Finite Group Databasebeta

← #23980

Subgroups — #23980

Order 256.

611 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 611 conjugacy classes of subgroups — too many to draw. Only its 133 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×#556C211yesmaximal, characteristic
2C2×#451C211yesmaximal, characteristic, complemented
3C2×#419C211yesmaximal, characteristic, complemented
4#2409C211yesmaximal, characteristic
5#2024C212yesmaximal, complemented
6#2001C212yesmaximal, complemented
7#1894C211yesmaximal, characteristic, complemented
8#1878C212yesmaximal, complemented
9#1873C212yesmaximal, complemented
10#1870C211yesmaximal, characteristic
11#1825C211yesmaximal, characteristic, complemented
12C23×Q8C2211yescharacteristic
13C22×Q16C2212yes—
14C22×M16C2211yescharacteristic
15#556—24no—
16#556—24no—
17C2×C4⋊Q8C2211yescharacteristic
18C2×#126—21no—
19C2×#126—21no—
20C2×#124—21no—
21C2×C22⋊Q8C2211yescharacteristic
22C2×C22⋊Q8C2212yes—
23C2×C22⋊Q8C2211yescharacteristic
24C2×C4×Q8—21no—
25C2×C4×Q8C2211yescharacteristic
26C2×C4×D4C2211yescharacteristic
27C2×#117C2211yescharacteristic
28C2×#117C2211yescharacteristic
29#451—24no—
30#419—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×C4C2311yescharacteristic
58C23×C4C2311yescharacteristic
59C2×Q16—24no—
60C2×Q16—24no—
61C2×Q16—42no—
62C2×M16—22no—
63C2×M16—42no—
64C22×C8C2312yes—
65C22×C8C2312yes—
66C4⋊Q8—22no—
67C4⋊Q8—22no—
68#126—28no—
69#126—28no—
70#124—28no—
71C22⋊Q8—22no—
72C22⋊Q8—22no—
73C22⋊Q8—22no—
74C22⋊Q8—24no—
75C22⋊Q8—42no—
76C22⋊Q8D414yes—
77C4×Q8—28no—
78C4×Q8—22no—
79C4×Q8—22no—
80C4×D4—22no—
81C4×D4—22no—
82#117—22no—
83#117—22no—
84#117—22no—
85#117D414yes—
86C2×C4⋊C4—21no—
87C2×C4⋊C4—21no—
88C2×C4⋊C4—21no—
89C2×C4⋊C4—21no—
90C2×C4⋊C4—22no—
91C2×C4⋊C4D411yescharacteristic
92C2×C4⋊C4C2312yes—
93C2×C4⋊C4C2311yescharacteristic
94C2×C4⋊C4C2311yescharacteristic
95C2×C4⋊C4C2311yescharacteristic
96C2×C22⋊C4—21no—
97C2×C22⋊C4—21no—
98C2×C22⋊C4—21no—
99C2×C22⋊C4—21no—
100C2×C22⋊C4—22no—
101C2×C22⋊C4—21no—
102C2×C22⋊C4D411yescharacteristic
103C2×C22⋊C4D411yescharacteristic
104C2×C42—21no—
105C2×C42—21no—
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—42no—
132C2×Q8—44no—
133C2×Q8—42no—
134C2×Q8—42no—
135C2×Q8C2×D412yes—
136C2×Q8#4114yes—
137C2×D4—22no—
138C2×D4—22no—
139C2×D4—41no—
140C22×C4—22no—
141C22×C4—22no—
142C22×C4—22no—
143C22×C4—21no—
144C22×C4—21no—
145C22×C4—21no—
146C22×C4—21no—
147C22×C4—21no—
148C22×C4—21no—
149C22×C4—22no—
150C22×C4—22no—
151C22×C4—22no—
152C22×C4—44no—
153C22×C4—42no—
154C22×C4C2×D411yescharacteristic
155C22×C4C2×D411yescharacteristic
