Finite Group Databasebeta

← #18383

Subgroups — #18383

Order 256.

493 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 493 conjugacy classes of subgroups — too many to draw. Only its 127 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#3501C211yesmaximal, characteristic
2#3189C212yesmaximal, complemented
3#2564C211yesmaximal, characteristic
4#1617C212yesmaximal, complemented
5#1614C212yesmaximal, complemented
6#1605C212yesmaximal, complemented
7#1602C212yesmaximal, complemented
8C2×#328C212yesmaximal, complemented
9#1459C211yesmaximal, characteristic
10#554—24no—
11#554—22no—
12D4×Q8—22no—
13#544C2212yes—
14#543—21no—
15#543C2212yes—
16#533—21no—
17C2×C22⋊Q8—21no—
18C2×C22⋊Q8C2212yes—
19C2×C22⋊Q8C2211yescharacteristic
20#518—21no—
21C2×C4×D4C2211yescharacteristic
22#484—22no—
23#484C2214yes—
24#467—24no—
25#451—22no—
26#451C2214yes—
27C2×C22⋊C8C2212yes—
28#400C2211yescharacteristic
29#399C2211yescharacteristic
30#395C2211yescharacteristic
31#393C2211yescharacteristic
32#387—21no—
33#386—21no—
34#385—21no—
35#380—21no—
36C2×#95C2211yescharacteristic
37#328C2214yes—
38#328C2214yes—
39#324C2212yes—
40#323C2212yes—
41C22×Q8—21no—
42C22×Q8D412yes—
43C22×Q8C2311yescharacteristic
44C22×D4D411yescharacteristic
45C23×C4—21no—
46C23×C4D411yescharacteristic
47C23×C4C2311yescharacteristic
48C2×Q16—24no—
49C22×C8—22no—
50C4⋊Q8—21no—
51C4⋊Q8—21no—
52C4⋊Q8—24no—
53#125—22no—
54#125—24no—
55C22⋊Q8—22no—
56C22⋊Q8—22no—
57C22⋊Q8—24no—
58C22⋊Q8—24no—
59C22⋊Q8—24no—
60C22⋊Q8—22no—
61C22⋊Q8—22no—
62C22⋊Q8—22no—
63C22⋊Q8—22no—
64C22⋊Q8—24no—
65C22⋊Q8—22no—
66C22⋊Q8—22no—
67C22⋊Q8D414yes—
68C22⋊Q8C2314yes—
69C4×Q8—22no—
70C4×Q8—22no—
71C4×D4—24no—
72C4×D4—24no—
73C4×D4—22no—
74C4×D4—22no—
75#117—22no—
76C2×C4⋊C4—21no—
77C2×C4⋊C4—22no—
78C2×C4⋊C4—22no—
79C2×C4⋊C4—21no—
80C2×C4⋊C4—21no—
81C2×C4⋊C4—21no—
82C2×C4⋊C4D412yes—
83C2×C4⋊C4C2311yescharacteristic
84C2×C4⋊C4C2311yescharacteristic
85C2×C22⋊C4—22no—
86C2×C22⋊C4—22no—
87C2×C22⋊C4—21no—
88C2×C22⋊C4—21no—
89C2×C22⋊C4—21no—
90C2×C22⋊C4D411yescharacteristic
91C2×C22⋊C4C2311yescharacteristic
92C2×C42D411yescharacteristic
93#107—24no—
94#103—24no—
95#103—24no—
96C22⋊C8—22no—
97C22⋊C8C2314yes—
98#95—21no—
99#95—21no—
100#95—21no—
101#95—21no—
102#95—21no—
103#95C2311yescharacteristic
104#95C2311yescharacteristic
105C24C2×D411yescharacteristic
106C24C2×D411yescharacteristic
107C2×Q8—22no—
108C2×Q8—22no—
109C2×Q8—22no—
110C2×Q8—22no—
111C2×Q8—81no—
112C2×Q8—82no—
113C2×Q8—82no—
114C2×Q8C2×D414yes—
115C2×D4—22no—
116C2×D4—22no—
117C2×D4Q1614yescomplemented
118C22×C4—21no—
119C22×C4—22no—
120C22×C4—22no—
121C22×C4—22no—
122C22×C4—22no—
123C22×C4—21no—
124C22×C4—21no—
125C22×C4—21no—
