Finite Group Databasebeta

← #24679

Subgroups — #24679

Order 256.

397 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 397 conjugacy classes of subgroups — too many to draw. Only its 131 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×#477C211yesmaximal, characteristic, complemented
2C2×C4×SD16C211yesmaximal, characteristic
3#2563C211yesmaximal, characteristic
4#2355C211yesmaximal, characteristic, complemented
5#2074C212yesmaximal, complemented
6#2055C212yesmaximal
7#1990C211yesmaximal, characteristic, complemented
8#1949C212yesmaximal, complemented
9#1939C211yesmaximal, characteristic, complemented
10#1925C212yesmaximal
11#1802C211yesmaximal, characteristic
12C22×SD16C2211yescharacteristic
13C2×C4⋊Q8C2211yescharacteristic
14C2×C4×Q8C2211yescharacteristic
15C2×C4×D4C2211yescharacteristic
16#477—24no—
17C4×SD16—24no—
18C2×#107C2212yes—
19C2×#106C2211yescharacteristic
20C2×C4⋊C8C2211yescharacteristic
21C2×C4⋊C8C2212yes—
22C2×#103C2212yes—
23C2×#103C2211yescharacteristic
24C2×#102C2212yes—
25C2×#102C2211yescharacteristic
26C2×C4×C8C2211yescharacteristic
27#393—22no—
28#391C2212yes—
29#390C2212yes—
30#387—22no—
31#384C2211yescharacteristic
32#382—21no—
33#382C2212yes—
34#380—21no—
35#380—21no—
36C4×C4⋊C4—21no—
37C4×C4⋊C4C2212yes—
38C4×C4⋊C4C2211yescharacteristic
39#340C2212yes—
40#340C2212yes—
41#340C2212yes—
42#336C2212yes—
43C22×Q8—21no—
44C22×Q8C2311yescharacteristic
45C22×D4C2311yescharacteristic
46C23×C4—21no—
47C2×SD16—22no—
48C2×SD16—22no—
49C2×SD16—22no—
50C22×C8C2311yescharacteristic
51C22×C8C2312yes—
52C22×C8C2311yescharacteristic
53C4⋊Q8—22no—
54C4⋊Q8—22no—
55C4×Q8—22no—
56C4×Q8D414yes—
57C4×D4—22no—
58C4×D4—22no—
59C2×C4⋊C4—21no—
60C2×C4⋊C4—21no—
61C2×C4⋊C4—21no—
62C2×C4⋊C4—21no—
63C2×C4⋊C4D411yescharacteristic
64C2×C4⋊C4D411yescharacteristic
65C2×C4⋊C4C2311yescharacteristic
66C2×C4⋊C4C2312yes—
67C2×C4⋊C4C2312yes—
68C2×C4⋊C4C2311yescharacteristic
69C2×C22⋊C4—21no—
70C2×C22⋊C4—22no—
71C2×C22⋊C4—22no—
72C2×C42—21no—
73C2×C42—22no—
74C2×C42C2312yes—
75C2×C42C2311yescharacteristic
76#107—24no—
77#106—22no—
78C4⋊C8—24no—
79C4⋊C8—22no—
80#103—22no—
81#103—24no—
82#102—24no—
83#102—22no—
84C4×C8—22no—
85#95—22no—
86#95—22no—
87#95—22no—
88#95—22no—
89#95—22no—
90#95D412yes—
91C24—21no—
92C2×Q8—21no—
93C2×Q8—21no—
94C2×Q8—21no—
95C2×Q8—21no—
96C2×Q8—21no—
97C2×Q8—42no—
98C2×Q8—81no—
99C2×Q8—81no—
100C2×Q8C2×D412yes—
101C2×D4—21no—
102C2×D4—21no—
103C2×D4—21no—
104C2×D4—21no—
105C2×D4—21no—
106C2×D4#4112yes—
107C22×C4—21no—
108C22×C4—21no—
109C22×C4—21no—
110C22×C4—22no—
111C22×C4—22no—
112C22×C4—42no—
113C22×C4—42no—
114C22×C4—42no—
115C22×C4C2×D412yes—
116C22×C4C2×D412yes—
