Finite Group Databasebeta

← #8376

Subgroups — #8376

Order 256.

517 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 517 conjugacy classes of subgroups — too many to draw. Only its 89 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×#547C211yesmaximal, characteristic
2#1823C212yesmaximal, complemented
3#1799C211yesmaximal, characteristic, complemented
4#1519C212yesmaximal, complemented
5#1459C211yesmaximal, characteristic, complemented
6C22×#41—21no—
7#547—24no—
8#547—24no—
9C2×C4⋊Q8C411yescharacteristic, complemented
10C2×#123—21no—
11C2×#123—21no—
12C2×C22⋊Q8—21no—
13C2×#121C411yescharacteristic, complemented
14C2×#121C2211yescharacteristic
15C2×C4×D4C411yescharacteristic, complemented
16C2×C4×D4C2211yescharacteristic
17C22×C4⋊C4C411yescharacteristic, complemented
18C22×C4⋊C4C2211yescharacteristic
19C2×#99—21no—
20C2×C22⋊C8C2212yes—
21#380C2212yes—
22#342—21no—
23#340—24no—
24#327—24no—
25#324—21no—
26#323—21no—
27C2×#41—28no—
28C2×#41—44no—
29C2×#41—42no—
30C2×#41—42no—
31C22×Q8—21no—
32C22×D4—21no—
33C22×D4—21no—
34C22×D4C2×C411yescharacteristic
35C22×D4D411yescharacteristic
36C23×C4—21no—
37C23×C4—21no—
38C23×C4—21no—
39C23×C4C2×C411yescharacteristic
40C23×C4C2311yescharacteristic
41C22×C8—22no—
42C4⋊Q8—22no—
43C4⋊Q8—22no—
44#123—28no—
45#123—28no—
46C22⋊Q8—28no—
47#121—22no—
48#121—22no—
49#121—22no—
50#121D414yes—
51C4×D4—22no—
52C4×D4—22no—
53C4×D4—22no—
54C4×D4—22no—
55C2×C4⋊C4—21no—
56C2×C4⋊C4—21no—
57C2×C4⋊C4—21no—
58C2×C4⋊C4—22no—
59C2×C4⋊C4—22no—
60C2×C4⋊C4—22no—
61C2×C4⋊C4—22no—
62C2×C4⋊C4C2×C411yescharacteristic
63C2×C4⋊C4C2×C411yescharacteristic
64C2×C4⋊C4C2×C411yescharacteristic
65C2×C4⋊C4D411yescharacteristic
66C2×C4⋊C4D414yes—
67C2×C22⋊C4—21no—
68C2×C22⋊C4—21no—
69C2×C22⋊C4—21no—
70C2×C22⋊C4—22no—
71C2×C22⋊C4D411yescharacteristic
72C2×C22⋊C4D411yescharacteristic
73C2×C42C2×C411yescharacteristic
74#99—42no—
75C22⋊C8—22no—
76C22⋊C8—22no—
77#95—22no—
78C24—21no—
79C24C22⋊C411yescharacteristic
80C24C2×D411yescharacteristic
81#41—416no—
82#41—416no—
83C2×Q8—24no—
84C2×Q8—42no—
85C2×Q8—42no—
86C2×D4—22no—
87C2×D4—22no—
88C2×D4—24no—
89C2×D4—24no—
90C2×D4—21no—
91C2×D4—42no—
92C2×D4—42no—
93C2×D4—42no—
94C2×D4—41no—
95C2×D4—41no—
96C2×D4—42no—
97C2×D4C22⋊C412yes—
98C2×D4D812yes—
99C2×D4SD1612yes—
100C22×C4—24no—
101C22×C4—24no—
102C22×C4—21no—
103C22×C4—21no—
104C22×C4—21no—
105C22×C4—24no—
106C22×C4—21no—
107C22×C4—21no—
108C22×C4—42no—
109C22×C4—42no—
110C22×C4—42no—
111C22×C4—42no—
112C22×C4—42no—
113C22×C4—41no—
114C22×C4—41no—
115C22×C4—42no—
116C22×C4—42no—
117C22×C4—42no—
118C22×C4C22⋊C411yescharacteristic