156C22×C4C2×D411yescharacteristic
157C22×C4C2×D412yes—
158C22×C4C2×D412yes—
159C22×C4C2×D411yescharacteristic
160C22×C4C2411yescharacteristic
161Q16—48no—
162M16—44no—
163C2×C8—22no—
164C2×C8—22no—
165C2×C8—22no—
166C2×C8—22no—
167C2×C8—22no—
168C2×C8—22no—
169C4⋊C4—24no—
170C4⋊C4—24no—
171C4⋊C4—24no—
172C4⋊C4—24no—
173C4⋊C4—22no—
174C4⋊C4—21no—
175C4⋊C4—21no—
176C4⋊C4—21no—
177C4⋊C4—22no—
178C4⋊C4—22no—
179C4⋊C4—44no—
180C4⋊C4C2×D412yes—
181C4⋊C4C2×D412yes—
182C4⋊C4C2×D412yes—
183C22⋊C4—24no—
184C22⋊C4—24no—
185C22⋊C4—24no—
186C22⋊C4—24no—
187C22⋊C4—22no—
188C22⋊C4—22no—
189C22⋊C4—44no—
190C22⋊C4—42no—
191C42—24no—
192C42—24no—
193C42—24no—
194C42—21no—
195C42—21no—
196C23—22no—
197C23—21no—
198C23—21no—
199C23—22no—
200C23—21no—
201C23—22no—
202C23—21no—
203C23—21no—
204C23—22no—
205C23#12012yes—
206C23C22×D411yescharacteristic
207Q8—24no—
208Q8—24no—
209Q8—24no—
210Q8—48no—
211Q8—48no—
212Q8—41no—
213Q8—41no—
214Q8—42no—
215Q8—42no—
216D4—44no—
217C2×C4—21no—
218C2×C4—22no—
219C2×C4—21no—
220C2×C4—22no—
221C2×C4—22no—
222C2×C4—24no—
223C2×C4—22no—
224C2×C4—21no—
225C2×C4—21no—
226C2×C4—22no—
227C2×C4—22no—
228C2×C4—22no—
229C2×C4—24no—
230C2×C4—22no—
231C2×C4—22no—
232C2×C4—48no—
233C2×C4—42no—
234C2×C4—42no—
235C2×C4—42no—
236C2×C4—41no—
237C2×C4—41no—
238C2×C4—41no—
239C2×C4—41no—
240C2×C4—42no—
241C2×C4—41no—
242C2×C4—41no—
243C2×C4—42no—
244C2×C4—42no—
245C2×C4—42no—
246C2×C4—42no—
247C2×C4#12012yes—
248C2×C4#13712yes—
249C2×C4C22×D411yescharacteristic
250C2×C4C22×D411yescharacteristic
251C2×C4C2×#4112yes—
252C2×C4H(5,𝔽2)12yes—
253C8—42no—
254C8—42no—
255C22—22no—
256C22—22no—
257C22—22no—
258C22—42no—
259C22—42no—
260C22—42no—
261C22—41no—
262C22—41no—
263C22—41no—
264C22—42no—
265C22C2×#12011yescharacteristic
266C22#54612yes—
267C22C2×#13711yescharacteristic
268C22#57511yescharacteristic
269C22#57812yes—
270C4—22no—
271C4—22no—
272C4—41no—
273C4—44no—
274C4—42no—
275C4—42no—
276C4—42no—
277C4—41no—
278C4—42no—
279C4—82no—
280C4#52511yescharacteristic
281C4#53511yescharacteristic
282C4#54612yes—
283C2—42no—
284C2—81no—
285C2#240411yescharacteristic
286C2#300711yescharacteristic
287C2#301111yescharacteristic
288C2#331312yes—
289C2#332812yes—
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. C2×#419C2semidirect
5. #2024C2semidirect
6. #2001C2semidirect
7. #1894C2semidirect
8. #1878C2semidirect
9. #1873C2semidirect
11. #1825C2semidirect

Maximal subgroups

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

subgroupindex
C2×#5562
C2×#4512
C2×#4192
#24092
#20242
#20012
#18942
#18782
#18732
#18702
#18252