Finite Group Databasebeta

← #18601

Subgroups — #18601

Order 256.

455 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 455 conjugacy classes of subgroups — too many to draw. Only its 117 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#3501C211yesmaximal, characteristic
2#3237C211yesmaximal, characteristic, complemented
3#3185C211yesmaximal, characteristic, complemented
4#3101C211yesmaximal, characteristic
5#1651C212yesmaximal, complemented
6#1643C212yesmaximal, complemented
7#1637C212yesmaximal, complemented
8#1632C212yesmaximal, complemented
9#1554C211yesmaximal, characteristic, complemented
10#1490C211yesmaximal, characteristic
11#1485C211yesmaximal, characteristic, complemented
12#554—21no—
13#554—22no—
14#554—22no—
15#554C2212yes—
16D4×Q8—21no—
17D4×Q8C2212yes—
18#544—21no—
19#543—22no—
20#533C2211yescharacteristic
21#533C2211yescharacteristic
22#533C2211yescharacteristic
23C2×C22⋊Q8—22no—
24#518—21no—
25C2×C4×D4C2211yescharacteristic
26#501—21no—
27#498C2211yescharacteristic, complemented
28#494C2211yescharacteristic, complemented
29#491—21no—
30#487C2211yescharacteristic, complemented
31#487C2211yescharacteristic, complemented
32#485—22no—
33#483—21no—
34#477C2212yescomplemented
35#476C2212yes—
36#475C2212yescomplemented
37#470—22no—
38#461C2212yes—
39#450—21no—
40#441—22no—
41C4×SD16—22no—
42#432C2211yescharacteristic
43#431C2211yescharacteristic
44C4×M16C2211yescharacteristic
45#333C2212yescomplemented
46#332C2212yescomplemented
47Q8⋊C8C2212yescomplemented
48Q8⋊C8C2212yescomplemented
49D4⋊C8C2212yes—
50D4⋊C8C2212yes—
51C22×Q8—22no—
52C22×D4D411yescharacteristic
53C23×C4—21no—
54C2×Q16—22no—
55C2×SD16—22no—
56C2×M16—21no—
57C4⋊Q8—22no—
58C4⋊Q8—22no—
59C4⋊Q8D411yescharacteristic
60C4⋊Q8D411yescharacteristic
61C4⋊Q8C2311yescharacteristic
62C4⋊Q8C2311yescharacteristic
63C4⋊Q8C2311yescharacteristic
64C4⋊Q8C2311yescharacteristic
65#125—22no—
66#125—22no—
67#125—21no—
68#125—21no—
69#125—21no—
70C22⋊Q8—24no—
71C22⋊Q8—24no—
72C22⋊Q8—24no—
73C22⋊Q8—24no—
74C22⋊Q8—22no—
75C22⋊Q8—24no—
76C22⋊Q8—21no—
77C22⋊Q8—21no—
78C22⋊Q8—22no—
79C22⋊Q8—24no—
80C22⋊Q8—21no—
81C22⋊Q8—21no—
82C22⋊Q8D412yes—
83C22⋊Q8D412yes—
84C4×Q8—21no—
85C4×Q8—21no—
86C4×Q8—21no—
87C4×Q8—21no—
88C4×Q8D412yes—
89C4×Q8C2312yes—
90C4×D4—22no—
91C4×D4—22no—
92C4×D4—22no—
93C4×D4—22no—
94C4×D4—21no—
95C4×D4—21no—
96C4×D4D412yes—
97C4×D4C2312yes—
98#117—21no—
99#117—21no—
100#117—21no—
101C2×C4⋊C4—24no—
102C2×C4⋊C4—22no—
103C2×C4⋊C4D411yescharacteristic
104C2×C22⋊C4—24no—
105C2×C22⋊C4—21no—
106C2×C42C2311yescharacteristic
107#107—22no—
108#106—22no—
109#106—22no—
110#106—22no—
111C4⋊C8C2312yes—
112C4⋊C8C2312yes—
113#103—22no—
114#103—22no—
115#103—22no—
116#103—22no—
117#103—22no—
118#102—22no—
119C22⋊C8—21no—
120C22⋊C8—21no—
121#97—21no—
122#97—21no—
123C4×C8—21no—
124C4×C8C2311yescharacteristic
125C4×C8C2311yescharacteristic
