Finite Group Databasebeta

← #29646

Subgroups — #29646

Order 256.

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

normal subgroups only

This group has 495 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
1C2×#544C211yesmaximal, characteristic, complemented
2C2×#482C211yesmaximal, characteristic
3C2×#480C211yesmaximal, characteristic
4#2624C211yesmaximal, characteristic
5#2059C212yesmaximal, complemented
6#2057C212yesmaximal, complemented
7#2025C212yesmaximal, complemented
8#2023C212yesmaximal, complemented
9#1825C211yesmaximal, characteristic, complemented
10#1824C211yesmaximal, characteristic, complemented
11#1818C211yesmaximal, characteristic, complemented
12#544—24no—
13#544—24no—
14C2×C4⋊Q8—21no—
15C2×C4⋊Q8C2212yes—
16C2×#125—21no—
17C2×#125C2212yes—
18C2×C22⋊Q8—22no—
19C2×#121C2211yescharacteristic
20C2×#117C2211yescharacteristic
21C2×#117C2211yescharacteristic
22C22×C4⋊C4C2211yescharacteristic
23#482—24no—
24#480—24no—
25C2×#107C2211yescharacteristic
26C2×#107C2212yes—
27C2×#106C2211yescharacteristic
28C2×#102C2212yes—
29C2×#102C2211yescharacteristic
30C2×#102C2211yescharacteristic
31C2×C22⋊C8C2211yescharacteristic
32C2×C22⋊C8C2211yescharacteristic
33#399—21no—
34#397—21no—
35#394—22no—
36#390C2212yes—
37#389C2212yes—
38#380C2211yescharacteristic
39#340C2212yes—
40#340C2212yes—
41#340C2212yes—
42#340C2212yes—
43#336C2212yes—
44#336C2212yes—
45C22×Q8—22no—
46C22×D4—21no—
47C22×D4C2311yescharacteristic
48C23×C4—21no—
49C23×C4C2311yescharacteristic
50C22×C8C2311yescharacteristic
51C22×C8C2311yescharacteristic
52C22×C8C2312yes—
53C4⋊Q8—28no—
54C4⋊Q8—24no—
55C4⋊Q8—24no—
56#125—28no—
57#125—24no—
58#125—24no—
59C22⋊Q8—216no—
60#121—22no—
61#121—22no—
62#117—22no—
63#117—22no—
64#117—22no—
65#117D414yes—
66C2×C4⋊C4—21no—
67C2×C4⋊C4—22no—
68C2×C4⋊C4—22no—
69C2×C4⋊C4—22no—
70C2×C4⋊C4—22no—
71C2×C4⋊C4—22no—
72C2×C4⋊C4D411yescharacteristic
73C2×C4⋊C4D411yescharacteristic
74C2×C4⋊C4D414yes—
75C2×C4⋊C4C2311yescharacteristic
76C2×C4⋊C4C2312yes—
77C2×C4⋊C4C2312yes—
78C2×C4⋊C4C2311yescharacteristic
79C2×C4⋊C4C2311yescharacteristic
80C2×C22⋊C4—21no—
81C2×C22⋊C4—21no—
82C2×C22⋊C4—21no—
83C2×C22⋊C4—22no—
84C2×C22⋊C4—22no—
85C2×C22⋊C4D412yes—
86C2×C42—21no—
87C2×C42C2312yes—
88#107—24no—
89#107—22no—
90#106—22no—
91#102—24no—
92#102—22no—
93#102—22no—
94C22⋊C8—22no—
95C22⋊C8—22no—
96#95—21no—
97#95—22no—
98C24—21no—
99C24C2×D411yescharacteristic
100C2×Q8—28no—
101C2×Q8—82no—
102C2×Q8—82no—
103C2×D4—21no—
104C2×D4—21no—
105C2×D4—22no—
106C2×D4—22no—
107C2×D4—42no—
108C2×D4—82no—
109C22×C4—22no—
110C22×C4—22no—
