Finite Group Databasebeta

← #29629

Subgroups — #29629

Order 256.

689 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 689 conjugacy classes of subgroups — too many to draw. Only its 131 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×#532C211yesmaximal, characteristic
2C2×#452C212yesmaximal, complemented
3#2855C211yesmaximal, characteristic, complemented
4#2043C211yesmaximal, characteristic, complemented
5#2042C212yesmaximal, complemented
6#2041C211yesmaximal, characteristic, complemented
7#2016C211yesmaximal, characteristic, complemented
8#2015C212yesmaximal, complemented
9#2014C211yesmaximal, characteristic, complemented
10#1817C211yesmaximal, characteristic, complemented
11#1788C212yesmaximal, complemented
12C22×#41C2212yes—
13C22×Q16C2212yes—
14C22×SD16C2212yes—
15#532—28no—
16C2×C4⋊Q8C2211yescharacteristic
17C2×#127C2211yescharacteristic
18C2×#124C2212yes—
19C2×C22⋊Q8C2211yescharacteristic
20C2×#121—21no—
21C2×#121—21no—
22C2×C4×D4—21no—
23C2×C4×D4—21no—
24C22×C42C2211yescharacteristic
25#452—28no—
26C2×#103C2212yes—
27C2×#103C2212yes—
28C2×#103C2212yes—
29C2×#102C2212yes—
30C2×#102C2212yes—
31C2×#102C2212yes—
32C2×C22⋊C8C2212yes—
33#391C2211yescharacteristic
34#391C2211yescharacteristic
35#389C2211yescharacteristic
36#389C2211yescharacteristic
37#387—21no—
38#387—21no—
39#385—21no—
40#381—21no—
41#380C2211yescharacteristic
42#336C2211yescharacteristic
43#336C2212yes—
44#336C2211yescharacteristic
45C2×#41—24no—
46C2×#41—42no—
47C2×#41—42no—
48C2×#41—42no—
49C2×#41—42no—
50C2×#41D418yes—
51C22×Q8C2311yescharacteristic
52C22×Q8C2312yes—
53C22×D4—22no—
54C22×D4—21no—
55C22×D4—21no—
56C22×D4C2312yes—
57C23×C4—22no—
58C23×C4D411yescharacteristic
59C23×C4D411yescharacteristic
60C23×C4C2311yescharacteristic
61C2×Q16—24no—
62C2×Q16—42no—
63C2×Q16—42no—
64C2×SD16—24no—
65C2×SD16—42no—
66C2×SD16—42no—
67C22×C8C2312yes—
68C22×C8C2312yes—
69C4⋊Q8—24no—
70#127—24no—
71#124—24no—
72#124—24no—
73C22⋊Q8—24no—
74#121—28no—
75#121—28no—
76C4×D4—28no—
77C4×D4—28no—
78C2×C4⋊C4—21no—
79C2×C4⋊C4—21no—
80C2×C4⋊C4—21no—
81C2×C4⋊C4—21no—
82C2×C4⋊C4—21no—
83C2×C4⋊C4C2311yescharacteristic
84C2×C4⋊C4C2311yescharacteristic
85C2×C4⋊C4C2311yescharacteristic
86C2×C22⋊C4—22no—
87C2×C22⋊C4—22no—
88C2×C22⋊C4—21no—
89C2×C22⋊C4—21no—
90C2×C42—24no—
91C2×C42D411yescharacteristic
92C2×C42D411yescharacteristic
93C2×C42C2311yescharacteristic
94C2×C42C2311yescharacteristic
95#103—24no—
96#103—42no—
97#103—42no—
98#102—24no—
99#102—42no—
100#102—42no—
101C22⋊C8—24no—
102#95—21no—
103#95—21no—
104#95—21no—
105#95—21no—
106#95—21no—
107#95—21no—
108#95—21no—
109C24—22no—
110C24C2×D411yescharacteristic
111#41—48no—
112#41—48no—
113#41—48no—
114#41—48no—
115C2×Q8—22no—
116C2×Q8—22no—
117C2×Q8—22no—
118C2×Q8—22no—
119C2×Q8—22no—
120C2×Q8—22no—
121C2×Q8—21no—
122C2×Q8—22no—
123C2×Q8—21no—
124C2×Q8C2×D414yes—
125C2×D4—24no—
126C2×D4—24no—
127C2×D4—28no—
128C2×D4—22no—
129C2×D4—22no—
130C2×D4—22no—
131C2×D4—22no—
132C2×D4—22no—
133C2×D4—42no—
134C2×D4—42no—
135C2×D4—44no—
136C2×D4—44no—
137C2×D4—42no—
138C2×D4—42no—
139C2×D4C2×D414yes—
140C22×C4—22no—
141C22×C4—21no—
142C22×C4—21no—
143C22×C4—22no—
