Finite Group Databasebeta

← #25604

Subgroups — #25604

Order 256.

589 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 589 conjugacy classes of subgroups — too many to draw. Only its 135 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×#529C211yesmaximal, characteristic
2C2×#497C211yesmaximal, characteristic, complemented
3C2×#491C211yesmaximal, characteristic, complemented
4#2857C211yesmaximal, characteristic, complemented
5#2001C212yesmaximal, complemented
6#2000C212yesmaximal
7#1992C211yesmaximal, characteristic, complemented
8#1959C211yesmaximal, characteristic, complemented
9#1878C212yesmaximal, complemented
10#1877C212yesmaximal, complemented
11#1749C211yesmaximal, characteristic, complemented
12C22×Q16C2212yes—
13C22×SD16C2212yes—
14C2×C4⋊Q8C2211yescharacteristic
15#529—24no—
16#529—24no—
17C2×#126—21no—
18C2×#125C2211yescharacteristic
19C2×#124C2211yescharacteristic
20C2×#123—21no—
21C2×C22⋊Q8C2211yescharacteristic
22C2×C22⋊Q8C2212yes—
23C2×#121C2211yescharacteristic
24C2×C4×Q8C2211yescharacteristic
25C2×C4×D4—21no—
26C2×C4×D4C2211yescharacteristic
27C2×#117—21no—
28C22×C42C2211yescharacteristic
29#497—24no—
30#491—24no—
31C2×#103C2212yes—
32C2×#103C2212yes—
33C2×#103C2212yes—
34C2×#102C2212yes—
35C2×C22⋊C8C2212yes—
36C2×#97C2211yescharacteristic
37C2×#97C2211yescharacteristic
38#391C2212yes—
39#389—21no—
40#389—21no—
41#384C2211yescharacteristic
42#380—21no—
43#380—21no—
44#340C2212yes—
45#340C2212yes—
46#336C2212yes—
47C22×Q8C2312yes—
48C22×Q8C2311yescharacteristic
49C22×D4—21no—
50C22×D4C2311yescharacteristic
51C23×C4—21no—
52C23×C4D411yescharacteristic
53C23×C4D411yescharacteristic
54C23×C4C2311yescharacteristic
55C2×Q16—24no—
56C2×Q16—24no—
57C2×Q16—42no—
58C2×SD16—24no—
59C2×SD16—24no—
60C2×SD16—42no—
61C22×C8C2312yes—
62C22×C8C2312yes—
63C4⋊Q8—22no—
64C4⋊Q8—22no—
65#126—28no—
66#125—22no—
67#125D414yes—
68#124—22no—
69#124D414yes—
70#123—28no—
71C22⋊Q8—22no—
72C22⋊Q8—22no—
73C22⋊Q8—42no—
74C22⋊Q8—42no—
75#121—22no—
76#121—22no—
77C4×Q8—22no—
78C4×Q8—22no—
79C4×D4—28no—
80C4×D4—22no—
81C4×D4—22no—
82#117—28no—
83C2×C4⋊C4—21no—
84C2×C4⋊C4—21no—
85C2×C4⋊C4—21no—
86C2×C4⋊C4—21no—
87C2×C4⋊C4—22no—
88C2×C4⋊C4—21no—
89C2×C4⋊C4—21no—
90C2×C4⋊C4—21no—
91C2×C4⋊C4—21no—
92C2×C4⋊C4C2312yes—
93C2×C4⋊C4C2311yescharacteristic
94C2×C4⋊C4C2311yescharacteristic
95C2×C22⋊C4—21no—
96C2×C22⋊C4—21no—
97C2×C22⋊C4—21no—
98C2×C22⋊C4—21no—
99C2×C22⋊C4—21no—
100C2×C22⋊C4—22no—
101C2×C42—21no—
102C2×C42—22no—
103C2×C42—22no—
104C2×C42D411yescharacteristic
105C2×C42D411yescharacteristic
106C2×C42C2311yescharacteristic
107C2×C42C2311yescharacteristic
108#103—24no—
109#103—24no—
110#103—42no—
111#102—42no—
112C22⋊C8—24no—
113#97—22no—
114#97—22no—
115#95—22no—
116#95—22no—
117C24—21no—
118C24C2×D411yescharacteristic
119C2×Q8—21no—
120C2×Q8—22no—
121C2×Q8—22no—
122C2×Q8—22no—
123C2×Q8—22no—
124C2×Q8—22no—
125C2×Q8—22no—
126C2×Q8—22no—
127C2×Q8—22no—
128C2×Q8C2×D412yes—
129C2×D4—24no—
130C2×D4—21no—
131C2×D4—22no—
132C2×D4—22no—
133C2×D4—42no—
134C2×D4—42no—
135C2×D4C2×D412yes—
136C22×C4—24no—
137C22×C4—22no—
138C22×C4—21no—
139C22×C4—21no—
140C22×C4—21no—
141C22×C4—22no—
142C22×C4—22no—
143C22×C4—22no—
