Finite Group Databasebeta

← #18122

Subgroups — #18122

Order 256.

646 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 646 conjugacy classes of subgroups — too many to draw. Only its 127 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
1#3449C211yesmaximal, characteristic, complemented
2#3100C212yesmaximal, complemented
3#2392C211yesmaximal, characteristic, complemented
4#1610C212yesmaximal, complemented
5#1607C212yesmaximal, complemented
6#1598C212yesmaximal, complemented
7#1595C212yesmaximal, complemented
8#1502C212yesmaximal, complemented
9#1458C211yesmaximal, characteristic, complemented
10C22×#41—21no—
11#537—21no—
12#537C2212yes—
13#536—22no—
14#535—22no—
15#534—21no—
16#534C2212yes—
17#532—21no—
18#532C2212yes—
19#525—24no—
20C2×C4×D4—21no—
21C2×C4×D4C2211yescharacteristic, complemented
22C2×#117C2211yescharacteristic
23#480C2214yescomplemented
24#474—22no—
25#459—22no—
26#447C2214yescomplemented
27C4×D8—24no—
28#431C2212yescomplemented
29#398C2211yescharacteristic, complemented
30#397C2211yescharacteristic
31#396—21no—
32#394—21no—
33#394C2211yescharacteristic, complemented
34#392C2211yescharacteristic
35C4×C4⋊C4C2211yescharacteristic, complemented
36C4×C22⋊C4—21no—
37#327C2214yescomplemented
38#327C2214yescomplemented
39#324C2212yescomplemented
40#323C2212yes—
41C2×#41—22no—
42C2×#41—22no—
43C2×#41—22no—
44C2×#41—22no—
45C2×#41—44no—
46C2×#41—44no—
47C22×Q8—21no—
48C22×D4—21no—
49C22×D4—22no—
50C22×D4—21no—
51C22×D4C2311yescharacteristic
52C23×C4—22no—
53C23×C4—21no—
54C23×C4D411yescharacteristic
55C2×D8—24no—
56C4⋊Q8—21no—
57C4⋊Q8D412yescomplemented
58#127—21no—
59#127D412yescomplemented
60#124—22no—
61#124—22no—
62#123—24no—
63#123—24no—
64C22⋊Q8—24no—
65C22⋊Q8—24no—
66#121—24no—
67#121—24no—
68#121—22no—
69#121—22no—
70#121—22no—
71#121C2314yes—
72#120—24no—
73#120—24no—
74C4×D4—22no—
75C4×D4—22no—
76C4×D4—22no—
77C4×D4—22no—
78C4×D4—22no—
79C4×D4—44no—
80C4×D4D414yescomplemented
81#117—22no—
82#117—22no—
83C2×C4⋊C4—21no—
84C2×C4⋊C4—21no—
85C2×C4⋊C4D411yescharacteristic
86C2×C4⋊C4C2311yescharacteristic
87C2×C4⋊C4C2311yescharacteristic
88C2×C22⋊C4—21no—
89C2×C22⋊C4—21no—
90C2×C22⋊C4—21no—
91C2×C22⋊C4—21no—
92C2×C22⋊C4—21no—
93C2×C22⋊C4D411yescharacteristic
94C2×C22⋊C4C2311yescharacteristic
95C2×C42—21no—
96C2×C42D411yescharacteristic
97C2×C42C2311yescharacteristic
98#107—24no—
99C4⋊C8—22no—
100#102—24no—
101#102—24no—
102C22⋊C8C2314yes—
103C4×C8—22no—
104#95—21no—
105#95—21no—
106#95C2311yescharacteristic
107#95C2311yescharacteristic
108C24—22no—
109C24—21no—
110C24C2×D411yescharacteristic
111#41—44no—
112#41—48no—
113#41—48no—
114#41—44no—
115#41—44no—
116#41—44no—
117C2×Q8—22no—
118C2×Q8—22no—
119C2×Q8—44no—
120C2×D4—24no—
121C2×D4—22no—
122C2×D4—24no—
123C2×D4—22no—
124C2×D4—22no—
125C2×D4—22no—
126C2×D4—22no—
127C2×D4—22no—
128C2×D4—42no—
129C2×D4—44no—
130C2×D4—42no—
131C2×D4—42no—
132C2×D4—44no—
133C2×D4—42no—
134C2×D4—44no—
135C2×D4C2×D414yes—
136C22×C4—21no—
137C22×C4—21no—
138C22×C4—22no—
139C22×C4—24no—
140C22×C4—22no—
141C22×C4—21no—
142C22×C4—24no—
143C22×C4—21no—
144C22×C4—21no—
145C22×C4—48no—
146C22×C4—41no—
147C22×C4—41no—
148C22×C4—42no—