126C24—21no—
127C2×Q8—24no—
128C2×Q8—21no—
129C2×Q8—21no—
130C2×Q8—21no—
131C2×Q8—21no—
132C2×Q8—84no—
133C2×Q8C2×D412yes—
134C2×Q8C2×D412yes—
135C2×D4—24no—
136C2×D4—21no—
137C2×D4C2×D412yes—
138C22×C4—24no—
139C22×C4—24no—
140C22×C4—21no—
141C22×C4—22no—
142C22×C4—22no—
143C22×C4—82no—
144C22×C4C2×D411yescharacteristic
145C22×C4C2×D411yescharacteristic
146C22×C4C2×D411yescharacteristic
147Q16—44no—
148SD16—44no—
149M16—42no—
150C2×C8—21no—
151C2×C8—22no—
152C2×C8—21no—
153C2×C8—22no—
154C4⋊C4—24no—
155C4⋊C4—24no—
156C4⋊C4—24no—
157C4⋊C4—22no—
158C4⋊C4—22no—
159C4⋊C4—21no—
160C4⋊C4—21no—
161C4⋊C4—21no—
162C4⋊C4—21no—
163C4⋊C4—22no—
164C4⋊C4—21no—
165C4⋊C4—21no—
166C4⋊C4—21no—
167C4⋊C4—22no—
168C4⋊C4—21no—
169C4⋊C4—21no—
170C4⋊C4—21no—
171C4⋊C4—21no—
172C4⋊C4—41no—
173C4⋊C4—41no—
174C4⋊C4C2×D412yes—
175C4⋊C4C2×D412yes—
176C4⋊C4C2×D412yes—
177C22⋊C4—24no—
178C22⋊C4—24no—
179C22⋊C4—22no—
180C22⋊C4—22no—
181C22⋊C4—22no—
182C22⋊C4—22no—
183C22⋊C4—41no—
184C42—22no—
185C42—22no—
186C42—41no—
187C42C2×D411yescharacteristic
188C42C2×D411yescharacteristic
189C42C2×D411yescharacteristic
190C42C2411yescharacteristic
191C23—24no—
192C23—22no—
193C23—81no—
194C23—81no—
195C23#12011yescharacteristic
196Q8—24no—
197Q8—42no—
198Q8—42no—
199Q8—42no—
200Q8—42no—
201Q8—81no—
202Q8—84no—
203D4—24no—
204D4—22no—
205D4#13714yes—
206C2×C4—22no—
207C2×C4—22no—
208C2×C4—22no—
209C2×C4—22no—
210C2×C4—22no—
211C2×C4—22no—
212C2×C4—22no—
213C2×C4—41no—
214C2×C4—44no—
215C2×C4—41no—
216C2×C4—41no—
217C2×C4—82no—
218C2×C4—81no—
219C2×C4—84no—
220C2×C4—84no—
221C2×C4#12011yescharacteristic
222C2×C4#12011yescharacteristic
223C2×C4#12011yescharacteristic
224C2×C4C22×D411yescharacteristic
225C2×C4C22×D411yescharacteristic
226C2×C4C22×D411yescharacteristic
227C8—22no—
228C8—41no—
229C8—42no—
230C8—42no—
231C22—21no—
232C22—24no—
233C22—21no—
234C22—21no—
235C22—42no—
236C22—42no—
237C22—84no—
238C22C2×#12011yescharacteristic
239C4—21no—
240C4—21no—
241C4—21no—
242C4—21no—
243C4—42no—
244C4—42no—
245C4—42no—
246C4—42no—
247C4—42no—
248C4—81no—
249C4—81no—
250C4C2×#13712yes—
251C4#57712yes—
252C2—41no—
253C2—44no—
254C2#300411yescharacteristic
255C2#300911yescharacteristic
256C2#301811yescharacteristic
subgroupG/H
center, socleC22C2×#120
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
2. #3237C2semidirect
3. #3185C2semidirect
5. #1651C2semidirect
6. #1643C2semidirect
7. #1637C2semidirect
8. #1632C2semidirect
9. #1554C2semidirect
11. #1485C2semidirect
27. #498C22semidirect
28. #494C22semidirect
30. #487C22semidirect
31. #487C22semidirect
34. #477C22semidirect
36. #475C22semidirect
45. #333C22semidirect
46. #332C22semidirect
47. Q8⋊C8C22semidirect
48. Q8⋊C8C22semidirect

Maximal subgroups

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

subgroupindex
#35012
#32372
#31852
#31012
#16512
#16432
#16372
#16322
#15542
#14902
#14852