Finite Group Databasebeta

← #21814

Subgroups — #21814

Order 256.

625 conjugacy classes of subgroups; only the 191 normal subgroups are shown.

normal subgroups only

This group has 625 conjugacy classes of subgroups — too many to draw. Only its 191 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×#536C211yesmaximal, characteristic
2C2×#433C211yesmaximal, characteristic, complemented
3#2314C211yesmaximal, characteristic, complemented
4#1983C212yesmaximal, complemented
5#1981C212yesmaximal, complemented
6#1960C211yesmaximal, characteristic, complemented
7#1959C211yesmaximal, characteristic, complemented
8#1958C211yesmaximal, characteristic, complemented
9#1956C211yesmaximal, characteristic, complemented
10#1788C212yesmaximal
11#1787C212yesmaximal, complemented
12C22×#41C2211yescharacteristic
13C23×Q8C2211yescharacteristic
14#536—24no—
15#536C418yescomplemented
16C2×C4⋊Q8C2211yescharacteristic
17C2×C4⋊Q8C2211yescharacteristic
18C2×#124C2211yescharacteristic
19C2×#124C2211yescharacteristic
20C2×#123—22no—
21C2×C22⋊Q8—22no—
22C2×#117C2211yescharacteristic
23#433—24no—
24C2×C4⋊C8C2212yes—
25C2×#103C2212yes—
26C2×#103C2212yes—
27C2×#103C2212yes—
28C2×#103C2212yes—
29C2×#103C2212yes—
30C2×#103C2212yes—
31C2×#102C2212yes—
32C2×#102C2212yes—
33C2×C22⋊C8C2212yes—
34C2×#97C2211yescharacteristic
35C2×C4×C8C2211yescharacteristic
36#388—22no—
37#387—22no—
38#384C2211yescharacteristic
39#382C2211yescharacteristic
40#380C2212yes—
41C4×C4⋊C4C2212yes—
42C2×#41—22no—
43C2×#41—42no—
44C2×#41—42no—
45C2×#41C2×C414yes—
46C22×Q8—21no—
47C22×Q8—22no—
48C22×Q8—42no—
49C22×Q8—42no—
50C22×Q8C2×C414yes—
51C22×Q8C2311yescharacteristic
52C22×Q8C2311yescharacteristic
53C22×Q8C2311yescharacteristic
54C22×D4—21no—
55C22×D4C2311yescharacteristic
56C23×C4—21no—
57C23×C4—21no—
58C23×C4C2311yescharacteristic
59C22×C8C2312yes—
60C22×C8C2312yes—
61C4⋊Q8—22no—
62C4⋊Q8—22no—
63C4⋊Q8C2×C414yes—
64C4⋊Q8C2×C414yes—
65#124—22no—
66#124—22no—
67#124C2×C414yes—
68#124C2×C414yes—
69#123—216no—
70C22⋊Q8—216no—
71#117C2×C414yes—
72#117D414yes—
73C2×C4⋊C4—22no—
74C2×C4⋊C4—22no—
75C2×C4⋊C4—22no—
76C2×C4⋊C4D412yes—
77C2×C4⋊C4C2312yes—
78C2×C4⋊C4C2312yes—
79C2×C22⋊C4—22no—
80C2×C22⋊C4—22no—
81C2×C22⋊C4—22no—
82C2×C22⋊C4D412yes—
83C2×C42—22no—
84C2×C42C2311yescharacteristic
85C2×C42C2311yescharacteristic
86C4⋊C8—24no—
87#103—42no—
88#103—42no—
89#103—42no—
90#103—42no—
91#103—42no—
92#103—42no—
93#102—42no—
94#102—42no—
95C22⋊C8—24no—
96#97—22no—
97C4×C8—22no—
98#95—22no—
99#95—22no—
100#95—22no—
101C24—21no—
102C24C2×D411yescharacteristic
103#41—48no—
104#41—48no—
105C2×Q8—24no—
106C2×Q8—21no—
107C2×Q8—21no—
108C2×Q8—21no—
109C2×Q8—22no—
110C2×Q8—22no—
111C2×Q8—22no—
112C2×Q8—22no—
113C2×Q8—22no—
114C2×Q8—22no—
115C2×Q8—48no—
116C2×Q8—42no—
117C2×Q8—42no—
118C2×Q8—48no—
119C2×Q8C22×C412yes—
120C2×Q8C22×C412yes—
