Finite Group Databasebeta

← #10420

Subgroups — #10420

Order 256.

278 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 278 conjugacy classes of subgroups — too many to draw. Only its 75 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#2542C211yesmaximal, characteristic
2#1990C211yesmaximal, characteristic, complemented
3#1948C211yesmaximal, characteristic, complemented
4#1802C211yesmaximal, characteristic
5C2×#332C211yesmaximal, characteristic, complemented
6C2×D4⋊C8C211yesmaximal, characteristic
7#1279C211yesmaximal, characteristic, complemented
8C2×C4⋊Q8C2211yescharacteristic
9C2×C22⋊Q8—21no—
10C2×C22⋊Q8—21no—
11C2×C4×D4C2211yescharacteristic
12C2×#106—21no—
13C2×C4⋊C8C2211yescharacteristic
14C2×C4⋊C8C2211yescharacteristic
15C2×#102—21no—
16C2×C4×C8C2211yescharacteristic
17#399—22no—
18#390—21no—
19#384—21no—
20#384C2211yescharacteristic
21C4×C4⋊C4C2211yescharacteristic
22#340—21no—
23#340—21no—
24#340—22no—
25#332—22no—
26D4⋊C8—22no—
27C22×Q8—21no—
28C22×D4D411yescharacteristic
29C23×C4—21no—
30C22×C8—21no—
31C22×C8—21no—
32C22×C8—21no—
33C4⋊Q8—21no—
34C4⋊Q8—21no—
35C4⋊Q8—21no—
36C4⋊Q8—21no—
37C22⋊Q8—42no—
38C22⋊Q8—42no—
39C22⋊Q8—42no—
40C22⋊Q8—42no—
41C4×D4—21no—
42C4×D4—21no—
43C4×D4—21no—
44C4×D4—21no—
45C2×C4⋊C4—22no—
46C2×C4⋊C4—22no—
47C2×C4⋊C4—21no—
48C2×C4⋊C4—21no—
49C2×C4⋊C4—21no—
50C2×C4⋊C4—21no—
51C2×C4⋊C4D411yescharacteristic
52C2×C4⋊C4D412yes—
53C2×C4⋊C4D411yescharacteristic
54C2×C4⋊C4D411yescharacteristic
55C2×C22⋊C4—22no—
56C2×C22⋊C4—22no—
57C2×C22⋊C4—21no—
58C2×C42—21no—
59C2×C42C2311yescharacteristic
60#106—42no—
61C4⋊C8—21no—
62C4⋊C8—21no—
63C4⋊C8—21no—
64C4⋊C8D412yes—
65#102—42no—
66C4×C8—21no—
67C4×C8—21no—
68#95—21no—
69C24—21no—
70C2×Q8—41no—
71C2×Q8—41no—
72C2×Q8—41no—
73C2×Q8—41no—
74C2×Q8—41no—
75C2×Q8—41no—
76C2×D4—21no—
77C2×D4—21no—
78C2×D4—21no—
79C2×D4—21no—
80C2×D4—21no—
81C2×D4SD1612yes—
82C22×C4—21no—
83C22×C4—21no—
84C22×C4—22no—
85C22×C4—21no—
86C22×C4—41no—
87C22×C4—41no—
88C22×C4—41no—
89C22×C4—41no—
90C22×C4—41no—
91C22×C4—41no—
92C22×C4—41no—
93C22×C4C2×D411yescharacteristic
94C22×C4C2×D411yescharacteristic
95C22×C4C2×D411yescharacteristic
96C2×C8—22no—
97C2×C8—22no—
98C2×C8—41no—
99C2×C8—41no—
100C2×C8—41no—
101C2×C8—41no—
102C2×C8—41no—
103C2×C8—41no—
104C2×C8—41no—
105C4⋊C4—21no—
106C4⋊C4—21no—
107C4⋊C4—21no—
108C4⋊C4—22no—
109C4⋊C4—21no—
110C4⋊C4—22no—
111C4⋊C4—44no—
112C4⋊C4—44no—
113C4⋊C4—42no—
114C4⋊C4—41no—
115C4⋊C4—41no—
116C4⋊C4—41no—
117C4⋊C4—41no—
118C4⋊C4—41no—
119C4⋊C4—41no—
120C4⋊C4D812yes—
