Finite Group Databasebeta

← #28178

Subgroups — #28178

Order 256.

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

normal subgroups only

This group has 407 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#3037C211yesmaximal, characteristic, complemented
2#3036C211yesmaximal, characteristic
3C2×#440C211yesmaximal, characteristic, complemented
4#2932C211yesmaximal, characteristic, complemented
5#2078C212yesmaximal, complemented
6#2019C212yesmaximal, complemented
7#1908C212yesmaximal, complemented
8#1907C212yesmaximal, complemented
9#1838C211yesmaximal, characteristic, complemented
10#1801C211yesmaximal, characteristic, complemented
11#1800C211yesmaximal, characteristic, complemented
12C2×#137C2212yes—
13C2×AGL(1,ℤ/8)C2212yes—
14C22×SD16C2211yescharacteristic
15C2×#131C2212yes—
16#532C2211yescharacteristic
17C2×C4×Q8C2211yescharacteristic
18C2×C4×D4C2211yescharacteristic
19#464—22no—
20#463—22no—
21#462—21no—
22#461—21no—
23#460—21no—
24#459—21no—
25#440—22no—
26#440—22no—
27#435—22no—
28#435—22no—
29C2×#107C2211yescharacteristic
30#423C2211yescharacteristic
31#423C2211yescharacteristic
32#423C2211yescharacteristic
33#417C2212yes—
34#417C2212yes—
35C2×#103C2211yescharacteristic
36C2×#102C2211yescharacteristic
37C2×#101C2212yes—
38C2×#100C2212yes—
39#408C2212yes—
40C2×#97C2211yescharacteristic
41#343C2212yes—
42#343C2212yes—
43#341C2212yes—
44#341C2212yes—
45C2×#41C2312yes—
46C22×Q8C2311yescharacteristic
47C22×D4C2311yescharacteristic
48C23×C4—21no—
49#137—24no—
50#137—42no—
51AGL(1,ℤ/8)—24no—
52AGL(1,ℤ/8)—42no—
53C2×Q16—22no—
54C2×SD16—22no—
55C2×SD16—22no—
56C2×SD16—22no—
57C2×SD16—22no—
58C2×SD16—22no—
59C2×D8—22no—
60#131—42no—
61#131—42no—
62C2×M16—22no—
63C2×M16C2312yes—
64C2×M16C2312yes—
65C2×M16C2312yes—
66C22×C8—22no—
67C22×C8C2311yescharacteristic
68C22×C8C2311yescharacteristic
69C4⋊Q8D411yescharacteristic
70#127D411yescharacteristic
71#124D412yes—
72#121—21no—
73#121—21no—
74C4×Q8—21no—
75C4×Q8—21no—
76C4×Q8—21no—
77C4×Q8—21no—
78C4×D4—21no—
79C4×D4—21no—
80C4×D4—21no—
81C4×D4—21no—
82C4×D4—21no—
83C4×D4—21no—
84C2×C4⋊C4—21no—
85C2×C4⋊C4C2311yescharacteristic
86C2×C4⋊C4C2311yescharacteristic
87C2×C22⋊C4—21no—
88C2×C42—21no—
89C2×C42C2311yescharacteristic
90#107—22no—
91C4⋊C8—21no—
92C4⋊C8—21no—
93C4⋊C8—21no—
94C4⋊C8—21no—
95C4⋊C8—21no—
96C4⋊C8—21no—
97#103—22no—
98#103—22no—
99#103—22no—
100#102—22no—
101#102—22no—
102#102—22no—
103#101—42no—
104#100—42no—
105C22⋊C8—24no—
106#97D412yes—
107#97D412yes—
108C24—21no—
109#41—22no—
110#41—22no—
111#41—22no—
112#41—22no—
113C2×Q8—21no—
114C2×Q8—21no—
115C2×Q8—21no—
116C2×Q8—21no—
117C2×Q8—21no—
118C2×Q8C2×D412yes—
119C2×Q8#4112yes—
120C2×D4—22no—
121C2×D4—21no—
122C2×D4—21no—
123C2×D4—21no—
124C2×D4—21no—
125C2×D4—21no—
126C2×D4—21no—
127C2×D4—21no—
128C2×D4C2×D412yes—
129C2×D4#4112yes—
130C22×C4—21no—
131C22×C4—22no—
