Finite Group Databasebeta

← #25581

Subgroups — #25581

Order 256.

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

normal subgroups only

This group has 663 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×#553C211yesmaximal, characteristic, complemented
2C2×#490C211yesmaximal, characteristic, complemented
3C2×#459C211yesmaximal, characteristic, complemented
4#2541C211yesmaximal, characteristic, complemented
5#1999C212yesmaximal, complemented
6#1999C212yesmaximal, complemented
7#1989C211yesmaximal, characteristic, complemented
8#1955C211yesmaximal, characteristic, complemented
9#1875C212yesmaximal, complemented
10#1874C212yesmaximal, complemented
11#1762C211yesmaximal, characteristic, complemented
12C22×D8C2212yes—
13#553—24no—
14#553—24no—
15C2×#127—21no—
16C2×#127C2211yescharacteristic
17C2×#126—21no—
18C2×#125C2211yescharacteristic
19C2×#123—21no—
20C2×#121—21no—
21C2×#121C2211yescharacteristic
22C2×#121C2211yescharacteristic
23C2×#121C2212yes—
24C2×#121C2212yes—
25C2×C4×D4—21no—
26C2×C4×D4C2211yescharacteristic
27C2×C4×D4C2211yescharacteristic
28C2×#117C2211yescharacteristic
29C22×C4⋊C4C2211yescharacteristic
30#490—24no—
31#459—24no—
32C2×C4⋊C8C2211yescharacteristic
33C2×#102C2212yes—
34C2×#102C2212yes—
35C2×#102C2212yes—
36C2×#102C2212yes—
37C2×C22⋊C8C2212yes—
38C2×C22⋊C8C2212yes—
39C2×#97C2211yescharacteristic
40#392—21no—
41#392—21no—
42#390—21no—
43C4×C4⋊C4C2211yescharacteristic
44#340C2212yes—
45#340C2212yes—
46#340C2212yes—
47C22×D4—21no—
48C22×D4—21no—
49C22×D4—22no—
50C22×D4—22no—
51C22×D4—21no—
52C22×D4—21no—
53C22×D4D411yescharacteristic
54C22×D4C2312yes—
55C22×D4C2311yescharacteristic
56C23×C4—21no—
57C23×C4—21no—
58C23×C4C2311yescharacteristic
59C23×C4C2311yescharacteristic
60C2×D8—24no—
61C2×D8—42no—
62C2×D8—42no—
63C22×C8C2312yes—
64C22×C8C2312yes—
65#127—22no—
66#127—22no—
67#127—44no—
68#126—28no—
69#125—22no—
70#125D414yescomplemented
71#123—28no—
72#121—28no—
73#121—22no—
74#121—22no—
75#121—22no—
76#121—22no—
77#121—24no—
78#121—24no—
79#121—42no—
80#121—42no—
81C4×D4—28no—
82C4×D4—22no—
83C4×D4—22no—
84C4×D4—22no—
85C4×D4D414yescomplemented
86#117—22no—
87#117—22no—
88C2×C4⋊C4—21no—
89C2×C4⋊C4—21no—
90C2×C4⋊C4—21no—
91C2×C4⋊C4—22no—
92C2×C4⋊C4—22no—
93C2×C4⋊C4D411yescharacteristic
94C2×C4⋊C4C2311yescharacteristic
95C2×C4⋊C4C2311yescharacteristic
96C2×C4⋊C4C2311yescharacteristic
97C2×C4⋊C4C2312yes—
98C2×C22⋊C4—21no—
99C2×C22⋊C4—21no—
100C2×C22⋊C4—21no—
101C2×C22⋊C4—21no—
102C2×C22⋊C4—22no—
103C2×C22⋊C4—22no—
104C2×C22⋊C4D411yescharacteristic
105C2×C22⋊C4D411yescharacteristic
106C2×C42—21no—
107C2×C42—21no—
108C2×C42C2311yescharacteristic
109C4⋊C8—22no—
110#102—24no—
111#102—24no—
112#102—24no—
113#102—24no—
114C22⋊C8—42no—
115C22⋊C8—42no—
116#97—22no—
117#95—21no—
118C24—21no—
119C24—22no—
120C24C2×D411yescharacteristic
121C24C2×D411yescharacteristic
122C2×D4—24no—
123C2×D4—22no—
124C2×D4—24no—
125C2×D4—22no—
126C2×D4—21no—
127C2×D4—22no—
128C2×D4—22no—
129C2×D4—22no—
130C2×D4—22no—
131C2×D4—22no—
132C2×D4—22no—
133C2×D4—22no—
134C2×D4—22no—
135C2×D4—41no—
136C2×D4—44no—
137C2×D4—44no—
138C2×D4—42no—
139C2×D4—42no—
140C2×D4—42no—
141C2×D4—42no—
142C2×D4—44no—
143C2×D4—44no—
144C2×D4—42no—
145C2×D4—42no—
146C2×D4—82no—
147C2×D4—82no—
148C2×D4—82no—
149C2×D4—82no—
150C2×D4C2×D412yes—
151C22×C4—22no—
