Finite Group Databasebeta

← #31192

Subgroups — #31192

Order 256.

460 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 460 conjugacy classes of subgroups — too many to draw. Only its 117 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×#482C212yesmaximal, complemented
2#2414C212yesmaximal, complemented
3#2065C212yesmaximal, complemented
4#2033C212yesmaximal, complemented
5#1999C212yesmaximal, complemented
6#1970C211yesmaximal, characteristic, complemented
7#1954C211yesmaximal, characteristic, complemented
8#1942C211yesmaximal, characteristic, complemented
9#1854C212yesmaximal, complemented
10C22×D8C2211yescharacteristic
11C2×#121C2212yes—
12C2×#121C2212yes—
13C2×C4×D4C2211yescharacteristic
14C2×#117C2212yes—
15#482—28no—
16C2×#107C2211yescharacteristic
17C2×#107C2211yescharacteristic
18C2×#107C2211yescharacteristic
19C2×#106C2212yes—
20C2×C4⋊C8C2211yescharacteristic
21C2×#102C2212yes—
22C2×#102C2212yes—
23C2×#102C2212yes—
24C2×C22⋊C8C2212yes—
25C2×C22⋊C8C2212yes—
26C2×C4×C8C2211yescharacteristic
27#394—22no—
28#392—22no—
29#390—22no—
30#390C2212yes—
31#389C2212yes—
32#388—22no—
33#380C2212yes—
34#340C2212yes—
35#340C2212yes—
36C22×D4—22no—
37C22×D4—22no—
38C22×D4D411yescharacteristic
39C22×D4C2312yes—
40C23×C4C2312yes—
41C2×D8—22no—
42C2×D8—41no—
43C2×D8—41no—
44C22×C8C2312yes—
45C22×C8C2311yescharacteristic
46C22×C8C2311yescharacteristic
47#121—24no—
48#121—42no—
49#121—42no—
50#121—42no—
51C4×D4—41no—
52C4×D4—41no—
53#117—24no—
54#117—42no—
55C2×C4⋊C4—22no—
56C2×C4⋊C4D411yescharacteristic
57C2×C4⋊C4C2312yes—
58C2×C4⋊C4C2312yes—
59C2×C4⋊C4C2312yes—
60C2×C22⋊C4—22no—
61C2×C22⋊C4—22no—
62C2×C22⋊C4—22no—
63C2×C22⋊C4—22no—
64C2×C22⋊C4—22no—
65C2×C22⋊C4—22no—
66C2×C22⋊C4D412yes—
67C2×C42—22no—
68C2×C42C2311yescharacteristic
69#107—41no—
70#107—41no—
71#107D414yescomplemented
72#106—24no—
73C4⋊C8—41no—
74#102—24no—
75#102—24no—
76#102—42no—
77C22⋊C8—24no—
78C22⋊C8—24no—
79C4×C8—41no—
80#95—22no—
81#95—22no—
82C24—22no—
83C24C2×D412yes—
84C2×D4—22no—
85C2×D4—22no—
86C2×D4—22no—
87C2×D4—22no—
88C2×D4—22no—
89C2×D4—22no—
90C2×D4—22no—
91C2×D4—41no—
92C2×D4—41no—
93C2×D4—44no—
94C2×D4—44no—
95C2×D4—44no—
96C2×D4—82no—
97C2×D4—82no—
98C2×D4—82no—
99C22×C4—22no—
100C22×C4—22no—
101C22×C4—22no—
102C22×C4—22no—
103C22×C4—22no—
104C22×C4—22no—
105C22×C4—22no—
106C22×C4—22no—
107C22×C4—22no—
108C22×C4C2×D412yes—
109C22×C4C2×D411yescharacteristic
110C22×C4C2×D411yescharacteristic
111C22×C4C2411yescharacteristic
112D8—44no—
113C2×C8—21no—
114C2×C8—21no—
115C2×C8—21no—
116C2×C8—21no—
117C2×C8—22no—
118C2×C8—22no—
119C2×C8—22no—
120C2×C8C2×D412yes—
121C2×C8#4112yes—
122C4⋊C4—22no—
123C4⋊C4—22no—
124C4⋊C4—22no—
125C4⋊C4—22no—
126C4⋊C4—22no—
127C4⋊C4—41no—
128C4⋊C4—82no—
129C4⋊C4C2×D414yes—
130C22⋊C4—42no—
131C22⋊C4—44no—
132C22⋊C4—44no—
133C22⋊C4—44no—
134C22⋊C4—44no—
135C22⋊C4—82no—
136C22⋊C4—82no—
137C42—21no—
138C42—21no—
139C42—44no—
140C23—22no—
141C23—22no—
142C23—22no—
143C23—22no—
144C23—22no—
145C23—22no—
146C23—22no—
147C23—42no—
148C23—42no—
149C23—42no—
150C23—42no—
151C23—42no—
152C23—42no—
153C23—42no—
154C23C22×D411yescharacteristic
155D4—44no—
156D4—42no—
157D4—42no—
158D4—42no—
159D4—42no—
160D4—88no—
161D4—88no—
162C2×C4—21no—
163C2×C4—21no—
164C2×C4—21no—
165C2×C4—22no—
166C2×C4—22no—
167C2×C4—24no—
168C2×C4—22no—
169C2×C4—21no—
170C2×C4—21no—
171C2×C4—42no—
172C2×C4—42no—
173C2×C4—42no—
174C2×C4—42no—
175C2×C4—42no—
176C2×C4—42no—
177C2×C4—42no—
178C2×C4—42no—
179C2×C4—42no—
180C2×C4—42no—
181C2×C4—42no—
182C2×C4—42no—
183C2×C4#13512yes—
184C2×C4C22×D411yescharacteristic
185C2×C4C2×#4111yescharacteristic
186C2×C4H(5,𝔽2)11yescharacteristic
187C2×C4H(5,𝔽2)11yescharacteristic
188C2×C4H(5,𝔽2)11yescharacteristic
189C2×C42−1+411yescharacteristic
190C8—22no—
191C8—22no—
192C8—42no—
193C22—42no—
194C22—42no—
195C22—42no—
196C22—42no—
197C22—42no—
198C22—42no—
199C22—42no—
200C22—82no—
201C22—82no—
202C22—82no—
203C22—82no—
204C22—82no—
205C22—82no—
206C22—82no—
207C22#53411yescharacteristic
208C22D4211yescharacteristic
209C22#54711yescharacteristic
210C22C2×#13511yescharacteristic
211C22#57511yescharacteristic
212C22#57611yescharacteristic
213C22#57611yescharacteristic
214C4—22no—
215C4—42no—
216C4—41no—
217C4—44no—
218C4—82no—
219C4—82no—
220C4#53711yescharacteristic
221C4#53811yescharacteristic
222C4#55011yescharacteristic
223C4#56211yescharacteristic
224C2—82no—
225C2—162no—
226C2#268211yescharacteristic
227C2#319111yescharacteristic
228C2#319911yescharacteristic
229C2#327311yescharacteristic
230C2#328711yescharacteristic
231C2#334111yescharacteristic
232C2#335511yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#482C2semidirect
2. #2414C2semidirect
3. #2065C2semidirect
4. #2033C2semidirect
5. #1999C2semidirect
6. #1970C2semidirect
7. #1954C2semidirect
8. #1942C2semidirect
9. #1854C2semidirect
71. #107D4semidirect

Maximal subgroups

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

subgroupindex
C2×#4822
#24142
#20652
#20332
#19992
#19702
#19542
#19422
#18542