132C22×C4—21no—
133C22×C4—21no—
134C22×C4—41no—
135C22×C4—41no—
136C22×C4—82no—
137C22×C4#4111yescharacteristic
138C22×C4#4111yescharacteristic
139C22×C4C2411yescharacteristic
140Q16—44no—
141SD16—28no—
142SD16—44no—
143SD16—44no—
144D8—44no—
145M16—22no—
146M16—22no—
147M16—22no—
148M16—22no—
149M16—22no—
150M16—22no—
151M16—44no—
152C2×C8—21no—
153C2×C8—21no—
154C2×C8—21no—
155C2×C8—21no—
156C2×C8—22no—
157C2×C8—22no—
158C2×C8—22no—
159C2×C8—44no—
160C2×C8C2×D412yes—
161C2×C8C2×D412yes—
162C4⋊C4—21no—
163C4⋊C4—21no—
164C4⋊C4—21no—
165C4⋊C4—21no—
166C4⋊C4—21no—
167C4⋊C4—21no—
168C4⋊C4—41no—
169C4⋊C4—41no—
170C22⋊C4—22no—
171C22⋊C4—22no—
172C22⋊C4—41no—
173C22⋊C4—41no—
174C42—41no—
175C42—41no—
176C42C2×D411yescharacteristic
177C42C2×D411yescharacteristic
178C42C2×D411yescharacteristic
179C42C2×D411yescharacteristic
180C23—22no—
181C23—22no—
182C23—41no—
183C23—41no—
184C23—81no—
185C23—81no—
186C23C2×#4111yescharacteristic
187Q8—22no—
188Q8—22no—
189Q8—22no—
190Q8—22no—
191Q8—41no—
192Q8—41no—
193D4—22no—
194D4—22no—
195D4—22no—
196D4—22no—
197D4—42no—
198D4—42no—
199D4—41no—
200D4—41no—
201D4—81no—
202D4—81no—
203C2×C4—22no—
204C2×C4—22no—
205C2×C4—41no—
206C2×C4—41no—
207C2×C4—41no—
208C2×C4—41no—
209C2×C4—42no—
210C2×C4—42no—
211C2×C4—41no—
212C2×C4—41no—
213C2×C4—82no—
214C2×C4—82no—
215C2×C4—81no—
216C2×C4—81no—
217C2×C4#12111yescharacteristic
218C2×C4#12111yescharacteristic
219C2×C4#12111yescharacteristic
220C2×C4#12111yescharacteristic
221C2×C4C22×D411yescharacteristic
222C2×C4C22×D411yescharacteristic
223C2×C4C2×#4111yescharacteristic
224C2×C4C2×#4111yescharacteristic
225C2×C4H(5,𝔽2)11yescharacteristic
226C2×C4H(5,𝔽2)11yescharacteristic
227C8—22no—
228C8—22no—
229C8—42no—
230C8—42no—
231C8—42no—
232C22—21no—
233C22—21no—
234C22—42no—
235C22—42no—
236C22—42no—
237C22—42no—
238C22—82no—
239C22—82no—
240C22—81no—
241C22—81no—
242C22C2×#12111yescharacteristic
243C22#55311yescharacteristic
244C22#55311yescharacteristic
245C4—21no—
246C4—21no—
247C4—21no—
248C4—21no—
249C4—42no—
250C4—42no—
251C4—81no—
252C4—81no—
253C4#52511yescharacteristic
254C4#53511yescharacteristic
255C4#54611yescharacteristic
256C4#54611yescharacteristic
257C2—21no—
258C2—21no—
259C2—82no—
260C2—82no—
261C2#253511yescharacteristic
262C2#306811yescharacteristic
263C2#306911yescharacteristic
subgroupG/H
center, socleC22C2×#121
commutator, Frattini, cosocleC22×C4C24
prime radical#3036C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #3037C2semidirect
3. C2×#440C2semidirect
4. #2932C2semidirect
5. #2078C2semidirect
6. #2019C2semidirect
7. #1908C2semidirect
8. #1907C2semidirect
9. #1838C2semidirect
10. #1801C2semidirect
11. #1800C2semidirect

Maximal subgroups

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

subgroupindex
#30372
#30362
C2×#4402
#29322
#20782
#20192
#19082
#19072
#18382
#18012
#18002