Finite Group Databasebeta

← #9044

Subgroups — #9044

Order 256.

297 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 297 conjugacy classes of subgroups — too many to draw. Only its 77 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×#483C211yesmaximal, characteristic
2#2329C211yesmaximal, characteristic
3#1823C211yesmaximal, characteristic
4C2×#328C211yesmaximal, characteristic
5#1445C211yesmaximal, characteristic
6#1294C211yesmaximal, characteristic
7#1280C211yesmaximal, characteristic
8C2×C22⋊Q8C2211yescharacteristic
9C22×C4⋊C4C2211yescharacteristic
10#483—22no—
11#483C414yescomplemented
12C2×#106—21no—
13C2×#103—21no—
14C2×C22⋊C8C2211yescharacteristic
15C2×C22⋊C8C2211yescharacteristic
16#391—21no—
17#389—21no—
18#387—21no—
19#387—21no—
20#385—21no—
21#380C2211yescharacteristic
22C4×C22⋊C4—21no—
23C2×#95—21no—
24C2×#95—21no—
25C2×#95C2211yescharacteristic
26C2×#95C2211yescharacteristic
27#340—21no—
28#340—21no—
29#328—22no—
30C22×Q8D411yescharacteristic
31C23×C4—21no—
32C23×C4—21no—
33C23×C4D411yescharacteristic
34C23×C4D411yescharacteristic
35C23×C4C2311yescharacteristic
36C22×C8—21no—
37C22×C8—21no—
38C22⋊Q8—21no—
39C22⋊Q8—21no—
40C22⋊Q8C2×C412yes—
41C22⋊Q8D412yes—
42C2×C4⋊C4—21no—
43C2×C4⋊C4—21no—
44C2×C4⋊C4—21no—
45C2×C4⋊C4—21no—
46C2×C4⋊C4—21no—
47C2×C4⋊C4—21no—
48C2×C4⋊C4—21no—
49C2×C4⋊C4—21no—
50C2×C4⋊C4C2×C412yes—
51C2×C4⋊C4D411yescharacteristic
52C2×C4⋊C4D411yescharacteristic
53C2×C4⋊C4D411yescharacteristic
54C2×C22⋊C4—21no—
55C2×C22⋊C4—21no—
56C2×C42—21no—
57C2×C42—21no—
58#106—24no—
59#103—24no—
60C22⋊C8—21no—
61C22⋊C8—21no—
62C22⋊C8—21no—
63C22⋊C8C2×C412yes—
64#95—21no—
65#95—21no—
66#95—21no—
67#95—21no—
68#95—21no—
69#95—21no—
70#95—21no—
71#95—21no—
72#95—21no—
73#95—42no—
74#95—42no—
75C24C2×D411yescharacteristic
76C2×Q8—21no—
77C2×Q8—41no—
78C2×Q8—41no—
79C2×Q8C22⋊C412yes—
80C22×C4—21no—
81C22×C4—21no—
82C22×C4—21no—
83C22×C4—21no—
84C22×C4—22no—
85C22×C4—22no—
86C22×C4—21no—
87C22×C4—21no—
88C22×C4—21no—
89C22×C4—21no—
90C22×C4—21no—
91C22×C4—41no—
92C22×C4—41no—
93C22×C4—41no—
94C22×C4—41no—
95C22×C4—41no—
96C22×C4—41no—
97C22×C4—41no—
98C22×C4—41no—
99C22×C4—41no—
100C22×C4—41no—
101C22×C4—41no—
102C22×C4—41no—
103C22×C4—41no—
104C22×C4—41no—
105C22×C4—41no—
106C22×C4—41no—
107C22×C4—41no—
108C22×C4—41no—
109C22×C4SD1612yes—
110C22×C4C22×C411yescharacteristic
111C22×C4C2×D411yescharacteristic
112C22×C4C2×D411yescharacteristic
113C22×C4C2×D411yescharacteristic
114C22×C4#4111yescharacteristic
115C22×C4#4111yescharacteristic
116C2×C8—22no—
117C2×C8—41no—
118C2×C8—41no—
119C2×C8—41no—
120C2×C8—41no—
121C2×C8—41no—