117C22×C4C2×D411yescharacteristic
118C22×C4C2×D411yescharacteristic
119C22×C4C2411yescharacteristic
120SD16—28no—
121C2×C8—21no—
122C2×C8—21no—
123C2×C8—21no—
124C2×C8—22no—
125C2×C8—22no—
126C2×C8—22no—
127C2×C8—21no—
128C2×C8—21no—
129C2×C8#4112yes—
130C4⋊C4—24no—
131C4⋊C4—22no—
132C4⋊C4—22no—
133C4⋊C4—21no—
134C4⋊C4—21no—
135C4⋊C4—22no—
136C4⋊C4—22no—
137C4⋊C4—21no—
138C4⋊C4—22no—
139C4⋊C4—22no—
140C4⋊C4—42no—
141C4⋊C4—42no—
142C4⋊C4—81no—
143C4⋊C4C2×D412yes—
144C22⋊C4—42no—
145C22⋊C4—82no—
146C22⋊C4—82no—
147C42—24no—
148C42—22no—
149C42—22no—
150C42—82no—
151C42C2×D412yes—
152C42#4112yes—
153C23—22no—
154C23—41no—
155C23—41no—
156C23—41no—
157C23—41no—
158C23—41no—
159C23—41no—
160C23C22×D411yescharacteristic
161Q8—22no—
162Q8—22no—
163Q8—41no—
164Q8—41no—
165Q8—84no—
166D4—22no—
167D4—22no—
168D4—41no—
169D4—41no—
170C2×C4—21no—
171C2×C4—21no—
172C2×C4—22no—
173C2×C4—22no—
174C2×C4—22no—
175C2×C4—22no—
176C2×C4—22no—
177C2×C4—22no—
178C2×C4—22no—
179C2×C4—22no—
180C2×C4—41no—
181C2×C4—41no—
182C2×C4—41no—
183C2×C4—41no—
184C2×C4—41no—
185C2×C4—42no—
186C2×C4—42no—
187C2×C4—42no—
188C2×C4—41no—
189C2×C4—41no—
190C2×C4—42no—
191C2×C4—42no—
192C2×C4—42no—
193C2×C4—84no—
194C2×C4#12112yes—
195C2×C4#12112yes—
196C2×C4#13512yes—
197C2×C4#13712yes—
198C2×C4C22×D411yescharacteristic
199C2×C4C2×#4111yescharacteristic
200C2×C4C2×#4111yescharacteristic
201C2×C4C2×#4111yescharacteristic
202C2×C42−1+411yescharacteristic
203C2×C42−1+411yescharacteristic
204C8—22no—
205C8—42no—
206C8—41no—
207C22—42no—
208C22—42no—
209C22—42no—
210C22—81no—
211C22—81no—
212C22—81no—
213C22—81no—
214C22C2×#12111yescharacteristic
215C22#54711yescharacteristic
216C22#54811yescharacteristic
217C22C2×#13511yescharacteristic
218C22C2×#13711yescharacteristic
219C22#57611yescharacteristic
220C22#57711yescharacteristic
221C4—22no—
222C4—42no—
223C4—42no—
224C4—42no—
225C4—42no—
226C4—82no—
227C4—81no—
228C4—82no—
229C4#46414yes—
230C4#52511yescharacteristic
231C4#53611yescharacteristic
232C4#55211yescharacteristic
233C4#55611yescharacteristic
234C2—82no—
235C2#241511yescharacteristic
236C2C2×#46411yescharacteristic
237C2#304511yescharacteristic
238C2#334011yescharacteristic
239C2#335711yescharacteristic
240C2#336111yescharacteristic
241C2#337211yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC22×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#477C2semidirect
4. #2355C2semidirect
5. #2074C2semidirect
7. #1990C2semidirect
8. #1949C2semidirect
9. #1939C2semidirect

Maximal subgroups

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

subgroupindex
C2×#4772
C2×C4×SD162
#25632
#23552
#20742
#20552
#19902
#19492
#19392
#19252
#18022