Finite Group Databasebeta

← #28159

Subgroups — #28159

Order 256.

452 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 452 conjugacy classes of subgroups — too many to draw. Only its 133 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×#462C211yesmaximal, characteristic, complemented
2C2×#461C211yesmaximal, characteristic
3C4×C4⋊Q8C211yesmaximal, characteristic, complemented
4C4×#124C211yesmaximal, characteristic
5#2074C212yesmaximal, complemented
6#2015C212yesmaximal
7#1880C212yesmaximal
8#1878C212yesmaximal, complemented
9#1831C211yesmaximal, characteristic, complemented
10#1803C211yesmaximal, characteristic, complemented
11#1802C211yesmaximal, characteristic
12C22×Q16C2212yes—
13C22×SD16C2212yes—
14C2×C4⋊Q8C2211yescharacteristic
15C2×#124C2211yescharacteristic
16C2×C4×Q8C2211yescharacteristic
17C2×C4×Q8C2212yes—
18C2×C4×D4C2211yescharacteristic
19#462—24no—
20#461—24no—
21C2×C4⋊C8C2211yescharacteristic
22C2×C4⋊C8C2211yescharacteristic
23C2×#103C2212yes—
24C2×#103C2212yes—
25C2×#103C2212yes—
26C2×#102C2212yes—
27#391C2212yes—
28#391C2211yescharacteristic
29#390C2211yescharacteristic
30#389—21no—
31#389—21no—
32#388—21no—
33#388—21no—
34#384C2211yescharacteristic
35#382C2211yescharacteristic
36C4×C4⋊C4—21no—
37C4×C4⋊C4—21no—
38C4×C22⋊C4—21no—
39C4×C22⋊C4—21no—
40C43C2211yescharacteristic
41#340C2212yes—
42#340C2212yes—
43#336C2212yes—
44#336C2212yes—
45C22×Q8C2312yes—
46C22×Q8C2311yescharacteristic
47C22×D4C2311yescharacteristic
48C23×C4—21no—
49C2×Q16—24no—
50C2×Q16—24no—
51C2×Q16—42no—
52C2×SD16—24no—
53C2×SD16—24no—
54C2×SD16—42no—
55C22×C8C2312yes—
56C22×C8C2312yes—
57C4⋊Q8—22no—
58C4⋊Q8D414yes—
59#124—22no—
60#124—22no—
61C4×Q8—22no—
62C4×Q8—22no—
63C4×Q8—42no—
64C4×Q8—42no—
65C4×D4—22no—
66C4×D4—22no—
67C2×C4⋊C4—21no—
68C2×C4⋊C4—21no—
69C2×C4⋊C4—21no—
70C2×C4⋊C4—22no—
71C2×C4⋊C4—21no—
72C2×C4⋊C4—21no—
73C2×C4⋊C4C2311yescharacteristic
74C2×C4⋊C4C2311yescharacteristic
75C2×C4⋊C4C2312yes—
76C2×C22⋊C4—21no—
77C2×C22⋊C4—21no—
78C2×C22⋊C4—21no—
79C2×C22⋊C4—21no—
80C2×C22⋊C4—21no—
81C2×C42—21no—
82C2×C42—22no—
83C2×C42D411yescharacteristic
84C2×C42D411yescharacteristic
85C2×C42D411yescharacteristic
86C2×C42D411yescharacteristic
87C2×C42C2311yescharacteristic
88C2×C42C2311yescharacteristic
89C2×C42C2311yescharacteristic
90C4⋊C8—22no—
91C4⋊C8—22no—
92#103—24no—
93#103—24no—
94#103—42no—
95#102—42no—
96#95—21no—
97#95—21no—
98#95—21no—
99#95—21no—
100#95—22no—
101#95—22no—
102C24—21no—
103C2×Q8—21no—
104C2×Q8—22no—
105C2×Q8—21no—
106C2×Q8—21no—
107C2×Q8—22no—
108C2×Q8—22no—
109C2×Q8—21no—
110C2×Q8—22no—
111C2×Q8—22no—
112C2×Q8—21no—
113C2×Q8C2×D414yes—
114C2×Q8#4112yes—
115C2×D4—21no—
116C2×D4—21no—
117C2×D4—21no—
118C2×D4—21no—
119C2×D4—21no—
120C2×D4#4112yes—
121C22×C4—21no—
122C22×C4—22no—
123C22×C4—21no—
124C22×C4—21no—
125C22×C4—22no—
126C22×C4—42no—
127C22×C4—42no—
128C22×C4—42no—
129C22×C4C2×D411yescharacteristic
