Finite Group Databasebeta

← #24199

Subgroups — #24199

Order 256.

542 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 542 conjugacy classes of subgroups — too many to draw. Only its 135 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×#557C211yesmaximal, characteristic, complemented
2C2×#461C211yesmaximal, characteristic
3C2×#440C211yesmaximal, characteristic
4#2580C211yesmaximal, characteristic
5#2043C212yesmaximal, complemented
6#2013C212yesmaximal
7#1985C211yesmaximal, characteristic, complemented
8#1960C211yesmaximal, characteristic, complemented
9#1880C212yesmaximal
10#1863C212yesmaximal, complemented
11#1805C211yesmaximal, characteristic, complemented
12C22×SD16C2211yescharacteristic
13C22×SD16C2212yes—
14#557—24no—
15#557—24no—
16C2×C4⋊Q8C2212yes—
17C2×C4⋊Q8C2211yescharacteristic
18C2×#125—22no—
19C2×#125—21no—
20C2×C4×Q8—21no—
21C2×C4×Q8C2212yes—
22C2×C4×Q8C2211yescharacteristic
23C2×C4×Q8C2211yescharacteristic
24C2×C4×D4C2211yescharacteristic
25#461—24no—
26#440—24no—
27C2×#107C2211yescharacteristic
28C2×C4⋊C8C2211yescharacteristic
29C2×#103C2212yes—
30C2×#103C2212yes—
31C2×#103C2211yescharacteristic
32C2×#103C2212yes—
33C2×#103C2212yes—
34C2×#102C2211yescharacteristic
35C2×#97C2211yescharacteristic
36#399—21no—
37#393—21no—
38#390C2212yes—
39#389C2212yes—
40#385—22no—
41#382C2211yescharacteristic
42#340C2212yes—
43#336C2212yes—
44#336C2212yes—
45C22×Q8D411yescharacteristic
46C22×Q8C2312yes—
47C22×Q8C2311yescharacteristic
48C22×D4C2311yescharacteristic
49C23×C4—21no—
50C2×SD16—22no—
51C2×SD16—22no—
52C2×SD16—24no—
53C2×SD16—24no—
54C2×SD16—22no—
55C2×SD16—42no—
56C22×C8C2312yes—
57C22×C8C2311yescharacteristic
58C22×C8C2311yescharacteristic
59C4⋊Q8—24no—
60C4⋊Q8—24no—
61C4⋊Q8—22no—
62C4⋊Q8D414yes—
63#125—216no—
64#125—28no—
65C4×Q8—28no—
66C4×Q8—24no—
67C4×Q8—24no—
68C4×Q8—22no—
69C4×Q8—22no—
70C4×Q8—22no—
71C4×Q8D414yes—
72C4×D4—22no—
73C4×D4—22no—
74C2×C4⋊C4—22no—
75C2×C4⋊C4—21no—
76C2×C4⋊C4—22no—
77C2×C4⋊C4—21no—
78C2×C4⋊C4—22no—
79C2×C4⋊C4—21no—
80C2×C4⋊C4—22no—
81C2×C4⋊C4D412yes—
82C2×C4⋊C4D411yescharacteristic
83C2×C4⋊C4C2311yescharacteristic
84C2×C4⋊C4C2312yes—
85C2×C4⋊C4C2311yescharacteristic
86C2×C22⋊C4—21no—
87C2×C22⋊C4—22no—
88C2×C22⋊C4—22no—
89C2×C42—22no—
90C2×C42—21no—
91C2×C42C2312yes—
92C2×C42C2311yescharacteristic
93#107—22no—
94C4⋊C8—22no—
95#103—22no—
96#103—24no—
97#103—24no—
98#103—42no—
99#103—42no—
100#102—22no—
101#97—22no—
102#95—21no—
103#95—21no—
104#95—22no—
105C24—21no—
106C2×Q8—22no—
107C2×Q8—24no—
108C2×Q8—22no—
109C2×Q8—21no—
110C2×Q8—22no—
111C2×Q8—21no—
112C2×Q8—22no—
113C2×Q8—21no—
114C2×Q8—21no—
115C2×Q8—22no—
116C2×Q8—21no—
117C2×Q8—22no—
118C2×Q8C2×D414yes—
119C2×Q8C2×D412yes—
120C2×D4—21no—
