Finite Group Databasebeta

← #32909

Subgroups — #32909

Order 256.

731 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 731 conjugacy classes of subgroups — too many to draw. Only its 165 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
1C22×C4×Q8C211yesmaximal, characteristic, complemented
2C2×#464C212yesmaximal, complemented
3C2×#451C211yesmaximal, characteristic, complemented
4C2×#448C211yesmaximal, characteristic
5#2850C211yesmaximal, characteristic
6#1880C212yesmaximal, complemented
7#1878C212yesmaximal, complemented
8#1874C211yesmaximal, characteristic
9#1872C211yesmaximal, characteristic, complemented
10#1805C212yesmaximal, complemented
11#1797C211yesmaximal, characteristic, complemented
12C23×Q8C2211yescharacteristic
13C22×Q16C2212yes—
14C22×SD16C2212yes—
15C2×#124C2211yescharacteristic
16C2×C22⋊Q8C2211yescharacteristic
17C2×#121C2211yescharacteristic
18C2×C4×Q8—28no—
19C2×C4×Q8—21no—
20C2×C4×Q8—21no—
21C2×C4×Q8C2212yes—
22C2×C4×Q8C2212yes—
23C22×C4⋊C4—21no—
24C22×C4⋊C4C2211yescharacteristic
25C22×C42—21no—
26C22×C42C2211yescharacteristic
27#464—28no—
28#451—24no—
29#448—24no—
30C2×C4⋊C8C2212yes—
31C2×#103C2212yes—
32C2×#103C2212yes—
33C2×#103C2212yes—
34C2×#102C2212yes—
35C2×C22⋊C8C2211yescharacteristic
36C2×C22⋊C8C2211yescharacteristic
37#391C2211yescharacteristic
38#390C2211yescharacteristic
39#388—21no—
40#388—21no—
41#386—21no—
42#386—21no—
43#382C2211yescharacteristic
44#340C2212yes—
45#340C2212yes—
46#336C2212yes—
47C22×Q8—24no—
48C22×Q8—24no—
49C22×Q8—21no—
50C22×Q8—22no—
51C22×Q8D414yes—
52C22×Q8C2311yescharacteristic
53C22×Q8C2312yes—
54C22×D4—21no—
55C22×D4C2311yescharacteristic
56C23×C4—21no—
57C23×C4—21no—
58C23×C4D411yescharacteristic
59C23×C4D411yescharacteristic
60C23×C4C2311yescharacteristic
61C2×Q16—24no—
62C2×Q16—24no—
63C2×Q16—24no—
64C2×SD16—24no—
65C2×SD16—24no—
66C2×SD16—24no—
67C22×C8C2312yes—
68C22×C8C2312yes—
69#124—24no—
70C22⋊Q8—22no—
71C22⋊Q8—22no—
72#121—22no—
73#121—22no—
74C4×Q8—28no—
75C4×Q8—28no—
76C4×Q8—24no—
77C4×Q8—24no—
78C4×Q8—24no—
79C4×Q8D418yescomplemented
80C2×C4⋊C4—28no—
81C2×C4⋊C4—22no—
82C2×C4⋊C4—22no—
83C2×C4⋊C4—21no—
84C2×C4⋊C4—24no—
85C2×C4⋊C4—21no—
86C2×C4⋊C4C2311yescharacteristic
87C2×C4⋊C4C2311yescharacteristic
88C2×C4⋊C4C2312yes—
89C2×C22⋊C4—21no—
90C2×C22⋊C4—21no—
91C2×C22⋊C4—21no—
92C2×C22⋊C4—21no—
93C2×C22⋊C4—21no—
94C2×C22⋊C4—21no—
95C2×C42—28no—
96C2×C42—22no—
97C2×C42—22no—
98C2×C42—24no—
99C2×C42D411yescharacteristic
100C2×C42D411yescharacteristic
101C2×C42C2311yescharacteristic
102C2×C42C2311yescharacteristic
103C4⋊C8—24no—
104#103—24no—
105#103—24no—
106#103—24no—
107#102—24no—
108C22⋊C8—22no—
109C22⋊C8—22no—
110#95—21no—
111#95—21no—
112#95—21no—
113#95—21no—
114C24—21no—
115C24C2×D411yescharacteristic
116C2×Q8—24no—
117C2×Q8—216no—
118C2×Q8—24no—
119C2×Q8—216no—
120C2×Q8—24no—
121C2×Q8—22no—
122C2×Q8—22no—
123C2×Q8—22no—
124C2×Q8—22no—
125C2×Q8—21no—
126C2×Q8—21no—
127C2×Q8—22no—
128C2×Q8C2×D414yes—
129C2×Q8C2×D414yes—
130C2×Q8#4114yes—
131C2×D4—22no—
132C2×D4—21no—
133C2×D4—21no—
134C2×D4—22no—
135C2×D4—42no—
136C2×D4—42no—
137C2×D4—42no—
138C22×C4—24no—
139C22×C4—24no—
140C22×C4—24no—
141C22×C4—22no—
142C22×C4—22no—
