Finite Group Databasebeta

← #32973

Subgroups — #32973

Order 256.

643 conjugacy classes of subgroups; only the 147 normal subgroups are shown.

normal subgroups only

This group has 643 conjugacy classes of subgroups — too many to draw. Only its 147 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
1C4×H(5,𝔽2)C211yesmaximal, characteristic
2#3080C212yesmaximal, complemented
3#2566C211yesmaximal, characteristic, complemented
4#1860C212yesmaximal, complemented
5#1859C212yesmaximal, complemented
6#1856C211yesmaximal, characteristic, complemented
7#1856C211yesmaximal, characteristic, complemented
8#1806C212yesmaximal, complemented
9#1800C211yesmaximal, characteristic, complemented
10#1792C211yesmaximal, characteristic
11#1792C211yesmaximal, characteristic
12C2×H(5,𝔽2)C2211yescharacteristic
13C2×#125C2211yescharacteristic, complemented
14#520—22no—
15#520C2212yes—
16#518—22no—
17#518—22no—
18#518C2211yescharacteristic
19C4×#41—22no—
20C4×#41C2212yes—
21C2×C4×D4—22no—
22C2×C4×D4—22no—
23C2×C4×D4C2211yescharacteristic
24#479—22no—
25#479—22no—
26#478—22no—
27#478—22no—
28#428C2212yescomplemented
29#428C2212yescomplemented
30#423C2212yescomplemented
31C2×#104C2212yes—
32C2×#104C2212yes—
33#417C2212yes—
34#417C2212yes—
35C2×#100C2211yescharacteristic
36C2×#100C2211yescharacteristic
37#396—22no—
38#390C2211yescharacteristic
39#390C2211yescharacteristic
40#385—22no—
41#384C2211yescharacteristic, complemented
42#382C2211yescharacteristic, complemented
43C4×C4⋊C4C2211yescharacteristic, complemented
44#343C2212yescomplemented
45#339C2212yescomplemented
46#339C2212yescomplemented
47H(5,𝔽2)—24no—
48H(5,𝔽2)—22no—
49H(5,𝔽2)Q814yes—
50C2×#41—22no—
51C2×#41C2312yes—
52C22×D4—22no—
53C22×D4—22no—
54C22×D4C2311yescharacteristic
55C23×C4—22no—
56C23×C4D412yes—
57C2×M16C2312yes—
58C2×M16C2312yes—
59#125—22no—
60#125—22no—
61C4×Q8—22no—
62C4×Q8D412yes—
63C4×Q8D412yes—
64C4×D4—24no—
65C4×D4—24no—
66C4×D4—24no—
67C4×D4—24no—
68C4×D4—22no—
69C4×D4—22no—
70C4×D4—24no—
71C4×D4—22no—
72C4×D4—22no—
73C4×D4—22no—
74C4×D4—22no—
75C4×D4—22no—
76C4×D4D412yes—
77C4×D4D412yes—
78#117—24no—
79#117—22no—
80#117—22no—
81#117C2312yes—
82C2×C4⋊C4—22no—
83C2×C4⋊C4—22no—
84C2×C4⋊C4—21no—
85C2×C4⋊C4—21no—
86C2×C4⋊C4—21no—
87C2×C4⋊C4—22no—
88C2×C4⋊C4C2311yescharacteristic
89C2×C4⋊C4C2311yescharacteristic
90C2×C4⋊C4C2311yescharacteristic
91C2×C22⋊C4—22no—
92C2×C22⋊C4—24no—
93C2×C22⋊C4—22no—
94C2×C22⋊C4—22no—
95C2×C22⋊C4—22no—
96C2×C22⋊C4D412yes—
97C2×C42—22no—
98C2×C42—22no—
99C2×C42C2311yescharacteristic
100C2×C42C2311yescharacteristic
101C2×C42C2311yescharacteristic
102#107—22no—
103#107—22no—
104#106—22no—
105#106—22no—
106C4⋊C8—22no—
107C4⋊C8—22no—
108#104—24no—
109#104—24no—
110#103—22no—
111#103—22no—
112#102—22no—
113#102—22no—
114#100—22no—
115#100—22no—
116#95—21no—
117#95—22no—
118C24—22no—
119C24C2×D412yes—
120#41—28no—
121#41—28no—
122#41—22no—
123#41—22no—
124#41C2×Q814yes—
125#41#4114yes—
126C2×Q8C2×D412yes—
127C2×D4—24no—
128C2×D4—28no—
129C2×D4—28no—
130C2×D4—28no—
131C2×D4—22no—
132C2×D4—28no—
133C2×D4—22no—
134C2×D4—22no—
135C2×D4C2×D412yes—
