Finite Group Databasebeta

← #28619

Subgroups — #28619

Order 256.

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

normal subgroups only

This group has 428 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×#497C211yesmaximal, characteristic
2C2×#491C211yesmaximal, characteristic, complemented
3#2861C211yesmaximal, characteristic
4C4×C4⋊Q8C211yesmaximal, characteristic, complemented
5#2056C212yesmaximal, complemented
6#2055C212yesmaximal
7#1992C211yesmaximal, characteristic, complemented
8#1959C211yesmaximal, characteristic
9#1880C212yesmaximal
10#1878C212yesmaximal, complemented
11#1767C211yesmaximal, characteristic, complemented
12C22×Q16C2212yes—
13C22×SD16C2212yes—
14C2×C4⋊Q8C2211yescharacteristic
15C2×#125C2211yescharacteristic
16C2×#124C2211yescharacteristic
17C2×C4×Q8C2212yes—
18#497—24no—
19#491—24no—
20C2×C4⋊C8C2212yes—
21C2×#103C2212yes—
22C2×#103C2212yes—
23C2×#103C2212yes—
24C2×#102C2212yes—
25C2×#97C2211yescharacteristic
26C2×#97C2211yescharacteristic
27#391C2212yes—
28#391C2211yescharacteristic
29#391C2211yescharacteristic
30#390C2211yescharacteristic
31#390C2211yescharacteristic
32#389—21no—
33#389—21no—
34#388—21no—
35#388—21no—
36#388—21no—
37#388—21no—
38#384C2211yescharacteristic
39C4×C4⋊C4—21no—
40C4×C4⋊C4—21no—
41C43C2211yescharacteristic
42#340C2212yes—
43#340C2212yes—
44#336C2212yes—
45C22×Q8C2311yescharacteristic
46C22×Q8C2312yes—
47C22×D4C2311yescharacteristic
48C2×Q16—24no—
49C2×Q16—24no—
50C2×Q16—42no—
51C2×SD16—24no—
52C2×SD16—24no—
53C2×SD16—42no—
54C22×C8C2312yes—
55C22×C8C2312yes—
56C4⋊Q8—22no—
57C4⋊Q8D414yes—
58#125—22no—
59#125—22no—
60#124—22no—
61#124—22no—
62C4×Q8—42no—
63C4×Q8—42no—
64C2×C4⋊C4—21no—
65C2×C4⋊C4—21no—
66C2×C4⋊C4—22no—
67C2×C4⋊C4—21no—
68C2×C4⋊C4—21no—
69C2×C4⋊C4—21no—
70C2×C4⋊C4—21no—
71C2×C4⋊C4C2312yes—
72C2×C4⋊C4C2311yescharacteristic
73C2×C4⋊C4C2311yescharacteristic
74C2×C22⋊C4—21no—
75C2×C22⋊C4—21no—
76C2×C22⋊C4—21no—
77C2×C22⋊C4—21no—
78C2×C22⋊C4—21no—
79C2×C22⋊C4—21no—
80C2×C42—22no—
81C2×C42D411yescharacteristic
82C2×C42D411yescharacteristic
83C2×C42D411yescharacteristic
84C2×C42D411yescharacteristic
85C2×C42C2311yescharacteristic
86C2×C42C2311yescharacteristic
87C2×C42C2311yescharacteristic
88C4⋊C8—24no—
89#103—24no—
90#103—24no—
91#103—42no—
92#102—42no—
93#97—22no—
94#97—22no—
95#95—22no—
96#95—22no—
97#95—21no—
98#95—21no—
99#95—21no—
100#95—21no—
101C24—21no—
102C2×Q8—22no—
103C2×Q8—21no—
104C2×Q8—22no—
105C2×Q8—22no—
106C2×Q8—21no—
107C2×Q8—22no—
108C2×Q8—22no—
109C2×Q8—22no—
110C2×Q8—22no—
111C2×Q8C2×D414yes—
112C2×D4—21no—
113C2×D4—21no—
114C2×D4—22no—
115C2×D4—22no—
116C22×C4—22no—
117C22×C4—21no—
118C22×C4—22no—
119C22×C4—21no—
120C22×C4—21no—
