Finite Group Databasebeta

← #19167

Subgroups — #19167

Order 256.

337 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 337 conjugacy classes of subgroups — too many to draw. Only its 115 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#3139C211yesmaximal, characteristic, complemented
2#3101C211yesmaximal, characteristic
3C2×#474C211yesmaximal, characteristic
4D4×M16C211yesmaximal, characteristic
5#1703C212yesmaximal, complemented
6#1699C212yesmaximal, complemented
7#1683C212yesmaximal, complemented
8#1679C212yesmaximal, complemented
9#1565C211yesmaximal, characteristic, complemented
10#1550C211yesmaximal, characteristic, complemented
11#1546C211yesmaximal, characteristic, complemented
12C22×M16—21no—
13#533C2211yescharacteristic
14C2×C4⋊Q8C2211yescharacteristic
15C2×C4×D4C2211yescharacteristic
16#491—22no—
17#487C2212yescomplemented
18#487C2212yescomplemented
19#483—21no—
20#476C2211yescharacteristic
21#476C2211yescharacteristic
22#474—21no—
23#474—21no—
24#474C2212yes—
25#474C2212yes—
26#461C2211yescharacteristic
27#461C2211yescharacteristic
28#450—21no—
29C4×SD16—21no—
30C4×SD16—21no—
31#436C2212yes—
32#435—22no—
33C8×D4C2212yes—
34#431C2211yescharacteristic
35C2×#107—21no—
36#423C2211yescharacteristic
37C2×C4⋊C8C2211yescharacteristic
38#419—21no—
39#419—21no—
40C2×#102—21no—
41#407—21no—
42C4×M16C2211yescharacteristic
43#335C2212yescomplemented
44#334C2212yescomplemented
45#333C2212yescomplemented
46#332C2212yescomplemented
47#332C2212yescomplemented
48#332C2212yescomplemented
49C22×Q8—21no—
50C22×D4D411yescharacteristic
51C23×C4—21no—
52C2×SD16—21no—
53C2×SD16—21no—
54C2×M16—21no—
55C2×M16—22no—
56C2×M16—44no—
57C2×M16D411yescharacteristic
58C2×M16D411yescharacteristic
59C22×C8—21no—
60C22×C8—22no—
61C4⋊Q8—21no—
62C4⋊Q8—21no—
63C4⋊Q8—21no—
64C4⋊Q8C2312yes—
65C4⋊Q8C2311yescharacteristic
66C4⋊Q8C2311yescharacteristic
67#125—21no—
68C22⋊Q8—21no—
69C22⋊Q8—21no—
70C4×Q8—21no—
71C4×Q8—21no—
72C4×D4—21no—
73C4×D4—21no—
74C4×D4D411yescharacteristic
75C4×D4D411yescharacteristic
76C4×D4C2311yescharacteristic
77C4×D4C2311yescharacteristic
78#117—21no—
79C2×C4⋊C4—21no—
80C2×C4⋊C4—21no—
81C2×C4⋊C4D411yescharacteristic
82C2×C22⋊C4—21no—
83C2×C42C2311yescharacteristic
84#107—22no—
85#107—22no—
86#106—21no—
87#106—21no—
88C4⋊C8—21no—
89C4⋊C8—21no—
90C4⋊C8C2312yes—
91C4⋊C8C2311yescharacteristic
92C4⋊C8C2311yescharacteristic
93C4⋊C8C2312yes—
94#103—22no—
95#103—22no—
96#103—22no—
97#103—22no—
98#103—21no—
99#103—21no—
100#102—22no—
101#102—22no—
102#102—21no—
103#102—21no—
104C22⋊C8—21no—
105C22⋊C8—22no—
106C22⋊C8—22no—
107#97D412yes—
108C4×C8—21no—
109C4×C8C2312yes—
110C24—21no—
111C2×Q8—22no—
112C2×Q8—21no—
113C2×Q8—21no—
114C2×Q8—41no—
115C2×Q8—42no—
116C2×Q8—42no—
117C2×D4—22no—
118C2×D4—21no—
119C2×D4—41no—
120C2×D4C2×D411yescharacteristic
121C2×D4C2×D411yescharacteristic
122C22×C4—21no—
123C22×C4—21no—
124C22×C4—21no—
