Finite Group Databasebeta

← #23735

Subgroups — #23735

Order 256.

605 conjugacy classes of subgroups; only the 175 normal subgroups are shown.

normal subgroups only

This group has 605 conjugacy classes of subgroups — too many to draw. Only its 175 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×#428C211yesmaximal, characteristic, complemented
2C2×#427C211yesmaximal, characteristic, complemented
3C2×#418C211yesmaximal, characteristic
4C22×#102C211yesmaximal, characteristic
5#2317C211yesmaximal, characteristic, complemented
6#2309C211yesmaximal, characteristic
7#2094C212yesmaximal, complemented
8#2093C212yesmaximal, complemented
9#2092C212yesmaximal, complemented
10#2092C212yesmaximal, complemented
11#1729C211yesmaximal, characteristic, complemented
12C23×D4C2211yescharacteristic
13C22×M16C2211yescharacteristic
14C23×C8C2211yescharacteristic
15C2×#117C2211yescharacteristic
16C2×#117C2211yescharacteristic
17C2×#117C2211yescharacteristic
18C22×C4⋊C4C2211yescharacteristic
19#428—24no—
20#427—24no—
21C2×#107C2211yescharacteristic, complemented
22C2×#107C2212yescomplemented
23C2×#107C2211yescharacteristic, complemented
24C2×#106C2211yescharacteristic, complemented
25C2×#106C2212yescomplemented
26C2×#106C2211yescharacteristic, complemented
27#418—24no—
28C2×#102—24no—
29C2×#102C2212yes—
30C2×#102C2212yes—
31C2×#102C2212yes—
32C2×#102C2212yes—
33#390—22no—
34#390C2212yes—
35#390C2212yes—
36#389—22no—
37#385—22no—
38#384C2212yescomplemented
39#382C2212yescomplemented
40#379—22no—
41#336C2212yescomplemented
42#336C2212yescomplemented
43C25—21no—
44C22×D4—22no—
45C22×D4—22no—
46C22×D4—21no—
47C22×D4—22no—
48C22×D4—22no—
49C22×D4—22no—
50C22×D4Q814yes—
51C22×D4C2312yes—
52C23×C4—21no—
53C23×C4C2311yescharacteristic
54C2×M16—22no—
55C2×M16—22no—
56C2×M16D414yes—
57C22×C8—22no—
58C22×C8—22no—
59C22×C8D414yes—
60C22×C8C2311yescharacteristic
61C22×C8C2311yescharacteristic
62C22×C8C2312yes—
63#117—22no—
64#117—22no—
65#117—22no—
66#117—22no—
67#117—22no—
68#117—22no—
69C2×C4⋊C4—22no—
70C2×C4⋊C4—22no—
71C2×C4⋊C4—21no—
72C2×C4⋊C4—24no—
73C2×C4⋊C4D412yes—
74C2×C4⋊C4C2312yes—
75C2×C4⋊C4C2311yescharacteristic
76C2×C4⋊C4C2311yescharacteristic
77C2×C4⋊C4C2311yescharacteristic
78C2×C4⋊C4C2311yescharacteristic
79C2×C22⋊C4—21no—
80C2×C22⋊C4—21no—
81C2×C22⋊C4—24no—
82C2×C22⋊C4—24no—
83C2×C22⋊C4D412yes—
84C2×C42—21no—
85C2×C42—21no—
86C2×C42C2312yes—
87#107—22no—
88#107—22no—
89#107—24no—
90#106—22no—
91#106—22no—
92#106—24no—
93#102—24no—
94#102—24no—
95#102—24no—
96#102—24no—
97#95—24no—
98C24—22no—
99C24—24no—
100C24—42no—
101C24—42no—
102C24—82no—
103C24—82no—
104C24C2×D411yescharacteristic
105C2×D4—28no—
106C2×D4—28no—
107C2×D4—24no—
108C2×D4—22no—
109C2×D4—22no—
110C2×D4—22no—
111C2×D4—22no—
112C2×D4—22no—
113C2×D4—44no—
114C2×D4—44no—
115C2×D4—42no—
116C2×D4—42no—
117C2×D4C2×Q814yes—
118C22×C4—21no—
119C22×C4—22no—
120C22×C4—21no—
121C22×C4—21no—
122C22×C4—21no—
123C22×C4—42no—
124C22×C4—82no—