144C22×C4—21no—
145C22×C4—21no—
146C22×C4—22no—
147C22×C4—22no—
148C22×C4—22no—
149C22×C4—22no—
150C22×C4—41no—
151C22×C4—41no—
152C22×C4—42no—
153C22×C4—42no—
154C22×C4C2×D411yescharacteristic
155C22×C4C2×D411yescharacteristic
156C22×C4C2×D411yescharacteristic
157C22×C4C2×D411yescharacteristic
158C22×C4C2×D411yescharacteristic
159C22×C4C2411yescharacteristic
160Q16—48no—
161SD16—48no—
162C2×C8—22no—
163C2×C8—22no—
164C2×C8—22no—
165C2×C8—22no—
166C2×C8—22no—
167C2×C8#4114yes—
168C4⋊C4—24no—
169C4⋊C4—24no—
170C4⋊C4—24no—
171C4⋊C4—24no—
172C4⋊C4—21no—
173C4⋊C4—21no—
174C4⋊C4—22no—
175C4⋊C4—22no—
176C4⋊C4—42no—
177C4⋊C4—42no—
178C4⋊C4—44no—
179C4⋊C4—42no—
180C4⋊C4—42no—
181C4⋊C4C2×D412yes—
182C4⋊C4C2×D412yes—
183C22⋊C4—24no—
184C22⋊C4—24no—
185C22⋊C4—24no—
186C22⋊C4—24no—
187C22⋊C4—24no—
188C22⋊C4—44no—
189C42—24no—
190C42—22no—
191C42—22no—
192C42—21no—
193C42—21no—
194C42C2×D412yes—
195C42C2×D412yes—
196C23—21no—
197C23—21no—
198C23—21no—
199C23—21no—
200C23—21no—
201C23—21no—
202C23—22no—
203C23—41no—
204C23—41no—
205C23—41no—
206C23—41no—
207C23—41no—
208C23—41no—
209C23#13712yes—
210C23C22×D411yescharacteristic
211Q8—24no—
212Q8—42no—
213Q8—42no—
214Q8—42no—
215Q8—42no—
216Q8—42no—
217D4—24no—
218D4—48no—
219D4—42no—
220C2×C4—21no—
221C2×C4—21no—
222C2×C4—21no—
223C2×C4—21no—
224C2×C4—21no—
225C2×C4—21no—
226C2×C4—21no—
227C2×C4—21no—
228C2×C4—22no—
229C2×C4—22no—
230C2×C4—22no—
231C2×C4—21no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—44no—
236C2×C4—44no—
237C2×C4—48no—
238C2×C4—41no—
239C2×C4—41no—
240C2×C4—41no—
241C2×C4—41no—
242C2×C4—41no—
243C2×C4—41no—
244C2×C4—42no—
245C2×C4—42no—
246C2×C4—42no—
247C2×C4—42no—
248C2×C4—42no—
249C2×C4—42no—
250C2×C4—42no—
251C2×C4—42no—
252C2×C4#12012yes—
253C2×C4#12012yes—
254C2×C4#13712yes—
255C2×C4C22×D411yescharacteristic
256C2×C4C22×D411yescharacteristic
257C2×C4C2×#4112yes—
258C2×C4H(5,𝔽2)12yes—
259C8—24no—
260C8—42no—
261C22—22no—
262C22—22no—
263C22—22no—
264C22—41no—
265C22—41no—
266C22—41no—
267C22—41no—
268C22—42no—
269C22—42no—
270C22—42no—
271C22—81no—
272C22—81no—
273C22—81no—
274C22—81no—
275C22C2×#12011yescharacteristic
276C22#53211yescharacteristic
277C22#53511yescharacteristic
278C22C2×#13711yescharacteristic
279C22C2×#13711yescharacteristic
280C22#57511yescharacteristic
281C22#57511yescharacteristic
282C4—22no—
283C4—22no—
284C4—22no—
285C4—41no—
286C4—42no—
287C4—41no—
288C4—42no—
289C4—42no—
290C4—82no—
291C4—82no—
292C4#54612yes—
293C4#55012yes—
294C2—42no—
295C2—82no—
296C2#243111yescharacteristic
297C2#300711yescharacteristic
298C2#300711yescharacteristic
299C2#317211yescharacteristic
300C2#318111yescharacteristic
301C2#321811yescharacteristic
302C2#321911yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC2×#121C22
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×#497C2semidirect
3. C2×#491C2semidirect
4. #2857C2semidirect
5. #2001C2semidirect
7. #1992C2semidirect
8. #1959C2semidirect
9. #1878C2semidirect
10. #1877C2semidirect
11. #1749C2semidirect

Maximal subgroups

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

subgroupindex
C2×#5292
C2×#4972
C2×#4912
#28572
#20012
#20002
#19922
#19592
#18782
#18772
#17492