119C22×C4C22⋊C411yescharacteristic
120C22×C4C22⋊C411yescharacteristic
121C22×C4C22×C411yescharacteristic
122C22×C4C2×D412yes—
123C22×C4C2×D412yes—
124C22×C4C2×D411yescharacteristic
125C2×C8—44no—
126C2×C8—42no—
127C4⋊C4—24no—
128C4⋊C4—24no—
129C4⋊C4—21no—
130C4⋊C4—21no—
131C4⋊C4—21no—
132C4⋊C4—24no—
133C4⋊C4—22no—
134C4⋊C4—44no—
135C4⋊C4C22⋊C412yes—
136C4⋊C4C22⋊C412yes—
137C4⋊C4C22⋊C412yes—
138C22⋊C4—24no—
139C22⋊C4—24no—
140C22⋊C4—22no—
141C22⋊C4—24no—
142C22⋊C4—22no—
143C22⋊C4—42no—
144C22⋊C4—42no—
145C42—21no—
146C42—21no—
147C23—21no—
148C23—21no—
149C23—22no—
150C23—41no—
151C23—41no—
152C23—41no—
153C23—41no—
154C23—42no—
155C23—42no—
156C23—41no—
157C23—41no—
158C23#10212yes—
159C23C2×C22⋊C411yescharacteristic
160C23#12012yes—
161C23C2×D811yescharacteristic
162C23C2×SD1611yescharacteristic
163C23AGL(1,ℤ/8)12yes—
164Q8—48no—
165D4—24no—
166D4—24no—
167D4—42no—
168D4—48no—
169D4—44no—
170D4—44no—
171D4—42no—
172C2×C4—21no—
173C2×C4—21no—
174C2×C4—24no—
175C2×C4—22no—
176C2×C4—21no—
177C2×C4—21no—
178C2×C4—22no—
179C2×C4—22no—
180C2×C4—22no—
181C2×C4—41no—
182C2×C4—42no—
183C2×C4—42no—
184C2×C4—48no—
185C2×C4—48no—
186C2×C4—41no—
187C2×C4—41no—
188C2×C4—41no—
189C2×C4—41no—
190C2×C4—41no—
191C2×C4—41no—
192C2×C4—44no—
193C2×C4—44no—
194C2×C4—42no—
195C2×C4—41no—
196C2×C4—41no—
197C2×C4—82no—
198C2×C4—82no—
199C2×C4—82no—
200C2×C4#10212yes—
201C2×C4C2×C22⋊C411yescharacteristic
202C2×C4C2×C22⋊C411yescharacteristic
203C2×C4#12012yes—
204C8—82no—
205C22—21no—
206C22—22no—
207C22—21no—
208C22—22no—
209C22—22no—
210C22—41no—
211C22—41no—
212C22—42no—
213C22—42no—
214C22—42no—
215C22—41no—
216C22—41no—
217C22—42no—
218C22—42no—
219C22—42no—
220C22—82no—
221C22—82no—
222C22#37911yescharacteristic
223C22C2×#10211yescharacteristic
224C22#41811yescharacteristic
225C22#44712yes—
226C22#45012yes—
227C4—22no—
228C4—21no—
229C4—22no—
230C4—41no—
231C4—41no—
232C4—42no—
233C4—42no—
234C4—42no—
235C4—162no—
236C4#41012yes—
237C2—22no—
238C2—21no—
239C2—21no—
240C2—22no—
241C2—42no—
242C2—82no—
243C2#178211yescharacteristic
244C2#178611yescharacteristic
245C2#179611yescharacteristic
subgroupG/H
center, socleC22#379
commutatorC22×C4C22×C4
Frattini, cosocleC23×C4C23
prime radicalC2×#547C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #1823C2semidirect
3. #1799C2semidirect
4. #1519C2semidirect
5. #1459C2semidirect
9. C2×C4⋊Q8C4semidirect
13. C2×#121C4semidirect
15. C2×C4×D4C4semidirect
17. C22×C4⋊C4C4semidirect

Maximal subgroups

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

subgroupindex
C2×#5472
#18232
#17992
#15192
#14592