125C22×C4C2×D414yes—
126C22×C4C2×D411yescharacteristic
127C22×C4C2×D412yes—
128C22×C4C2×D412yes—
129C22×C4C2×Q812yes—
130C22×C4#4112yes—
131C22×C4#4112yes—
132C22×C4#4112yes—
133C22×C4C2411yescharacteristic
134M16—28no—
135C2×C8—28no—
136C2×C8—21no—
137C2×C8—21no—
138C2×C8—21no—
139C2×C8—21no—
140C2×C8—22no—
141C2×C8—22no—
142C2×C8C2×D412yes—
143C2×C8C2×D414yes—
144C2×C8C2×D412yes—
145C4⋊C4—24no—
146C4⋊C4—22no—
147C4⋊C4—21no—
148C4⋊C4—21no—
149C4⋊C4—21no—
150C4⋊C4—22no—
151C4⋊C4—21no—
152C4⋊C4—21no—
153C4⋊C4—21no—
154C4⋊C4—21no—
155C4⋊C4—21no—
156C4⋊C4—42no—
157C4⋊C4—84no—
158C22⋊C4—24no—
159C22⋊C4—42no—
160C22⋊C4—42no—
161C22⋊C4—84no—
162C22⋊C4—84no—
163C42—22no—
164C42—22no—
165C42—42no—
166C42—42no—
167C23—22no—
168C23—22no—
169C23—24no—
170C23—42no—
171C23—42no—
172C23—44no—
173C23—44no—
174C23—48no—
175C23—88no—
176C23—88no—
177C23—84no—
178C23—84no—
179C23C22⋊Q812yes—
180C23#12312yes—
181C23#12712yes—
182C23C22×D411yescharacteristic
183D4—24no—
184D4—24no—
185D4—42no—
186D4—42no—
187D4—48no—
188C2×C4—24no—
189C2×C4—24no—
190C2×C4—24no—
191C2×C4—41no—
192C2×C4—41no—
193C2×C4—41no—
194C2×C4—41no—
195C2×C4—42no—
196C2×C4—42no—
197C2×C4—42no—
198C2×C4—44no—
199C2×C4—42no—
200C2×C4—42no—
201C2×C4—84no—
202C2×C4C22⋊Q814yes—
203C2×C4C22⋊Q814yes—
204C2×C4C22⋊Q812yes—
205C2×C4#12314yes—
206C2×C4#12314yes—
207C2×C4#12312yes—
208C2×C4#12712yes—
209C2×C4C22×D411yescharacteristic
210C2×C4C22×D411yescharacteristic
211C2×C4C22×Q811yescharacteristic
212C2×C4C2×#4111yescharacteristic
213C2×C4C2×#4111yescharacteristic
214C2×C4C2×#4111yescharacteristic
215C8—22no—
216C8—22no—
217C8—24no—
218C22—24no—
219C22—24no—
220C22—22no—
221C22—22no—
222C22—48no—
223C22—44no—
224C22—816no—
225C22—84no—
226C22—82no—
227C22—82no—
228C22#39914yes—
229C22C2×C22⋊Q811yescharacteristic
230C22C2×#12311yescharacteristic
231C22C2×#12711yescharacteristic
232C22#57612yes—
233C22#57712yes—
234C4—44no—
235C4—81no—
236C4—81no—
237C4—81no—
238C4—81no—
239C4—82no—
240C4#39914yes—
241C4C2×C22⋊Q811yescharacteristic
242C4C2×C22⋊Q811yescharacteristic
243C4C2×#12311yescharacteristic
244C4C2×#12311yescharacteristic
245C2—24no—
246C2—84no—
247C2C2×#39911yescharacteristic
248C2#308112yes—
249C2#309612yes—
250C2#315411yescharacteristic
251C2#315511yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC23×D4C22
LCSG ▷ C22×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
1. C2×#428C2semidirect
2. C2×#427C2semidirect
5. #2317C2semidirect
7. #2094C2semidirect
8. #2093C2semidirect
9. #2092C2semidirect
10. #2092C2semidirect
11. #1729C2semidirect
21. C2×#107C22semidirect
22. C2×#107C22semidirect
23. C2×#107C22semidirect
24. C2×#106C22semidirect
25. C2×#106C22semidirect
26. C2×#106C22semidirect
38. #384C22semidirect
39. #382C22semidirect
41. #336C22semidirect
42. #336C22semidirect

Maximal subgroups

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

subgroupindex
C2×#4282
C2×#4272
C2×#4182
C22×#1022
#23172
#23092
#20942
#20932
#20922
#17292