136C2×D4C2×Q812yes—
137C2×D4#4112yes—
138C22×C4—22no—
139C22×C4—22no—
140C22×C4—21no—
141C22×C4—22no—
142C22×C4—24no—
143C22×C4—22no—
144C22×C4—22no—
145C22×C4—22no—
146C22×C4—22no—
147C22×C4—22no—
148C22×C4—21no—
149C22×C4—21no—
150C22×C4—21no—
151C22×C4—82no—
152C22×C4—84no—
153C22×C4C2×D412yes—
154C22×C4C2×D411yescharacteristic
155C22×C4C2×D411yescharacteristic
156C22×C4C2411yescharacteristic
157M16—22no—
158M16—22no—
159M16—22no—
160M16—22no—
161C2×C8—22no—
162C2×C8—22no—
163C4⋊C4—24no—
164C4⋊C4—22no—
165C4⋊C4—22no—
166C4⋊C4—22no—
167C4⋊C4—22no—
168C4⋊C4—22no—
169C4⋊C4—42no—
170C4⋊C4—41no—
171C4⋊C4—42no—
172C4⋊C4—41no—
173C4⋊C4—82no—
174C4⋊C4C2×D412yes—
175C4⋊C4C2×D412yes—
176C22⋊C4—24no—
177C22⋊C4—24no—
178C22⋊C4—22no—
179C22⋊C4—22no—
180C22⋊C4—22no—
181C22⋊C4—24no—
182C22⋊C4—44no—
183C22⋊C4—82no—
184C42—24no—
185C42—22no—
186C42—22no—
187C42—22no—
188C42—22no—
189C42—22no—
190C42C2×D412yes—
191C42C2×D412yes—
192C23—28no—
193C23—22no—
194C23—22no—
195C23—22no—
196C23—42no—
197C23—42no—
198C23—82no—
199C23—82no—
200C23C22⋊Q814yes—
201C23C22×D411yescharacteristic
202Q8—22no—
203Q8C22⋊Q814yes—
204D4—216no—
205D4—24no—
206D4—24no—
207D4—24no—
208D4—22no—
209D4—24no—
210D4C22⋊Q814yes—
211C2×C4—28no—
212C2×C4—22no—
213C2×C4—24no—
214C2×C4—24no—
215C2×C4—22no—
216C2×C4—22no—
217C2×C4—22no—
218C2×C4—22no—
219C2×C4—41no—
220C2×C4—41no—
221C2×C4—44no—
222C2×C4—44no—
223C2×C4—41no—
224C2×C4—41no—
225C2×C4—42no—
226C2×C4—42no—
227C2×C4—41no—
228C2×C4—42no—
229C2×C4—41no—
230C2×C4—41no—
231C2×C4—41no—
232C2×C4—41no—
233C2×C4—82no—
234C2×C4—82no—
235C2×C4—82no—
236C2×C4—84no—
237C2×C4—82no—
238C2×C4#12011yescharacteristic
239C2×C4#12012yes—
240C2×C4#12011yescharacteristic
241C2×C4C22×D412yes—
242C2×C4C22×Q811yescharacteristic
243C2×C4C2×#4112yes—
244C2×C4C2×#4111yescharacteristic
245C8—42no—
246C8—42no—
247C22—21no—
248C22—21no—
249C22—24no—
250C22—24no—
251C22—44no—
252C22—42no—
253C22—42no—
254C22—42no—
255C22—42no—
256C22—82no—
257C22—88no—
258C22C2×#12011yescharacteristic
259C22C2×C22⋊Q811yescharacteristic
260C22#52511yescharacteristic
261C4—24no—
262C4—22no—
263C4—21no—
264C4—21no—
265C4—42no—
266C4—42no—
267C4—42no—
268C4—42no—
269C4—81no—
270C4—81no—
271C4—81no—
272C4C2×C22⋊Q812yes—
273C4#52512yes—
274C2—21no—
275C2—21no—
276C2—44no—
277C2—44no—
278C2#284811yescharacteristic
279C2#301811yescharacteristic
280C2#301911yescharacteristic
subgroupG/H
center, socleC22C2×#120
commutator, Frattini, cosocleC22×C4C24
prime radicalC2×H(5,𝔽2)C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #3080C2semidirect
3. #2566C2semidirect
4. #1860C2semidirect
5. #1859C2semidirect
6. #1856C2semidirect
7. #1856C2semidirect
8. #1806C2semidirect
9. #1800C2semidirect
13. C2×#125C22semidirect
28. #428C22semidirect
29. #428C22semidirect
30. #423C22semidirect
41. #384C22semidirect
42. #382C22semidirect
43. C4×C4⋊C4C22semidirect
44. #343C22semidirect
45. #339C22semidirect
46. #339C22semidirect

Maximal subgroups

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

subgroupindex
C4×H(5,𝔽2)2
#30802
#25662
#18602
#18592
#18562
#18062
#18002
#17922