130C22×C4C2×D411yescharacteristic
131C22×C4C2×D411yescharacteristic
132C22×C4C2×D411yescharacteristic
133C22×C4C2×D411yescharacteristic
134C22×C4C2×D411yescharacteristic
135C22×C4C2411yescharacteristic
136Q16—48no—
137SD16—48no—
138C2×C8—22no—
139C2×C8—22no—
140C2×C8—22no—
141C2×C8—22no—
142C2×C8—22no—
143C2×C8—22no—
144C4⋊C4—24no—
145C4⋊C4—24no—
146C4⋊C4—21no—
147C4⋊C4—21no—
148C4⋊C4—21no—
149C4⋊C4—21no—
150C4⋊C4—22no—
151C4⋊C4—22no—
152C4⋊C4—42no—
153C4⋊C4—44no—
154C4⋊C4—42no—
155C4⋊C4—42no—
156C22⋊C4—24no—
157C22⋊C4—24no—
158C22⋊C4—42no—
159C22⋊C4—42no—
160C22⋊C4—42no—
161C42—22no—
162C42—22no—
163C42—22no—
164C42—21no—
165C42—21no—
166C42—21no—
167C42—21no—
168C42—41no—
169C42—44no—
170C42—42no—
171C42C2×D412yes—
172C42#4112yes—
173C23—22no—
174C23—41no—
175C23—41no—
176C23—41no—
177C23—41no—
178C23—41no—
179C23—41no—
180C23C22×D411yescharacteristic
181Q8—24no—
182Q8—22no—
183Q8—24no—
184Q8—22no—
185Q8—41no—
186Q8—41no—
187Q8—42no—
188Q8—42no—
189D4—22no—
190D4—22no—
191D4—41no—
192D4—41no—
193C2×C4—21no—
194C2×C4—21no—
195C2×C4—21no—
196C2×C4—21no—
197C2×C4—24no—
198C2×C4—21no—
199C2×C4—21no—
200C2×C4—22no—
201C2×C4—21no—
202C2×C4—21no—
203C2×C4—21no—
204C2×C4—21no—
205C2×C4—21no—
206C2×C4—21no—
207C2×C4—21no—
208C2×C4—41no—
209C2×C4—42no—
210C2×C4—42no—
211C2×C4—41no—
212C2×C4—41no—
213C2×C4—41no—
214C2×C4—42no—
215C2×C4—42no—
216C2×C4—41no—
217C2×C4—41no—
218C2×C4—41no—
219C2×C4—41no—
220C2×C4—42no—
221C2×C4—84no—
222C2×C4#12112yes—
223C2×C4#12112yes—
224C2×C4#13512yes—
225C2×C4#13712yes—
226C2×C4#13712yes—
227C2×C4C22×D411yescharacteristic
228C2×C4C2×#4111yescharacteristic
229C2×C4C2×#4111yescharacteristic
230C2×C4C2×#4111yescharacteristic
231C2×C4H(5,𝔽2)11yescharacteristic
232C2×C4H(5,𝔽2)11yescharacteristic
233C8—42no—
234C8—42no—
235C22—42no—
236C22—42no—
237C22—42no—
238C22—81no—
239C22—81no—
240C22—81no—
241C22—81no—
242C22C2×#12111yescharacteristic
243C22#52511yescharacteristic
244C22#53511yescharacteristic
245C22C2×#13511yescharacteristic
246C22C2×#13711yescharacteristic
247C22C2×#13711yescharacteristic
248C22#57511yescharacteristic
249C4—22no—
250C4—22no—
251C4—41no—
252C4—41no—
253C4—41no—
254C4—44no—
255C4—42no—
256C4—82no—
257C4—81no—
258C4—81no—
259C4#46414yes—
260C4#54611yescharacteristic
261C4#54611yescharacteristic
262C4#55311yescharacteristic
263C4#55311yescharacteristic
264C2—82no—
265C2#253511yescharacteristic
266C2C2×#46411yescharacteristic
267C2#303711yescharacteristic
268C2#310711yescharacteristic
269C2#311611yescharacteristic
270C2#321211yescharacteristic
271C2#321811yescharacteristic
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×#462C2semidirect
3. C4×C4⋊Q8C2semidirect
5. #2074C2semidirect
8. #1878C2semidirect
9. #1831C2semidirect
10. #1803C2semidirect

Maximal subgroups

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

subgroupindex
C2×#4622
C2×#4612
C4×C4⋊Q82
C4×#1242
#20742
#20152
#18802
#18782
#18312
#18032
#18022