Finite Group Databasebeta

← #29667

Subgroups — #29667

Order 256.

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

normal subgroups only

This group has 719 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×#532C211yesmaximal, characteristic, complemented
2C2×#463C212yesmaximal, complemented
3#2850C211yesmaximal, characteristic, complemented
4#2024C211yesmaximal, characteristic, complemented
5#2023C211yesmaximal, characteristic, complemented
6#2014C211yesmaximal, characteristic, complemented
7#2011C211yesmaximal, characteristic, complemented
8#2000C212yesmaximal, complemented
9#1999C212yesmaximal, complemented
10#1804C212yesmaximal, complemented
11#1797C211yesmaximal, characteristic, complemented
12C22×#41—21no—
13C22×SD16C2212yes—
14C22×D8C2212yes—
15#532—28no—
16C2×C4⋊Q8C2211yescharacteristic
17C2×#127C2211yescharacteristic
18C2×#124—21no—
19C2×#124C2211yescharacteristic
20C2×C22⋊Q8C2211yescharacteristic
21C2×#121—21no—
22C2×#121C2212yes—
23C2×#121C2211yescharacteristic
24C2×C4×D4—21no—
25C2×C4×D4C2212yes—
26C22×C42C2211yescharacteristic
27#463—28no—
28C2×C4⋊C8C2212yes—
29C2×#103C2212yes—
30C2×#102C2212yes—
31C2×#102C2212yes—
32C2×#102C2212yes—
33C2×C22⋊C8C2212yes—
34#391C2211yescharacteristic
35#390C2211yescharacteristic
36#388—21no—
37#388—21no—
38#386—21no—
39#386—21no—
40#382C2211yescharacteristic
41#340C2212yes—
42#340C2212yes—
43#336C2211yescharacteristic
44#336C2211yescharacteristic
45C2×#41—28no—
46C2×#41—44no—
47C2×#41—44no—
48C22×Q8—21no—
49C22×Q8C2311yescharacteristic
50C22×D4—22no—
51C22×D4—21no—
52C22×D4—21no—
53C22×D4—21no—
54C22×D4C2311yescharacteristic
55C22×D4C2312yes—
56C23×C4—22no—
57C23×C4D411yescharacteristic
58C23×C4D411yescharacteristic
59C23×C4C2311yescharacteristic
60C2×SD16—24no—
61C2×SD16—42no—
62C2×SD16—42no—
63C2×D8—24no—
64C2×D8—42no—
65C2×D8—42no—
66C22×C8C2312yes—
67C22×C8C2312yes—
68C4⋊Q8—24no—
69#127—24no—
70#124—28no—
71#124—24no—
72C22⋊Q8—42no—
73#121—28no—
74#121—24no—
75#121—24no—
76#121—42no—
77C4×D4—28no—
78C4×D4—24no—
79C4×D4D418yescomplemented
80C2×C4⋊C4—21no—
81C2×C4⋊C4—21no—
82C2×C4⋊C4C2311yescharacteristic
83C2×C4⋊C4C2311yescharacteristic
84C2×C4⋊C4C2312yes—
85C2×C22⋊C4—22no—
86C2×C22⋊C4—22no—
87C2×C22⋊C4—21no—
88C2×C22⋊C4—21no—
89C2×C22⋊C4—21no—
90C2×C22⋊C4—21no—
91C2×C22⋊C4—21no—
92C2×C22⋊C4—21no—
93C2×C42—24no—
94C2×C42D411yescharacteristic
95C2×C42D411yescharacteristic
96C2×C42C2311yescharacteristic
97C2×C42C2311yescharacteristic
98C4⋊C8—24no—
99#103—24no—
100#102—24no—
101#102—42no—
102#102—42no—
103C22⋊C8—42no—
104#95—21no—
105#95—21no—
106#95—21no—
107#95—21no—
108C24—22no—
109C24—21no—
110C24C2×D411yescharacteristic
111#41—432no—
112C2×Q8—24no—
113C2×Q8—22no—
114C2×Q8—22no—
115C2×Q8—21no—
116C2×Q8—21no—
117C2×Q8—42no—
118C2×Q8—42no—
119C2×D4—24no—
120C2×D4—28no—
121C2×D4—24no—
122C2×D4—22no—
123C2×D4—22no—
124C2×D4—22no—
125C2×D4—22no—
126C2×D4—22no—
127C2×D4—22no—
128C2×D4—21no—
129C2×D4—22no—
130C2×D4—21no—
131C2×D4—42no—
132C2×D4—42no—
133C2×D4—44no—
134C2×D4—44no—
135C2×D4—42no—
136C2×D4—42no—
137C2×D4—42no—
138C2×D4—42no—
139C2×D4—42no—
140C2×D4C2×D414yes—
141C22×C4—22no—
142C22×C4—22no—
143C22×C4—22no—
144C22×C4—22no—
145C22×C4—22no—
146C22×C4—28no—
147C22×C4—22no—
148C22×C4—22no—
149C22×C4—21no—
150C22×C4—22no—
