Finite Group Databasebeta

← #18575

Subgroups — #18575

Order 256.

447 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 447 conjugacy classes of subgroups — too many to draw. Only its 119 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#3503C211yesmaximal, characteristic
2#3255C211yesmaximal, characteristic, complemented
3#3139C211yesmaximal, characteristic, complemented
4#3107C211yesmaximal, characteristic
5#1645C212yesmaximal, complemented
6#1645C212yesmaximal, complemented
7#1644C212yesmaximal, complemented
8#1643C212yesmaximal, complemented
9#1548C211yesmaximal, characteristic, complemented
10#1489C211yesmaximal, characteristic
11#1485C211yesmaximal, characteristic, complemented
12Q82C2212yes—
13#557—21no—
14#554—22no—
15#554C2212yes—
16D4×Q8—22no—
17D4×Q8—21no—
18#544—22no—
19#544—21no—
20#533—21no—
21#533C2211yescharacteristic
22#533C2211yescharacteristic
23C2×C4⋊Q8—21no—
24C2×C4⋊Q8C2211yescharacteristic
25#520—21no—
26C4×#41C2211yescharacteristic
27#491—22no—
28#487C2212yes—
29#487C2212yes—
30#485—21no—
31#475—21no—
32#475C2211yescharacteristic
33#475C2212yes—
34#474C2211yescharacteristic
35#470—22no—
36#462—21no—
37#462C2211yescharacteristic
38#462C2212yes—
39#461C2211yescharacteristic
40#452—21no—
41C4×Q16—21no—
42C4×SD16—21no—
43#431C2211yescharacteristic
44#423C2211yescharacteristic
45#419—22no—
46C4×M16C2211yescharacteristic
47#333C2212yes—
48#333C2212yes—
49Q8⋊C8C2212yes—
50Q8⋊C8C2212yes—
51Q8⋊C8C2212yes—
52D4⋊C8C2212yes—
53C2×#41D411yescharacteristic
54C22×Q8—22no—
55C22×Q8D412yes—
56C2×Q16—21no—
57C2×Q16—22no—
58C2×SD16—21no—
59C2×M16—21no—
60C2×M16—21no—
61C4⋊Q8—22no—
62C4⋊Q8—22no—
63C4⋊Q8—24no—
64C4⋊Q8—22no—
65C4⋊Q8—22no—
66C4⋊Q8—21no—
67C4⋊Q8—21no—
68C4⋊Q8D412yes—
69C4⋊Q8D411yescharacteristic
70C4⋊Q8D411yescharacteristic
71C4⋊Q8C2311yescharacteristic
72C4⋊Q8C2311yescharacteristic
73C4⋊Q8C2312yes—
74#125—22no—
75#125—24no—
76#125—22no—
77#125—21no—
78#125—21no—
79C22⋊Q8—22no—
80C22⋊Q8—24no—
81C22⋊Q8—24no—
82C22⋊Q8—22no—
83C22⋊Q8—24no—
84C22⋊Q8—21no—
85C22⋊Q8—21no—
86C22⋊Q8—21no—
87C22⋊Q8D412yes—
88C4×Q8—22no—
89C4×Q8—22no—
90C4×Q8—22no—
91C4×Q8—21no—
92C4×Q8—21no—
93C4×Q8—21no—
94C4×Q8D411yescharacteristic
95C4×Q8C2311yescharacteristic
96C4×Q8C2312yes—
97C4×D4—24no—
98C4×D4—22no—
99C4×D4—21no—
100C4×D4—21no—
101C4×D4D411yescharacteristic
102C4×D4C2311yescharacteristic
103#117—22no—
104#117—21no—
105#117—21no—
106#117—21no—
107#117D411yescharacteristic
108C2×C4⋊C4—22no—
109C2×C4⋊C4—21no—
110C2×C4⋊C4—21no—
111C2×C4⋊C4—22no—
112C2×C42—21no—
113C2×C42C2311yescharacteristic
114#107—21no—
115#106—21no—
116#106—22no—
117C4⋊C8—21no—
118C4⋊C8C2312yes—
119C4⋊C8C2311yescharacteristic
120C4⋊C8C2311yescharacteristic
121#103—24no—
122#103—24no—
123#103—21no—
124#103—22no—
125#103—21no—
126#103—21no—
127#103—22no—
128#102—21no—
129C22⋊C8—21no—
130#97—21no—
131C4×C8—21no—
132C4×C8C2312yes—
133#41—22no—
134#41—22no—
135C2×Q8—24no—
136C2×Q8—22no—
137C2×Q8—21no—
