Finite Group Databasebeta

← #10256

Subgroups — #10256

Order 256.

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

normal subgroups only

This group has 290 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#3246C211yesmaximal, characteristic, complemented
2#3077C211yesmaximal, characteristic, complemented
3#3036C211yesmaximal, characteristic
4#1954C211yesmaximal, characteristic, complemented
5#1493C211yesmaximal, characteristic, complemented
6#1492C211yesmaximal, characteristic, complemented
7#1292C211yesmaximal, characteristic, complemented
8C2×AGL(1,ℤ/8)—21no—
9#533C2211yescharacteristic
10#532C2211yescharacteristic
11C2×C4×D4C2211yescharacteristic
12#478—22no—
13#476—21no—
14#476—21no—
15#475—21no—
16#474—21no—
17#465—22no—
18#463—22no—
19#461—21no—
20#461—21no—
21#460—21no—
22#459—21no—
23#428—21no—
24#423C2211yescharacteristic
25#423C2211yescharacteristic
26C2×C4⋊C8C2211yescharacteristic
27#418—21no—
28#417—21no—
29C2×C22⋊C8—22no—
30C2×C4×C8C2211yescharacteristic
31Q8⋊C8—22no—
32D4⋊C8—22no—
33C2×#41—21no—
34C22×D4D411yescharacteristic
35C23×C4—21no—
36AGL(1,ℤ/8)—42no—
37AGL(1,ℤ/8)—42no—
38C2×SD16—22no—
39C2×SD16—22no—
40C2×D8—22no—
41C2×M16—21no—
42C2×M16—21no—
43C22×C8—21no—
44C22×C8D412yes—
45C4⋊Q8D411yescharacteristic
46C4⋊Q8D411yescharacteristic
47C4⋊Q8D411yescharacteristic
48#127D411yescharacteristic
49#125—21no—
50#124—21no—
51C22⋊Q8—21no—
52C22⋊Q8—21no—
53#121—21no—
54#121—21no—
55C4×Q8—21no—
56C4×Q8—21no—
57C4×D4—21no—
58C4×D4—21no—
59C4×D4—21no—
60C4×D4—21no—
61C4×D4—21no—
62C4×D4—21no—
63#117—21no—
64C2×C4⋊C4D411yescharacteristic
65C2×C22⋊C4—21no—
66C2×C42C2311yescharacteristic
67#107—22no—
68#106—22no—
69#106—22no—
70C4⋊C8—21no—
71C4⋊C8—21no—
72C4⋊C8—21no—
73C4⋊C8—21no—
74C4⋊C8D412yes—
75C4⋊C8D412yes—
76#103—22no—
77#103—22no—
78#102—22no—
79#102—22no—
80#102—22no—
81#102—22no—
82C22⋊C8—44no—
83C4×C8—21no—
84C4×C8—21no—
85C24—21no—
86#41—41no—
87#41—41no—
88#41—41no—
89#41—41no—
90C2×Q8—21no—
91C2×Q8—21no—
92C2×Q8—21no—
93C2×D4—21no—
94C2×D4—21no—
95C2×D4—21no—
96C2×D4—21no—
97C2×D4—21no—
98C2×D4—21no—
99C2×D4—21no—
100C2×D4—21no—
101C2×D4—41no—
102C2×D4SD1612yescomplemented
103C22×C4—21no—
104C22×C4—42no—
105C22×C4—42no—
106C22×C4—41no—
107C22×C4—41no—
108C22×C4—41no—
109C22×C4C2×D411yescharacteristic
110C22×C4C2×D411yescharacteristic
111C22×C4#4111yescharacteristic
112SD16—44no—
113SD16—82no—
114D8—44no—
115M16—41no—
116M16—41no—
117M16—41no—
118M16—41no—
119C2×C8—22no—
120C2×C8—22no—
121C2×C8—22no—
122C2×C8—22no—
123C2×C8—41no—
124C2×C8—41no—
125C2×C8—42no—
126C4⋊C4—21no—
127C4⋊C4—21no—
128C4⋊C4—21no—
129C4⋊C4—21no—
130C4⋊C4—21no—
131C4⋊C4—21no—
132C4⋊C4—21no—
133C4⋊C4—21no—
134C4⋊C4—41no—
135C4⋊C4—41no—
136C22⋊C4—41no—
137C22⋊C4—41no—
138C22⋊C4—41no—
139C22⋊C4—41no—
140C22⋊C4—41no—
141C42—41no—
142C42C2×D411yescharacteristic
143C42C2×D411yescharacteristic
144C42C2×D411yescharacteristic
145C42C2×D411yescharacteristic
146C23—22no—
147C23—42no—
148C23—42no—
149C23—41no—
150C23—41no—
151C23—41no—
152C23#12111yescharacteristic
153Q8—42no—
154Q8—41no—
155Q8—41no—
156Q8—41no—
157Q8—41no—
158D4—22no—
159D4—22no—
160D4—42no—
161D4—41no—
162D4—41no—
163D4—41no—
164D4—41no—
165D4—41no—
166D4—41no—
167D4—81no—
168D4—81no—
169C2×C4—21no—
170C2×C4—21no—
171C2×C4—41no—
172C2×C4—42no—
173C2×C4—41no—
174C2×C4—41no—
175C2×C4—41no—
176C2×C4—41no—
177C2×C4—82no—
178C2×C4—81no—
179C2×C4—81no—
180C2×C4—81no—
181C2×C4—81no—
182C2×C4#12011yescharacteristic
183C2×C4#12011yescharacteristic
184C2×C4#12011yescharacteristic
185C2×C4#12111yescharacteristic
186C2×C4#12111yescharacteristic
187C2×C4#12711yescharacteristic
188C2×C4C2×SD1611yescharacteristic
189C2×C4#13511yescharacteristic
190C2×C4AGL(1,ℤ/8)11yescharacteristic
191C2×C4#13711yescharacteristic
192C8—42no—
193C8—42no—
194C8—42no—
195C8—42no—
196C22—21no—
197C22—21no—
198C22—44no—
199C22—42no—
200C22—82no—
201C22—81no—
202C22—81no—
203C22—81no—
204C22—81no—
205C22—81no—
206C22#39211yescharacteristic
207C22#46511yescharacteristic
208C22#46811yescharacteristic
209C4—21no—
210C4—21no—
211C4—21no—
212C4—81no—
213C4—81no—
214C4—81no—
215C4—81no—
216C4#44811yescharacteristic
217C4#44911yescharacteristic
218C4#45011yescharacteristic
219C4#45211yescharacteristic
220C4#45311yescharacteristic
221C4#45411yescharacteristic
222C2—21no—
223C2—21no—
224C2—82no—
225C2—161no—
226C2#200011yescharacteristic
227C2#200611yescharacteristic
228C2#201711yescharacteristic
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. #3246C2semidirect
2. #3077C2semidirect
4. #1954C2semidirect
5. #1493C2semidirect
6. #1492C2semidirect
7. #1292C2semidirect
102. C2×D4SD16semidirect

Maximal subgroups

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

subgroupindex
#32462
#30772
#30362
#19542
#14932
#14922
#12922