151C22×C4—21no—
152C22×C4—21no—
153C22×C4—21no—
154C22×C4—21no—
155C22×C4—41no—
156C22×C4—44no—
157C22×C4—44no—
158C22×C4C2×D411yescharacteristic
159C22×C4C2×D411yescharacteristic
160C22×C4C2×D411yescharacteristic
161C22×C4C2×D411yescharacteristic
162C22×C4C2×D411yescharacteristic
163C22×C4C2411yescharacteristic
164SD16—48no—
165D8—48no—
166C2×C8—22no—
167C2×C8—22no—
168C2×C8—22no—
169C2×C8—22no—
170C2×C8—22no—
171C2×C8—22no—
172C4⋊C4—24no—
173C4⋊C4—22no—
174C4⋊C4—22no—
175C4⋊C4—22no—
176C4⋊C4—42no—
177C4⋊C4C2×D414yes—
178C22⋊C4—28no—
179C22⋊C4—28no—
180C22⋊C4—42no—
181C22⋊C4—42no—
182C22⋊C4—42no—
183C22⋊C4—42no—
184C22⋊C4—42no—
185C22⋊C4—42no—
186C42—22no—
187C42—22no—
188C42—22no—
189C42C2×D414yes—
190C23—21no—
191C23—21no—
192C23—21no—
193C23—21no—
194C23—21no—
195C23—21no—
196C23—24no—
197C23—42no—
198C23—42no—
199C23—42no—
200C23—42no—
201C23—42no—
202C23—42no—
203C23—41no—
204C23—41no—
205C23—41no—
206C23—41no—
207C23—41no—
208C23—41no—
209C23—41no—
210C23AGL(1,ℤ/8)12yes—
211C23C22×D411yescharacteristic
212Q8—48no—
213Q8—42no—
214Q8—42no—
215D4—24no—
216D4—24no—
217D4—48no—
218D4—48no—
219D4—416no—
220D4—42no—
221D4—42no—
222D4—42no—
223D4—42no—
224D4—84no—
225C2×C4—28no—
226C2×C4—21no—
227C2×C4—21no—
228C2×C4—21no—
229C2×C4—21no—
230C2×C4—21no—
231C2×C4—21no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—24no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—44no—
239C2×C4—416no—
240C2×C4—42no—
241C2×C4—42no—
242C2×C4—42no—
243C2×C4—42no—
244C2×C4—41no—
245C2×C4—41no—
246C2×C4—41no—
247C2×C4—41no—
248C2×C4—41no—
249C2×C4—41no—
250C2×C4—41no—
251C2×C4—41no—
252C2×C4—41no—
253C2×C4#12012yes—
254C2×C4#12012yes—
255C2×C4#13512yes—
256C2×C4AGL(1,ℤ/8)12yes—
257C2×C4C22×D412yes—
258C2×C4H(5,𝔽2)11yescharacteristic
259C2×C4H(5,𝔽2)12yes—
260C2×C42−1+411yescharacteristic
261C8—42no—
262C8—42no—
263C22—22no—
264C22—22no—
265C22—22no—
266C22—41no—
267C22—41no—
268C22—41no—
269C22—41no—
270C22—44no—
271C22—44no—
272C22—44no—
273C22—82no—
274C22—82no—
275C22—82no—
276C22—82no—
277C22—81no—
278C22—81no—
279C22—81no—
280C22—81no—
281C22—81no—
282C22—81no—
283C22—81no—
284C22C2×#12011yescharacteristic
285C22#53511yescharacteristic
286C22#53711yescharacteristic
287C22C2×#13511yescharacteristic
288C22C2×AGL(1,ℤ/8)11yescharacteristic
289C22C2×AGL(1,ℤ/8)11yescharacteristic
290C22#57511yescharacteristic
291C4—22no—
292C4—22no—
293C4—22no—
294C4—22no—
295C4—41no—
296C4—44no—
297C4—81no—
298C4—81no—
299C4—81no—
300C4#53512yes—
301C4#53712yes—
302C2—42no—
303C2—84no—
304C2—161no—
305C2#261511yescharacteristic
306C2#300211yescharacteristic
307C2#300311yescharacteristic
308C2#320711yescharacteristic
309C2#321311yescharacteristic
310C2#324711yescharacteristic
311C2#325111yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
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
1. C2×#532C2semidirect
2. C2×#463C2semidirect
3. #2850C2semidirect
4. #2024C2semidirect
5. #2023C2semidirect
6. #2014C2semidirect
7. #2011C2semidirect
8. #2000C2semidirect
9. #1999C2semidirect
10. #1804C2semidirect
11. #1797C2semidirect
79. C4×D4D4semidirect

Maximal subgroups

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

subgroupindex
C2×#5322
C2×#4632
#28502
#20242
#20232
#20142
#20112
#20002
#19992
#18042
#17972