Finite Group Databasebeta

← #28266

Subgroups — #28266

Order 256.

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

normal subgroups only

This group has 407 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×C8×Q8C211yesmaximal, characteristic, complemented
2C2×C4×SD16C211yesmaximal, characteristic
3#2581C211yesmaximal, characteristic, complemented
4#2580C211yesmaximal, characteristic
5#2068C212yesmaximal, complemented
6#2067C212yesmaximal
7#1968C212yesmaximal
8#1941C212yesmaximal, complemented
9#1846C211yesmaximal, characteristic, complemented
10#1805C211yesmaximal, characteristic, complemented
11#1804C211yesmaximal, characteristic
12C22×SD16C2211yescharacteristic
13C2×C4×Q8C2211yescharacteristic
14C2×C4×Q8C2211yescharacteristic
15C2×C4×D4C2211yescharacteristic
16C8×Q8—24no—
17C4×SD16—24no—
18C2×#107C2212yes—
19C2×#106C2212yes—
20C2×#106C2211yescharacteristic
21C2×C4⋊C8C2211yescharacteristic
22C2×C4⋊C8C2212yes—
23C2×#103C2211yescharacteristic
24C2×#103C2212yes—
25C2×#102C2211yescharacteristic
26C2×#102C2212yes—
27C2×C4×C8C2212yes—
28C2×C4×C8C2211yescharacteristic
29#400—21no—
30#399—21no—
31#395—21no—
32#393—21no—
33#391C2212yes—
34#390C2212yes—
35#389C2212yes—
36#389C2212yes—
37#387—22no—
38#385—22no—
39#382C2211yescharacteristic
40#382C2211yescharacteristic
41#340C2212yes—
42#340C2212yes—
43C22×Q8D411yescharacteristic
44C22×Q8C2311yescharacteristic
45C22×D4C2311yescharacteristic
46C23×C4—21no—
47C2×SD16—22no—
48C2×SD16—22no—
49C2×SD16Q814yescomplemented
50C22×C8C2312yes—
51C22×C8C2311yescharacteristic
52C22×C8C2311yescharacteristic
53C4×Q8—22no—
54C4×Q8—22no—
55C4×Q8—22no—
56C4×Q8—22no—
57C4×D4—22no—
58C4×D4—22no—
59C2×C4⋊C4—21no—
60C2×C4⋊C4—22no—
61C2×C4⋊C4—22no—
62C2×C4⋊C4D412yes—
63C2×C4⋊C4D411yescharacteristic
64C2×C4⋊C4C2311yescharacteristic
65C2×C4⋊C4C2311yescharacteristic
66C2×C4⋊C4C2312yes—
67C2×C4⋊C4C2312yes—
68C2×C22⋊C4—21no—
69C2×C22⋊C4—22no—
70C2×C22⋊C4—22no—
71C2×C42—21no—
72C2×C42C2312yes—
73C2×C42C2311yescharacteristic
74#107—42no—
75#106—22no—
76#106—24no—
77C4⋊C8—24no—
78C4⋊C8—22no—
79#103—22no—
80#103—42no—
81#102—22no—
82#102—42no—
83C4×C8—24no—
84C4×C8D414yes—
85#95—21no—
86#95—21no—
87#95—21no—
88#95—21no—
89#95—22no—
90#95—22no—
91#95—22no—
92#95—22no—
93C24—21no—
94C2×Q8—22no—
95C2×Q8—24no—
96C2×Q8—21no—
97C2×Q8—21no—
98C2×Q8—21no—
99C2×Q8—21no—
100C2×Q8—21no—
101C2×Q8C2×Q812yes—
102C2×D4—21no—
103C2×D4—21no—
104C2×D4—21no—
105C2×D4—21no—
106C2×D4—21no—
107C2×D4C2×Q812yes—
108C22×C4—21no—
109C22×C4—22no—
110C22×C4—22no—
