Finite Group Databasebeta

← #28181

Subgroups — #28181

Order 256.

396 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 396 conjugacy classes of subgroups — too many to draw. Only its 113 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#3067C211yesmaximal, characteristic, complemented
2#3054C211yesmaximal, characteristic, complemented
3#2957C211yesmaximal, characteristic, complemented
4#2930C211yesmaximal, characteristic, complemented
5#2080C212yesmaximal, complemented
6#2010C212yesmaximal, complemented
7#1891C212yesmaximal, complemented
8#1881C212yesmaximal, complemented
9#1833C211yesmaximal, characteristic, complemented
10#1814C211yesmaximal, characteristic, complemented
11#1811C211yesmaximal, characteristic, complemented
12#575C2212yes—
13C2×#135C2211yescharacteristic
14#568C2212yes—
15C2×#131C2211yescharacteristic
16C22×M16C2211yescharacteristic
17#525C2211yescharacteristic
18#525C2212yes—
19#473—21no—
20#472—21no—
21#471—22no—
22#470—21no—
23#469—21no—
24#468—22no—
25#444—22no—
26#443—22no—
27#436—22no—
28#435—22no—
29C2×#108C2211yescharacteristic
30C2×#108C2211yescharacteristic
31#427C2211yescharacteristic
32#424C2211yescharacteristic
33#421C2212yes—
34#421C2212yes—
35#416C2212yes—
36#413C2212yes—
37#408C2211yescharacteristic
38#407C2211yescharacteristic
39#407C2212yes—
40#405C2211yescharacteristic
41#344C2212yes—
42#344C2212yes—
43#341C2212yes—
44H(3,ℤ/4)C2212yes—
45C2×#41C2312yes—
46C2×#41C2311yescharacteristic
47C23×C4C2311yescharacteristic
48#137—22no—
49#137—22no—
50AGL(1,ℤ/8)—22no—
51AGL(1,ℤ/8)—22no—
52#135—21no—
53#135—21no—
54#135—21no—
55#135—21no—
56#135—42no—
57C2×Q16D411yescharacteristic, complemented
58C2×SD16D412yescomplemented
59C2×D8D411yescharacteristic, complemented
60#131—21no—
61#131—21no—
62#131—24no—
63#131—21no—
64#131—21no—
65#131—42no—
66C2×M16—21no—
67C2×M16—21no—
68C2×M16—21no—
69C2×M16—21no—
70C2×M16—21no—
71C2×M16—21no—
72C2×M16—21no—
73C2×M16—42no—
74C2×M16C2311yescharacteristic
75C2×M16C2312yes—
76C2×M16C2312yes—
77C2×M16C2311yescharacteristic
78C22×C8—21no—
79C22×C8C2311yescharacteristic
80C22×C8C2311yescharacteristic
81#123—21no—
82#123—22no—
83C22⋊Q8—21no—
84C22⋊Q8—22no—
85#121—21no—
86#121—22no—
87#120—21no—
88#120—22no—
89C4×D4—22no—
90C4×D4—42no—
91#117C2312yes—
92#117C2311yescharacteristic
93#108—21no—
94#108—21no—
95#108—21no—
96#108—21no—
97#107—21no—
98#106—21no—
99C4⋊C8—22no—
100#104—24no—
101#104—42no—
102#103—22no—
103#102—22no—
104#101—22no—
105#100—22no—
106C22⋊C8—22no—
107C22⋊C8—22no—
108C22⋊C8—42no—
109#97D412yescomplemented
110C4×C8D412yescomplemented
111C24#4111yescharacteristic
112#41—22no—
113#41—21no—
114#41—21no—
115#41—22no—
116#41—22no—
117#41—21no—
118#41—22no—
119#41—21no—
120C2×Q8C2×D412yes—
121C2×Q8#4111yescharacteristic
122C2×D4—21no—
123C2×D4—22no—
124C2×D4—22no—
125C2×D4—42no—
126C2×D4C2×D412yes—
127C2×D4#4111yescharacteristic
128C22×C4—21no—
129C22×C4—21no—
130C22×C4—21no—
131C22×C4—21no—
132C22×C4—21no—
133C22×C4—21no—
134C22×C4—21no—
135C22×C4—22no—
