Finite Group Databasebeta

← #28659

Subgroups — #28659

Order 256.

396 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 396 conjugacy classes of subgroups — too many to draw. Only its 127 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×#501C211yesmaximal, characteristic, complemented
2C2×#489C211yesmaximal, characteristic
3#2861C211yesmaximal, characteristic
4C4×C4⋊Q8C211yesmaximal, characteristic, complemented
5#2068C212yesmaximal, complemented
6#2065C212yesmaximal
7#1992C211yesmaximal, characteristic, complemented
8#1959C211yesmaximal, characteristic
9#1927C212yesmaximal, complemented
10#1925C212yesmaximal
11#1767C211yesmaximal, characteristic, complemented
12C2×C4⋊Q8C2211yescharacteristic
13C2×#125C2211yescharacteristic
14C2×#124C2211yescharacteristic
15C2×C4×Q8—21no—
16#501—24no—
17#489—24no—
18C2×#107C2212yes—
19C2×#106C2212yes—
20C2×C4⋊C8C2212yes—
21C2×#103C2212yes—
22C2×#103C2212yes—
23C2×#102C2212yes—
24C2×#102C2212yes—
25C2×#97C2211yescharacteristic
26C2×#97C2211yescharacteristic
27#391—21no—
28#391C2211yescharacteristic
29#391C2211yescharacteristic
30#390C2211yescharacteristic
31#390C2211yescharacteristic
32#389—21no—
33#389C2212yes—
34#388—21no—
35#388—21no—
36#388—21no—
37#388—21no—
38#384C2211yescharacteristic
39C4×C4⋊C4—21no—
40C4×C4⋊C4C2212yes—
41C43C2211yescharacteristic
42#340C2212yes—
43#340C2212yes—
44#336C2212yes—
45C22×Q8—21no—
46C22×Q8C2311yescharacteristic
47C22×D4C2311yescharacteristic
48C22×C8C2312yes—
49C22×C8C2312yes—
50C4⋊Q8—22no—
51C4⋊Q8D414yes—
52#125—22no—
53#125—22no—
54#124—22no—
55#124—22no—
56C4×Q8—44no—
57C2×C4⋊C4—21no—
58C2×C4⋊C4—21no—
59C2×C4⋊C4—22no—
60C2×C4⋊C4—21no—
61C2×C4⋊C4—21no—
62C2×C4⋊C4—21no—
63C2×C4⋊C4C2311yescharacteristic
64C2×C4⋊C4C2311yescharacteristic
65C2×C4⋊C4C2312yes—
66C2×C4⋊C4C2312yes—
67C2×C22⋊C4—21no—
68C2×C22⋊C4—21no—
69C2×C22⋊C4—21no—
70C2×C22⋊C4—21no—
71C2×C22⋊C4—21no—
72C2×C22⋊C4—21no—
73C2×C42—22no—
74C2×C42D411yescharacteristic
75C2×C42D411yescharacteristic
76C2×C42D411yescharacteristic
77C2×C42D411yescharacteristic
78C2×C42C2311yescharacteristic
79C2×C42C2311yescharacteristic
80C2×C42C2311yescharacteristic
81#107—24no—
82#106—24no—
83C4⋊C8—24no—
84#103—24no—
85#103—24no—
86#102—24no—
87#102—24no—
88#97—22no—
89#97—22no—
90#95—22no—
91#95—22no—
92#95—21no—
93#95—21no—
94#95—21no—
95#95—21no—
96C24—21no—
97C2×Q8—24no—
98C2×Q8—21no—
99C2×Q8—22no—
100C2×Q8—21no—
101C2×Q8—22no—
102C2×Q8—42no—
103C2×Q8—42no—
104C2×D4—21no—
105C2×D4—22no—
106C2×D4—21no—
107C2×D4—22no—
108C22×C4—22no—
109C22×C4—21no—
110C22×C4—22no—
111C22×C4—21no—
112C22×C4—21no—
113C22×C4C2×D411yescharacteristic
114C22×C4C2×D411yescharacteristic
