Finite Group Databasebeta

← #29613

Subgroups — #29613

Order 256.

628 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 628 conjugacy classes of subgroups — too many to draw. Only its 121 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
1C2×#536C211yesmaximal, characteristic
2#3095C211yesmaximal, characteristic, complemented
3#3007C211yesmaximal, characteristic
4#2434C211yesmaximal, characteristic, complemented
5#2038C214yesmaximal, complemented
6#2005C214yesmaximal, complemented
7#1826C211yesmaximal, characteristic, complemented
8#1789C211yesmaximal, characteristic
9#1784C211yesmaximal, characteristic, complemented
10C22×#41C2211yescharacteristic
11C23×Q8C2211yescharacteristic
12C2×#137C2212yes—
13#536—22no—
14#536—22no—
15#536—22no—
16#536C2214yes—
17C2×C4⋊Q8—21no—
18C2×C4⋊Q8C2212yescomplemented
19C2×#124—21no—
20C2×#123—22no—
21C2×C22⋊Q8—21no—
22C2×C22⋊Q8—21no—
23C2×C22⋊Q8C2211yescharacteristic
24C2×#117C2211yescharacteristic
25#485—22no—
26#484—21no—
27#483—21no—
28#452—22no—
29#451—21no—
30#450—21no—
31#428C2212yescomplemented
32C2×#104C2214yes—
33#419C2212yescomplemented
34#417C2212yes—
35C2×#101C2214yescomplemented
36#407C2211yescharacteristic
37#407C2211yescharacteristic
38#395—22no—
39#391C2212yescomplemented
40#385—22no—
41C4×C22⋊C4C2211yescharacteristic
42#339C2214yescomplemented
43C2×#41—21no—
44C2×#41—21no—
45C2×#41—44no—
46C2×#41D412yes—
47C2×#41C2312yes—
48C22×Q8—21no—
49C22×Q8—21no—
50C22×Q8—21no—
51C22×Q8—21no—
52C22×Q8—44no—
53C22×Q8D411yescharacteristic
54C22×Q8D412yes—
55C22×Q8C2311yescharacteristic
56C22×Q8C2312yes—
57C22×D4—21no—
58C22×D4D411yescharacteristic
59C23×C4—21no—
60C23×C4D412yes—
61C23×C4C2311yescharacteristic
62#137—44no—
63C2×Q16—22no—
64C2×SD16—22no—
65C2×M16C2312yes—
66C2×M16C2312yes—
67C4⋊Q8—24no—
68C4⋊Q8—24no—
69C4⋊Q8—44no—
70#124—24no—
71#124—24no—
72#123—28no—
73#123—28no—
74C22⋊Q8—24no—
75C22⋊Q8—24no—
76C22⋊Q8—24no—
77C22⋊Q8—24no—
78C22⋊Q8—21no—
79C22⋊Q8—21no—
80C22⋊Q8—21no—
81C22⋊Q8—21no—
82#117—21no—
83#117—21no—
84#117D412yes—
85#117C2312yes—
86C2×C4⋊C4—22no—
87C2×C4⋊C4—22no—
88C2×C4⋊C4—21no—
89C2×C4⋊C4—24no—
90C2×C4⋊C4D411yescharacteristic
91C2×C4⋊C4D411yescharacteristic
92C2×C4⋊C4C2311yescharacteristic
93C2×C22⋊C4—22no—
94C2×C22⋊C4—22no—
95C2×C22⋊C4—21no—
96C2×C22⋊C4—21no—
97C2×C22⋊C4—21no—
98C2×C42—21no—
99C2×C42C2312yes—
100#107—22no—
101#106—22no—
102#104—44no—
103#103—22no—
104#103—22no—
105#103—22no—
106#102—22no—
107#101—44no—
108C22⋊C8—21no—
109C22⋊C8—21no—
110C22⋊C8—21no—
111C22⋊C8—21no—
112#95—21no—
113#95—24no—
114C24—21no—
115C24C2×D411yescharacteristic
116#41—24no—
117#41—24no—
118#41—42no—
119#41—42no—
120#41—44no—
121#41—44no—
122C2×Q8—22no—
123C2×Q8—22no—
124C2×Q8—21no—
125C2×Q8—22no—
126C2×Q8—22no—
127C2×Q8—24no—
128C2×Q8—24no—
129C2×Q8—21no—
130C2×Q8—21no—
131C2×Q8—24no—
132C2×Q8—44no—
133C2×Q8—42no—
134C2×Q8—44no—
135C2×Q8—44no—
136C2×Q8—44no—
137C2×Q8—44no—
138C2×Q8C2×D412yes—
