Finite Group Databasebeta

← #22273

Subgroups — #22273

Order 256.

453 conjugacy classes of subgroups; only the 163 normal subgroups are shown.

normal subgroups only

This group has 453 conjugacy classes of subgroups — too many to draw. Only its 163 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
2C4×C4⋊Q8C211yesmaximal, characteristic, complemented
3C4×#124C211yesmaximal, characteristic
4#1960C211yesmaximal, characteristic, complemented
5#1959C211yesmaximal, characteristic
6#1880C212yesmaximal
7#1878C212yesmaximal, complemented
8#1878C212yesmaximal, complemented
9#1877C212yesmaximal
10#1836C211yesmaximal, characteristic, complemented
11#1745C211yesmaximal, characteristic, complemented
12C22×Q16C2212yes—
13C22×SD16C2212yes—
14C2×C4⋊Q8C2211yescharacteristic
15C2×#124C2211yescharacteristic
16C2×C4×Q8C2212yes—
17C2×C4×Q8C2211yescharacteristic
18C2×C4×D4C2211yescharacteristic
19#497C418yescomplemented
20C2×#103C2212yes—
21C2×#103C2212yes—
22C2×#103C2212yes—
23C2×#102C2212yes—
24C2×#97C2211yescharacteristic
25C2×#97C2211yescharacteristic
26#391C2212yes—
27#391C2211yescharacteristic
28#390C2211yescharacteristic
29#389—21no—
30#389—21no—
31#388—21no—
32#388—21no—
33#384C2211yescharacteristic
34#382C2211yescharacteristic
35C4×C4⋊C4—21no—
36C4×C4⋊C4—21no—
37C4×C22⋊C4—21no—
38C4×C22⋊C4—21no—
39C43C2211yescharacteristic
40#340C2212yes—
41#340C2212yes—
42#336C2212yes—
43#336C2212yes—
44C22×Q8C2311yescharacteristic
45C22×Q8C2312yes—
46C22×D4C2311yescharacteristic
47C23×C4—21no—
48C2×Q16—42no—
49C2×Q16—42no—
50C2×Q16C2×C418yescomplemented
51C2×SD16—42no—
52C2×SD16—42no—
53C2×SD16C2×C418yes—
54C22×C8C2312yes—
55C22×C8C2312yes—
56C4⋊Q8—22no—
57C4⋊Q8C2×C414yescomplemented
58#124—22no—
59#124C2×C414yes—
60C4×Q8—42no—
61C4×Q8—42no—
62C4×Q8—42no—
63C4×D4—42no—
64C2×C4⋊C4—21no—
65C2×C4⋊C4—21no—
66C2×C4⋊C4—21no—
67C2×C4⋊C4—22no—
68C2×C4⋊C4—21no—
69C2×C4⋊C4—21no—
70C2×C4⋊C4C2311yescharacteristic
71C2×C4⋊C4C2311yescharacteristic
72C2×C4⋊C4C2312yes—
73C2×C22⋊C4—21no—
74C2×C22⋊C4—21no—
75C2×C22⋊C4—21no—
76C2×C22⋊C4—21no—
77C2×C22⋊C4—21no—
78C2×C42—21no—
79C2×C42—22no—
80C2×C42D411yescharacteristic
81C2×C42D411yescharacteristic
82C2×C42D411yescharacteristic
83C2×C42D411yescharacteristic
84C2×C42C2311yescharacteristic
85C2×C42C2311yescharacteristic
86C2×C42C2311yescharacteristic
87#103—42no—
88#103—42no—
89#103—42no—
90#102—42no—
91#97—22no—
92#97C2×C414yescomplemented
93#95—21no—
94#95—21no—
95#95—21no—
96#95—21no—
97#95—22no—
98#95—22no—
99C24—21no—
100C2×Q8—21no—
101C2×Q8—22no—
102C2×Q8—22no—
103C2×Q8—22no—
104C2×Q8—22no—
105C2×Q8—22no—
106C2×Q8—22no—
107C2×Q8—22no—
108C2×Q8C22×C412yes—
109C2×Q8C22×C414yes—
110C2×D4—21no—
111C2×D4—22no—
112C2×D4—22no—
113C2×D4C22×C412yes—
114C22×C4—21no—
115C22×C4—22no—
116C22×C4—21no—
117C22×C4—21no—
118C22×C4—22no—
119C22×C4—42no—
120C22×C4—42no—
