Finite Group Databasebeta

← #23970

Subgroups — #23970

Order 256.

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

normal subgroups only

This group has 639 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×#554C211yesmaximal, characteristic
2C2×#451C211yesmaximal, characteristic, complemented
3C22×#103C211yesmaximal, characteristic, complemented
4#2409C211yesmaximal, characteristic
5#2024C212yesmaximal, complemented
6#2001C212yesmaximal, complemented
7#1892C211yesmaximal, characteristic, complemented
8#1878C212yesmaximal, complemented
9#1872C212yesmaximal, complemented
10#1869C211yesmaximal, characteristic
11#1824C211yesmaximal, characteristic, complemented
12C23×Q8C2211yescharacteristic
13C22×Q16C2212yes—
14C23×C8C2211yescharacteristic
15#554—24no—
16#554—24no—
17C2×C4⋊Q8C2211yescharacteristic
18C2×#125—21no—
19C2×C22⋊Q8—21no—
20C2×C22⋊Q8—21no—
21C2×C22⋊Q8C2211yescharacteristic
22C2×C22⋊Q8C2212yes—
23C2×C22⋊Q8C2211yescharacteristic
24C2×C4×Q8C2211yescharacteristic
25C2×C4×D4—21no—
26C2×C4×D4C2211yescharacteristic
27C22×C4⋊C4C2211yescharacteristic
28C22×C4⋊C4C2211yescharacteristic
29#451—24no—
30C2×#103—24no—
31C2×#103C2212yes—
32C2×#103C2212yes—
33C2×#103C2212yes—
34C2×#103C2212yes—
35C2×C22⋊C8C2211yescharacteristic
36C2×C22⋊C8C2212yes—
37#395—21no—
38#393—21no—
39#391C2212yes—
40#386—21no—
41#385—21no—
42#380C2211yescharacteristic
43#340C2212yes—
44#340C2212yes—
45#340C2212yes—
46#336C2212yes—
47C22×Q8—21no—
48C22×Q8—21no—
49C22×Q8—24no—
50C22×Q8—22no—
51C22×Q8—22no—
52C22×Q8—42no—
53C22×Q8C2311yescharacteristic
54C22×Q8C2312yes—
55C22×D4D411yescharacteristic
56C23×C4—21no—
57C23×C4—21no—
58C23×C4—21no—
59C23×C4C2311yescharacteristic
60C23×C4C2311yescharacteristic
61C2×Q16—24no—
62C2×Q16—24no—
63C2×Q16—42no—
64C22×C8—22no—
65C22×C8—42no—
66C22×C8C2312yes—
67C22×C8C2312yes—
68C4⋊Q8—22no—
69C4⋊Q8—22no—
70#125—28no—
71C22⋊Q8—28no—
72C22⋊Q8—28no—
73C22⋊Q8—22no—
74C22⋊Q8—22no—
75C22⋊Q8—22no—
76C22⋊Q8—24no—
77C22⋊Q8—42no—
78C22⋊Q8D414yes—
79C4×Q8—22no—
80C4×Q8—22no—
81C4×D4—28no—
82C4×D4—22no—
83C4×D4—22no—
84C2×C4⋊C4—21no—
85C2×C4⋊C4—21no—
86C2×C4⋊C4—22no—
87C2×C4⋊C4—21no—
88C2×C4⋊C4—21no—
89C2×C4⋊C4—21no—
90C2×C4⋊C4—21no—
91C2×C4⋊C4—22no—
92C2×C4⋊C4—22no—
93C2×C4⋊C4—22no—
94C2×C4⋊C4D411yescharacteristic
95C2×C4⋊C4D414yes—
96C2×C4⋊C4C2312yes—
97C2×C4⋊C4C2311yescharacteristic
98C2×C4⋊C4C2311yescharacteristic
99C2×C4⋊C4C2311yescharacteristic
100C2×C22⋊C4—21no—
101C2×C22⋊C4—21no—
102C2×C22⋊C4—22no—
103C2×C22⋊C4—21no—
104C2×C22⋊C4D411yescharacteristic
105C2×C22⋊C4D411yescharacteristic
106C2×C42—21no—
107C2×C42C2311yescharacteristic
108#103—24no—
109#103—24no—
110#103—24no—
111#103—42no—
112C22⋊C8—22no—
113C22⋊C8—42no—
114#95—21no—
115#95—22no—
116#95—22no—
117C24C2×D411yescharacteristic
118C24C2×D411yescharacteristic
119C2×Q8—24no—
120C2×Q8—21no—
121C2×Q8—22no—
122C2×Q8—28no—
123C2×Q8—22no—
124C2×Q8—22no—
125C2×Q8—22no—
126C2×Q8—22no—
127C2×Q8—22no—
128C2×Q8—22no—
129C2×Q8—44no—
130C2×Q8—48no—
131C2×Q8—44no—
132C2×Q8—42no—
133C2×Q8—82no—
134C2×Q8C2×D412yes—
135C2×Q8#4114yes—
136C2×D4—22no—
137C2×D4—22no—
