Finite Group Databasebeta

← #9237

Subgroups — #9237

Order 256.

294 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 294 conjugacy classes of subgroups — too many to draw. Only its 71 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#3100C211yesmaximal, characteristic
2#2293C211yesmaximal, characteristic, complemented
3#1962C211yesmaximal, characteristic, complemented
4#1502C211yesmaximal, characteristic, complemented
5#1486C211yesmaximal, characteristic, complemented
6#1303C211yesmaximal, characteristic, complemented
7#1276C211yesmaximal, characteristic, complemented
8#532C2211yescharacteristic
9C2×C4×D4C2211yescharacteristic
10C2×#117—21no—
11#480—21no—
12#474C412yescomplemented
13#459C412yescomplemented
14#447—21no—
15C4×D8—22no—
16#431C2211yescharacteristic
17#431C2211yescharacteristic
18#423C2211yescharacteristic
19C2×#104—21no—
20C2×#104—21no—
21#390—21no—
22#389—21no—
23#388—22no—
24C4×C4⋊C4C2211yescharacteristic
25C4×C22⋊C4—21no—
26C43C2211yescharacteristic
27#327—22no—
28D4⋊C8—22no—
29C2×#41—21no—
30C22×D4D411yescharacteristic
31C23×C4—21no—
32C2×D8—22no—
33C2×M16—21no—
34C4⋊Q8C2×C411yescharacteristic
35#127C2×C411yescharacteristic
36#124—21no—
37#121—21no—
38#121—21no—
39C4×D4—21no—
40C4×D4—21no—
41C4×D4—21no—
42C4×D4—21no—
43C4×D4C2×C412yes—
44C4×D4D412yes—
45#117—42no—
46#117—42no—
47C2×C4⋊C4—22no—
48C2×C4⋊C4—21no—
49C2×C4⋊C4D411yescharacteristic
50C2×C22⋊C4—21no—
51C2×C22⋊C4—22no—
52C2×C22⋊C4—22no—
53C2×C42—21no—
54C2×C42—21no—
55C2×C42—21no—
56C2×C42D411yescharacteristic
57C2×C42D411yescharacteristic
58C2×C42C2311yescharacteristic
59#107—22no—
60C4⋊C8—21no—
61C4⋊C8—21no—
62C4⋊C8—21no—
63C4⋊C8C2×C412yes—
64#104—42no—
65#104—42no—
66#102—22no—
67#102—22no—
68C22⋊C8—21no—
69C22⋊C8—21no—
70C4×C8—21no—
71C4×C8—21no—
72#95—22no—
73#95D412yes—
74C24—21no—
75#41—41no—
76#41—41no—
77#41—41no—
78#41—41no—
79C2×Q8—21no—
80C2×D4—21no—
81C2×D4—21no—
82C2×D4—21no—
83C2×D4—21no—
84C2×D4—42no—
85C2×D4—41no—
86C2×D4C22⋊C412yes—
87C22×C4—21no—
88C22×C4—21no—
89C22×C4—21no—
90C22×C4—22no—
91C22×C4—22no—
92C22×C4—22no—
93C22×C4—41no—
94C22×C4—41no—
95C22×C4—41no—
96C22×C4—41no—
97C22×C4—41no—
98C22×C4C2×D411yescharacteristic
99C22×C4C2×D411yescharacteristic
100C22×C4C2×D411yescharacteristic
101D8—44no—
102M16—41no—
103M16—41no—
104C2×C8—22no—
105C2×C8—41no—
106C2×C8—41no—
107C4⋊C4—21no—
108C4⋊C4—21no—
109C4⋊C4—21no—
110C4⋊C4—42no—
111C4⋊C4—44no—
112C4⋊C4C22⋊C412yes—
113C22⋊C4—22no—
114C22⋊C4—22no—
115C22⋊C4—44no—
116C22⋊C4—44no—
117C22⋊C4—41no—
118C22⋊C4—41no—
119C42—21no—
120C42—21no—
121C42—21no—
122C42—42no—
123C42—42no—
124C42—41no—
125C42—41no—
126C42D812yescomplemented
127C42C22×C411yescharacteristic
128C42C2×D411yescharacteristic
129C42#4111yescharacteristic
130C42#4111yescharacteristic
131C23—22no—
132C23—42no—
133C23—42no—
134C23—41no—
135C23—41no—
136C23—41no—
137C23#12011yescharacteristic
138Q8—41no—
139Q8—41no—
140D4—22no—
141D4—22no—
142D4—42no—
143D4—42no—
144D4—41no—
145D4—41no—
146D4—81no—
147D4—81no—
148D4—81no—
149C2×C4—22no—
150C2×C4—22no—
151C2×C4—22no—
152C2×C4—22no—
153C2×C4—41no—
154C2×C4—41no—
155C2×C4—41no—
156C2×C4—41no—
157C2×C4—41no—
158C2×C4—41no—
159C2×C4—41no—
160C2×C4—42no—
161C2×C4—42no—
162C2×C4—42no—
163C2×C4—42no—
164C2×C4—42no—
165C2×C4—42no—
166C2×C4—41no—
167C2×C4—42no—
168C2×C4—81no—
169C2×C4—81no—
170C2×C4C2×C22⋊C411yescharacteristic
171C2×C4C4×D411yescharacteristic
172C2×C4C4×D411yescharacteristic
173C2×C4#12111yescharacteristic
174C2×C4#12111yescharacteristic
175C2×C4#12311yescharacteristic
176C2×C4C2×D811yescharacteristic
177C2×C4#13511yescharacteristic
178C2×C4AGL(1,ℤ/8)11yescharacteristic
179C2×C4AGL(1,ℤ/8)11yescharacteristic
180C8—42no—
181C8—81no—
182C8—81no—
183C22—21no—
184C22—21no—
185C22—42no—
186C22—42no—
187C22—42no—
188C22—82no—
189C22—82no—
190C22—81no—
191C22—81no—
192C22—81no—
193C22#38611yescharacteristic
194C22#44711yescharacteristic
195C22#44911yescharacteristic
196C4—21no—
197C4—21no—
198C4—21no—
199C4—42no—
200C4—42no—
201C4—42no—
202C4—82no—
203C4—81no—
204C4#42112yes—
205C4C4×D811yescharacteristic
206C4#44211yescharacteristic
207C4#45911yescharacteristic
208C4#46311yescharacteristic
209C2—21no—
210C2—21no—
211C2—82no—
212C2—161no—
213C2#187911yescharacteristic
214C2#189711yescharacteristic
215C2#190611yescharacteristic
subgroupG/H
centerC2×C4#121
commutatorC42C22×C4
Frattini, cosocleC2×C42C23
socleC22#386
prime radical#3100C2
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
2. #2293C2semidirect
3. #1962C2semidirect
4. #1502C2semidirect
5. #1486C2semidirect
6. #1303C2semidirect
7. #1276C2semidirect
12. #474C4semidirect
13. #459C4semidirect
126. C42D8semidirect

Maximal subgroups

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

subgroupindex
#31002
#22932
#19622
#15022
#14862
#13032
#12762