Finite Group Databasebeta

← #10242

Subgroups — #10242

Order 256.

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

normal subgroups only

This group has 329 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#3205C211yesmaximal, characteristic
2#3037C211yesmaximal, characteristic, complemented
3#2548C211yesmaximal, characteristic, complemented
4#1545C211yesmaximal, characteristic, complemented
5#1502C211yesmaximal, characteristic, complemented
6#1479C211yesmaximal, characteristic, complemented
7#1303C211yesmaximal, characteristic, complemented
8C2×#137—21no—
9#532C2211yescharacteristic
10C2×#127C2211yescharacteristic
11C2×#124—21no—
12C2×C4×Q8C2211yescharacteristic
13#493—21no—
14#486—21no—
15#464—22no—
16#462—21no—
17#460—21no—
18#459—22no—
19#447—22no—
20#431C2211yescharacteristic
21#431C2211yescharacteristic
22#423C2211yescharacteristic
23#417—21no—
24#394—22no—
25#390—22no—
26#390—21no—
27C4×C4⋊C4C2211yescharacteristic
28#331—22no—
29#327—22no—
30Q8⋊C8—22no—
31C2×#41—21no—
32C22×Q8D411yescharacteristic
33C22×D4—21no—
34C22×D4—21no—
35C22×D4D412yes—
36#137—42no—
37#137—42no—
38C2×Q16—22no—
39C2×SD16—22no—
40C2×D8—24no—
41C2×M16—21no—
42C4⋊Q8D411yescharacteristic
43#127—21no—
44#127—21no—
45#127—21no—
46#127—21no—
47#127D411yescharacteristic
48#124—21no—
49#124—42no—
50#124—42no—
51#121—21no—
52#121—21no—
53C4×Q8—21no—
54C4×Q8—21no—
55C4×Q8—21no—
56C4×Q8—21no—
57C4×D4—21no—
58C4×D4—21no—
59C2×C4⋊C4—21no—
60C2×C4⋊C4—21no—
61C2×C4⋊C4D411yescharacteristic
62C2×C22⋊C4—24no—
63C2×C22⋊C4—24no—
64C2×C42—21no—
65C2×C42—21no—
66C2×C42C2311yescharacteristic
67C4⋊C8—21no—
68C4⋊C8—21no—
69C4⋊C8D412yes—
70C4⋊C8D412yes—
71#103—22no—
72#102—22no—
73#102—24no—
74C22⋊C8—21no—
75C22⋊C8—21no—
76C4×C8—21no—
77C4×C8—21no—
78#95D412yes—
79C24—22no—
80#41—41no—
81#41—41no—
82#41—41no—
83#41—41no—
84C2×Q8—21no—
85C2×Q8—21no—
86C2×Q8—21no—
87C2×Q8—21no—
88C2×Q8—21no—
89C2×Q8—21no—
90C2×Q8D812yescomplemented
91C2×D4—21no—
92C2×D4—22no—
93C2×D4—21no—
94C2×D4—21no—
95C2×D4—22no—
96C2×D4—44no—
97C2×D4—41no—
98C2×D4—41no—
99C2×D4—41no—
100C2×D4—41no—
101C2×D4—41no—
102C2×D4—41no—
103C2×D4—41no—
104C2×D4—41no—
105C2×D4—41no—
106C2×D4—82no—
107C22×C4—21no—
108C22×C4—22no—
109C22×C4—21no—
110C22×C4—41no—
111C22×C4C2×D411yescharacteristic
112C22×C4C2×D411yescharacteristic
113C22×C4C2×D411yescharacteristic
114Q16—44no—
115SD16—44no—
116D8—84no—
117M16—41no—
118M16—41no—
119C2×C8—22no—
120C2×C8—22no—
121C2×C8—41no—
122C4⋊C4—21no—
123C4⋊C4—21no—
124C4⋊C4—21no—
125C4⋊C4—21no—
126C4⋊C4—42no—
127C4⋊C4—41no—
128C4⋊C4—41no—
129C22⋊C4—48no—
130C22⋊C4—41no—
131C22⋊C4—41no—
132C22⋊C4—84no—
133C42—41no—
134C42—41no—
135C42—41no—
136C42—41no—
137C42C2×D411yescharacteristic
138C42C2×D411yescharacteristic
139C42C2×D411yescharacteristic
140C42#4111yescharacteristic
141C23—44no—
142C23—44no—
143C23—41no—
144C23—42no—
145C23—42no—
146C23—42no—
147C23#12011yescharacteristic
148Q8—22no—
149Q8—22no—
150Q8—41no—
151Q8—41no—
152Q8—41no—
153Q8—41no—
154D4—41no—
155D4—41no—
156D4—42no—
157D4—44no—
158D4—81no—
159D4—82no—
160D4—82no—
161D4—82no—
162D4—81no—
163D4—81no—
164D4—81no—
165D4—81no—
166D4—81no—
167D4—81no—
168C2×C4—22no—
169C2×C4—22no—
170C2×C4—41no—
171C2×C4—41no—
172C2×C4—41no—
173C2×C4—42no—
174C2×C4—42no—
175C2×C4—42no—
176C2×C4—41no—
177C2×C4—41no—
178C2×C4—42no—
179C2×C4—81no—
180C2×C4—81no—
181C2×C4#12011yescharacteristic
182C2×C4#12011yescharacteristic
183C2×C4#12111yescharacteristic
184C2×C4#12111yescharacteristic
185C2×C4#12111yescharacteristic
186C2×C4#12711yescharacteristic
187C2×C4C2×D811yescharacteristic
188C2×C4#13511yescharacteristic
189C2×C4AGL(1,ℤ/8)11yescharacteristic
190C2×C4AGL(1,ℤ/8)11yescharacteristic
191C8—42no—
192C8—42no—
193C8—81no—
194C22—21no—
195C22—21no—
196C22—84no—
197C22—84no—
198C22—81no—
199C22—82no—
200C22—82no—
201C22—81no—
202C22—82no—
203C22—81no—
204C22#39211yescharacteristic
205C22#44711yescharacteristic
206C22#44911yescharacteristic
207C4—21no—
208C4—21no—
209C4—21no—
210C4—42no—
211C4—82no—
212C4—81no—
213C4—81no—
214C4#44711yescharacteristic
215C4#44911yescharacteristic
216C4#45512yes—
217C4#46611yescharacteristic
218C4#46911yescharacteristic
219C2—21no—
220C2—21no—
221C2—161no—
222C2—162no—
223C2#199911yescharacteristic
224C2#200411yescharacteristic
225C2#200811yescharacteristic
subgroupG/H
center, socleC22#392
commutator, Frattini, cosocleC2×C42C23
prime radical#3205C2
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
2. #3037C2semidirect
3. #2548C2semidirect
4. #1545C2semidirect
5. #1502C2semidirect
6. #1479C2semidirect
7. #1303C2semidirect
90. C2×Q8D8semidirect

Maximal subgroups

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

subgroupindex
#32052
#30372
#25482
#15452
#15022
#14792
#13032