Finite Group Databasebeta

← #10184

Subgroups — #10184

Order 256.

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

normal subgroups only

This group has 291 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
3C2×#461C211yesmaximal, characteristic, complemented
4#1546C211yesmaximal, characteristic, complemented
5#1480C211yesmaximal, characteristic, complemented
6#1460C211yesmaximal, characteristic, complemented
7#1301C211yesmaximal, characteristic, complemented
8C2×AGL(1,ℤ/8)—21no—
9C22×SD16—21no—
10#532C2211yescharacteristic
11C2×C4⋊Q8C2211yescharacteristic
12C2×C4×D4C2211yescharacteristic
13#500—21no—
14#486—21no—
15#480—22no—
16#474—22no—
17#463—22no—
18#461—22no—
19#461—21no—
20#461—22no—
21#459—21no—
22#431C2211yescharacteristic
23#431C2211yescharacteristic
24#423C2211yescharacteristic
25C2×C4⋊C8C2211yescharacteristic
26#417—21no—
27C2×#103—21no—
28#332—22no—
29D4⋊C8—22no—
30#323—22no—
31C2×#41—21no—
32C22×Q8—21no—
33C22×D4D411yescharacteristic
34C23×C4—21no—
35AGL(1,ℤ/8)—42no—
36AGL(1,ℤ/8)—42no—
37C2×SD16—24no—
38C2×SD16—22no—
39C2×SD16—42no—
40C2×SD16—42no—
41C2×D8—22no—
42C2×M16—21no—
43C22×C8—21no—
44C4⋊Q8—21no—
45C4⋊Q8—21no—
46C4⋊Q8—21no—
47C4⋊Q8D412yes—
48C4⋊Q8D411yescharacteristic
49#127D411yescharacteristic
50#124—21no—
51#121—21no—
52#121—21no—
53C4×D4—21no—
54C4×D4—21no—
55C4×D4—21no—
56C4×D4—21no—
57C4×D4—21no—
58C4×D4—21no—
59C2×C4⋊C4—21no—
60C2×C4⋊C4—21no—
61C2×C4⋊C4D411yescharacteristic
62C2×C22⋊C4—21no—
63C2×C42C2311yescharacteristic
64#107—24no—
65C4⋊C8—21no—
66C4⋊C8—21no—
67C4⋊C8—21no—
68C4⋊C8—21no—
69C4⋊C8D412yes—
70C4⋊C8D412yes—
71#103—24no—
72#103—22no—
73#102—22no—
74#102—24no—
75C22⋊C8—21no—
76C22⋊C8D412yes—
77C4×C8—21no—
78C4×C8—21no—
79C24—21no—
80#41—41no—
81#41—41no—
82#41—41no—
83#41—41no—
84C2×Q8—21no—
85C2×Q8—22no—
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—41no—
103C22×C4—41no—
104C22×C4—41no—
105C22×C4—41no—
106C22×C4—82no—
107C22×C4C2×D411yescharacteristic
108C22×C4C2×D411yescharacteristic
109C22×C4#4111yescharacteristic
110SD16—48no—
111SD16—44no—
112D8—44no—
113M16—41no—
114M16—41no—
115C2×C8—22no—
116C2×C8—22no—
117C2×C8—22no—
118C2×C8—41no—
119C2×C8—41no—
120C2×C8—41no—
121C4⋊C4—21no—
122C4⋊C4—22no—
123C4⋊C4—21no—
124C4⋊C4—21no—
125C4⋊C4—22no—
126C4⋊C4—21no—
127C4⋊C4—41no—
128C4⋊C4—41no—
129C22⋊C4—41no—
130C22⋊C4—41no—
131C22⋊C4—41no—
132C22⋊C4—41no—
133C42C2×D411yescharacteristic
134C42C2×D411yescharacteristic
135C42C2×D411yescharacteristic
136C42C2×D411yescharacteristic
137C23—22no—
138C23—41no—
139C23—41no—
140C23—41no—
141C23—81no—
142C23—81no—
143C23#12111yescharacteristic
144Q8—42no—
145Q8—41no—
146Q8—42no—
147Q8—41no—
148Q8—81no—
149Q8—81no—
150D4—22no—
151D4—22no—
152D4—42no—
153D4—41no—
154D4—41no—
155D4—41no—
156D4—41no—
157D4—42no—
158D4—81no—
159D4—81no—
160C2×C4—22no—
161C2×C4—41no—
162C2×C4—41no—
163C2×C4—41no—
164C2×C4—42no—
165C2×C4—82no—
166C2×C4—81no—
167C2×C4—81no—
168C2×C4—81no—
169C2×C4—81no—
170C2×C4—82no—
171C2×C4—81no—
172C2×C4#12011yescharacteristic
173C2×C4#12011yescharacteristic
174C2×C4#12011yescharacteristic
175C2×C4#12111yescharacteristic
176C2×C4#12111yescharacteristic
177C2×C4#12711yescharacteristic
178C2×C4C2×D811yescharacteristic
179C2×C4#13511yescharacteristic
180C2×C4AGL(1,ℤ/8)11yescharacteristic
181C2×C4AGL(1,ℤ/8)11yescharacteristic
182C8—42no—
183C8—42no—
184C8—42no—
185C8—81no—
186C22—21no—
187C22—21no—
188C22—42no—
189C22—82no—
190C22—82no—
191C22—81no—
192C22—81no—
193C22—81no—
194C22—81no—
195C22—81no—
196C22#39211yescharacteristic
197C22#46611yescharacteristic
198C22#46911yescharacteristic
199C4—21no—
200C4—21no—
201C4—21no—
202C4—81no—
203C4—82no—
204C4—81no—
205C4#44711yescharacteristic
206C4#44711yescharacteristic
207C4#44911yescharacteristic
208C4#44911yescharacteristic
209C4#47112yes—
210C2—21no—
211C2—21no—
212C2—82no—
213C2—161no—
214C2#199911yescharacteristic
215C2#200911yescharacteristic
216C2#201911yescharacteristic
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. C2×#461C2semidirect
4. #1546C2semidirect
5. #1480C2semidirect
6. #1460C2semidirect
7. #1301C2semidirect

Maximal subgroups

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

subgroupindex
#32262
#30362
C2×#4612
#15462
#14802
#14602
#13012