Finite Group Databasebeta

← #24155

Subgroups — #24155

Order 256.

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

normal subgroups only

This group has 557 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×#533C211yesmaximal, characteristic
2C2×#477C211yesmaximal, characteristic, complemented
3C2×#462C211yesmaximal, characteristic, complemented
4C4×C22⋊Q8C211yesmaximal, characteristic
5#2036C212yesmaximal
6#2001C212yesmaximal
7#1987C211yesmaximal, characteristic, complemented
8#1981C211yesmaximal, characteristic, complemented
9#1878C212yesmaximal, complemented
10#1863C212yesmaximal, complemented
11#1797C211yesmaximal, characteristic
12C22×Q16C2212yes—
13#533—24no—
14#533—24no—
15C2×C4⋊Q8C2211yescharacteristic
16C2×C4⋊Q8C2211yescharacteristic
17C2×#125—21no—
18C2×C22⋊Q8—21no—
19C2×C22⋊Q8C2212yes—
20C2×C22⋊Q8C2211yescharacteristic
21C2×C4×Q8—21no—
22C2×C4×Q8C2211yescharacteristic
23C2×C4×Q8C2212yes—
24C2×#117—21no—
25C22×C42C2211yescharacteristic
26#477—24no—
27#462—24no—
28C2×#107C2212yes—
29C2×C4⋊C8C2211yescharacteristic
30C2×C4⋊C8C2211yescharacteristic
31C2×#103C2212yes—
32C2×#103C2212yes—
33C2×#103C2212yes—
34C2×#103C2212yes—
35C2×C22⋊C8C2212yes—
36#391C2211yescharacteristic
37#389C2211yescharacteristic
38#387—21no—
39#385—21no—
40#380C2211yescharacteristic
41C4×C4⋊C4—21no—
42C4×C4⋊C4C2211yescharacteristic
43C4×C22⋊C4—21no—
44#340C2212yes—
45#340C2212yes—
46#336C2212yes—
47C22×Q8—21no—
48C22×Q8C2312yes—
49C22×Q8C2311yescharacteristic
50C23×C4D411yescharacteristic
51C23×C4D411yescharacteristic
52C23×C4C2311yescharacteristic
53C2×Q16—24no—
54C2×Q16—24no—
55C2×Q16—42no—
56C22×C8C2312yes—
57C22×C8C2312yes—
58C4⋊Q8—22no—
59C4⋊Q8—22no—
60C4⋊Q8D414yes—
61C4⋊Q8D414yes—
62#125—28no—
63C22⋊Q8—28no—
64C22⋊Q8—24no—
65C22⋊Q8—22no—
66C22⋊Q8—24no—
67C22⋊Q8—22no—
68C4×Q8—28no—
69C4×Q8—24no—
70C4×Q8—24no—
71C4×Q8—22no—
72C4×Q8—22no—
73#117—28no—
74C2×C4⋊C4—21no—
75C2×C4⋊C4—21no—
76C2×C4⋊C4—21no—
77C2×C4⋊C4—22no—
78C2×C4⋊C4—22no—
79C2×C4⋊C4—21no—
80C2×C4⋊C4—21no—
81C2×C4⋊C4—21no—
82C2×C4⋊C4C2311yescharacteristic
83C2×C4⋊C4C2311yescharacteristic
84C2×C4⋊C4C2312yes—
85C2×C4⋊C4C2311yescharacteristic
86C2×C22⋊C4—22no—
87C2×C22⋊C4—21no—
88C2×C22⋊C4—21no—
89C2×C42—22no—
90C2×C42—22no—
91C2×C42—22no—
92C2×C42—21no—
93C2×C42—21no—
94C2×C42D411yescharacteristic
95C2×C42D411yescharacteristic
96C2×C42C2311yescharacteristic
97C2×C42C2311yescharacteristic
98#107—24no—
99C4⋊C8—22no—
100C4⋊C8—22no—
101#103—24no—
102#103—24no—
103#103—24no—
104#103—42no—
105C22⋊C8—42no—
106#95—21no—
107#95—21no—
108#95—21no—
109#95—21no—
110#95—21no—
111C24C2×D411yescharacteristic
112C2×Q8—24no—
113C2×Q8—22no—
114C2×Q8—21no—
115C2×Q8—21no—
116C2×Q8—21no—
117C2×Q8—22no—
118C2×Q8—22no—
119C2×Q8—22no—
120C2×Q8—21no—
121C2×Q8—22no—
122C2×Q8—21no—
123C2×Q8—42no—
124C2×Q8—42no—
125C2×Q8C2×D414yes—
126C2×Q8#4112yes—
127C22×C4—22no—
128C22×C4—22no—
129C22×C4—21no—
130C22×C4—21no—
131C22×C4—22no—
132C22×C4—22no—
133C22×C4—22no—
134C22×C4—21no—
135C22×C4—21no—
136C22×C4—21no—
137C22×C4—21no—
138C22×C4—21no—
139C22×C4—21no—
140C22×C4—21no—
