Finite Group Databasebeta

← #8976

Subgroups — #8976

Order 256.

296 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 296 conjugacy classes of subgroups — too many to draw. Only its 85 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
1C2×#476C211yesmaximal, characteristic
2#2354C211yesmaximal, characteristic
3#1987C211yesmaximal, characteristic, complemented
4#1956C211yesmaximal, characteristic, complemented
5C2×D4⋊C8C211yesmaximal, characteristic
6#1296C211yesmaximal, characteristic, complemented
7#1279C211yesmaximal, characteristic, complemented
8C2×C4⋊Q8C2211yescharacteristic
9C2×C4×D4C2211yescharacteristic
10C2×#117—21no—
11#476—22no—
12#476C414yescomplemented
13C2×#106—21no—
14C2×C4⋊C8C2211yescharacteristic
15C2×C4⋊C8C2211yescharacteristic
16C2×#103—22no—
17C2×#102—21no—
18C2×C4×C8C2211yescharacteristic
19#390—21no—
20#388—22no—
21#384C2211yescharacteristic
22#380—21no—
23C4×C4⋊C4—21no—
24C4×C4⋊C4C2211yescharacteristic
25#340—21no—
26#340—21no—
27D4⋊C8—22no—
28C22×Q8—21no—
29C22×D4D411yescharacteristic
30C23×C4—21no—
31C22×C8—21no—
32C22×C8—21no—
33C22×C8—21no—
34C4⋊Q8—21no—
35C4⋊Q8—21no—
36C4⋊Q8—21no—
37C4⋊Q8C2×C412yes—
38C4×D4—21no—
39C4×D4—21no—
40C4×D4C2×C412yes—
41C4×D4D412yes—
42#117—42no—
43#117—42no—
44C2×C4⋊C4—21no—
45C2×C4⋊C4—21no—
46C2×C4⋊C4—21no—
47C2×C4⋊C4—21no—
48C2×C4⋊C4D411yescharacteristic
49C2×C4⋊C4D411yescharacteristic
50C2×C4⋊C4D411yescharacteristic
51C2×C4⋊C4D412yes—
52C2×C22⋊C4—21no—
53C2×C22⋊C4—22no—
54C2×C22⋊C4—22no—
55C2×C42—22no—
56C2×C42—21no—
57C2×C42C2311yescharacteristic
58#106—24no—
59C4⋊C8—21no—
60C4⋊C8—21no—
61C4⋊C8—21no—
62C4⋊C8C2×C412yes—
63#103—44no—
64#102—24no—
65C4×C8—21no—
66C4×C8—21no—
67#95—22no—
68#95—21no—
69C24—21no—
70C2×Q8—22no—
71C2×Q8—42no—
72C2×Q8—42no—
73C2×Q8—41no—
74C2×D4—21no—
75C2×D4—21no—
76C2×D4—21no—
77C2×D4—21no—
78C2×D4—21no—
79C2×D4C22⋊C412yes—
80C22×C4—21no—
81C22×C4—21no—
82C22×C4—22no—
83C22×C4—22no—
84C22×C4—21no—
85C22×C4—22no—
86C22×C4—41no—
87C22×C4—41no—
88C22×C4—41no—
89C22×C4—41no—
90C22×C4—41no—
91C22×C4C2×D411yescharacteristic
92C22×C4C2×D411yescharacteristic
93C22×C4C2×D411yescharacteristic
94C2×C8—22no—
95C2×C8—22no—
96C2×C8—41no—
97C2×C8—41no—
98C2×C8—41no—
99C2×C8—41no—
100C2×C8—41no—
101C2×C8—41no—
102C2×C8—41no—
103C4⋊C4—21no—
104C4⋊C4—21no—
105C4⋊C4—22no—
106C4⋊C4—22no—
107C4⋊C4—21no—
108C4⋊C4—21no—
109C4⋊C4—42no—
110C4⋊C4—41no—
111C4⋊C4—41no—
112C4⋊C4—41no—
113C4⋊C4—41no—
114C4⋊C4—41no—
115C4⋊C4—41no—
116C4⋊C4C22⋊C412yes—
117C4⋊C4SD1612yes—
118C22⋊C4—22no—
119C22⋊C4—22no—
120C22⋊C4—44no—
121C22⋊C4—82no—
122C42—44no—
123C42—41no—
124C42—41no—
125C42C22×C411yescharacteristic
126C42C2×D411yescharacteristic
127C42#4111yescharacteristic
128C42#4111yescharacteristic
129C23—22no—
130C23—41no—
131C23—41no—
132C23—41no—
133C23—41no—
134C23—41no—
135C23—41no—
136C23#12011yescharacteristic
137Q8—44no—
138Q8—82no—
139D4—22no—
140D4—22no—
141D4—41no—
142D4—41no—
143C2×C4—21no—
144C2×C4—21no—
145C2×C4—22no—
146C2×C4—22no—
147C2×C4—41no—
148C2×C4—41no—
149C2×C4—41no—
150C2×C4—41no—
151C2×C4—42no—
152C2×C4—42no—
153C2×C4—42no—
154C2×C4—42no—
155C2×C4—42no—
156C2×C4—42no—
157C2×C4—41no—
158C2×C4—41no—
159C2×C4—42no—
160C2×C4—41no—
161C2×C4—42no—
162C2×C4—81no—
163C2×C4—81no—
164C2×C4C2×C22⋊C411yescharacteristic
165C2×C4C4×D411yescharacteristic
166C2×C4C4×D411yescharacteristic
167C2×C4#12111yescharacteristic
168C2×C4#12111yescharacteristic
169C2×C4#12311yescharacteristic
170C2×C4C2×SD1611yescharacteristic
171C2×C4#13511yescharacteristic
172C2×C4#13511yescharacteristic
173C2×C4#13511yescharacteristic
174C2×C4AGL(1,ℤ/8)11yescharacteristic
175C2×C4#13711yescharacteristic
176C2×C4#13711yescharacteristic
177C2×C4#13711yescharacteristic
178C8—42no—
179C8—42no—
180C8—81no—
181C22—42no—
182C22—42no—
183C22—42no—
184C22—81no—
185C22—81no—
186C22—81no—
187C22—81no—
188C22#38611yescharacteristic
189C22#45011yescharacteristic
190C22#45211yescharacteristic
191C22#45211yescharacteristic
192C22#45211yescharacteristic
193C22#45311yescharacteristic
194C22#45411yescharacteristic
195C4—21no—
196C4—21no—
197C4—42no—
198C4—42no—
199C4—82no—
200C4—82no—
201C4—81no—
202C4#41611yescharacteristic
203C4#41911yescharacteristic
204C4C4×SD1611yescharacteristic
205C4#44011yescharacteristic
206C4#46111yescharacteristic
207C4#46311yescharacteristic
208C4#48511yescharacteristic
209C4#48511yescharacteristic
210C2—82no—
211C2#163111yescharacteristic
212C2#163411yescharacteristic
213C2#163511yescharacteristic
214C2#163811yescharacteristic
215C2#187311yescharacteristic
216C2#187711yescharacteristic
217C2#188311yescharacteristic
subgroupG/H
center, socleC23#120
commutatorC42C22×C4
Frattini, cosocleC2×C42C23
prime radicalC22×D4D4
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
3. #1987C2semidirect
4. #1956C2semidirect
6. #1296C2semidirect
7. #1279C2semidirect
12. #476C4semidirect

Maximal subgroups

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

subgroupindex
C2×#4762
#23542
#19872
#19562
C2×D4⋊C82
#12962
#12792