharacteristic, complemented
13#489C411yescharacteristic, complemented
14#470—21no—
15#468—21no—
16#446—21no—
17C4×Q16—21no—
18C4×SD16—21no—
19#436—21no—
20#435—21no—
21#435—21no—
22#433—21no—
23#432C2211yescharacteristic
24#431C2211yescharacteristic
25#424—21no—
26#423C2211yescharacteristic
27C4×M16C2211yescharacteristic
28#376C2211yescharacteristic
29#339—21no—
30#339—21no—
31#328—21no—
32#327—21no—
33Q8⋊C8—21no—
34Q8⋊C8—21no—
35C2×Q16—21no—
36C2×SD16—21no—
37C2×M16—21no—
38C2×M16—21no—
39C2×M16D411yescharacteristic
40C2×M16D411yescharacteristic
41C22×C8—21no—
42C4⋊Q8C2×C411yescharacteristic
43C4⋊Q8C2×C411yescharacteristic
44#126—21no—
45#125—21no—
46#125C2×C411yescharacteristic
47#124C2×C411yescharacteristic
48#123—21no—
49C22⋊Q8—21no—
50C22⋊Q8—21no—
51C22⋊Q8D411yescharacteristic
52#121D411yescharacteristic
53C4×Q8—21no—
54C4×Q8—21no—
55C4×Q8D411yescharacteristic
56C4×D4—21no—
57C4×D4D411yescharacteristic
58#117—21no—
59#117—21no—
60C2×C42—21no—
61C2×C42C2311yescharacteristic
62#107—21no—
63#106—21no—
64C4⋊C8—21no—
65C4⋊C8—21no—
66C4⋊C8—21no—
67C4⋊C8—21no—
68C4⋊C8—21no—
69C4⋊C8D411yescharacteristic
70C4⋊C8D411yescharacteristic
71#103—21no—
72#103—21no—
73#103—21no—
74#102—21no—
75C22⋊C8—21no—
76C22⋊C8—21no—
77C22⋊C8—21no—
78C22⋊C8—21no—
79#97—21no—
80#97—21no—
81#97C2×C411yescharacteristic
82#97C2×C411yescharacteristic
83C4×C8—21no—
84C4×C8—21no—
85C4×C8—21no—
86#95—21no—
87#95—21no—
88C2×Q8—21no—
89C2×Q8—21no—
90C2×Q8C22⋊C411yescharacteristic
91C2×D4—21no—
92C2×D4C22⋊C411yescharacteristic
93C22×C4—21no—
94C22×C4—41no—
95C22×C4C2×D411yescharacteristic
96C22×C4C2×D411yescharacteristic
97C22×C4#4111yescharacteristic
98Q16—42no—
99SD16—42no—
100M16—42no—
101M16—41no—
102M16—41no—
103M16—81no—
104C2×C8—21no—
105C2×C8—21no—
106C2×C8—21no—
107C2×C8—21no—
108C2×C8—21no—
109C2×C8—21no—
110C2×C8—41no—
111C2×C8—81no—
112C4⋊C4—21no—
113C4⋊C4—21no—
114C4⋊C4—21no—
115C4⋊C4—21no—
116C4⋊C4—21no—
117C4⋊C4—21no—
118C4⋊C4—21no—
119C4⋊C4—21no—
120C4⋊C4—41no—
121C4⋊C4—41no—
122C4⋊C4C22⋊C411yescharacteristic
123C4⋊C4C22⋊C411yescharacteristic
124C22⋊C4—21no—
125C22⋊C4—21no—
126C22⋊C4—21no—
127C22⋊C4—21no—
128C22⋊C4—21no—
129C22⋊C4—41no—
130C42—21no—
131C42—22no—
132C42—41no—
133C42—41no—
134C42C22×C411yescharacteristic
135C42C2×D411yescharacteristic
136C42C2×D411yescharacteristic
137C42#4111yescharacteristic
138C23—21no—
139C23#12111yescharacteristic
140Q8—41no—
141Q8—41no—
142Q8—41no—
143Q8—42no—
144D4—41no—
145D4—81no—
146C2×C4—21no—
147C2×C4—21no—
148C2×C4—21no—
149C2×C4—21no—
150C2×C4—21no—
151C2×C4—41no—
152C2×C4—41no—
153C2×C4—41no—
154C2×C4—41no—
155C2×C4—42no—
156C2×C4—41no—
157C2×C4—41no—
158C2×C4—81no—
159C2×C4—81no—
160C2×C4C2×C22⋊C411yescharacteristic
161C2×C4C4×D411yescharacteristic
162C2×C4C4×D411yescharacteristic
163C2×C4#12011yescharacteristic
164C2×C4#12111yescharacteristic
165C2×C4#12311yescharacteristic
166C8—22no—
167C8—41no—
168C8—41no—
169C8—41no—
170C8—41no—
171C8—41no—
172C8—81no—
173C22—21no—
174C22—21no—
175C22—21no—
176C22—41no—
177C22—41no—
178C22—41no—
179C22#38611yescharacteristic
180C4—21no—
181C4—21no—
182C4—21no—
183C4—21no—
184C4—21no—
185C4—41no—
186C4—41no—
187C4—41no—
188C4—82no—
189C4—81no—
190C4—81no—
191C4#45511yescharacteristic
192C4#45611yescharacteristic
193C2—41no—
194C2—81no—
195C2#188111yescharacteristic
196C2#188211yescharacteristic
197C2#188411yescharacteristic
subgroupG/H
centerC2×C4#120
commutatorC42C22×C4
Frattini, cosocleC2×C42C23
socleC22#386
prime radical#121D4
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
3. #1995C2semidirect
5. #1497C2semidirect
6. #1300C2semidirect
7. #1299C2semidirect
10. #501C4semidirect
11. #497C4semidirect
12. #491C4semidirect
13. #489C4semidirect

Maximal subgroups

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

subgroupindex
#31722
#29952
#19952
#15002
#14972
#13002
#12992