121C22×C4—42no—
122C22×C4C2×D411yescharacteristic
123C22×C4C2×D411yescharacteristic
124C22×C4C2×D411yescharacteristic
125C22×C4C2×D411yescharacteristic
126C22×C4C2×D411yescharacteristic
127C22×C4C2×D411yescharacteristic
128C22×C4C2411yescharacteristic
129Q16—48no—
130SD16—48no—
131C2×C8—22no—
132C2×C8—22no—
133C2×C8—22no—
134C2×C8—22no—
135C2×C8—22no—
136C2×C8C22×C414yes—
137C4⋊C4—24no—
138C4⋊C4—24no—
139C4⋊C4—22no—
140C4⋊C4—22no—
141C4⋊C4—22no—
142C4⋊C4—22no—
143C4⋊C4—42no—
144C4⋊C4—42no—
145C4⋊C4—42no—
146C4⋊C4—44no—
147C22⋊C4—24no—
148C22⋊C4—24no—
149C22⋊C4—42no—
150C22⋊C4—42no—
151C22⋊C4—42no—
152C42—22no—
153C42—22no—
154C42—22no—
155C42—22no—
156C42—41no—
157C42—41no—
158C42—44no—
159C42—42no—
160C42C22×C412yes—
161C42#4112yes—
162C23—22no—
163C23—41no—
164C23—41no—
165C23—41no—
166C23—41no—
167C23—41no—
168C23—41no—
169C23C22×D411yescharacteristic
170Q8—24no—
171Q8—24no—
172Q8—24no—
173Q8—42no—
174Q8—42no—
175Q8—42no—
176D4—24no—
177D4—42no—
178C2×C4—21no—
179C2×C4—21no—
180C2×C4—21no—
181C2×C4—21no—
182C2×C4—21no—
183C2×C4—21no—
184C2×C4—21no—
185C2×C4—21no—
186C2×C4—22no—
187C2×C4—24no—
188C2×C4—21no—
189C2×C4—21no—
190C2×C4—21no—
191C2×C4—21no—
192C2×C4—21no—
193C2×C4—21no—
194C2×C4—41no—
195C2×C4—42no—
196C2×C4—42no—
197C2×C4—41no—
198C2×C4—41no—
199C2×C4—42no—
200C2×C4—41no—
201C2×C4—41no—
202C2×C4—41no—
203C2×C4—41no—
204C2×C4—42no—
205C2×C4—42no—
206C2×C4—41no—
207C2×C4—84no—
208C2×C4C4×D412yes—
209C2×C4C4×D412yes—
210C2×C4#13712yes—
211C2×C4#13712yes—
212C2×C4C23×C411yescharacteristic
213C2×C4C2×#4111yescharacteristic
214C2×C4H(5,𝔽2)12yes—
215C2×C4H(5,𝔽2)12yes—
216C8—24no—
217C8—42no—
218C22—42no—
219C22—42no—
220C22—42no—
221C22—81no—
222C22—81no—
223C22—81no—
224C22—81no—
225C22C2×C4×D411yescharacteristic
226C22#53511yescharacteristic
227C22#53511yescharacteristic
228C22C2×#13711yescharacteristic
229C22C2×#13711yescharacteristic
230C22#57511yescharacteristic
231C22#57511yescharacteristic
232C4—22no—
233C4—22no—
234C4—22no—
235C4—22no—
236C4—42no—
237C4—44no—
238C4—41no—
239C4—41no—
240C4—81no—
241C4—81no—
242C4—82no—
243C4#51812yes—
244C4#54012yes—
245C2—82no—
246C2#232811yescharacteristic
247C2C4×#13711yescharacteristic
248C2C4×#13711yescharacteristic
249C2#321511yescharacteristic
250C2#321611yescharacteristic
251C2#321811yescharacteristic
252C2#321911yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutatorC2×C4C23×C4
Frattini, cosocleC22×C4C24
prime radicalC22×D4C23
LCSG ▷ C2×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. C4×C4⋊Q8C2semidirect
4. #1960C2semidirect
7. #1878C2semidirect
8. #1878C2semidirect
10. #1836C2semidirect
11. #1745C2semidirect
19. #497C4semidirect
50. C2×Q16C2×C4semidirect
57. C4⋊Q8C2×C4semidirect
92. #97C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#4972
C4×C4⋊Q82
C4×#1242
#19602
#19592
#18802
#18782
#18772
#18362
#17452