Finite Group Databasebeta

← #24600

Subgroups — #24600

Order 256.

471 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 471 conjugacy classes of subgroups — too many to draw. Only its 141 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×#480C211yesmaximal, characteristic
2C2×#431C211yesmaximal, characteristic, complemented
3#2854C211yesmaximal, characteristic
4C2×#389C211yesmaximal, characteristic, complemented
5#2086C212yesmaximal, complemented
6#2067C212yesmaximal, complemented
7#1942C212yesmaximal, complemented
8#1934C212yesmaximal, complemented
9#1874C211yesmaximal, characteristic
10#1852C211yesmaximal, characteristic, complemented
11#1823C211yesmaximal, characteristic, complemented
12C2×#123—21no—
13C2×#121C2211yescharacteristic
14C22×C4⋊C4—21no—
15C22×C4⋊C4C2211yescharacteristic
16C22×C4⋊C4C2211yescharacteristic
17C22×C42C2211yescharacteristic
18#480—24no—
19#431—24no—
20C2×#107C2212yes—
21C2×#107C2212yes—
22C2×C4⋊C8C2212yes—
23C2×#102C2212yes—
24C2×#102C2212yes—
25C2×C22⋊C8C2211yescharacteristic
26C2×C22⋊C8C2211yescharacteristic
27C2×C4×C8C2212yes—
28#390C2211yescharacteristic
29#390C2211yescharacteristic
30#389—21no—
31#389—21no—
32#389C2212yes—
33#389C2211yescharacteristic
34#389C2211yescharacteristic
35#389C2212yes—
36#388—21no—
37#388—21no—
38#386—21no—
39#380C2211yescharacteristic
40C2×#95—21no—
41#340C2212yes—
42#340C2212yes—
43#340C2212yes—
44#340C2212yes—
45C22×D4—21no—
46C22×D4C2311yescharacteristic
47C23×C4—21no—
48C23×C4—21no—
49C23×C4D411yescharacteristic
50C23×C4D411yescharacteristic
51C23×C4C2311yescharacteristic
52C22×C8C2312yes—
53C22×C8C2312yes—
54#123—44no—
55#121—22no—
56#121—22no—
57C2×C4⋊C4—21no—
58C2×C4⋊C4—24no—
59C2×C4⋊C4—22no—
60C2×C4⋊C4—22no—
61C2×C4⋊C4—22no—
62C2×C4⋊C4—22no—
63C2×C4⋊C4—21no—
64C2×C4⋊C4—21no—
65C2×C4⋊C4—21no—
66C2×C4⋊C4—44no—
67C2×C4⋊C4C2312yes—
68C2×C4⋊C4C2311yescharacteristic
69C2×C4⋊C4C2311yescharacteristic
70C2×C4⋊C4C2311yescharacteristic
71C2×C4⋊C4C2312yes—
72C2×C22⋊C4—21no—
73C2×C22⋊C4—21no—
74C2×C22⋊C4—21no—
75C2×C22⋊C4—21no—
76C2×C22⋊C4—21no—
77C2×C22⋊C4—21no—
78C2×C42—22no—
79C2×C42—22no—
80C2×C42D411yescharacteristic
81C2×C42D411yescharacteristic
82C2×C42C2311yescharacteristic
83C2×C42C2311yescharacteristic
84#107—24no—
85#107—24no—
86C4⋊C8—24no—
87#102—24no—
88#102—24no—
89C22⋊C8—22no—
90C22⋊C8D414yes—
91C4×C8—24no—
92#95—22no—
93#95—22no—
94#95—21no—
95#95—21no—
96#95—21no—
97C24—21no—
98C24C2×D411yescharacteristic
99C2×D4—21no—
100C2×D4—22no—
101C2×D4—21no—
102C2×D4—22no—
103C2×D4—42no—
104C2×D4—42no—
105C2×D4—42no—
106C22×C4—22no—
107C22×C4—22no—
108C22×C4—22no—
109C22×C4—22no—
110C22×C4—24no—
111C22×C4—22no—
112C22×C4—22no—
113C22×C4—22no—
114C22×C4—21no—
115C22×C4—21no—
116C22×C4—21no—
117C22×C4—22no—
118C22×C4—41no—
119C22×C4—42no—
120C22×C4—42no—
121C22×C4—42no—
122C22×C4—42no—
123C22×C4—42no—
124C22×C4D814yes—
125C22×C4C2×D411yescharacteristic
