Finite Group Databasebeta

← #28398

Subgroups — #28398

Order 256.

485 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 485 conjugacy classes of subgroups — too many to draw. Only its 139 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×#487C211yesmaximal, characteristic, complemented
2C2×#476C211yesmaximal, characteristic, complemented
3C2×#461C211yesmaximal, characteristic, complemented
4C2×C4×D8C211yesmaximal, characteristic
5#2563C211yesmaximal, characteristic, complemented
6#2547C211yesmaximal, characteristic, complemented
7#2097C212yesmaximal, complemented
8#2012C212yesmaximal
9#1968C212yesmaximal, complemented
10#1958C212yesmaximal, complemented
11C8×C4⋊C4C211yesmaximal, characteristic, complemented
12C22×SD16C2212yes—
13C22×D8C2211yescharacteristic
14C2×C4⋊Q8C2211yescharacteristic
15C2×C4⋊Q8C2211yescharacteristic
16C2×#124C2212yes—
17C2×C4×D4C2211yescharacteristic
18C2×C4×D4C2211yescharacteristic
19#487—24no—
20#476—24no—
21#461—24no—
22C4×D8—24no—
23C2×#107C2211yescharacteristic
24C2×#106C2212yes—
25C2×C4⋊C8C2211yescharacteristic
26C2×#103C2212yes—
27C2×#103C2212yes—
28C2×#103C2212yes—
29C2×#102C2211yescharacteristic
30C2×#102C2211yescharacteristic
31C2×#102C2212yes—
32C2×C4×C8C2212yes—
33C2×C4×C8C2211yescharacteristic
34#397—22no—
35#393—22no—
36#391C2212yes—
37#390C2212yes—
38#382C2212yes—
39#380—22no—
40#380—22no—
41C4×C4⋊C4C2211yescharacteristic
42#336C2212yes—
43C22×Q8—21no—
44C22×Q8C2312yes—
45C22×D4C2311yescharacteristic
46C22×D4C2311yescharacteristic
47C23×C4—21no—
48C23×C4—21no—
49C2×SD16—24no—
50C2×SD16—24no—
51C2×SD16—42no—
52C2×D8—22no—
53C2×D8—22no—
54C2×D8—22no—
55C22×C8C2311yescharacteristic
56C22×C8C2312yes—
57C22×C8C2311yescharacteristic
58C4⋊Q8—22no—
59C4⋊Q8—22no—
60C4⋊Q8—22no—
61C4⋊Q8—22no—
62#124—24no—
63#124—42no—
64C4×D4—22no—
65C4×D4—22no—
66C4×D4—22no—
67C4×D4—22no—
68C2×C4⋊C4—21no—
69C2×C4⋊C4—21no—
70C2×C4⋊C4—21no—
71C2×C4⋊C4D412yes—
72C2×C4⋊C4C2311yescharacteristic
73C2×C4⋊C4C2311yescharacteristic
74C2×C4⋊C4C2312yes—
75C2×C22⋊C4—21no—
76C2×C22⋊C4—21no—
77C2×C22⋊C4—24no—
78C2×C22⋊C4—24no—
79C2×C42C2312yes—
80C2×C42C2311yescharacteristic
81#107—22no—
82#106—24no—
83C4⋊C8D414yes—
84#103—24no—
85#103—24no—
86#103—24no—
87#102—22no—
88#102—22no—
89#102—24no—
90C4×C8—42no—
91C4×C8D414yes—
92#95—24no—
93#95—24no—
94#95D412yes—
95C24—21no—
96C24—21no—
97C2×Q8—22no—
98C2×Q8—22no—
99C2×Q8—22no—
100C2×Q8—22no—
101C2×Q8—22no—
102C2×Q8—22no—
103C2×Q8—42no—
104C2×Q8—82no—
105C2×D4—21no—
106C2×D4—21no—
107C2×D4—21no—
108C2×D4—21no—
109C2×D4—21no—
110C2×D4—21no—
111C2×D4—21no—
112C2×D4—21no—
113C2×D4—21no—
114C2×D4—21no—
115C2×D4#4112yes—
116C2×D4#4112yes—