126C22×C4—21no—
127C22×C4—21no—
128C22×C4—42no—
129C22×C4—42no—
130C22×C4—81no—
131C22×C4—81no—
132C22×C4—81no—
133C22×C4C2×D411yescharacteristic
134C22×C4C2×D411yescharacteristic
135C22×C4C2×D411yescharacteristic
136C22×C4C2×D411yescharacteristic
137C22×C4C2×D412yes—
138C22×C4C2×D412yes—
139C22×C4C2411yescharacteristic
140Q16—84no—
141C2×C8—24no—
142C2×C8—82no—
143C4⋊C4—22no—
144C4⋊C4—24no—
145C4⋊C4—22no—
146C4⋊C4—22no—
147C4⋊C4—24no—
148C4⋊C4—22no—
149C4⋊C4—24no—
150C4⋊C4—22no—
151C4⋊C4—24no—
152C4⋊C4—22no—
153C4⋊C4—22no—
154C4⋊C4—42no—
155C4⋊C4—41no—
156C4⋊C4—41no—
157C4⋊C4—81no—
158C4⋊C4C2×D414yes—
159C22⋊C4—24no—
160C22⋊C4—24no—
161C22⋊C4—22no—
162C22⋊C4—24no—
163C22⋊C4—22no—
164C22⋊C4—42no—
165C22⋊C4—42no—
166C22⋊C4—81no—
167C42—24no—
168C42—21no—
169C42—21no—
170C23—21no—
171C23—21no—
172C23—41no—
173C23—41no—
174C23—41no—
175C23—41no—
176C23#12012yes—
177C23C2×Q1612yes—
178C23C2×Q1612yes—
179C23#13712yes—
180C23C22×D411yescharacteristic
181Q8—44no—
182Q8—44no—
183Q8—44no—
184Q8—82no—
185Q8—82no—
186D4—28no—
187C2×C4—21no—
188C2×C4—21no—
189C2×C4—22no—
190C2×C4—21no—
191C2×C4—24no—
192C2×C4—24no—
193C2×C4—21no—
194C2×C4—42no—
195C2×C4—41no—
196C2×C4—42no—
197C2×C4—41no—
198C2×C4—42no—
199C2×C4—41no—
200C2×C4—41no—
201C2×C4—41no—
202C2×C4—41no—
203C2×C4—41no—
204C2×C4—41no—
205C2×C4—41no—
206C2×C4—41no—
207C2×C4—41no—
208C2×C4—41no—
209C2×C4—41no—
210C2×C4—41no—
211C2×C4—41no—
212C2×C4—81no—
213C2×C4—82no—
214C2×C4—82no—
215C2×C4—82no—
216C2×C4—82no—
217C2×C4—84no—
218C2×C4#12011yescharacteristic
219C2×C4#12011yescharacteristic
220C2×C4C2×Q1612yes—
221C2×C4C22×D412yes—
222C8—44no—
223C22—21no—
224C22—22no—
225C22—21no—
226C22—22no—
227C22—22no—
228C22—42no—
229C22—42no—
230C22—42no—
231C22—41no—
232C22—41no—
233C22—81no—
234C22#45114yes—
235C22C2×#12011yescharacteristic
236C22C22×Q1611yescharacteristic
237C22C2×#13711yescharacteristic
238C4—22no—
239C4—22no—
240C4—21no—
241C4—21no—
242C4—42no—
243C4—41no—
244C4—44no—
245C4—41no—
246C4—44no—
247C4—81no—
248C4—81no—
249C4—81no—
250C2—21no—
251C2—21no—
252C2—24no—
253C2—42no—
254C2C2×#45111yescharacteristic
255C2#301811yescharacteristic
256C2#302611yescharacteristic
subgroupG/H
center, socleC22C2×#120
commutator, Frattini, cosocleC22×C4C24
prime radicalC22×D4D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ G
LECSG ▷ C22×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #3189C2semidirect
4. #1617C2semidirect
5. #1614C2semidirect
6. #1605C2semidirect
7. #1602C2semidirect
8. C2×#328C2semidirect
117. C2×D4Q16semidirect

Maximal subgroups

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

subgroupindex
#35012
#31892
#25642
#16172
#16142
#16052
#16022
C2×#3282
#14592