138C2×D4Q1614yescomplemented
139C22×C4—22no—
140C22×C4—22no—
141C22×C4—22no—
142C22×C4—21no—
143C22×C4—21no—
144C22×C4—21no—
145C22×C4—21no—
146C22×C4—24no—
147C22×C4—24no—
148C22×C4—21no—
149C22×C4—21no—
150C22×C4—22no—
151C22×C4—22no—
152C22×C4—22no—
153C22×C4—44no—
154C22×C4—42no—
155C22×C4—82no—
156C22×C4—82no—
157C22×C4C2×D411yescharacteristic
158C22×C4C2×D411yescharacteristic
159C22×C4C2×D411yescharacteristic
160C22×C4C2×D412yes—
161C22×C4C2×D412yes—
162C22×C4C2×D411yescharacteristic
163C22×C4C2411yescharacteristic
164Q16—48no—
165C2×C8—22no—
166C2×C8—22no—
167C2×C8—22no—
168C2×C8—22no—
169C2×C8—22no—
170C2×C8—22no—
171C2×C8—44no—
172C4⋊C4—24no—
173C4⋊C4—24no—
174C4⋊C4—24no—
175C4⋊C4—24no—
176C4⋊C4—24no—
177C4⋊C4—24no—
178C4⋊C4—22no—
179C4⋊C4—21no—
180C4⋊C4—21no—
181C4⋊C4—21no—
182C4⋊C4—22no—
183C4⋊C4—22no—
184C4⋊C4—44no—
185C4⋊C4C2×D412yes—
186C4⋊C4C2×D412yes—
187C4⋊C4C2×D412yes—
188C22⋊C4—24no—
189C22⋊C4—24no—
190C22⋊C4—22no—
191C22⋊C4—22no—
192C22⋊C4—44no—
193C22⋊C4—42no—
194C42—24no—
195C42—21no—
196C42—21no—
197C23—22no—
198C23—22no—
199C23—21no—
200C23—22no—
201C23—21no—
202C23—22no—
203C23#12012yes—
204C23C2×Q1612yes—
205C23C2×Q1612yes—
206C23#13712yes—
207C23C22×D411yescharacteristic
208Q8—24no—
209Q8—24no—
210Q8—24no—
211Q8—48no—
212Q8—41no—
213Q8—41no—
214Q8—42no—
215Q8—42no—
216Q8—84no—
217D4—28no—
218C2×C4—21no—
219C2×C4—22no—
220C2×C4—21no—
221C2×C4—22no—
222C2×C4—22no—
223C2×C4—24no—
224C2×C4—22no—
225C2×C4—21no—
226C2×C4—22no—
227C2×C4—21no—
228C2×C4—22no—
229C2×C4—22no—
230C2×C4—24no—
231C2×C4—22no—
232C2×C4—22no—
233C2×C4—48no—
234C2×C4—42no—
235C2×C4—42no—
236C2×C4—42no—
237C2×C4—42no—
238C2×C4—42no—
239C2×C4—42no—
240C2×C4—41no—
241C2×C4—41no—
242C2×C4—42no—
243C2×C4—42no—
244C2×C4—42no—
245C2×C4—42no—
246C2×C4—84no—
247C2×C4—84no—
248C2×C4#12012yes—
249C2×C4C2×Q1612yes—
250C2×C4C22×D411yescharacteristic
251C2×C4C22×D411yescharacteristic
252C2×C4C2×#4112yes—
253C2×C4H(5,𝔽2)12yes—
254C8—42no—
255C8—42no—
256C22—22no—
257C22—22no—
258C22—24no—
259C22—24no—
260C22—22no—
261C22—24no—
262C22—41no—
263C22—41no—
264C22—42no—
265C22#45114yes—
266C22C2×#12011yescharacteristic
267C22#54612yes—
268C22C22×Q1611yescharacteristic
269C22C2×#13711yescharacteristic
270C22#57712yes—
271C4—22no—
272C4—22no—
273C4—41no—
274C4—44no—
275C4—42no—
276C4—41no—
277C4—42no—
278C4—42no—
279C4—42no—
280C4—82no—
281C4#52511yescharacteristic
282C4#53511yescharacteristic
283C4#54612yes—
284C2—24no—
285C2—42no—
286C2#240411yescharacteristic
287C2C2×#45111yescharacteristic
288C2#300911yescharacteristic
289C2#329912yes—
290C2#331612yes—
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC22×D4D4
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
2. C2×#451C2semidirect
3. C22×#103C2semidirect
5. #2024C2semidirect
6. #2001C2semidirect
7. #1892C2semidirect
8. #1878C2semidirect
9. #1872C2semidirect
11. #1824C2semidirect
138. C2×D4Q16semidirect

Maximal subgroups

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

subgroupindex
C2×#5542
C2×#4512
C22×#1032
#24092
#20242
#20012
#18922
#18782
#18722
#18692
#18242