144C22×C4—22no—
145C22×C4—22no—
146C22×C4—28no—
147C22×C4—22no—
148C22×C4—22no—
149C22×C4—21no—
150C22×C4—21no—
151C22×C4—21no—
152C22×C4—21no—
153C22×C4—41no—
154C22×C4—41no—
155C22×C4—44no—
156C22×C4—44no—
157C22×C4C2×D411yescharacteristic
158C22×C4C2×D411yescharacteristic
159C22×C4C2×D411yescharacteristic
160C22×C4C2×D414yes—
161C22×C4C2×D411yescharacteristic
162C22×C4C2×D411yescharacteristic
163C22×C4C2411yescharacteristic
164Q16—48no—
165SD16—48no—
166C2×C8—22no—
167C2×C8—22no—
168C2×C8—22no—
169C2×C8—22no—
170C2×C8—22no—
171C2×C8—22no—
172C4⋊C4—24no—
173C4⋊C4—24no—
174C4⋊C4—22no—
175C4⋊C4—21no—
176C4⋊C4—22no—
177C4⋊C4—21no—
178C4⋊C4—42no—
179C4⋊C4—42no—
180C4⋊C4—42no—
181C22⋊C4—28no—
182C22⋊C4—28no—
183C22⋊C4—42no—
184C22⋊C4—42no—
185C42—22no—
186C42—22no—
187C42—22no—
188C42—22no—
189C23—21no—
190C23—21no—
191C23—21no—
192C23—21no—
193C23—21no—
194C23—21no—
195C23—24no—
196C23—42no—
197C23—42no—
198C23—42no—
199C23—42no—
200C23—42no—
201C23—42no—
202C23#12012yes—
203C23C22×D411yescharacteristic
204Q8—24no—
205Q8—24no—
206Q8—42no—
207Q8—42no—
208Q8—42no—
209Q8—42no—
210D4—24no—
211D4—24no—
212D4—48no—
213D4—48no—
214D4—416no—
215D4—42no—
216D4—42no—
217C2×C4—21no—
218C2×C4—21no—
219C2×C4—21no—
220C2×C4—21no—
221C2×C4—21no—
222C2×C4—21no—
223C2×C4—24no—
224C2×C4—24no—
225C2×C4—21no—
226C2×C4—21no—
227C2×C4—24no—
228C2×C4—21no—
229C2×C4—21no—
230C2×C4—21no—
231C2×C4—21no—
232C2×C4—44no—
233C2×C4—44no—
234C2×C4—416no—
235C2×C4—42no—
236C2×C4—42no—
237C2×C4—41no—
238C2×C4—41no—
239C2×C4—41no—
240C2×C4—41no—
241C2×C4—41no—
242C2×C4—41no—
243C2×C4—41no—
244C2×C4—41no—
245C2×C4—41no—
246C2×C4—41no—
247C2×C4—41no—
248C2×C4—41no—
249C2×C4#12012yes—
250C2×C4#13512yes—
251C2×C4#13712yes—
252C2×C4C22×D412yes—
253C2×C4H(5,𝔽2)12yes—
254C2×C4H(5,𝔽2)11yescharacteristic
255C2×C4H(5,𝔽2)11yescharacteristic
256C8—42no—
257C8—42no—
258C22—22no—
259C22—22no—
260C22—22no—
261C22—41no—
262C22—41no—
263C22—41no—
264C22—41no—
265C22—44no—
266C22—44no—
267C22—44no—
268C22—82no—
269C22—82no—
270C22—82no—
271C22—82no—
272C22C2×#12011yescharacteristic
273C22#53511yescharacteristic
274C22#53511yescharacteristic
275C22C2×#13511yescharacteristic
276C22C2×#13711yescharacteristic
277C22#57511yescharacteristic
278C22#57511yescharacteristic
279C4—22no—
280C4—22no—
281C4—22no—
282C4—44no—
283C4—41no—
284C4—41no—
285C4—81no—
286C4—81no—
287C4—81no—
288C4—81no—
289C4#53412yes—
290C4#53412yes—
291C2—42no—
292C2—84no—
293C2#261311yescharacteristic
294C2#300211yescharacteristic
295C2#300711yescharacteristic
296C2#321011yescharacteristic
297C2#321211yescharacteristic
298C2#321611yescharacteristic
299C2#321911yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC2×#532C2
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×#452C2semidirect
3. #2855C2semidirect
4. #2043C2semidirect
5. #2042C2semidirect
6. #2041C2semidirect
7. #2016C2semidirect
8. #2015C2semidirect
9. #2014C2semidirect
10. #1817C2semidirect
11. #1788C2semidirect

Maximal subgroups

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

subgroupindex
C2×#5322
C2×#4522
#28552
#20432
#20422
#20412
#20162
#20152
#20142
#18172
#17882