122C4⋊C4—22no—
123C4⋊C4—22no—
124C4⋊C4—21no—
125C4⋊C4—22no—
126C4⋊C4—22no—
127C4⋊C4—41no—
128C4⋊C4—41no—
129C4⋊C4—41no—
130C4⋊C4—41no—
131C4⋊C4—81no—
132C4⋊C4—81no—
133C4⋊C4C22⋊C412yes—
134C22⋊C4—22no—
135C22⋊C4—22no—
136C22⋊C4—41no—
137C22⋊C4—41no—
138C42—81no—
139C42—81no—
140C23—21no—
141C23—21no—
142C23—21no—
143C23—21no—
144C23C4×D411yescharacteristic
145C23#12011yescharacteristic
146C23#12111yescharacteristic
147C23C2×SD1611yescharacteristic
148C23#13511yescharacteristic
149C23AGL(1,ℤ/8)11yescharacteristic
150C23#13711yescharacteristic
151Q8—42no—
152Q8—81no—
153C2×C4—21no—
154C2×C4—21no—
155C2×C4—21no—
156C2×C4—21no—
157C2×C4—22no—
158C2×C4—22no—
159C2×C4—22no—
160C2×C4—41no—
161C2×C4—41no—
162C2×C4—41no—
163C2×C4—41no—
164C2×C4—41no—
165C2×C4—41no—
166C2×C4—41no—
167C2×C4—41no—
168C2×C4—41no—
169C2×C4—41no—
170C2×C4—42no—
171C2×C4—42no—
172C2×C4—42no—
173C2×C4—42no—
174C2×C4—42no—
175C2×C4—42no—
176C2×C4—42no—
177C2×C4—42no—
178C2×C4—41no—
179C2×C4—41no—
180C2×C4—41no—
181C2×C4—41no—
182C2×C4—41no—
183C2×C4—41no—
184C2×C4—41no—
185C2×C4—41no—
186C2×C4—41no—
187C2×C4—81no—
188C2×C4—81no—
189C2×C4—81no—
190C2×C4—81no—
191C2×C4—81no—
192C2×C4—81no—
193C2×C4—81no—
194C2×C4—81no—
195C2×C4—81no—
196C2×C4—81no—
197C2×C4C2×C22⋊C411yescharacteristic
198C2×C4C4×D411yescharacteristic
199C2×C4#12111yescharacteristic
200C2×C4#12311yescharacteristic
201C8—42no—
202C8—81no—
203C22—21no—
204C22—21no—
205C22—21no—
206C22—21no—
207C22—21no—
208C22—21no—
209C22—21no—
210C22—21no—
211C22—21no—
212C22—21no—
213C22—21no—
214C22—21no—
215C22#38611yescharacteristic
216C22C4×SD1611yescharacteristic
217C22#44011yescharacteristic
218C22#44811yescharacteristic
219C22#44911yescharacteristic
220C22#45511yescharacteristic
221C22#45611yescharacteristic
222C22#45811yescharacteristic
223C22#45811yescharacteristic
224C22#46011yescharacteristic
225C22#46411yescharacteristic
226C4—21no—
227C4—21no—
228C4—21no—
229C4—21no—
230C4—42no—
231C4—42no—
232C4—42no—
233C4—82no—
234C4—81no—
235C4—81no—
236C4—81no—
237C4—82no—
238C2—21no—
239C2—21no—
240C2—21no—
241C2—21no—
242C2#160811yescharacteristic
243C2#160911yescharacteristic
244C2#161111yescharacteristic
245C2#161211yescharacteristic
246C2#188011yescharacteristic
247C2#188211yescharacteristic
248C2#189011yescharacteristic
subgroupG/H
center, socleC23#120
commutatorC22×C4C22×C4
Frattini, cosocleC23×C4C23
prime radicalC24C2×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
11. #483C4semidirect

Maximal subgroups

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

subgroupindex
C2×#4832
#23292
#18232
C2×#3282
#14452
#12942
#12802