Finite Group Databasebeta

← #9998

Subgroups — #9998

Order 256.

301 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 301 conjugacy classes of subgroups — too many to draw. Only its 59 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#3103C211yesmaximal, characteristic, complemented
2#3036C211yesmaximal, characteristic
3#2580C211yesmaximal, characteristic, complemented
4#1517C211yesmaximal, characteristic, complemented
5#1487C211yesmaximal, characteristic, complemented
6#1478C211yesmaximal, characteristic, complemented
7#1286C211yesmaximal, characteristic, complemented
8C2×AGL(1,ℤ/8)—21no—
9#532C2211yescharacteristic
10C2×C4×Q8C2211yescharacteristic
11C2×C4×D4C2211yescharacteristic
12#481—21no—
13#475—21no—
14#463—22no—
15#461—21no—
16#460—21no—
17#459—21no—
18#448—21no—
19C4×SD16—22no—
20#431C2211yescharacteristic
21#431C2211yescharacteristic
22#423C2211yescharacteristic
23#417—21no—
24#399—21no—
25#393—21no—
26#390—21no—
27#389—21no—
28#385—22no—
29#382C2211yescharacteristic
30#327—22no—
31Q8⋊C8—22no—
32D4⋊C8—22no—
33C2×#41—21no—
34C22×Q8D411yescharacteristic
35C22×D4D411yescharacteristic
36C23×C4—21no—
37AGL(1,ℤ/8)—42no—
38AGL(1,ℤ/8)—42no—
39C2×SD16—22no—
40C2×SD16—22no—
41C2×D8—22no—
42C2×M16—21no—
43C4⋊Q8D411yescharacteristic
44#127D411yescharacteristic
45#124—21no—
46#121—21no—
47#121—21no—
48C4×Q8—21no—
49C4×Q8—21no—
50C4×Q8—21no—
51C4×Q8—21no—
52C4×D4—21no—
53C4×D4—21no—
54C4×D4—21no—
55C4×D4—21no—
56C4×D4—21no—
57C4×D4—21no—
58C2×C4⋊C4—21no—
59C2×C4⋊C4—21no—
60C2×C4⋊C4—22no—
61C2×C4⋊C4D411yescharacteristic
62C2×C4⋊C4D411yescharacteristic
63C2×C22⋊C4—22no—
64C2×C22⋊C4—21no—
65C2×C22⋊C4—22no—
66C2×C42—21no—
67C2×C42C2311yescharacteristic
68#106—22no—
69C4⋊C8—21no—
70C4⋊C8—21no—
71C4⋊C8—21no—
72C4⋊C8D412yes—
73#103—22no—
74#103—22no—
75#102—22no—
76#102—22no—
77C22⋊C8—21no—
78C22⋊C8—21no—
79C4×C8—21no—
80C4×C8D412yes—
81#95—22no—
82#95—21no—
83#95D412yes—
84C24—21no—
85#41—41no—
86#41—41no—
87#41—41no—
88#41—41no—
89C2×Q8—21no—
90C2×Q8—21no—
91C2×Q8—21no—
92C2×Q8—42no—
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—22no—
105C22×C4—21no—
106C22×C4—21no—
107C22×C4—41no—
108C22×C4—41no—
109C22×C4—41no—
110C22×C4—82no—
111C22×C4C2×D411yescharacteristic
112C22×C4C2×D411yescharacteristic
113C22×C4C2×D411yescharacteristic
114SD16—44no—
115SD16—44no—
116D8—44no—
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—21no—
127C4⋊C4—21no—
128C4⋊C4—41no—
129C4⋊C4—41no—
130C4⋊C4—41no—
131C4⋊C4—41no—
132C4⋊C4—82no—
133C22⋊C4—44no—
134C22⋊C4—41no—
135C22⋊C4—41no—
136C22⋊C4—41no—
137C22⋊C4—41no—
138C22⋊C4—82no—
139C42—41no—
140C42—41no—
141C42C2×D411yescharacteristic
142C42C2×D411yescharacteristic
143C42C2×D411yescharacteristic
144C42#4111yescharacteristic
145C23—22no—
146C23—42no—
147C23—42no—
148C23—41no—
149C23—41no—
150C23—41no—
151C23#12011yescharacteristic
152Q8—22no—
153Q8—22no—
154Q8—41no—
155Q8—41no—
156Q8—41no—
157Q8—41no—
158D4—22no—
159D4—22no—
160D4—42no—
161D4—41no—
162D4—41no—
163D4—41no—
164D4—41no—
165D4—81no—
166D4—81no—
167D4—81no—
168C2×C4—22no—
169C2×C4—41no—
170C2×C4—41no—
171C2×C4—41no—
172C2×C4—41no—
173C2×C4—42no—
174C2×C4—41no—
175C2×C4—42no—
176C2×C4—41no—
177C2×C4—42no—
178C2×C4—41no—
179C2×C4—41no—
180C2×C4—41no—
181C2×C4—42no—
182C2×C4—82no—
183C2×C4—82no—
184C2×C4—81no—
185C2×C4—81no—
186C2×C4#12011yescharacteristic
187C2×C4#12011yescharacteristic
188C2×C4#12111yescharacteristic
189C2×C4#12111yescharacteristic
190C2×C4#12111yescharacteristic
191C2×C4#12711yescharacteristic
192C2×C4C2×SD1611yescharacteristic
193C2×C4#13511yescharacteristic
194C2×C4AGL(1,ℤ/8)11yescharacteristic
195C2×C4#13711yescharacteristic
196C8—42no—
197C8—42no—
198C8—81no—
199C22—21no—
200C22—21no—
201C22—44no—
202C22—42no—
203C22—82no—
204C22—81no—
205C22—81no—
206C22—81no—
207C22—81no—
208C22—81no—
209C22#39211yescharacteristic
210C22#44811yescharacteristic
211C22#44911yescharacteristic
212C4—21no—
213C4—21no—
214C4—21no—
215C4—21no—
216C4—42no—
217C4—82no—
218C4—81no—
219C4—81no—
220C4—81no—
221C4#45011yescharacteristic
222C4#45211yescharacteristic
223C4#46511yescharacteristic
224C4#46811yescharacteristic
225C2—21no—
226C2—21no—
227C2—82no—
228C2—161no—
229C2#200011yescharacteristic
230C2#200811yescharacteristic
231C2#201011yescharacteristic
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. #3103C2semidirect
3. #2580C2semidirect
4. #1517C2semidirect
5. #1487C2semidirect
6. #1478C2semidirect
7. #1286C2semidirect
102. C2×D4SD16semidirect

Maximal subgroups

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

subgroupindex
#31032
#30362
#25802
#15172
#14872
#14782
#12862