111C22×C4—21no—
112C22×C4—21no—
113C22×C4—42no—
114C22×C4—82no—
115C22×C4C2×D412yes—
116C22×C4C2×D411yescharacteristic
117C22×C4C2×D411yescharacteristic
118C22×C4C2×D412yes—
119C22×C4C2411yescharacteristic
120SD16—28no—
121C2×C8—21no—
122C2×C8—21no—
123C2×C8—22no—
124C2×C8—22no—
125C2×C8—22no—
126C2×C8C2×D412yes—
127C2×C8C2×D412yes—
128C2×C8C2×Q812yes—
129C2×C8#4112yes—
130C4⋊C4—24no—
131C4⋊C4—22no—
132C4⋊C4—21no—
133C4⋊C4—21no—
134C4⋊C4—22no—
135C4⋊C4—22no—
136C4⋊C4—21no—
137C4⋊C4—21no—
138C4⋊C4—22no—
139C4⋊C4—22no—
140C4⋊C4—42no—
141C4⋊C4—44no—
142C4⋊C4—82no—
143C22⋊C4—42no—
144C22⋊C4—44no—
145C22⋊C4—82no—
146C42—22no—
147C42—22no—
148C42—42no—
149C42C2×D412yes—
150C42#4112yes—
151C23—22no—
152C23—41no—
153C23—41no—
154C23—42no—
155C23—42no—
156C23C22×D411yescharacteristic
157Q8—28no—
158Q8—22no—
159Q8—22no—
160Q8—41no—
161Q8—41no—
162D4—22no—
163D4—22no—
164D4—41no—
165D4—41no—
166C2×C4—22no—
167C2×C4—22no—
168C2×C4—22no—
169C2×C4—22no—
170C2×C4—22no—
171C2×C4—22no—
172C2×C4—22no—
173C2×C4—22no—
174C2×C4—41no—
175C2×C4—41no—
176C2×C4—41no—
177C2×C4—41no—
178C2×C4—42no—
179C2×C4—42no—
180C2×C4—42no—
181C2×C4—42no—
182C2×C4—42no—
183C2×C4—42no—
184C2×C4—42no—
185C2×C4—42no—
186C2×C4—84no—
187C2×C4#12112yes—
188C2×C4#12112yes—
189C2×C4#13514yes—
190C2×C4C22×D411yescharacteristic
191C2×C4C22×Q811yescharacteristic
192C2×C4C2×#4111yescharacteristic
193C2×C4C2×#4111yescharacteristic
194C2×C4H(5,𝔽2)11yescharacteristic
195C2×C42−1+411yescharacteristic
196C8—22no—
197C8—42no—
198C8C22⋊Q814yes—
199C22—44no—
200C22—42no—
201C22—82no—
202C22—81no—
203C22—81no—
204C22C2×#12111yescharacteristic
205C22D4×Q811yescharacteristic
206C22#55011yescharacteristic
207C22C2×#13512yes—
208C22#57712yes—
209C4—22no—
210C4—24no—
211C4—22no—
212C4—42no—
213C4—42no—
214C4—82no—
215C4—82no—
216C4—81no—
217C4—81no—
218C4C2×C22⋊Q811yescharacteristic
219C4#53711yescharacteristic
220C4#55411yescharacteristic
221C4#55611yescharacteristic
222C2—82no—
223C2#253811yescharacteristic
224C2#305011yescharacteristic
225C2#305711yescharacteristic
226C2#338112yes—
227C2#339412yes—
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC22×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×C8×Q8C2semidirect
3. #2581C2semidirect
5. #2068C2semidirect
8. #1941C2semidirect
9. #1846C2semidirect
10. #1805C2semidirect
49. C2×SD16Q8semidirect

Maximal subgroups

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

subgroupindex
C2×C8×Q82
C2×C4×SD162
#25812
#25802
#20682
#20672
#19682
#19412
#18462
#18052
#18042