Finite Group Databasebeta

← #24187

Subgroups — #24187

Order 256.

559 conjugacy classes of subgroups; only the 147 normal subgroups are shown.

normal subgroups only

This group has 559 conjugacy classes of subgroups — too many to draw. Only its 147 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×C4×SD16C211yesmaximal, characteristic
4#2538C211yesmaximal, characteristic
5#2043C212yesmaximal, complemented
6#2013C212yesmaximal
7#1984C211yesmaximal, characteristic, complemented
8#1956C211yesmaximal, characteristic, complemented
9#1880C212yesmaximal
10#1865C212yesmaximal, complemented
11#1803C211yesmaximal, characteristic, complemented
12C22×SD16C2211yescharacteristic
13C22×SD16C2212yes—
14#557—24no—
15#557—24no—
16C2×C4⋊Q8C2212yes—
17C2×C4⋊Q8C2211yescharacteristic
18C2×#125—21no—
19C2×#125—22no—
20C2×C22⋊Q8—21no—
21C2×C4×Q8—21no—
22C2×C4×Q8C2212yes—
23C2×C4×Q8C2211yescharacteristic
24C2×C4×Q8C2211yescharacteristic
25C2×C4×D4C2211yescharacteristic
26#461—24no—
27C4×SD16—24no—
28C2×#106C2211yescharacteristic
29C2×C4⋊C8C2211yescharacteristic
30C2×#103C2212yes—
31C2×#103C2212yes—
32C2×#103C2211yescharacteristic
33C2×#103C2212yes—
34C2×#103C2212yes—
35C2×#102C2211yescharacteristic
36C2×C4×C8C2211yescharacteristic
37#396—22no—
38#390C2212yes—
39#389C2212yes—
40#384C2211yescharacteristic
41#380—21no—
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—
65C22⋊Q8—44no—
66C4×Q8—28no—
67C4×Q8—24no—
68C4×Q8—24no—
69C4×Q8—22no—
70C4×Q8—22no—
71C4×Q8—22no—
72C4×Q8D414yes—
73C4×D4—22no—
74C4×D4—22no—
75C2×C4⋊C4—22no—
76C2×C4⋊C4—21no—
77C2×C4⋊C4—22no—
78C2×C4⋊C4—21no—
79C2×C4⋊C4—22no—
80C2×C4⋊C4—21no—
81C2×C4⋊C4—21no—
82C2×C4⋊C4—22no—
83C2×C4⋊C4—21no—
84C2×C4⋊C4D411yescharacteristic
85C2×C4⋊C4D412yes—
86C2×C4⋊C4C2311yescharacteristic
87C2×C4⋊C4C2312yes—
88C2×C4⋊C4C2311yescharacteristic
89C2×C22⋊C4—21no—
90C2×C22⋊C4—22no—
91C2×C22⋊C4—22no—
92C2×C42—22no—
93C2×C42—21no—
94C2×C42C2312yes—
95C2×C42C2311yescharacteristic
96#106—22no—
97C4⋊C8—22no—
98#103—22no—
99#103—24no—
100#103—24no—
101#103—42no—
102#103—42no—
103#102—22no—
104C4×C8—22no—
105#95—22no—
106C24—21no—
107C2×Q8—22no—
108C2×Q8—22no—
109C2×Q8—22no—
110C2×Q8—21no—
111C2×Q8—22no—
112C2×Q8—21no—
113C2×Q8—22no—
114C2×Q8—21no—
115C2×Q8—21no—
116C2×Q8—22no—
117C2×Q8—21no—
118C2×Q8—22no—
119C2×Q8SD1614yescomplemented
120C2×Q8C2×D414yes—
121C2×Q8C2×D412yes—
122C2×D4—21no—
123C2×D4—21no—
124C2×D4—21no—
125C2×D4—21no—
126C2×D4—21no—
127C2×D4#4112yes—
128C22×C4—21no—
129C22×C4—22no—
130C22×C4—21no—
131C22×C4—21no—
132C22×C4—22no—
133C22×C4—42no—
134C22×C4—42no—
135C22×C4—42no—
136C22×C4C2×D412yes—
137C22×C4C2×D411yescharacteristic
138C22×C4C2×D412yes—