136C22×C4#4111yescharacteristic
137C22×C4C2411yescharacteristic
138Q16—21no—
139Q16—21no—
140Q16—42no—
141SD16—22no—
142SD16—22no—
143SD16—42no—
144SD16—42no—
145D8—21no—
146D8—21no—
147D8—42no—
148M16—21no—
149M16—21no—
150M16—21no—
151M16—22no—
152M16—22no—
153M16—21no—
154M16—22no—
155M16—22no—
156M16—44no—
157M16—41no—
158M16—41no—
159M16—41no—
160M16—41no—
161M16—41no—
162M16—41no—
163C2×C8—21no—
164C2×C8—21no—
165C2×C8—21no—
166C2×C8—21no—
167C2×C8—22no—
168C2×C8—22no—
169C2×C8—44no—
170C2×C8—41no—
171C2×C8—41no—
172C2×C8C2×D412yes—
173C2×C8C2×D411yescharacteristic
174C2×C8C2×D411yescharacteristic
175C2×C8C2×D411yescharacteristic
176C2×C8C2×D411yescharacteristic
177C2×C8#4111yescharacteristic
178C2×C8#4111yescharacteristic
179C4⋊C4—22no—
180C4⋊C4—21no—
181C4⋊C4—22no—
182C4⋊C4—42no—
183C22⋊C4—22no—
184C22⋊C4—22no—
185C22⋊C4—21no—
186C22⋊C4—22no—
187C22⋊C4—42no—
188C22⋊C4—42no—
189C42—22no—
190C42C2×D412yes—
191C23—21no—
192C23—21no—
193C23—21no—
194C23—21no—
195C23—22no—
196C23—81no—
197C23C2×#4111yescharacteristic
198Q8—22no—
199Q8—21no—
200Q8—22no—
201Q8—21no—
202D4—22no—
203D4—21no—
204D4—22no—
205D4—21no—
206D4—42no—
207D4—41no—
208D4—41no—
209D4—42no—
210D4—82no—
211D4—84no—
212C2×C4—22no—
213C2×C4—21no—
214C2×C4—21no—
215C2×C4—21no—
216C2×C4—22no—
217C2×C4—21no—
218C2×C4—41no—
219C2×C4—41no—
220C2×C4—42no—
221C2×C4—41no—
222C2×C4—41no—
223C2×C4—41no—
224C2×C4—41no—
225C2×C4—42no—
226C2×C4—42no—
227C2×C4C22×D411yescharacteristic
228C2×C4C22×D411yescharacteristic
229C2×C4C2×#4111yescharacteristic
230C2×C4C2×#4111yescharacteristic
231C2×C4H(5,𝔽2)11yescharacteristic
232C2×C4H(5,𝔽2)11yescharacteristic
233C8—22no—
234C8—41no—
235C8—41no—
236C8—42no—
237C8—42no—
238C8#12111yescharacteristic
239C8#12111yescharacteristic
240C8#12111yescharacteristic
241C8#12111yescharacteristic
242C22—41no—
243C22—41no—
244C22—42no—
245C22—42no—
246C22—41no—
247C22—42no—
248C22—41no—
249C22—41no—
250C22—42no—
251C22—82no—
252C22#52511yescharacteristic
253C22#54611yescharacteristic
254C22#55311yescharacteristic
255C4—42no—
256C4—41no—
257C4—41no—
258C4—42no—
259C4—82no—
260C4C2×#12111yescharacteristic
261C4#53511yescharacteristic
262C4#54611yescharacteristic
263C4#55311yescharacteristic
264C2—21no—
265C2—21no—
266C2—21no—
267C2—81no—
268C2—81no—
269C2—82no—
270C2#253511yescharacteristic
subgroupG/H
centerC4#535
commutator, Frattini, cosocleC22×C4C24
socleC2#2535
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C4 ◁ C22×C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #3067C2semidirect
2. #3054C2semidirect
3. #2957C2semidirect
4. #2930C2semidirect
5. #2080C2semidirect
6. #2010C2semidirect
7. #1891C2semidirect
8. #1881C2semidirect
9. #1833C2semidirect
10. #1814C2semidirect
11. #1811C2semidirect
57. C2×Q16D4semidirect
58. C2×SD16D4semidirect
59. C2×D8D4semidirect
109. #97D4semidirect
110. C4×C8D4semidirect

Maximal subgroups

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

subgroupindex
#30672
#30542
#29572
#29302
#20802
#20102
#18912
#18812
#18332
#18142
#18112