111C22×C4—21no—
112C22×C4—21no—
113C22×C4—22no—
114C22×C4—24no—
115C22×C4—22no—
116C22×C4—21no—
117C22×C4—82no—
118C22×C4C2×D412yes—
119C22×C4C2×D412yes—
120C22×C4C2×D411yescharacteristic
121C22×C4C2×D412yes—
122C22×C4C2×D412yes—
123C22×C4C2411yescharacteristic
124C2×C8—21no—
125C2×C8—22no—
126C2×C8—21no—
127C2×C8—22no—
128C2×C8—22no—
129C2×C8—22no—
130C2×C8—22no—
131C4⋊C4—28no—
132C4⋊C4—28no—
133C4⋊C4—28no—
134C4⋊C4—24no—
135C4⋊C4—22no—
136C4⋊C4—22no—
137C4⋊C4—22no—
138C4⋊C4—21no—
139C4⋊C4—21no—
140C4⋊C4—21no—
141C4⋊C4—21no—
142C4⋊C4—22no—
143C4⋊C4—22no—
144C4⋊C4—82no—
145C4⋊C4C2×D412yes—
146C4⋊C4C2×D414yes—
147C4⋊C4C2×D412yes—
148C22⋊C4—24no—
149C22⋊C4—24no—
150C22⋊C4—42no—
151C22⋊C4—82no—
152C22⋊C4—82no—
153C22⋊C4—81no—
154C42—24no—
155C42—22no—
156C42—22no—
157C23—22no—
158C23—22no—
159C23—21no—
160C23—21no—
161C23—41no—
162C23—42no—
163C23—42no—
164C23—41no—
165C23—41no—
166C23#12012yes—
167C23C22×D411yescharacteristic
168Q8—88no—
169D4—42no—
170D4—41no—
171D4—41no—
172D4—84no—
173C2×C4—22no—
174C2×C4—22no—
175C2×C4—22no—
176C2×C4—24no—
177C2×C4—22no—
178C2×C4—22no—
179C2×C4—22no—
180C2×C4—22no—
181C2×C4—24no—
182C2×C4—22no—
183C2×C4—22no—
184C2×C4—22no—
185C2×C4—22no—
186C2×C4—42no—
187C2×C4—42no—
188C2×C4—42no—
189C2×C4—41no—
190C2×C4—41no—
191C2×C4—41no—
192C2×C4—42no—
193C2×C4—42no—
194C2×C4—42no—
195C2×C4—42no—
196C2×C4—84no—
197C2×C4#12012yes—
198C2×C4C22×D411yescharacteristic
199C2×C4C22×D411yescharacteristic
200C2×C42−1+412yes—
201C2×C42−1+412yes—
202C8—41no—
203C8—42no—
204C8—41no—
205C22—22no—
206C22—22no—
207C22—22no—
208C22—42no—
209C22—42no—
210C22—81no—
211C22—82no—
212C22—82no—
213C22—81no—
214C22—81no—
215C22C2×#12011yescharacteristic
216C22#53612yes—
217C22#57612yes—
218C22#57711yescharacteristic
219C22#57811yescharacteristic
220C4—22no—
221C4—42no—
222C4—44no—
223C4—42no—
224C4—42no—
225C4—42no—
226C4—82no—
227C4—81no—
228C4#53611yescharacteristic
229C4#53611yescharacteristic
230C4#53612yes—
231C2—42no—
232C2—161no—
233C2#261411yescharacteristic
234C2#300811yescharacteristic
235C2#301311yescharacteristic
236C2#324112yes—
237C2#324312yes—
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC2×#121C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#544C2semidirect
5. #2059C2semidirect
6. #2057C2semidirect
7. #2025C2semidirect
8. #2023C2semidirect
9. #1825C2semidirect
10. #1824C2semidirect
11. #1818C2semidirect

Maximal subgroups

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

subgroupindex
C2×#5442
C2×#4822
C2×#4802
#26242
#20592
#20572
#20252
#20232
#18252
#18242
#18182