Finite Group Databasebeta

← #28121

Subgroups — #28121

Order 256.

497 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 497 conjugacy classes of subgroups — too many to draw. Only its 139 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×#484C211yesmaximal, characteristic
2C2×#451C211yesmaximal, characteristic
3C4×C22⋊Q8C211yesmaximal, characteristic
4C2×#391C211yesmaximal, characteristic
5#2068C212yesmaximal, complemented
6#2056C212yesmaximal, complemented
7#1928C212yesmaximal, complemented
8#1878C212yesmaximal, complemented
9#1824C211yesmaximal, characteristic
10#1797C211yesmaximal, characteristic
11#1787C211yesmaximal, characteristic
12C23×Q8C2211yescharacteristic
13C22×Q16C2212yes—
14C2×C22⋊Q8C2211yescharacteristic
15C2×C4×Q8C2211yescharacteristic
16C22×C4⋊C4C2211yescharacteristic
17C22×C42C2211yescharacteristic
18#484—24no—
19#451—24no—
20C2×#107C2212yes—
21C2×C4⋊C8C2212yes—
22C2×#103C2212yes—
23C2×#103C2212yes—
24C2×#103C2212yes—
25C2×#103C2212yes—
26C2×C22⋊C8C2211yescharacteristic
27C2×C22⋊C8C2211yescharacteristic
28#391—21no—
29#391—21no—
30#391C2211yescharacteristic
31#391C2212yes—
32#391C2212yes—
33#389C2211yescharacteristic
34#387—21no—
35#385—21no—
36#380C2211yescharacteristic
37C4×C4⋊C4—21no—
38C4×C4⋊C4C2211yescharacteristic
39C4×C22⋊C4—21no—
40C2×#95—21no—
41C2×#95—21no—
42#340C2212yes—
43#340C2212yes—
44#336C2212yes—
45C22×Q8—24no—
46C22×Q8—21no—
47C22×Q8—22no—
48C22×Q8—22no—
49C22×Q8—42no—
50C22×Q8C2311yescharacteristic
51C22×Q8C2312yes—
52C23×C4—21no—
53C23×C4—21no—
54C23×C4D411yescharacteristic
55C23×C4D411yescharacteristic
56C23×C4C2311yescharacteristic
57C2×Q16—24no—
58C2×Q16—24no—
59C2×Q16—42no—
60C22×C8C2312yes—
61C22×C8C2312yes—
62C22⋊Q8—22no—
63C22⋊Q8D414yes—
64C4×Q8—42no—
65C2×C4⋊C4—21no—
66C2×C4⋊C4—22no—
67C2×C4⋊C4—22no—
68C2×C4⋊C4—21no—
69C2×C4⋊C4—21no—
70C2×C4⋊C4—21no—
71C2×C4⋊C4C2312yes—
72C2×C4⋊C4C2311yescharacteristic
73C2×C4⋊C4C2311yescharacteristic
74C2×C4⋊C4C2311yescharacteristic
75C2×C22⋊C4—21no—
76C2×C22⋊C4—21no—
77C2×C42—21no—
78C2×C42—21no—
79C2×C42—42no—
80C2×C42D411yescharacteristic
81C2×C42D411yescharacteristic
82C2×C42C2311yescharacteristic
83C2×C42C2311yescharacteristic
84#107—24no—
85C4⋊C8—42no—
86#103—24no—
87#103—24no—
88#103—24no—
89#103—42no—
90C22⋊C8—22no—
91C22⋊C8—22no—
92#95—22no—
93#95—22no—
94#95—22no—
95#95—22no—
96#95—21no—
97#95—21no—
98#95—21no—
99#95—21no—
100#95—21no—
101C24C2×D411yescharacteristic
102C2×Q8—24no—
103C2×Q8—21no—
104C2×Q8—22no—
105C2×Q8—22no—
106C2×Q8—22no—
107C2×Q8—22no—
108C2×Q8—22no—
109C2×Q8—22no—
110C2×Q8—22no—
111C2×Q8—48no—
112C2×Q8—48no—
113C2×Q8—42no—
114C2×Q8—42no—
115C2×Q8C2×D412yes—
116C2×Q8#4114yes—
117C22×C4—22no—
118C22×C4—22no—
119C22×C4—24no—
120C22×C4—22no—
121C22×C4—22no—
122C22×C4—21no—
123C22×C4—21no—
124C22×C4—21no—
125C22×C4—22no—
126C22×C4—21no—
127C22×C4—41no—