117C22×C4—21no—
118C22×C4—21no—
119C22×C4—22no—
120C22×C4—22no—
121C22×C4—42no—
122C22×C4—42no—
123C22×C4—44no—
124C22×C4—44no—
125C22×C4C2×D414yes—
126C22×C4C2×D411yescharacteristic
127C22×C4C2×D411yescharacteristic
128C22×C4C2411yescharacteristic
129SD16—48no—
130D8—28no—
131C2×C8—21no—
132C2×C8—21no—
133C2×C8—22no—
134C2×C8—21no—
135C2×C8—22no—
136C2×C8—21no—
137C2×C8C2×D412yes—
138C2×C8C2×D412yes—
139C2×C8C2×D414yes—
140C4⋊C4—24no—
141C4⋊C4—22no—
142C4⋊C4—21no—
143C4⋊C4—21no—
144C4⋊C4—21no—
145C4⋊C4—22no—
146C4⋊C4—21no—
147C4⋊C4—42no—
148C4⋊C4—42no—
149C4⋊C4—42no—
150C22⋊C4—48no—
151C22⋊C4—42no—
152C22⋊C4—42no—
153C22⋊C4—84no—
154C42—22no—
155C42—22no—
156C42C2×D412yes—
157C42C2×D412yes—
158C23—22no—
159C23—22no—
160C23—41no—
161C23—41no—
162C23—41no—
163C23—41no—
164C23—42no—
165C23—42no—
166C23—42no—
167C23—42no—
168C23C22×D411yescharacteristic
169Q8—42no—
170Q8—42no—
171Q8—42no—
172Q8—42no—
173Q8—84no—
174D4—22no—
175D4—22no—
176D4—22no—
177D4—22no—
178D4—41no—
179D4—41no—
180D4—41no—
181D4—41no—
182C2×C4—22no—
183C2×C4—24no—
184C2×C4—24no—
185C2×C4—24no—
186C2×C4—41no—
187C2×C4—41no—
188C2×C4—42no—
189C2×C4—42no—
190C2×C4—42no—
191C2×C4—42no—
192C2×C4—42no—
193C2×C4—42no—
194C2×C4—44no—
195C2×C4—84no—
196C2×C4—84no—
197C2×C4#12712yes—
198C2×C4#12712yes—
199C2×C4#13514yes—
200C2×C4C22×D411yescharacteristic
201C2×C4C22×D411yescharacteristic
202C2×C4C2×#4111yescharacteristic
203C2×C4C2×#4111yescharacteristic
204C2×C4H(5,𝔽2)11yescharacteristic
205C2×C42−1+411yescharacteristic
206C8—22no—
207C8—22no—
208C8—24no—
209C22—44no—
210C22—44no—
211C22—42no—
212C22—42no—
213C22—82no—
214C22—81no—
215C22—81no—
216C22—81no—
217C22—81no—
218C22—82no—
219C22C2×#12711yescharacteristic
220C22#54611yescharacteristic
221C22#54811yescharacteristic
222C22C2×#13512yes—
223C22#57712yes—
224C4—22no—
225C4—44no—
226C4—81no—
227C4—81no—
228C4—82no—
229C4—82no—
230C4#49514yes—
231C4#52511yescharacteristic
232C4#53711yescharacteristic
233C4#54611yescharacteristic
234C4#54811yescharacteristic
235C2—82no—
236C2—82no—
237C2#254011yescharacteristic
238C2C2×#49511yescharacteristic
239C2#315411yescharacteristic
240C2#330312yes—
241C2#336012yes—
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC22×D8C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#487C2semidirect
2. C2×#476C2semidirect
3. C2×#461C2semidirect
5. #2563C2semidirect
6. #2547C2semidirect
7. #2097C2semidirect
9. #1968C2semidirect
10. #1958C2semidirect
11. C8×C4⋊C4C2semidirect

Maximal subgroups

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

subgroupindex
C2×#4872
C2×#4762
C2×#4612
C2×C4×D82
#25632
#25472
#20972
#20122
#19682
#19582
C8×C4⋊C42