138C2×Q8—21no—
139C2×Q8—21no—
140C2×Q8—44no—
141C2×Q8—84no—
142C2×Q8C2×D412yes—
143C2×Q8C2×D411yescharacteristic
144C2×Q8C2×D414yes—
145C2×D4—21no—
146C2×D4C2×D411yescharacteristic
147C22×C4—21no—
148C22×C4—24no—
149C22×C4—22no—
150C22×C4—21no—
151C22×C4—21no—
152C22×C4C2×D411yescharacteristic
153C22×C4C2×D411yescharacteristic
154C22×C4C2×D411yescharacteristic
155Q16—42no—
156Q16—82no—
157SD16—42no—
158M16—42no—
159M16—81no—
160C2×C8—21no—
161C2×C8—21no—
162C2×C8—22no—
163C2×C8—22no—
164C4⋊C4—24no—
165C4⋊C4—24no—
166C4⋊C4—24no—
167C4⋊C4—24no—
168C4⋊C4—22no—
169C4⋊C4—21no—
170C4⋊C4—22no—
171C4⋊C4—21no—
172C4⋊C4—21no—
173C4⋊C4—21no—
174C4⋊C4—22no—
175C4⋊C4—22no—
176C4⋊C4—22no—
177C4⋊C4—21no—
178C4⋊C4—22no—
179C4⋊C4—21no—
180C4⋊C4—21no—
181C4⋊C4—21no—
182C4⋊C4—21no—
183C4⋊C4—41no—
184C4⋊C4—41no—
185C4⋊C4C2×D412yes—
186C4⋊C4C2×D411yescharacteristic
187C4⋊C4C2×D411yescharacteristic
188C22⋊C4—24no—
189C22⋊C4—22no—
190C22⋊C4—21no—
191C22⋊C4—21no—
192C22⋊C4—21no—
193C22⋊C4—41no—
194C42—24no—
195C42—21no—
196C42—21no—
197C42—21no—
198C42—22no—
199C42—41no—
200C42C2×D411yescharacteristic
201C42C2×D411yescharacteristic
202C42C2×D411yescharacteristic
203C42C2411yescharacteristic
204C23—21no—
205C23#12011yescharacteristic
206Q8—21no—
207Q8—22no—
208Q8—22no—
209Q8—44no—
210Q8—44no—
211Q8—42no—
212Q8—41no—
213Q8—41no—
214Q8—41no—
215Q8—41no—
216Q8—84no—
217Q8#13712yes—
218D4—24no—
219D4—21no—
220D4#13712yes—
221C2×C4—24no—
222C2×C4—21no—
223C2×C4—21no—
224C2×C4—21no—
225C2×C4—21no—
226C2×C4—21no—
227C2×C4—22no—
228C2×C4—22no—
229C2×C4—21no—
230C2×C4—21no—
231C2×C4—24no—
232C2×C4—22no—
233C2×C4—41no—
234C2×C4—41no—
235C2×C4—42no—
236C2×C4—41no—
237C2×C4—41no—
238C2×C4—82no—
239C2×C4—84no—
240C2×C4#12011yescharacteristic
241C2×C4#12011yescharacteristic
242C2×C4#12011yescharacteristic
243C2×C4C22×D411yescharacteristic
244C2×C4C22×D411yescharacteristic
245C2×C4C22×D411yescharacteristic
246C8—42no—
247C8—41no—
248C8—41no—
249C8—42no—
250C22—21no—
251C22—22no—
252C22—21no—
253C22—21no—
254C22—41no—
255C22—41no—
256C22C2×#12011yescharacteristic
257C4—21no—
258C4—22no—
259C4—21no—
260C4—21no—
261C4—41no—
262C4—42no—
263C4—41no—
264C4—44no—
265C4—42no—
266C4—81no—
267C4—81no—
268C4#45512yes—
269C4C2×#13711yescharacteristic
270C4C2×#13711yescharacteristic
271C4#57711yescharacteristic
272C4#57811yescharacteristic
273C2—41no—
274C2—42no—
275C2#300411yescharacteristic
276C2#301311yescharacteristic
277C2C2×#45511yescharacteristic
subgroupG/H
center, socleC22C2×#120
commutator, Frattini, cosocleC42C24
prime radicalC2×#41D4
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. #3255C2semidirect
3. #3139C2semidirect
5. #1645C2semidirect
6. #1645C2semidirect
7. #1644C2semidirect
8. #1643C2semidirect
9. #1548C2semidirect
11. #1485C2semidirect

Maximal subgroups

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

subgroupindex
#35032
#32552
#31392
#31072
#16452
#16442
#16432
#15482
#14892
#14852