125C22×C4—21no—
126C22×C4—21no—
127C22×C4—42no—
128C22×C4—42no—
129C22×C4—41no—
130C22×C4C2×D411yescharacteristic
131C22×C4C2×D411yescharacteristic
132C22×C4#4111yescharacteristic
133SD16—42no—
134SD16—42no—
135M16—22no—
136M16—22no—
137M16—44no—
138M16—42no—
139C2×C8—21no—
140C2×C8—21no—
141C2×C8—22no—
142C2×C8—22no—
143C2×C8—44no—
144C2×C8—42no—
145C2×C8C2×D412yes—
146C2×C8C2×D412yes—
147C4⋊C4—21no—
148C4⋊C4—21no—
149C4⋊C4—22no—
150C4⋊C4—21no—
151C4⋊C4—21no—
152C4⋊C4—22no—
153C4⋊C4—21no—
154C4⋊C4—41no—
155C4⋊C4—41no—
156C4⋊C4—41no—
157C4⋊C4—41no—
158C4⋊C4C2×D411yescharacteristic
159C4⋊C4C2×D411yescharacteristic
160C22⋊C4—21no—
161C22⋊C4—21no—
162C22⋊C4—21no—
163C22⋊C4—21no—
164C22⋊C4—41no—
165C42—41no—
166C42C2×D411yescharacteristic
167C42C2×D411yescharacteristic
168C42#4111yescharacteristic
169C42C2411yescharacteristic
170C23—21no—
171C23—21no—
172C23—42no—
173C23—42no—
174C23—81no—
175C23#12111yescharacteristic
176Q8—42no—
177Q8—42no—
178Q8—41no—
179Q8—41no—
180Q8—81no—
181Q8—82no—
182D4—21no—
183D4—21no—
184D4—21no—
185D4—42no—
186D4#13712yes—
187C2×C4—21no—
188C2×C4—21no—
189C2×C4—21no—
190C2×C4—21no—
191C2×C4—41no—
192C2×C4—42no—
193C2×C4—41no—
194C2×C4—41no—
195C2×C4—41no—
196C2×C4—41no—
197C2×C4—42no—
198C2×C4—42no—
199C2×C4—41no—
200C2×C4—41no—
201C2×C4—42no—
202C2×C4—82no—
203C2×C4#12111yescharacteristic
204C2×C4#12111yescharacteristic
205C2×C4#12111yescharacteristic
206C2×C4#13512yes—
207C2×C4AGL(1,ℤ/8)12yes—
208C2×C4C22×D411yescharacteristic
209C2×C4C22×D411yescharacteristic
210C2×C4C2×#4111yescharacteristic
211C8—22no—
212C8—22no—
213C8—42no—
214C8—41no—
215C8—42no—
216C8—41no—
217C22—21no—
218C22—21no—
219C22—22no—
220C22—41no—
221C22—41no—
222C22—41no—
223C22—41no—
224C22—41no—
225C22—84no—
226C22#46312yes—
227C22C2×#12111yescharacteristic
228C4—21no—
229C4—21no—
230C4—21no—
231C4—41no—
232C4—41no—
233C4—82no—
234C4—81no—
235C4—81no—
236C4#46312yes—
237C4C2×#13511yescharacteristic
238C4C2×AGL(1,ℤ/8)11yescharacteristic
239C4C2×#13711yescharacteristic
240C4#57511yescharacteristic
241C2—22no—
242C2—42no—
243C2—81no—
244C2C2×#46311yescharacteristic
245C2#305611yescharacteristic
246C2#306911yescharacteristic
subgroupG/H
center, socleC22C2×#121
commutator, Frattini, cosocleC42C24
prime radicalC22×D4D4
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ G
LECSG ▷ C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #3139C2semidirect
5. #1703C2semidirect
6. #1699C2semidirect
7. #1683C2semidirect
8. #1679C2semidirect
9. #1565C2semidirect
10. #1550C2semidirect
11. #1546C2semidirect
17. #487C22semidirect
18. #487C22semidirect
43. #335C22semidirect
44. #334C22semidirect
45. #333C22semidirect
46. #332C22semidirect
47. #332C22semidirect
48. #332C22semidirect

Maximal subgroups

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

subgroupindex
#31392
#31012
C2×#4742
D4×M162
#17032
#16992
#16832
#16792
#15652
#15502
#15462