Finite Group Databasebeta

← #10434

Subgroups — #10434

Order 256.

273 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 273 conjugacy classes of subgroups — too many to draw. Only its 73 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#2538C211yesmaximal, characteristic
2#1845C211yesmaximal, characteristic, complemented
3#1803C211yesmaximal, characteristic, complemented
4#1802C211yesmaximal, characteristic
5C2×Q8⋊C8C211yesmaximal, characteristic, complemented
6C2×D4⋊C8C211yesmaximal, characteristic
7#1296C211yesmaximal, characteristic, complemented
8C2×C22⋊Q8—21no—
9C2×C4×Q8C2211yescharacteristic
10C2×C4×D4C2211yescharacteristic
11C2×#107—22no—
12C2×C4⋊C8C2211yescharacteristic
13C2×C4⋊C8C2211yescharacteristic
14C2×#103—21no—
15C2×#102—21no—
16C2×C4×C8C2211yescharacteristic
17#396—22no—
18#390—21no—
19#389—21no—
20#384C2211yescharacteristic
21#384C2211yescharacteristic
22#380—21no—
23#340—21no—
24#340—21no—
25Q8⋊C8—22no—
26D4⋊C8—22no—
27C22×Q8D411yescharacteristic
28C22×D4D411yescharacteristic
29C23×C4—21no—
30C22×C8—21no—
31C22×C8—21no—
32C22×C8—21no—
33C22⋊Q8—42no—
34C22⋊Q8—42no—
35C4×Q8—21no—
36C4×Q8—21no—
37C4×Q8—21no—
38C4×Q8—21no—
39C4×D4—21no—
40C4×D4—21no—
41C4×D4—21no—
42C4×D4—21no—
43C2×C4⋊C4—22no—
44C2×C4⋊C4—21no—
45C2×C4⋊C4—21no—
46C2×C4⋊C4—21no—
47C2×C4⋊C4—21no—
48C2×C4⋊C4—21no—
49C2×C4⋊C4—21no—
50C2×C4⋊C4D411yescharacteristic
51C2×C4⋊C4D411yescharacteristic
52C2×C4⋊C4D412yes—
53C2×C22⋊C4—22no—
54C2×C22⋊C4—21no—
55C2×C22⋊C4—22no—
56C2×C42—21no—
57C2×C42C2311yescharacteristic
58#107—44no—
59C4⋊C8—21no—
60C4⋊C8—21no—
61C4⋊C8—21no—
62C4⋊C8—21no—
63#103—42no—
64#102—42no—
65C4×C8—21no—
66C4×C8D412yes—
67#95—22no—
68C24—21no—
69C2×Q8—21no—
70C2×Q8—21no—
71C2×Q8—21no—
72C2×Q8—21no—
73C2×Q8—21no—
74C2×Q8D812yes—
75C2×D4—21no—
76C2×D4—21no—
77C2×D4—21no—
78C2×D4—21no—
79C2×D4—21no—
80C2×D4Q1612yes—
81C22×C4—21no—
82C22×C4—21no—
83C22×C4—21no—
84C22×C4—22no—
85C22×C4—41no—
86C22×C4—41no—
87C22×C4—41no—
88C22×C4—41no—
89C22×C4—41no—
90C22×C4—41no—
91C22×C4—41no—
92C22×C4C2×D411yescharacteristic
93C22×C4C2×D411yescharacteristic
94C22×C4C2×D411yescharacteristic
95C2×C8—22no—
96C2×C8—22no—
97C2×C8—22no—
98C2×C8—41no—
99C2×C8—41no—
100C2×C8—41no—
101C2×C8—41no—
102C2×C8—41no—
103C2×C8—41no—
104C4⋊C4—21no—
105C4⋊C4—21no—
106C4⋊C4—21no—
107C4⋊C4—21no—
108C4⋊C4—42no—
109C4⋊C4—44no—
110C4⋊C4—42no—
111C4⋊C4—41no—
112C4⋊C4—41no—
113C4⋊C4—41no—
114C4⋊C4—41no—
115C4⋊C4—41no—
116C4⋊C4—41no—
117C4⋊C4—41no—
118C4⋊C4—41no—
119C4⋊C4—41no—
120C4⋊C4—41no—
121C22⋊C4—44no—
122C22⋊C4—41no—
123C22⋊C4—41no—
124C22⋊C4—82no—
125C42—41no—
126C42—41no—
127C42C2×D411yescharacteristic
128C42#4111yescharacteristic
129C42#4111yescharacteristic
130C42#4111yescharacteristic
131C23—22no—
132C23—41no—
133C23—41no—
134C23—41no—
135C23—41no—
136C23—41no—
137C23—41no—
138C23#12011yescharacteristic
139Q8—22no—
140Q8—22no—
141Q8—41no—
142Q8—41no—
143D4—22no—
144D4—22no—
145D4—41no—
146D4—41no—
147C2×C4—21no—
148C2×C4—21no—
149C2×C4—22no—
150C2×C4—41no—
151C2×C4—41no—
152C2×C4—41no—
153C2×C4—41no—
154C2×C4—42no—
155C2×C4—41no—
156C2×C4—42no—
157C2×C4—41no—
158C2×C4—41no—
159C2×C4—42no—
160C2×C4—41no—
161C2×C4—81no—
162C2×C4—81no—
163C2×C4—81no—
164C2×C4—81no—
165C2×C4—81no—
166C2×C4—81no—
167C2×C4—81no—
168C2×C4#12111yescharacteristic
169C2×C4#12111yescharacteristic
170C2×C4#12111yescharacteristic
171C2×C4#12311yescharacteristic
172C2×C4#12311yescharacteristic
173C2×C4#12411yescharacteristic
174C2×C4C2×D811yescharacteristic
175C2×C4C2×Q1611yescharacteristic
176C2×C4#13511yescharacteristic
177C2×C4#13511yescharacteristic
178C2×C4AGL(1,ℤ/8)11yescharacteristic
179C2×C4AGL(1,ℤ/8)11yescharacteristic
180C2×C4#13711yescharacteristic
181C2×C4#13711yescharacteristic
182C8—24no—
183C8—81no—
184C8—81no—
185C22—42no—
186C22—42no—
187C22—42no—
188C22—81no—
189C22—81no—
190C22—81no—
191C22—81no—
192C22#39411yescharacteristic
193C22#44711yescharacteristic
194C22#44911yescharacteristic
195C22#45111yescharacteristic
196C22#45211yescharacteristic
197C22#45311yescharacteristic
198C22#45611yescharacteristic
199C4—21no—
200C4—21no—
201C4—42no—
202C4—82no—
203C4—81no—
204C4—81no—
205C4—161no—
206C4#46611yescharacteristic
207C4#46711yescharacteristic
208C4#46811yescharacteristic
209C4#46811yescharacteristic
210C4#48011yescharacteristic
211C4#48211yescharacteristic
212C4#48411yescharacteristic
213C4#48511yescharacteristic
214C2—82no—
215C2#162111yescharacteristic
216C2#163211yescharacteristic
217C2#164911yescharacteristic
218C2#165011yescharacteristic
219C2#203311yescharacteristic
220C2#203611yescharacteristic
221C2#203911yescharacteristic
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. #1845C2semidirect
3. #1803C2semidirect
5. C2×Q8⋊C8C2semidirect
7. #1296C2semidirect

Maximal subgroups

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

subgroupindex
#25382
#18452
#18032
#18022
C2×Q8⋊C82
C2×D4⋊C82
#12962