Finite Group Databasebeta

← #8409

Subgroups — #8409

Order 256.

486 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 486 conjugacy classes of subgroups — too many to draw. Only its 87 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#3449C211yesmaximal, characteristic
2#1798C211yesmaximal, characteristic, complemented
3#1540C212yesmaximal, complemented
4#1480C212yesmaximal, complemented
5#1458C211yesmaximal, characteristic, complemented
6C22×#41—21no—
7#537—21no—
8#537C412yes—
9#536—22no—
10#535—22no—
11#534—21no—
12#534C412yes—
13#532—21no—
14#532C2212yes—
15#525—24no—
16C2×#117C2211yescharacteristic
17#431C2212yescomplemented
18C2×C4⋊C8C2211yescharacteristic, complemented
19C2×C4⋊C8C2211yescharacteristic, complemented
20C2×C22⋊C8—21no—
21#336—21no—
22#332—22no—
23#331—22no—
24D4⋊C8—24no—
25#324—21no—
26#323—21no—
27C2×#41—22no—
28C2×#41—22no—
29C2×#41—22no—
30C2×#41—22no—
31C2×#41—42no—
32C2×#41—42no—
33C2×#41—42no—
34C2×#41—42no—
35C22×Q8—21no—
36C22×D4—21no—
37C22×D4—22no—
38C23×C4—22no—
39C23×C4D411yescharacteristic
40C22×C8—21no—
41C22×C8—21no—
42C4⋊Q8—21no—
43C4⋊Q8D412yes—
44#127—21no—
45#127D412yes—
46#124—22no—
47#124—22no—
48#123—24no—
49#123—24no—
50C22⋊Q8—24no—
51C22⋊Q8—24no—
52#121—24no—
53#121—24no—
54#121—22no—
55#121—22no—
56#121—22no—
57#121C2×C414yes—
58#120—24no—
59#120—24no—
60C4×D4—22no—
61C4×D4—22no—
62C4×D4—22no—
63C4×D4D414yes—
64#117—22no—
65#117—22no—
66C2×C4⋊C4C2×C411yescharacteristic
67C2×C4⋊C4D411yescharacteristic
68C2×C22⋊C4C2×C411yescharacteristic
69C2×C22⋊C4D411yescharacteristic
70C2×C42D411yescharacteristic
71C2×C42C2311yescharacteristic
72C4⋊C8—22no—
73C4⋊C8—22no—
74C4⋊C8—22no—
75C22⋊C8—22no—
76C22⋊C8—42no—
77C4×C8—22no—
78C24—22no—
79C24C22⋊C411yescharacteristic
80#41—44no—
81#41—44no—
82#41—44no—
83#41—44no—
84#41—44no—
85#41—44no—
86#41—42no—
87#41—42no—
88#41—42no—
89#41—42no—
90C2×Q8—22no—
91C2×Q8—22no—
92C2×Q8—42no—
93C2×Q8—42no—
94C2×D4—24no—
95C2×D4—22no—
96C2×D4—24no—
97C2×D4—22no—
98C2×D4—22no—
99C2×D4—22no—
100C2×D4—22no—
101C2×D4—42no—
102C2×D4—42no—
103C2×D4—44no—
104C2×D4—44no—
105C2×D4C22⋊C414yes—
106C22×C4—21no—
107C22×C4—21no—
108C22×C4—21no—
109C22×C4—21no—
110C22×C4—24no—
111C22×C4—22no—
112C22×C4—24no—
113C22×C4—44no—
114C22×C4—44no—
115C22×C4C22⋊C411yescharacteristic
116C22×C4C22⋊C411yescharacteristic
117C22×C4C22⋊C411yescharacteristic
118C22×C4D812yes—
119C22×C4SD1612yes—
120C22×C4C22×C411yescharacteristic
121C22×C4C2×D411yescharacteristic
122C22×C4C2×D411yescharacteristic
123C2×C8—42no—
124C2×C8—41no—
125C2×C8—41no—
126C2×C8—41no—
127C2×C8—41no—
128C2×C8—41no—
129C2×C8—41no—
130C4⋊C4—22no—
131C4⋊C4—22no—
132C4⋊C4—22no—
133C4⋊C4—22no—
134C4⋊C4—22no—
135C4⋊C4C22⋊C414yes—
136C22⋊C4—24no—
137C22⋊C4—24no—
138C22⋊C4—22no—
139C22⋊C4—24no—
140C22⋊C4—22no—
141C22⋊C4—24no—
142C42—21no—
143C42—21no—
144C42C2×D412yes—
145C42C2×D412yes—
146C23—21no—
147C23—21no—
148C23—24no—
149C23—41no—
150C23—41no—
151C23—44no—
152C23—82no—
153C23—82no—
154C23#10212yes—
155C23C2×C22⋊C411yescharacteristic
156Q8—44no—
157Q8—42no—
158Q8—42no—
159D4—44no—
160D4—44no—
161D4—48no—
162D4—44no—
163D4—44no—
164D4—42no—
165D4—42no—
166D4—82no—
167D4—82no—
168D4—82no—
169C2×C4—21no—
170C2×C4—21no—
171C2×C4—22no—
172C2×C4—22no—
173C2×C4—22no—
174C2×C4—22no—
175C2×C4—21no—
176C2×C4—21no—
177C2×C4—24no—
178C2×C4—41no—
179C2×C4—41no—
180C2×C4—48no—
181C2×C4—44no—
182C2×C4—44no—
183C2×C4—44no—
184C2×C4—41no—
185C2×C4—41no—
186C2×C4#10212yes—
187C2×C4C2×C22⋊C412yes—
188C2×C4#12012yes—
189C2×C4#12011yescharacteristic
190C2×C4#12011yescharacteristic
191C2×C4C2×D811yescharacteristic
192C2×C4C2×SD1611yescharacteristic
193C2×C4AGL(1,ℤ/8)11yescharacteristic
194C2×C4AGL(1,ℤ/8)11yescharacteristic
195C8—82no—
196C8—81no—
197C8—81no—
198C22—21no—
199C22—21no—
200C22—22no—
201C22—42no—
202C22—41no—
203C22—41no—
204C22—44no—
205C22—44no—
206C22—44no—
207C22—81no—
208C22—88no—
209C22#37911yescharacteristic
210C22C2×#10211yescharacteristic
211C22#41811yescharacteristic
212C4—22no—
213C4—22no—
214C4—22no—
215C4—41no—
216C4—44no—
217C4—41no—
218C4#44711yescharacteristic
219C4#44711yescharacteristic
220C4#45011yescharacteristic
221C4#45011yescharacteristic
222C2—21no—
223C2—21no—
224C2—42no—
225C2—84no—
226C2#178611yescharacteristic
227C2#179111yescharacteristic
228C2#179411yescharacteristic
subgroupG/H
centerC2×C4#120
commutatorC22×C4C22×C4
Frattini, cosocleC2×C42C23
socleC22#379
prime radical#3449C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #1798C2semidirect
3. #1540C2semidirect
4. #1480C2semidirect
5. #1458C2semidirect
17. #431C22semidirect
18. C2×C4⋊C8C22semidirect
19. C2×C4⋊C8C22semidirect

Maximal subgroups

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

subgroupindex
#34492
#17982
#15402
#14802
#14582