121C4⋊C4SD1612yes—
122C22⋊C4—44no—
123C22⋊C4—44no—
124C22⋊C4—41no—
125C22⋊C4—41no—
126C42—41no—
127C42—41no—
128C42C2×D411yescharacteristic
129C42#4111yescharacteristic
130C42#4111yescharacteristic
131C42#4111yescharacteristic
132C23—22no—
133C23—41no—
134C23—41no—
135C23—41no—
136C23—41no—
137C23—41no—
138C23—41no—
139C23#12011yescharacteristic
140Q8—81no—
141Q8—81no—
142Q8—81no—
143Q8—81no—
144D4—22no—
145D4—22no—
146D4—41no—
147D4—41no—
148C2×C4—21no—
149C2×C4—21no—
150C2×C4—22no—
151C2×C4—22no—
152C2×C4—41no—
153C2×C4—41no—
154C2×C4—41no—
155C2×C4—41no—
156C2×C4—41no—
157C2×C4—42no—
158C2×C4—42no—
159C2×C4—41no—
160C2×C4—42no—
161C2×C4—41no—
162C2×C4—81no—
163C2×C4—81no—
164C2×C4—81no—
165C2×C4—81no—
166C2×C4—81no—
167C2×C4—81no—
168C2×C4—81no—
169C2×C4#12111yescharacteristic
170C2×C4#12111yescharacteristic
171C2×C4#12111yescharacteristic
172C2×C4#12311yescharacteristic
173C2×C4#12311yescharacteristic
174C2×C4#12411yescharacteristic
175C2×C4C2×D811yescharacteristic
176C2×C4C2×SD1611yescharacteristic
177C2×C4C2×SD1611yescharacteristic
178C2×C4#13511yescharacteristic
179C2×C4AGL(1,ℤ/8)11yescharacteristic
180C2×C4AGL(1,ℤ/8)11yescharacteristic
181C2×C4#13711yescharacteristic
182C2×C4#13711yescharacteristic
183C8—42no—
184C8—42no—
185C8—81no—
186C22—42no—
187C22—42no—
188C22—42no—
189C22—81no—
190C22—81no—
191C22—81no—
192C22—81no—
193C22#39411yescharacteristic
194C22#44711yescharacteristic
195C22#44811yescharacteristic
196C22#45011yescharacteristic
197C22#45211yescharacteristic
198C22#45411yescharacteristic
199C22#45611yescharacteristic
200C4—21no—
201C4—21no—
202C4—42no—
203C4—42no—
204C4—81no—
205C4—82no—
206C4—161no—
207C4#45911yescharacteristic
208C4#46011yescharacteristic
209C4#46511yescharacteristic
210C4#47011yescharacteristic
211C4#48311yescharacteristic
212C4#48511yescharacteristic
213C4#48611yescharacteristic
214C4#49011yescharacteristic
215C2—82no—
216C2#162711yescharacteristic
217C2#163511yescharacteristic
218C2#164711yescharacteristic
219C2#165111yescharacteristic
220C2#203511yescharacteristic
221C2#203811yescharacteristic
222C2#204111yescharacteristic
subgroupG/H
center, socleC23#120
commutator, Frattini, cosocleC2×C42C23
prime radicalC22×D4D4
LCSG ▷ C2×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
2. #1990C2semidirect
3. #1948C2semidirect
5. C2×#332C2semidirect
7. #1279C2semidirect

Maximal subgroups

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

subgroupindex
#25422
#19902
#19482
#18022
C2×#3322
C2×D4⋊C82
#12792