Finite Group Databasebeta

← #10177

Subgroups — #10177

Order 256.

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

normal subgroups only

This group has 307 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
3C2×#460C211yesmaximal, characteristic, complemented
4#1545C211yesmaximal, characteristic, complemented
5#1480C211yesmaximal, characteristic, complemented
6#1461C211yesmaximal, characteristic, complemented
7#1301C211yesmaximal, characteristic, complemented
8C2×#137—21no—
9C22×SD16—21no—
10#532C2211yescharacteristic
11C2×#127C2211yescharacteristic
12C2×C4×Q8C2211yescharacteristic
13#493—21no—
14#486—21no—
15#464—22no—
16#462—21no—
17#460—22no—
18#460—21no—
19#460—22no—
20#459—22no—
21#447—22no—
22#431C2211yescharacteristic
23#431C2211yescharacteristic
24#423C2211yescharacteristic
25C2×C4⋊C8C2211yescharacteristic
26#417—21no—
27C2×#102—21no—
28#331—22no—
29D4⋊C8—22no—
30#324—22no—
31C2×#41—21no—
32C22×Q8D411yescharacteristic
33C22×D4—21no—
34C22×D4—21no—
35C22×D4—21no—
36#137—42no—
37#137—42no—
38C2×Q16—22no—
39C2×SD16—24no—
40C2×SD16—22no—
41C2×SD16—42no—
42C2×SD16—42no—
43C2×D8—24no—
44C2×M16—21no—
45C22×C8—21no—
46C4⋊Q8D411yescharacteristic
47#127—21no—
48#127—21no—
49#127—21no—
50#127D412yes—
51#127D411yescharacteristic
52#124—21no—
53#121—21no—
54#121—21no—
55C4×Q8—21no—
56C4×Q8—21no—
57C4×Q8—21no—
58C4×Q8—21no—
59C4×D4—21no—
60C4×D4—21no—
61C2×C4⋊C4—21no—
62C2×C4⋊C4D411yescharacteristic
63C2×C42—21no—
64C2×C42C2311yescharacteristic
65C4⋊C8—21no—
66C4⋊C8—21no—
67C4⋊C8—21no—
68C4⋊C8—21no—
69C4⋊C8D412yes—
70C4⋊C8D412yes—
71#103—22no—
72#102—24no—
73#102—22no—
74#102—24no—
75C22⋊C8—21no—
76C22⋊C8D412yes—
77C4×C8—21no—
78C4×C8—21no—
79C24—41no—
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—22no—
94C2×D4—21no—
95C2×D4—21no—
96C2×D4—22no—
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—82no—
106C2×D4—82no—
107C22×C4—21no—
108C22×C4—21no—
109C22×C4—41no—
110C22×C4C2×D411yescharacteristic
111C22×C4C2×D411yescharacteristic
112C22×C4#4111yescharacteristic
113Q16—44no—
114SD16—48no—
115SD16—44no—
116D8—84no—
117M16—41no—
118M16—41no—
119C2×C8—22no—
120C2×C8—22no—
121C2×C8—22no—
122C2×C8—41no—
123C2×C8—41no—
124C2×C8—41no—
125C4⋊C4—21no—
126C4⋊C4—21no—
127C4⋊C4—21no—
128C4⋊C4—21no—
129C4⋊C4—41no—
130C4⋊C4—41no—
131C22⋊C4—41no—
132C22⋊C4—41no—
133C42—41no—
134C42—41no—
135C42C2×D411yescharacteristic
136C42C2×D411yescharacteristic
137C42C2×D411yescharacteristic
138C42C2×D411yescharacteristic
139C23—41no—
140C23—42no—
141C23—82no—
142C23—82no—
143C23—81no—
144C23—81no—
145C23#12111yescharacteristic
146Q8—22no—
147Q8—22no—
148Q8—41no—
149Q8—41no—
150Q8—41no—
151Q8—41no—
152D4—42no—
153D4—44no—
154D4—42no—
155D4—41no—
156D4—42no—
157D4—41no—
158D4—42no—
159D4—82no—
160D4—82no—
161D4—81no—
162D4—81no—
163D4—81no—
164D4—81no—
165D4—82no—
166C2×C4—22no—
167C2×C4—22no—
168C2×C4—41no—
169C2×C4—41no—
170C2×C4—41no—
171C2×C4—41no—
172C2×C4—41no—
173C2×C4—81no—
174C2×C4—81no—
175C2×C4—81no—
176C2×C4#12011yescharacteristic
177C2×C4#12011yescharacteristic
178C2×C4#12011yescharacteristic
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—42no—
189C8—81no—
190C22—21no—
191C22—21no—
192C22—81no—
193C22—82no—
194C22—82no—
195C22—81no—
196C22—81no—
197C22—82no—
198C22—162no—
199C22—162no—
200C22#39211yescharacteristic
201C22#46611yescharacteristic
202C22#46911yescharacteristic
203C4—21no—
204C4—21no—
205C4—21no—
206C4—42no—
207C4—81no—
208C4—81no—
209C4#44711yescharacteristic
210C4#44711yescharacteristic
211C4#44911yescharacteristic
212C4#44911yescharacteristic
213C4#47112yes—
214C2—21no—
215C2—21no—
216C2—161no—
217C2—162no—
218C2#199911yescharacteristic
219C2#200811yescharacteristic
220C2#201911yescharacteristic
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. C2×#460C2semidirect
4. #1545C2semidirect
5. #1480C2semidirect
6. #1461C2semidirect
7. #1301C2semidirect
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
C2×#4602
#15452
#14802
#14612
#13012