126C22×C4C2×D411yescharacteristic
127C22×C4C2×D412yes—
128C22×C4C2×D411yescharacteristic
129C22×C4C2×D411yescharacteristic
130C22×C4C2×D411yescharacteristic
131C22×C4#4112yes—
132C22×C4C2411yescharacteristic
133C2×C8—22no—
134C2×C8—22no—
135C2×C8—22no—
136C2×C8—22no—
137C2×C8—22no—
138C2×C8C2×D414yes—
139C4⋊C4—24no—
140C4⋊C4—22no—
141C4⋊C4—22no—
142C4⋊C4—21no—
143C4⋊C4—22no—
144C4⋊C4—22no—
145C4⋊C4—21no—
146C4⋊C4—22no—
147C4⋊C4—42no—
148C4⋊C4—44no—
149C4⋊C4—44no—
150C4⋊C4—42no—
151C4⋊C4—42no—
152C4⋊C4#4114yes—
153C22⋊C4—42no—
154C22⋊C4—42no—
155C22⋊C4—42no—
156C22⋊C4—42no—
157C22⋊C4—42no—
158C22⋊C4—42no—
159C42—22no—
160C42—22no—
161C42—21no—
162C42—21no—
163C42—21no—
164C42—21no—
165C23—21no—
166C23—21no—
167C23—21no—
168C23—21no—
169C23—41no—
170C23—41no—
171C23—41no—
172C23—41no—
173C23—41no—
174C23—41no—
175C23—41no—
176C23#12112yes—
177C23C2×D812yes—
178C23#13512yes—
179C23C22×D411yescharacteristic
180D4—42no—
181D4—42no—
182D4—84no—
183C2×C4—21no—
184C2×C4—21no—
185C2×C4—21no—
186C2×C4—21no—
187C2×C4—21no—
188C2×C4—21no—
189C2×C4—21no—
190C2×C4—21no—
191C2×C4—28no—
192C2×C4—24no—
193C2×C4—24no—
194C2×C4—21no—
195C2×C4—24no—
196C2×C4—21no—
197C2×C4—21no—
198C2×C4—44no—
199C2×C4—48no—
200C2×C4—42no—
201C2×C4—42no—
202C2×C4—41no—
203C2×C4—42no—
204C2×C4—41no—
205C2×C4—41no—
206C2×C4—41no—
207C2×C4—41no—
208C2×C4—42no—
209C2×C4—41no—
210C2×C4—42no—
211C2×C4—41no—
212C2×C4—41no—
213C2×C4—41no—
214C2×C4—84no—
215C2×C4#12112yes—
216C2×C4C2×D812yes—
217C2×C4C2×D812yes—
218C2×C4#13712yes—
219C2×C4C22×D411yescharacteristic
220C2×C4C2×#4112yes—
221C2×C4C2×#4111yescharacteristic
222C2×C42−1+412yes—
223C8—24no—
224C8—42no—
225C22—22no—
226C22—22no—
227C22—22no—
228C22—22no—
229C22—22no—
230C22—22no—
231C22—81no—
232C22—81no—
233C22—81no—
234C22—81no—
235C22—81no—
236C22—81no—
237C22—81no—
238C22#46614yes—
239C22C2×#12111yescharacteristic
240C22#52511yescharacteristic
241C22#53611yescharacteristic
242C22C22×D811yescharacteristic
243C22C2×#13511yescharacteristic
244C22C2×#13711yescharacteristic
245C22#57511yescharacteristic
246C4—22no—
247C4—22no—
248C4—22no—
249C4—22no—
250C4—44no—
251C4—41no—
252C4—81no—
253C4—81no—
254C4—82no—
255C4—81no—
256C4#54712yes—
257C4#55212yes—
258C2—24no—
259C2—161no—
260C2#241311yescharacteristic
261C2C2×#46611yescharacteristic
262C2#305611yescharacteristic
263C2#310011yescharacteristic
264C2#310811yescharacteristic
265C2#322611yescharacteristic
266C2#323211yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC2×#121C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#431C2semidirect
4. C2×#389C2semidirect
5. #2086C2semidirect
6. #2067C2semidirect
7. #1942C2semidirect
8. #1934C2semidirect
10. #1852C2semidirect
11. #1823C2semidirect

Maximal subgroups

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

subgroupindex
C2×#4802
C2×#4312
#28542
C2×#3892
#20862
#20672
#19422
#19342
#18742
#18522
#18232