Finite Group Databasebeta

← #10239

Subgroups — #10239

Order 256.

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

normal subgroups only

This group has 305 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#3226C211yesmaximal, characteristic, complemented
2#3036C211yesmaximal, characteristic
3#2547C211yesmaximal, characteristic, complemented
4#1546C211yesmaximal, characteristic, complemented
5#1502C211yesmaximal, characteristic, complemented
6#1478C211yesmaximal, characteristic, complemented
7#1303C211yesmaximal, characteristic, complemented
8C2×AGL(1,ℤ/8)—21no—
9#532C2211yescharacteristic
10C2×C4⋊Q8C2211yescharacteristic
11C2×#124—21no—
12C2×C4×D4C2211yescharacteristic
13#500—21no—
14#486—21no—
15#480—22no—
16#474—22no—
17#463—22no—
18#461—21no—
19#459—21no—
20#431C2211yescharacteristic
21#431C2211yescharacteristic
22#423C2211yescharacteristic
23#417—21no—
24#397—22no—
25#391—21no—
26#380—22no—
27C4×C4⋊C4C2211yescharacteristic
28#332—22no—
29#327—22no—
30D4⋊C8—22no—
31C2×#41—21no—
32C22×Q8—21no—
33C22×D4D411yescharacteristic
34C23×C4—21no—
35AGL(1,ℤ/8)—42no—
36AGL(1,ℤ/8)—42no—
37C2×SD16—22no—
38C2×D8—22no—
39C2×M16—21no—
40C4⋊Q8—21no—
41C4⋊Q8—21no—
42C4⋊Q8—21no—
43C4⋊Q8—21no—
44C4⋊Q8D411yescharacteristic
45#127D411yescharacteristic
46#124—21no—
47#124—42no—
48#124—42no—
49#121—21no—
50#121—21no—
51C4×D4—21no—
52C4×D4—21no—
53C4×D4—21no—
54C4×D4—21no—
55C4×D4—21no—
56C4×D4—21no—
57C2×C4⋊C4—21no—
58C2×C4⋊C4—21no—
59C2×C4⋊C4D411yescharacteristic
60C2×C4⋊C4D412yes—
61C2×C22⋊C4—24no—
62C2×C22⋊C4—21no—
63C2×C42—21no—
64C2×C42C2311yescharacteristic
65#107—24no—
66C4⋊C8—21no—
67C4⋊C8—21no—
68C4⋊C8D412yes—
69C4⋊C8D412yes—
70#103—22no—
71#102—22no—
72#102—24no—
73C22⋊C8—21no—
74C22⋊C8—21no—
75C4×C8—21no—
76C4×C8—21no—
77#95—24no—
78#95D412yes—
79C24—21no—
80#41—41no—
81#41—41no—
82#41—41no—
83#41—41no—
84C2×Q8—21no—
85C2×Q8—41no—
86C2×Q8—41no—
87C2×Q8—41no—
88C2×Q8—41no—
89C2×Q8—41no—
90C2×Q8—41no—
91C2×D4—21no—
92C2×D4—21no—
93C2×D4—21no—
94C2×D4—21no—
95C2×D4—21no—
96C2×D4—21no—
97C2×D4—21no—
98C2×D4—21no—
99C2×D4—41no—
100C2×D4D812yes—
101C22×C4—21no—
102C22×C4—22no—
103C22×C4—22no—
104C22×C4—44no—
105C22×C4—41no—
106C22×C4—41no—
107C22×C4—41no—
108C22×C4C2×D411yescharacteristic
109C22×C4C2×D411yescharacteristic
110C22×C4C2×D411yescharacteristic
111SD16—44no—
112D8—44no—
113M16—41no—
114M16—41no—
115C2×C8—22no—
116C2×C8—22no—
117C2×C8—41no—
118C4⋊C4—21no—
119C4⋊C4—22no—
120C4⋊C4—21no—
121C4⋊C4—21no—
122C4⋊C4—22no—
123C4⋊C4—21no—
124C4⋊C4—42no—
125C4⋊C4—41no—
126C4⋊C4—41no—
127C22⋊C4—48no—
128C22⋊C4—41no—
129C22⋊C4—41no—
130C22⋊C4—41no—
131C22⋊C4—41no—
132C42—41no—
133C42—41no—
134C42C2×D411yescharacteristic
135C42C2×D411yescharacteristic
136C42C2×D411yescharacteristic
137C42#4111yescharacteristic
138C23—22no—
139C23—42no—
140C23—42no—
141C23—41no—
142C23—41no—
143C23—41no—
144C23#12011yescharacteristic
145Q8—41no—
146Q8—41no—
147Q8—81no—
148Q8—81no—
149Q8—81no—
150Q8—81no—
151D4—22no—
152D4—22no—
153D4—41no—
154D4—41no—
155D4—41no—
156D4—42no—
157D4—41no—
158D4—81no—
159D4—81no—
160D4—81no—
161C2×C4—22no—
162C2×C4—41no—
163C2×C4—42no—
164C2×C4—42no—
165C2×C4—42no—
166C2×C4—41no—
167C2×C4—41no—
168C2×C4—42no—
169C2×C4—42no—
170C2×C4—42no—
171C2×C4—42no—
172C2×C4—82no—
173C2×C4—82no—
174C2×C4—81no—
175C2×C4—81no—
176C2×C4#12011yescharacteristic
177C2×C4#12011yescharacteristic
178C2×C4#12111yescharacteristic
179C2×C4#12111yescharacteristic
180C2×C4#12111yescharacteristic
181C2×C4#12711yescharacteristic
182C2×C4C2×D811yescharacteristic
183C2×C4#13511yescharacteristic
184C2×C4AGL(1,ℤ/8)11yescharacteristic
185C2×C4AGL(1,ℤ/8)11yescharacteristic
186C8—42no—
187C8—42no—
188C8—81no—
189C22—21no—
190C22—21no—
191C22—44no—
192C22—42no—
193C22—82no—
194C22—81no—
195C22—81no—
196C22—81no—
197C22—81no—
198C22—81no—
199C22#39211yescharacteristic
200C22#44711yescharacteristic
201C22#44911yescharacteristic
202C4—21no—
203C4—21no—
204C4—21no—
205C4—82no—
206C4—82no—
207C4—81no—
208C4—81no—
209C4#44711yescharacteristic
210C4#44911yescharacteristic
211C4#45412yes—
212C4#46611yescharacteristic
213C4#46911yescharacteristic
214C2—21no—
215C2—21no—
216C2—82no—
217C2—161no—
218C2#199911yescharacteristic
219C2#200311yescharacteristic
220C2#200911yescharacteristic
subgroupG/H
center, socleC22#392
commutator, Frattini, cosocleC2×C42C23
prime radical#3036C2
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
1. #3226C2semidirect
3. #2547C2semidirect
4. #1546C2semidirect
5. #1502C2semidirect
6. #1478C2semidirect
7. #1303C2semidirect

Maximal subgroups

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

subgroupindex
#32262
#30362
#25472
#15462
#15022
#14782
#13032