143C22×C4—22no—
144C22×C4—22no—
145C22×C4—22no—
146C22×C4—22no—
147C22×C4—24no—
148C22×C4—22no—
149C22×C4—22no—
150C22×C4—22no—
151C22×C4—21no—
152C22×C4—21no—
153C22×C4—21no—
154C22×C4—41no—
155C22×C4—42no—
156C22×C4—42no—
157C22×C4C2×D411yescharacteristic
158C22×C4C2×D411yescharacteristic
159C22×C4C2×D411yescharacteristic
160C22×C4C2×D411yescharacteristic
161C22×C4C2×D412yes—
162C22×C4C2×D411yescharacteristic
163C22×C4#4112yes—
164C22×C4C2411yescharacteristic
165Q16—48no—
166SD16—48no—
167C2×C8—22no—
168C2×C8—22no—
169C2×C8—22no—
170C2×C8—22no—
171C2×C8—22no—
172C2×C8—22no—
173C4⋊C4—28no—
174C4⋊C4—24no—
175C4⋊C4—28no—
176C4⋊C4—22no—
177C4⋊C4—21no—
178C4⋊C4—21no—
179C4⋊C4—21no—
180C4⋊C4—21no—
181C4⋊C4—42no—
182C4⋊C4C2×D414yes—
183C22⋊C4—42no—
184C22⋊C4—42no—
185C22⋊C4—42no—
186C22⋊C4—42no—
187C22⋊C4—42no—
188C22⋊C4—42no—
189C42—28no—
190C42—28no—
191C42—22no—
192C42—22no—
193C42—22no—
194C42C2×D414yes—
195C23—21no—
196C23—21no—
197C23—21no—
198C23—21no—
199C23—41no—
200C23—41no—
201C23—41no—
202C23—41no—
203C23—41no—
204C23—41no—
205C23—41no—
206C23#12112yes—
207C23#13512yes—
208C23#13712yes—
209C23C22×D411yescharacteristic
210Q8—216no—
211Q8—24no—
212Q8—24no—
213Q8—24no—
214Q8—42no—
215Q8—42no—
216Q8#12118yes—
217D4—42no—
218D4—42no—
219D4—84no—
220C2×C4—28no—
221C2×C4—216no—
222C2×C4—21no—
223C2×C4—24no—
224C2×C4—21no—
225C2×C4—24no—
226C2×C4—24no—
227C2×C4—21no—
228C2×C4—21no—
229C2×C4—21no—
230C2×C4—21no—
231C2×C4—21no—
232C2×C4—21no—
233C2×C4—24no—
234C2×C4—24no—
235C2×C4—21no—
236C2×C4—24no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—44no—
240C2×C4—48no—
241C2×C4—42no—
242C2×C4—42no—
243C2×C4—41no—
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#12012yes—
253C2×C4#12012yes—
254C2×C4#12112yes—
255C2×C4#13712yes—
256C2×C4C22×D411yescharacteristic
257C2×C4C22×D412yes—
258C2×C4C2×#4112yes—
259C2×C4C2×#4111yescharacteristic
260C8—42no—
261C8—42no—
262C22—22no—
263C22—22no—
264C22—22no—
265C22—22no—
266C22—22no—
267C22—22no—
268C22—81no—
269C22—81no—
270C22—81no—
271C22—81no—
272C22—81no—
273C22—81no—
274C22—81no—
275C22#46414yescomplemented
276C22C2×#12011yescharacteristic
277C22C2×#12111yescharacteristic
278C22#52511yescharacteristic
279C22C2×#13511yescharacteristic
280C22C2×#13711yescharacteristic
281C22C2×#13711yescharacteristic
282C22#57511yescharacteristic
283C4—28no—
284C4—22no—
285C4—22no—
286C4—22no—
287C4—22no—
288C4—41no—
289C4—44no—
290C4—81no—
291C4—81no—
292C4—81no—
293C4C2×#12112yes—
294C4#52512yes—
295C2—24no—
296C2—161no—
297C2#284711yescharacteristic
298C2#300211yescharacteristic
299C2#300411yescharacteristic
300C2C2×#46411yescharacteristic
301C2#303711yescharacteristic
302C2#310811yescharacteristic
303C2#311711yescharacteristic
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
1. C22×C4×Q8C2semidirect
2. C2×#464C2semidirect
3. C2×#451C2semidirect
6. #1880C2semidirect
7. #1878C2semidirect
9. #1872C2semidirect
10. #1805C2semidirect
11. #1797C2semidirect
79. C4×Q8D4semidirect
275. C22#464semidirect

Maximal subgroups

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

subgroupindex
C22×C4×Q82
C2×#4642
C2×#4512
C2×#4482
#28502
#18802
#18782
#18742
#18722
#18052
#17972