121C22×C4C2×D411yescharacteristic
122C22×C4C2×D411yescharacteristic
123C22×C4C2×D411yescharacteristic
124C22×C4C2×D411yescharacteristic
125C22×C4C2×D411yescharacteristic
126C22×C4C2×D411yescharacteristic
127C22×C4C2411yescharacteristic
128Q16—48no—
129SD16—48no—
130C2×C8—22no—
131C2×C8—22no—
132C2×C8—22no—
133C2×C8—22no—
134C2×C8—22no—
135C2×C8#4114yes—
136C4⋊C4—24no—
137C4⋊C4—24no—
138C4⋊C4—21no—
139C4⋊C4—21no—
140C4⋊C4—21no—
141C4⋊C4—21no—
142C4⋊C4—22no—
143C4⋊C4—22no—
144C4⋊C4—42no—
145C4⋊C4—42no—
146C4⋊C4—44no—
147C4⋊C4—42no—
148C4⋊C4—42no—
149C22⋊C4—42no—
150C22⋊C4—42no—
151C22⋊C4—42no—
152C22⋊C4—42no—
153C22⋊C4—42no—
154C22⋊C4—42no—
155C42—22no—
156C42—22no—
157C42—22no—
158C42—22no—
159C42—41no—
160C42—41no—
161C42—44no—
162C42C2×D412yes—
163C42#4112yes—
164C23—41no—
165C23—41no—
166C23—41no—
167C23—41no—
168C23—41no—
169C23—41no—
170C23—41no—
171C23C22×D411yescharacteristic
172Q8—24no—
173Q8—24no—
174Q8—42no—
175Q8—42no—
176Q8—42no—
177Q8—42no—
178D4—42no—
179D4—42no—
180C2×C4—21no—
181C2×C4—21no—
182C2×C4—21no—
183C2×C4—21no—
184C2×C4—21no—
185C2×C4—21no—
186C2×C4—21no—
187C2×C4—21no—
188C2×C4—24no—
189C2×C4—21no—
190C2×C4—21no—
191C2×C4—21no—
192C2×C4—21no—
193C2×C4—21no—
194C2×C4—21no—
195C2×C4—42no—
196C2×C4—42no—
197C2×C4—41no—
198C2×C4—41no—
199C2×C4—41no—
200C2×C4—41no—
201C2×C4—42no—
202C2×C4—41no—
203C2×C4—41no—
204C2×C4—41no—
205C2×C4—41no—
206C2×C4—41no—
207C2×C4—42no—
208C2×C4—42no—
209C2×C4#12112yes—
210C2×C4#12112yes—
211C2×C4#13712yes—
212C2×C4#13712yes—
213C2×C4C22×D411yescharacteristic
214C2×C4C2×#4112yes—
215C2×C4C2×#4111yescharacteristic
216C2×C4H(5,𝔽2)12yes—
217C8—24no—
218C8—42no—
219C22—81no—
220C22—81no—
221C22—81no—
222C22—81no—
223C22—81no—
224C22—81no—
225C22—81no—
226C22C2×#12111yescharacteristic
227C22#53211yescharacteristic
228C22#53511yescharacteristic
229C22C2×#13711yescharacteristic
230C22C2×#13711yescharacteristic
231C22#57511yescharacteristic
232C22#57511yescharacteristic
233C4—22no—
234C4—22no—
235C4—22no—
236C4—22no—
237C4—41no—
238C4—41no—
239C4—44no—
240C4—82no—
241C4—81no—
242C4—81no—
243C4—81no—
244C4#54612yes—
245C4#55912yes—
246C2—161no—
247C2#254411yescharacteristic
248C2#303711yescharacteristic
249C2#303711yescharacteristic
250C2#317211yescharacteristic
251C2#318111yescharacteristic
252C2#321811yescharacteristic
253C2#321911yescharacteristic
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
2. C2×#491C2semidirect
4. C4×C4⋊Q8C2semidirect
5. #2056C2semidirect
7. #1992C2semidirect
10. #1878C2semidirect
11. #1767C2semidirect

Maximal subgroups

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

subgroupindex
C2×#4972
C2×#4912
#28612
C4×C4⋊Q82
#20562
#20552
#19922
#19592
#18802
#18782
#17672