139C22×C4C2×D411yescharacteristic
140C22×C4C2411yescharacteristic
141SD16—28no—
142SD16—48no—
143C2×C8—21no—
144C2×C8—21no—
145C2×C8—21no—
146C2×C8—22no—
147C2×C8—22no—
148C2×C8—21no—
149C2×C8—22no—
150C2×C8—21no—
151C2×C8#4112yes—
152C4⋊C4—28no—
153C4⋊C4—28no—
154C4⋊C4—28no—
155C4⋊C4—24no—
156C4⋊C4—24no—
157C4⋊C4—24no—
158C4⋊C4—24no—
159C4⋊C4—22no—
160C4⋊C4—21no—
161C4⋊C4—21no—
162C4⋊C4—21no—
163C4⋊C4—22no—
164C4⋊C4—22no—
165C4⋊C4—42no—
166C4⋊C4—44no—
167C4⋊C4—81no—
168C4⋊C4C2×D412yes—
169C22⋊C4—42no—
170C22⋊C4—44no—
171C22⋊C4—82no—
172C42—28no—
173C42—24no—
174C42—22no—
175C42—22no—
176C42C2×D412yes—
177C42C2×D412yes—
178C23—22no—
179C23—41no—
180C23—41no—
181C23—41no—
182C23—41no—
183C23—41no—
184C23—41no—
185C23C22×D411yescharacteristic
186Q8—28no—
187Q8—22no—
188Q8—22no—
189Q8—24no—
190Q8—24no—
191Q8—42no—
192Q8—42no—
193Q8—41no—
194Q8—41no—
195D4—22no—
196D4—22no—
197D4—41no—
198D4—41no—
199C2×C4—22no—
200C2×C4—22no—
201C2×C4—21no—
202C2×C4—21no—
203C2×C4—22no—
204C2×C4—22no—
205C2×C4—24no—
206C2×C4—22no—
207C2×C4—22no—
208C2×C4—22no—
209C2×C4—41no—
210C2×C4—42no—
211C2×C4—42no—
212C2×C4—41no—
213C2×C4—41no—
214C2×C4—41no—
215C2×C4—41no—
216C2×C4—42no—
217C2×C4—42no—
218C2×C4—41no—
219C2×C4—41no—
220C2×C4—42no—
221C2×C4—84no—
222C2×C4#12012yes—
223C2×C4#12012yes—
224C2×C4C2×SD1614yes—
225C2×C4C2×SD1612yes—
226C2×C4#13712yes—
227C2×C4C22×D411yescharacteristic
228C2×C4C22×D411yescharacteristic
229C2×C4C2×#4111yescharacteristic
230C2×C4C2×#4111yescharacteristic
231C2×C4H(5,𝔽2)11yescharacteristic
232C2×C42−1+411yescharacteristic
233C8—22no—
234C8—41no—
235C8—42no—
236C22—42no—
237C22—42no—
238C22—42no—
239C22—81no—
240C22—81no—
241C22—81no—
242C22—81no—
243C22C2×#12011yescharacteristic
244C22#54811yescharacteristic
245C22#55011yescharacteristic
246C22C22×SD1611yescharacteristic
247C22C2×#13711yescharacteristic
248C22#57711yescharacteristic
249C22#57811yescharacteristic
250C4—24no—
251C4—22no—
252C4—42no—
253C4—44no—
254C4—42no—
255C4—42no—
256C4—82no—
257C4—81no—
258C4#44814yes—
259C4#52511yescharacteristic
260C4#53711yescharacteristic
261C4#54611yescharacteristic
262C4#54711yescharacteristic
263C2—82no—
264C2#240711yescharacteristic
265C2C2×#44811yescharacteristic
266C2#301311yescharacteristic
267C2#335911yescharacteristic
268C2#337211yescharacteristic
269C2#339211yescharacteristic
270C2#339811yescharacteristic
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. #1984C2semidirect
8. #1956C2semidirect
10. #1865C2semidirect
11. #1803C2semidirect
119. C2×Q8SD16semidirect

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×C4×SD162
#25382
#20432
#20132
#19842
#19562
#18802
#18652
#18032