152C22×C4—21no—
153C22×C4—21no—
154C22×C4—21no—
155C22×C4—24no—
156C22×C4—21no—
157C22×C4—21no—
158C22×C4—21no—
159C22×C4—21no—
160C22×C4—22no—
161C22×C4—22no—
162C22×C4—22no—
163C22×C4—22no—
164C22×C4—24no—
165C22×C4—82no—
166C22×C4—82no—
167C22×C4C2×D411yescharacteristic
168C22×C4C2×D411yescharacteristic
169C22×C4C2×D411yescharacteristic
170C22×C4C2×D411yescharacteristic
171C22×C4C2411yescharacteristic
172D8—48no—
173C2×C8—22no—
174C2×C8—22no—
175C2×C8—22no—
176C2×C8—22no—
177C2×C8—22no—
178C2×C8—22no—
179C4⋊C4—24no—
180C4⋊C4—24no—
181C4⋊C4—24no—
182C4⋊C4—22no—
183C4⋊C4—21no—
184C4⋊C4—21no—
185C4⋊C4—21no—
186C4⋊C4—22no—
187C4⋊C4C2×D412yes—
188C4⋊C4C2×D412yes—
189C4⋊C4C2×D412yes—
190C4⋊C4#4114yes—
191C22⋊C4—24no—
192C22⋊C4—24no—
193C22⋊C4—24no—
194C22⋊C4—24no—
195C22⋊C4—22no—
196C22⋊C4—22no—
197C22⋊C4—44no—
198C22⋊C4—44no—
199C42—24no—
200C42—42no—
201C42C2×D412yes—
202C42C2×D412yes—
203C23—21no—
204C23—22no—
205C23—21no—
206C23—22no—
207C23—21no—
208C23—22no—
209C23—22no—
210C23—21no—
211C23—21no—
212C23—22no—
213C23—42no—
214C23—41no—
215C23—41no—
216C23—42no—
217C23—42no—
218C23—42no—
219C23—42no—
220C23—42no—
221C23—42no—
222C23—42no—
223C23—42no—
224C23AGL(1,ℤ/8)12yes—
225C23C22×D411yescharacteristic
226D4—22no—
227D4—22no—
228D4—44no—
229D4—48no—
230D4—42no—
231D4—42no—
232D4—42no—
233D4—42no—
234D4—42no—
235D4—84no—
236D4—84no—
237D4—88no—
238D4—84no—
239D4—88no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—22no—
243C2×C4—21no—
244C2×C4—22no—
245C2×C4—22no—
246C2×C4—21no—
247C2×C4—22no—
248C2×C4—24no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—42no—
252C2×C4—41no—
253C2×C4—41no—
254C2×C4—42no—
255C2×C4—42no—
256C2×C4—42no—
257C2×C4—41no—
258C2×C4—41no—
259C2×C4—41no—
260C2×C4—41no—
261C2×C4—42no—
262C2×C4—42no—
263C2×C4—84no—
264C2×C4—84no—
265C2×C4#12012yes—
266C2×C4#12012yes—
267C2×C4C22×D411yescharacteristic
268C2×C4C22×D411yescharacteristic
269C2×C4C2×#4112yes—
270C2×C4H(5,𝔽2)12yes—
271C8—42no—
272C8—42no—
273C22—22no—
274C22—24no—
275C22—41no—
276C22—41no—
277C22—42no—
278C22—41no—
279C22—42no—
280C22—41no—
281C22—42no—
282C22—42no—
283C22—41no—
284C22—42no—
285C22—42no—
286C22—82no—
287C22—82no—
288C22—82no—
289C22—82no—
290C22—82no—
291C22—82no—
292C22—82no—
293C22—82no—
294C22—82no—
295C22C2×#12011yescharacteristic
296C22#54612yes—
297C22C2×AGL(1,ℤ/8)11yescharacteristic
298C22#57511yescharacteristic
299C22#57612yes—
300C4—22no—
301C4—22no—
302C4—41no—
303C4—41no—
304C4—42no—
305C4—42no—
306C4—42no—
307C4—44no—
308C4#53211yescharacteristic
309C4#53511yescharacteristic
310C4#55012yes—
311C2—42no—
312C2—81no—
313C2—82no—
314C2—162no—
315C2#243111yescharacteristic
316C2#300611yescharacteristic
317C2#300811yescharacteristic
318C2#330912yes—
319C2#331812yes—
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
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
1. C2×#553C2semidirect
2. C2×#490C2semidirect
3. C2×#459C2semidirect
4. #2541C2semidirect
5. #1999C2semidirect
6. #1999C2semidirect
7. #1989C2semidirect
8. #1955C2semidirect
9. #1875C2semidirect
10. #1874C2semidirect
11. #1762C2semidirect
70. #125D4semidirect
85. C4×D4D4semidirect

Maximal subgroups

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

subgroupindex
C2×#5532
C2×#4902
C2×#4592
#25412
#19992
#19892
#19552
#18752
#18742
#17622