121C2×Q8C22×C412yes—
122C2×D4—24no—
123C2×D4—21no—
124C2×D4—22no—
125C2×D4—22no—
126C2×D4—42no—
127C2×D4—42no—
128C2×D4C22×C412yes—
129C22×C4—22no—
130C22×C4—22no—
131C22×C4—24no—
132C22×C4—21no—
133C22×C4—21no—
134C22×C4—21no—
135C22×C4—24no—
136C22×C4—22no—
137C22×C4—22no—
138C22×C4—42no—
139C22×C4—42no—
140C22×C4—42no—
141C22×C4—42no—
142C22×C4C22×C412yes—
143C22×C4C2×D411yescharacteristic
144C22×C4C2×D412yes—
145C22×C4C2×D412yes—
146C22×C4C2×D412yes—
147C22×C4C2411yescharacteristic
148C2×C8—22no—
149C2×C8—22no—
150C2×C8—22no—
151C2×C8—22no—
152C2×C8—22no—
153C2×C8—22no—
154C4⋊C4—28no—
155C4⋊C4—28no—
156C4⋊C4—22no—
157C4⋊C4—22no—
158C4⋊C4—22no—
159C4⋊C4—22no—
160C4⋊C4—82no—
161C4⋊C4C22⋊C418yes—
162C22⋊C4—28no—
163C22⋊C4—28no—
164C22⋊C4—82no—
165C22⋊C4C22⋊C418yes—
166C42—82no—
167C42C22×C412yes—
168C42C22×C412yes—
169C42C2×D412yes—
170C42C2×D412yes—
171C23—21no—
172C23—21no—
173C23—22no—
174C23—22no—
175C23—22no—
176C23—41no—
177C23—41no—
178C23—42no—
179C23—42no—
180C23C2×C22⋊C412yes—
181C23C22×D411yescharacteristic
182Q8—24no—
183Q8—24no—
184Q8—24no—
185Q8—48no—
186Q8—42no—
187Q8—42no—
188Q8—42no—
189D4—24no—
190D4—48no—
191D4—42no—
192C2×C4—21no—
193C2×C4—21no—
194C2×C4—22no—
195C2×C4—22no—
196C2×C4—22no—
197C2×C4—22no—
198C2×C4—24no—
199C2×C4—24no—
200C2×C4—22no—
201C2×C4—22no—
202C2×C4—22no—
203C2×C4—48no—
204C2×C4—48no—
205C2×C4—41no—
206C2×C4—41no—
207C2×C4—41no—
208C2×C4—41no—
209C2×C4—41no—
210C2×C4—41no—
211C2×C4—42no—
212C2×C4—42no—
213C2×C4—42no—
214C2×C4—42no—
215C2×C4—42no—
216C2×C4—42no—
217C2×C4C2×C22⋊C414yes—
218C2×C4C2×C22⋊C414yes—
219C2×C4C2×C22⋊C412yes—
220C2×C4C23×C411yescharacteristic
221C2×C4C22×D411yescharacteristic
222C2×C4H(5,𝔽2)12yes—
223C2×C4H(5,𝔽2)12yes—
224C8—42no—
225C8—42no—
226C22—22no—
227C22—24no—
228C22—41no—
229C22—41no—
230C22—42no—
231C22—42no—
232C22—44no—
233C22—81no—
234C22—81no—
235C22—82no—
236C22C22×C22⋊C411yescharacteristic
237C22#53511yescharacteristic
238C22#53511yescharacteristic
239C22#57712yes—
240C22#57812yes—
241C4—22no—
242C4—24no—
243C4—24no—
244C4—42no—
245C4—42no—
246C4—42no—
247C4—82no—
248C4—82no—
249C4#51812yes—
250C4#53512yes—
251C2—42no—
252C2—82no—
253C2#230911yescharacteristic
254C2#289812yes—
255C2#322111yescharacteristic
256C2#322311yescharacteristic
257C2#322512yes—
subgroupG/H
center, socleC23C22×D4
commutatorC2×C4C23×C4
Frattini, cosocleC22×C4C24
prime radicalC22×#41C22
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#433C2semidirect
3. #2314C2semidirect
4. #1983C2semidirect
5. #1981C2semidirect
6. #1960C2semidirect
7. #1959C2semidirect
8. #1958C2semidirect
9. #1956C2semidirect
11. #1787C2semidirect
15. #536C4semidirect

Maximal subgroups

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

subgroupindex
C2×#5362
C2×#4332
#23142
#19832
#19812
#19602
#19592
#19582
#19562
#17882
#17872