149C22×C4—42no—
150C22×C4D814yescomplemented
151C22×C4C2×D411yescharacteristic
152C22×C4C2×D411yescharacteristic
153C22×C4C2×D411yescharacteristic
154C22×C4C2×D411yescharacteristic
155C22×C4C2×D411yescharacteristic
156C22×C4C2411yescharacteristic
157D8—48no—
158C2×C8—24no—
159C4⋊C4—22no—
160C4⋊C4—22no—
161C4⋊C4—22no—
162C4⋊C4—22no—
163C4⋊C4—22no—
164C4⋊C4—22no—
165C4⋊C4—42no—
166C4⋊C4—42no—
167C4⋊C4C2×D414yes—
168C22⋊C4—24no—
169C22⋊C4—24no—
170C22⋊C4—22no—
171C22⋊C4—24no—
172C22⋊C4—22no—
173C22⋊C4—24no—
174C22⋊C4—42no—
175C22⋊C4—42no—
176C22⋊C4—42no—
177C22⋊C4—42no—
178C22⋊C4—42no—
179C42—21no—
180C42—21no—
181C42—42no—
182C42C2×D412yes—
183C42C2×D412yes—
184C23—21no—
185C23—21no—
186C23—22no—
187C23—24no—
188C23—41no—
189C23—41no—
190C23—41no—
191C23—44no—
192C23—41no—
193C23—41no—
194C23—41no—
195C23—41no—
196C23—41no—
197C23—82no—
198C23—82no—
199C23C2×D812yes—
200C23C22×D411yescharacteristic
201Q8—44no—
202Q8—44no—
203D4—24no—
204D4—44no—
205D4—42no—
206D4—44no—
207D4—48no—
208D4—48no—
209D4—44no—
210D4—44no—
211D4—44no—
212D4—82no—
213D4—82no—
214D4—82no—
215C2×C4—21no—
216C2×C4—21no—
217C2×C4—24no—
218C2×C4—24no—
219C2×C4—22no—
220C2×C4—21no—
221C2×C4—21no—
222C2×C4—24no—
223C2×C4—41no—
224C2×C4—41no—
225C2×C4—48no—
226C2×C4—44no—
227C2×C4—41no—
228C2×C4—41no—
229C2×C4—48no—
230C2×C4—41no—
231C2×C4—41no—
232C2×C4—41no—
233C2×C4—41no—
234C2×C4—41no—
235C2×C4—41no—
236C2×C4—41no—
237C2×C4—41no—
238C2×C4—82no—
239C2×C4—82no—
240C2×C4#12012yes—
241C2×C4#12011yescharacteristic
242C2×C4#12011yescharacteristic
243C2×C4C2×D812yes—
244C2×C4C2×D812yes—
245C2×C4AGL(1,ℤ/8)12yes—
246C2×C4C22×D412yes—
247C8—44no—
248C22—21no—
249C22—21no—
250C22—22no—
251C22—42no—
252C22—42no—
253C22—42no—
254C22—41no—
255C22—41no—
256C22—44no—
257C22—44no—
258C22—42no—
259C22—44no—
260C22—81no—
261C22—81no—
262C22—81no—
263C22—88no—
264C22—81no—
265C22—81no—
266C22C2×#12011yescharacteristic
267C22C22×D811yescharacteristic
268C22C2×AGL(1,ℤ/8)11yescharacteristic
269C4—22no—
270C4—22no—
271C4—22no—
272C4—42no—
273C4—44no—
274C4—41no—
275C4—41no—
276C4—81no—
277C4—81no—
278C4#44712yes—
279C4#44712yes—
280C2—21no—
281C2—21no—
282C2—42no—
283C2—82no—
284C2—84no—
285C2C2×#44711yescharacteristic
286C2#301811yescharacteristic
287C2#302511yescharacteristic
subgroupG/H
centerC2×C4#120
commutator, Frattini, cosocleC22×C4C24
socleC22C2×#120
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ G
LECSG ▷ C22×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #3449C2semidirect
2. #3100C2semidirect
3. #2392C2semidirect
4. #1610C2semidirect
5. #1607C2semidirect
6. #1598C2semidirect
7. #1595C2semidirect
8. #1502C2semidirect
9. #1458C2semidirect
21. C2×C4×D4C22semidirect
23. #480C22semidirect
26. #447C22semidirect
28. #431C22semidirect
29. #398C22semidirect
33. #394C22semidirect
35. C4×C4⋊C4C22semidirect
37. #327C22semidirect
38. #327C22semidirect
39. #324C22semidirect
57. C4⋊Q8D4semidirect
59. #127D4semidirect
80. C4×D4D4semidirect
150. C22×C4D8semidirect

Maximal subgroups

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

subgroupindex
#34492
#31002
#23922
#16102
#16072
#15982
#15952
#15022
#14582