141C22×C4—21no—
142C22×C4—41no—
143C22×C4—41no—
144C22×C4Q1614yescomplemented
145C22×C4C2×D411yescharacteristic
146C22×C4C2×D411yescharacteristic
147C22×C4C2×D411yescharacteristic
148C22×C4C2×D411yescharacteristic
149C22×C4C2×D411yescharacteristic
150C22×C4C2411yescharacteristic
151Q16—48no—
152C2×C8—22no—
153C2×C8—22no—
154C2×C8—22no—
155C2×C8—22no—
156C2×C8—22no—
157C2×C8—22no—
158C4⋊C4—28no—
159C4⋊C4—28no—
160C4⋊C4—24no—
161C4⋊C4—24no—
162C4⋊C4—24no—
163C4⋊C4—24no—
164C4⋊C4—22no—
165C4⋊C4—21no—
166C4⋊C4—21no—
167C4⋊C4—21no—
168C4⋊C4—21no—
169C4⋊C4—21no—
170C4⋊C4—42no—
171C4⋊C4—42no—
172C4⋊C4C2×D414yes—
173C4⋊C4#4112yes—
174C22⋊C4—28no—
175C22⋊C4—24no—
176C22⋊C4—42no—
177C42—28no—
178C42—22no—
179C42—22no—
180C42—21no—
181C42—21no—
182C42—42no—
183C42—42no—
184C42C2×D412yes—
185C42C2×D412yes—
186C23—21no—
187C23—21no—
188C23—21no—
189C23—21no—
190C23—21no—
191C23—21no—
192C23C2×Q1612yes—
193C23C22×D411yescharacteristic
194Q8—22no—
195Q8—24no—
196Q8—24no—
197Q8—22no—
198Q8—48no—
199Q8—41no—
200Q8—42no—
201Q8—42no—
202Q8—41no—
203C2×C4—21no—
204C2×C4—21no—
205C2×C4—21no—
206C2×C4—21no—
207C2×C4—28no—
208C2×C4—21no—
209C2×C4—21no—
210C2×C4—24no—
211C2×C4—24no—
212C2×C4—22no—
213C2×C4—22no—
214C2×C4—21no—
215C2×C4—21no—
216C2×C4—21no—
217C2×C4—21no—
218C2×C4—44no—
219C2×C4—41no—
220C2×C4—42no—
221C2×C4—41no—
222C2×C4—42no—
223C2×C4—41no—
224C2×C4—41no—
225C2×C4—42no—
226C2×C4—42no—
227C2×C4—41no—
228C2×C4—41no—
229C2×C4—41no—
230C2×C4—41no—
231C2×C4—41no—
232C2×C4—41no—
233C2×C4—41no—
234C2×C4—41no—
235C2×C4—41no—
236C2×C4—41no—
237C2×C4#12012yes—
238C2×C4#12012yes—
239C2×C4C2×Q1612yes—
240C2×C4C2×Q1612yes—
241C2×C4#13712yes—
242C2×C4C22×D411yescharacteristic
243C2×C4C22×D411yescharacteristic
244C2×C4C2×#4111yescharacteristic
245C2×C4C2×#4111yescharacteristic
246C2×C4H(5,𝔽2)11yescharacteristic
247C2×C42−1+411yescharacteristic
248C8—42no—
249C8—42no—
250C22—22no—
251C22—22no—
252C22—22no—
253C22—41no—
254C22—41no—
255C22—41no—
256C22—41no—
257C22C2×#12011yescharacteristic
258C22#52511yescharacteristic
259C22#53711yescharacteristic
260C22C22×Q1611yescharacteristic
261C22C2×#13711yescharacteristic
262C22#57511yescharacteristic
263C22#57511yescharacteristic
264C4—22no—
265C4—22no—
266C4—42no—
267C4—41no—
268C4—44no—
269C4—42no—
270C4—41no—
271C4—44no—
272C4—81no—
273C4—81no—
274C4#45114yes—
275C4#54611yescharacteristic
276C4#54711yescharacteristic
277C4#54811yescharacteristic
278C4#55011yescharacteristic
279C2—42no—
280C2#240711yescharacteristic
281C2C2×#45111yescharacteristic
282C2#300511yescharacteristic
283C2#310411yescharacteristic
284C2#311611yescharacteristic
285C2#324911yescharacteristic
286C2#325211yescharacteristic
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
2. C2×#477C2semidirect
3. C2×#462C2semidirect
7. #1987C2semidirect
8. #1981C2semidirect
9. #1878C2semidirect
10. #1863C2semidirect
144. C22×C4Q16semidirect

Maximal subgroups

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

subgroupindex
C2×#5332
C2×#4772
C2×#4622
C4×C22⋊Q82
#20362
#20012
#19872
#19812
#18782
#18632
#17972