139C2×D4—22no—
140C2×D4—22no—
141C2×D4—21no—
142C2×D4—42no—
143C2×D4—44no—
144C2×D4C2×D412yes—
145C22×C4—24no—
146C22×C4—21no—
147C22×C4—21no—
148C22×C4—22no—
149C22×C4—21no—
150C22×C4—22no—
151C22×C4—22no—
152C22×C4—22no—
153C22×C4—22no—
154C22×C4—21no—
155C22×C4—21no—
156C22×C4—44no—
157C22×C4—44no—
158C22×C4C2×D414yes—
159C22×C4C2×D411yescharacteristic
160C22×C4C2×D411yescharacteristic
161C22×C4C2×D411yescharacteristic
162C22×C4C2×D411yescharacteristic
163C22×C4C2×D411yescharacteristic
164C22×C4C2411yescharacteristic
165Q16—44no—
166SD16—44no—
167M16—24no—
168M16—24no—
169C2×C8—22no—
170C2×C8—22no—
171C4⋊C4—28no—
172C4⋊C4—28no—
173C4⋊C4—21no—
174C4⋊C4—21no—
175C4⋊C4—21no—
176C4⋊C4—21no—
177C4⋊C4—48no—
178C4⋊C4—41no—
179C4⋊C4—41no—
180C4⋊C4C2×D412yes—
181C4⋊C4C2×D412yes—
182C22⋊C4—24no—
183C22⋊C4—28no—
184C22⋊C4—28no—
185C22⋊C4—22no—
186C22⋊C4—22no—
187C22⋊C4—41no—
188C22⋊C4—41no—
189C42—22no—
190C42—22no—
191C42—24no—
192C23—22no—
193C23—22no—
194C23—22no—
195C23—41no—
196C23—42no—
197C23—42no—
198C23—42no—
199C23#12011yescharacteristic
200C23#12011yescharacteristic
201C23C22×D411yescharacteristic
202Q8—22no—
203Q8—22no—
204Q8—44no—
205Q8—44no—
206Q8—42no—
207Q8—44no—
208Q8—44no—
209Q8—42no—
210Q8—42no—
211D4—22no—
212D4—22no—
213D4—42no—
214D4—44no—
215D4—44no—
216C2×C4—22no—
217C2×C4—22no—
218C2×C4—22no—
219C2×C4—22no—
220C2×C4—22no—
221C2×C4—24no—
222C2×C4—22no—
223C2×C4—22no—
224C2×C4—24no—
225C2×C4—22no—
226C2×C4—24no—
227C2×C4—24no—
228C2×C4—44no—
229C2×C4—42no—
230C2×C4—42no—
231C2×C4—42no—
232C2×C4—44no—
233C2×C4—44no—
234C2×C4—42no—
235C2×C4—42no—
236C2×C4—44no—
237C2×C4—42no—
238C2×C4—41no—
239C2×C4—41no—
240C2×C4—42no—
241C2×C4#12011yescharacteristic
242C2×C4#12011yescharacteristic
243C2×C4C22×D411yescharacteristic
244C2×C4C22×D411yescharacteristic
245C2×C4H(5,𝔽2)14yes—
246C8—42no—
247C8—42no—
248C22—21no—
249C22—21no—
250C22—21no—
251C22—21no—
252C22—21no—
253C22—21no—
254C22—21no—
255C22—44no—
256C22—42no—
257C22—42no—
258C22—42no—
259C22—84no—
260C22#45612yes—
261C22C2×#12011yescharacteristic
262C22#53412yes—
263C4—21no—
264C4—21no—
265C4—44no—
266C4—44no—
267C4—42no—
268C4—42no—
269C4—42no—
270C4—81no—
271C4—81no—
272C4#53412yes—
273C4#53512yes—
274C2—22no—
275C2—22no—
276C2—41no—
277C2—82no—
278C2#261311yescharacteristic
279C2C2×#45611yescharacteristic
280C2#302011yescharacteristic
subgroupG/H
center, socleC22C2×#120
commutator, Frattini, cosocleC22×C4C24
prime radicalC22×#41C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #3095C2semidirect
4. #2434C2semidirect
5. #2038C2semidirect
6. #2005C2semidirect
7. #1826C2semidirect
9. #1784C2semidirect
18. C2×C4⋊Q8C22semidirect
31. #428C22semidirect
33. #419C22semidirect
35. C2×#101C22semidirect
39. #391C22semidirect
42. #339C22semidirect

Maximal subgroups

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

subgroupindex
C2×#5362
#30952
#30072
#24342
#20382
#20052
#18262
#17892
#17842