121C2×D4—21no—
122C2×D4—21no—
123C2×D4—21no—
124C2×D4—21no—
125C2×D4#4112yes—
126C22×C4—21no—
127C22×C4—22no—
128C22×C4—21no—
129C22×C4—22no—
130C22×C4—21no—
131C22×C4—42no—
132C22×C4—82no—
133C22×C4C2×D412yes—
134C22×C4C2×D411yescharacteristic
135C22×C4C2×D411yescharacteristic
136C22×C4C2×D412yes—
137C22×C4C2411yescharacteristic
138SD16—28no—
139SD16—48no—
140C2×C8—21no—
141C2×C8—21no—
142C2×C8—21no—
143C2×C8—22no—
144C2×C8—22no—
145C2×C8—21no—
146C2×C8—22no—
147C2×C8—21no—
148C2×C8#4112yes—
149C4⋊C4—28no—
150C4⋊C4—28no—
151C4⋊C4—28no—
152C4⋊C4—24no—
153C4⋊C4—24no—
154C4⋊C4—24no—
155C4⋊C4—24no—
156C4⋊C4—22no—
157C4⋊C4—21no—
158C4⋊C4—21no—
159C4⋊C4—21no—
160C4⋊C4—22no—
161C4⋊C4—22no—
162C4⋊C4—82no—
163C4⋊C4C2×D412yes—
164C22⋊C4—42no—
165C22⋊C4—44no—
166C22⋊C4—82no—
167C42—28no—
168C42—24no—
169C42—22no—
170C42—22no—
171C42C2×D412yes—
172C42C2×D412yes—
173C23—22no—
174C23—41no—
175C23—41no—
176C23—42no—
177C23—42no—
178C23C22×D411yescharacteristic
179Q8—28no—
180Q8—22no—
181Q8—22no—
182Q8—24no—
183Q8—24no—
184Q8—42no—
185Q8—42no—
186Q8—41no—
187Q8—41no—
188D4—22no—
189D4—22no—
190D4—41no—
191D4—41no—
192C2×C4—22no—
193C2×C4—22no—
194C2×C4—22no—
195C2×C4—22no—
196C2×C4—22no—
197C2×C4—22no—
198C2×C4—24no—
199C2×C4—22no—
200C2×C4—22no—
201C2×C4—22no—
202C2×C4—41no—
203C2×C4—42no—
204C2×C4—42no—
205C2×C4—41no—
206C2×C4—42no—
207C2×C4—41no—
208C2×C4—42no—
209C2×C4—42no—
210C2×C4—42no—
211C2×C4—42no—
212C2×C4—84no—
213C2×C4#12012yes—
214C2×C4#12012yes—
215C2×C4#13714yes—
216C2×C4C22×D411yescharacteristic
217C2×C4C22×D411yescharacteristic
218C2×C4C2×#4111yescharacteristic
219C2×C4C2×#4111yescharacteristic
220C2×C4H(5,𝔽2)11yescharacteristic
221C2×C42−1+411yescharacteristic
222C8—22no—
223C8—41no—
224C8—42no—
225C22—44no—
226C22—42no—
227C22—82no—
228C22—81no—
229C22—81no—
230C22C2×#12011yescharacteristic
231C22#54811yescharacteristic
232C22#55011yescharacteristic
233C22C2×#13712yes—
234C22#57712yes—
235C4—22no—
236C4—24no—
237C4—22no—
238C4—42no—
239C4—44no—
240C4—42no—
241C4—42no—
242C4—82no—
243C4—81no—
244C4#52511yescharacteristic
245C4#53711yescharacteristic
246C4#54611yescharacteristic
247C4#54711yescharacteristic
248C2—82no—
249C2#240711yescharacteristic
250C2#300411yescharacteristic
251C2#301011yescharacteristic
252C2#337212yes—
253C2#340012yes—
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC22×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#557C2semidirect
5. #2043C2semidirect
7. #1985C2semidirect
8. #1960C2semidirect
10. #1863C2semidirect
11. #1805C2semidirect

Maximal subgroups

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

subgroupindex
C2×#5572
C2×#4612
C2×#4402
#25802
#20432
#20132
#19852
#19602
#18802
#18632
#18052