115C22×C4C2×D411yescharacteristic
116C22×C4C2×D411yescharacteristic
117C22×C4C2×D411yescharacteristic
118C22×C4C2×D411yescharacteristic
119C22×C4C2411yescharacteristic
120C2×C8—22no—
121C2×C8—22no—
122C2×C8—22no—
123C2×C8—22no—
124C2×C8—22no—
125C2×C8#4114yes—
126C4⋊C4—24no—
127C4⋊C4—22no—
128C4⋊C4—21no—
129C4⋊C4—21no—
130C4⋊C4—22no—
131C4⋊C4—22no—
132C4⋊C4—21no—
133C4⋊C4—21no—
134C4⋊C4—42no—
135C4⋊C4—42no—
136C4⋊C4—44no—
137C4⋊C4—42no—
138C4⋊C4—42no—
139C4⋊C4C2×D414yes—
140C22⋊C4—42no—
141C22⋊C4—42no—
142C22⋊C4—42no—
143C22⋊C4—42no—
144C22⋊C4—42no—
145C22⋊C4—42no—
146C42—22no—
147C42—22no—
148C42—22no—
149C42—22no—
150C42—41no—
151C42—41no—
152C42—44no—
153C42C2×D412yes—
154C42#4112yes—
155C23—41no—
156C23—41no—
157C23—41no—
158C23—41no—
159C23—41no—
160C23—41no—
161C23—41no—
162C23C22×D411yescharacteristic
163Q8—48no—
164Q8—42no—
165Q8—42no—
166D4—42no—
167D4—42no—
168C2×C4—21no—
169C2×C4—21no—
170C2×C4—21no—
171C2×C4—21no—
172C2×C4—21no—
173C2×C4—21no—
174C2×C4—21no—
175C2×C4—21no—
176C2×C4—24no—
177C2×C4—21no—
178C2×C4—21no—
179C2×C4—21no—
180C2×C4—21no—
181C2×C4—21no—
182C2×C4—21no—
183C2×C4—42no—
184C2×C4—42no—
185C2×C4—41no—
186C2×C4—41no—
187C2×C4—41no—
188C2×C4—42no—
189C2×C4—41no—
190C2×C4—41no—
191C2×C4—42no—
192C2×C4—41no—
193C2×C4—41no—
194C2×C4—41no—
195C2×C4—41no—
196C2×C4—42no—
197C2×C4#12112yes—
198C2×C4#12112yes—
199C2×C4AGL(1,ℤ/8)12yes—
200C2×C4#13712yes—
201C2×C4C22×D411yescharacteristic
202C2×C4C2×#4112yes—
203C2×C4C2×#4111yescharacteristic
204C2×C42−1+412yes—
205C8—24no—
206C8—42no—
207C22—81no—
208C22—81no—
209C22—81no—
210C22—81no—
211C22—81no—
212C22—81no—
213C22—81no—
214C22C2×#12111yescharacteristic
215C22#53211yescharacteristic
216C22#53611yescharacteristic
217C22C2×AGL(1,ℤ/8)11yescharacteristic
218C22C2×#13711yescharacteristic
219C22#57511yescharacteristic
220C22#57511yescharacteristic
221C4—22no—
222C4—22no—
223C4—22no—
224C4—22no—
225C4—41no—
226C4—41no—
227C4—44no—
228C4—82no—
229C4—81no—
230C4—81no—
231C4—81no—
232C4#54812yes—
233C4#55612yes—
234C2—161no—
235C2#254511yescharacteristic
236C2#303611yescharacteristic
237C2#303711yescharacteristic
238C2#317111yescharacteristic
239C2#318111yescharacteristic
240C2#323811yescharacteristic
241C2#324011yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC22×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#501C2semidirect
4. C4×C4⋊Q8C2semidirect
5. #2068C2semidirect
7. #1992C2semidirect
9. #1927C2semidirect
11. #1767C2semidirect

Maximal subgroups

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

subgroupindex
C2×#5012
C2×#4892
#28612
C4×C4⋊Q82
#20682
#20652
#19922
#19592
#19272
#19252
#17672