Finite Group Databasebeta

← #9051

Subgroups — #9051

Order 256.

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

normal subgroups only

This group has 218 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#3108C211yesmaximal, characteristic
2#2992C211yesmaximal, characteristic
3#1985C211yesmaximal, characteristic, complemented
4#1524C211yesmaximal, characteristic
5#1491C211yesmaximal, characteristic, complemented
6#1302C211yesmaximal, characteristic, complemented
7#1283C211yesmaximal, characteristic, complemented
8#529C2211yescharacteristic
9C2×C4×Q8C2211yescharacteristic
10#483C411yescharacteristic
11#480C411yescharacteristic
12#479—21no—
13#464—21no—
14#451C411yescharacteristic
15#448C411yescharacteristic
16#446—21no—
17C4×Q16—21no—
18C4×SD16—21no—
19#435—21no—
20#435—21no—
21C8×D4—21no—
22#433—21no—
23#432C2211yescharacteristic
24#431C2211yescharacteristic
25#424—21no—
26C2×C4⋊C8C2211yescharacteristic
27C4×M16C2211yescharacteristic
28C2×#97C2211yescharacteristic
29#336—21no—
30#336—21no—
31#330—21no—
32#329—21no—
33Q8⋊C8—21no—
34Q8⋊C8—21no—
35C22×Q8C2×C411yescharacteristic
36C2×Q16—21no—
37C2×SD16—21no—
38C2×M16—21no—
39C2×M16—21no—
40C22×C8—21no—
41C22×C8—21no—
42C22×C8D411yescharacteristic
43C22×C8D411yescharacteristic
44#126—21no—
45#125D411yescharacteristic
46#124D411yescharacteristic
47#123—21no—
48C22⋊Q8C2×C411yescharacteristic
49#121C2×C411yescharacteristic
50C4×Q8—21no—
51C4×Q8—21no—
52C4×Q8—21no—
53C4×Q8—21no—
54C4×Q8D411yescharacteristic
55C4×D4—21no—
56C4×D4D411yescharacteristic
57#117—21no—
58C2×C4⋊C4—21no—
59C2×C4⋊C4C2×C411yescharacteristic
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⋊C8C2×C411yescharacteristic
79C22⋊C8C2×C411yescharacteristic
80#97—21no—
81#97—21no—
82#97—21no—
83#97—21no—
84#97—21no—
85C4×C8—21no—
86C4×C8—21no—
87C4×C8—21no—
88C2×Q8—21no—
89C2×Q8—21no—
90C2×Q8—42no—
91C2×Q8C22⋊C411yescharacteristic
92C2×D4—21no—
93C2×D4C22⋊C411yescharacteristic
94C22×C4—21no—
95C22×C4—21no—
96C22×C4—21no—
97C22×C4C22×C411yescharacteristic
98C22×C4C2×D411yescharacteristic
99C22×C4#4111yescharacteristic
100Q16—42no—
101SD16—42no—
102M16—22no—
103M16—41no—
104M16—81no—
105C2×C8—22no—
106C2×C8—21no—
107C2×C8—21no—
108C2×C8—21no—
109C2×C8—21no—
110C2×C8—21no—
111C2×C8—21no—
112C2×C8—41no—
113C2×C8—41no—
114C2×C8—41no—
115C2×C8—41no—
116C2×C8—41no—
117C2×C8—81no—
118C4⋊C4—21no—
119C4⋊C4—21no—
120C4⋊C4—21no—
121C4⋊C4—21no—
122C4⋊C4—21no—
123C4⋊C4—21no—
124C4⋊C4—41no—
125C4⋊C4—41no—
126C4⋊C4C22⋊C411yescharacteristic
127C4⋊C4C22⋊C411yescharacteristic
128C22⋊C4—21no—
129C22⋊C4—21no—
130C22⋊C4—21no—
131C22⋊C4—21no—
132C22⋊C4—21no—
133C42—21no—
134C42—41no—
135C42—41no—
136C42C2×D411yescharacteristic
137C42C2×D411yescharacteristic
138C42C2×D411yescharacteristic
139C42#4111yescharacteristic
140C23—21no—
141C23C4×D411yescharacteristic
142Q8—22no—
143Q8—22no—
144Q8—41no—
145Q8—41no—
146Q8—41no—
147D4—41no—
148D4—41no—
149D4—41no—
150C2×C4—21no—
151C2×C4—21no—
152C2×C4—21no—
153C2×C4—21no—
154C2×C4—22no—
155C2×C4—21no—
156C2×C4—21no—
157C2×C4—21no—
158C2×C4—41no—
159C2×C4—41no—
160C2×C4—41no—
161C2×C4—41no—
162C2×C4—41no—
163C2×C4—41no—
164C2×C4—41no—
165C2×C4—41no—
166C2×C4C2×C22⋊C411yescharacteristic
167C2×C4C4×D411yescharacteristic
168C2×C4#12011yescharacteristic
169C2×C4#12111yescharacteristic
170C2×C4#12111yescharacteristic
171C2×C4#12311yescharacteristic
172C8—42no—
173C8—41no—
174C8—41no—
175C8—42no—
176C8—41no—
177C8—41no—
178C8—41no—
179C8—41no—
180C22—21no—
181C22—21no—
182C22—41no—
183C22—41no—
184C22—41no—
185C22#38611yescharacteristic
186C22#44311yescharacteristic
187C22#44411yescharacteristic
188C4—21no—
189C4—21no—
190C4—21no—
191C4—21no—
192C4—21no—
193C4—41no—
194C4—41no—
195C4—42no—
196C4—41no—
197C4—81no—
198C4#45511yescharacteristic
199C4#45611yescharacteristic
200C2—21no—
201C2—21no—
202C2—81no—
203C2#187611yescharacteristic
204C2#188211yescharacteristic
205C2#189111yescharacteristic
subgroupG/H
centerC2×C4#120
commutatorC22×C4C22×C4
Frattini, cosocleC2×C42C23
socleC22#386
prime radical#121C2×C4
LCSG ▷ C22×C4 ▷ 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. #1985C2semidirect
5. #1491C2semidirect
6. #1302C2semidirect
7. #1283C2semidirect

Maximal subgroups

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

subgroupindex
#31082
#29922
#19852
#15242
#14912
#13022
#12832