128C22×C4—41no—
129C22×C4—41no—
130C22×C4—41no—
131C22×C4—42no—
132C22×C4—42no—
133C22×C4—42no—
134C22×C4—42no—
135C22×C4—42no—
136C22×C4Q1614yescomplemented
137C22×C4C2×D411yescharacteristic
138C22×C4C2×D411yescharacteristic
139C22×C4C2×D411yescharacteristic
140C22×C4C2×D412yes—
141C22×C4C2×D411yescharacteristic
142C22×C4C2×D411yescharacteristic
143C22×C4#4112yes—
144C22×C4C2411yescharacteristic
145Q16—48no—
146C2×C8—22no—
147C2×C8—22no—
148C2×C8—22no—
149C2×C8—22no—
150C2×C8—22no—
151C2×C8—22no—
152C4⋊C4—24no—
153C4⋊C4—21no—
154C4⋊C4—22no—
155C4⋊C4—22no—
156C4⋊C4—22no—
157C4⋊C4—22no—
158C4⋊C4—42no—
159C4⋊C4—42no—
160C4⋊C4—42no—
161C4⋊C4C2×D412yes—
162C22⋊C4—24no—
163C22⋊C4—42no—
164C42—22no—
165C42—22no—
166C42—41no—
167C42—41no—
168C42—42no—
169C42—42no—
170C23—21no—
171C23—21no—
172C23—21no—
173C23—21no—
174C23#12112yes—
175C23C2×Q1612yes—
176C23#13712yes—
177C23C22×D411yescharacteristic
178Q8—24no—
179Q8—24no—
180Q8—24no—
181Q8—48no—
182Q8—42no—
183Q8—42no—
184Q8—42no—
185C2×C4—21no—
186C2×C4—21no—
187C2×C4—21no—
188C2×C4—21no—
189C2×C4—21no—
190C2×C4—21no—
191C2×C4—24no—
192C2×C4—24no—
193C2×C4—22no—
194C2×C4—22no—
195C2×C4—24no—
196C2×C4—21no—
197C2×C4—21no—
198C2×C4—21no—
199C2×C4—21no—
200C2×C4—21no—
201C2×C4—21no—
202C2×C4—44no—
203C2×C4—44no—
204C2×C4—48no—
205C2×C4—42no—
206C2×C4—41no—
207C2×C4—41no—
208C2×C4—42no—
209C2×C4—41no—
210C2×C4—41no—
211C2×C4—42no—
212C2×C4—41no—
213C2×C4—41no—
214C2×C4—41no—
215C2×C4—41no—
216C2×C4—41no—
217C2×C4—41no—
218C2×C4—42no—
219C2×C4—42no—
220C2×C4—84no—
221C2×C4#12112yes—
222C2×C4C2×Q1612yes—
223C2×C4C2×Q1612yes—
224C2×C4C22×D411yescharacteristic
225C2×C4C2×#4112yes—
226C2×C4C2×#4111yescharacteristic
227C2×C4H(5,𝔽2)12yes—
228C8—42no—
229C8—42no—
230C22—22no—
231C22—22no—
232C22—22no—
233C22—22no—
234C22—22no—
235C22—22no—
236C22#46214yes—
237C22C2×#12111yescharacteristic
238C22#52511yescharacteristic
239C22#53411yescharacteristic
240C22C22×Q1611yescharacteristic
241C22C2×#13711yescharacteristic
242C22#57511yescharacteristic
243C22#57511yescharacteristic
244C4—22no—
245C4—22no—
246C4—22no—
247C4—41no—
248C4—41no—
249C4—42no—
250C4—44no—
251C4—42no—
252C4—81no—
253C4—81no—
254C4—82no—
255C4#54612yes—
256C4#55112yes—
257C2—24no—
258C2#253411yescharacteristic
259C2C2×#46211yescharacteristic
260C2#304011yescharacteristic
261C2#310411yescharacteristic
262C2#311711yescharacteristic
263C2#318911yescharacteristic
264C2#319611yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC24C2×D4
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
5. #2068C2semidirect
6. #2056C2semidirect
7. #1928C2semidirect
8. #1878C2semidirect
136. C22×C4Q16semidirect

Maximal subgroups

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

subgroupindex
C2×#4842
C2×#4512
C4×C22⋊Q82
C2×#3912
#20682
#20562
#19282
#18782
#18242
#17972
#17872