Finite Group Databasebeta

← #29815

Subgroups — #29815

Order 256.

460 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 460 conjugacy classes of subgroups — too many to draw. Only its 123 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×#497C211yesmaximal, characteristic, complemented
2C2×#432C211yesmaximal, characteristic, complemented
3#2857C211yesmaximal, characteristic, complemented
4#2850C211yesmaximal, characteristic
5#2056C212yesmaximal, complemented
6#2055C212yesmaximal, complemented
7#2001C212yesmaximal, complemented
8#2000C212yesmaximal
9#1960C211yesmaximal, characteristic, complemented
10#1959C211yesmaximal, characteristic, complemented
11#1836C211yesmaximal, characteristic, complemented
12C22×Q16C2212yes—
13C22×SD16C2212yes—
14C2×C4⋊Q8C2211yescharacteristic
15C2×#124C2211yescharacteristic
16C2×C22⋊Q8C2211yescharacteristic
17C2×C22⋊Q8C2212yes—
18C2×#121C2211yescharacteristic
19C22×C42C2211yescharacteristic
20#497—24no—
21#432—24no—
22C2×C4⋊C8C2212yes—
23C2×#103C2212yes—
24C2×#103C2212yes—
25C2×#103C2212yes—
26C2×#102C2212yes—
27C2×C22⋊C8C2212yes—
28C2×#97C2211yescharacteristic
29C2×#97C2211yescharacteristic
30#391C2212yes—
31#391C2211yescharacteristic
32#390C2211yescharacteristic
33#389—21no—
34#389—21no—
35#388—21no—
36#388—21no—
37#386—21no—
38#386—21no—
39#384C2211yescharacteristic
40#382C2211yescharacteristic
41#380—21no—
42#380—21no—
43#340C2212yes—
44#340C2212yes—
45C22×Q8C2311yescharacteristic
46C22×Q8C2312yes—
47C22×D4—21no—
48C22×D4C2311yescharacteristic
49C23×C4D411yescharacteristic
50C23×C4D411yescharacteristic
51C23×C4C2311yescharacteristic
52C2×Q16—24no—
53C2×Q16—42no—
54C2×Q16—42no—
55C2×SD16—24no—
56C2×SD16—42no—
57C2×SD16—42no—
58C22×C8C2312yes—
59C22×C8C2312yes—
60C4⋊Q8—22no—
61C4⋊Q8—22no—
62#124—22no—
63#124—22no—
64C22⋊Q8—42no—
65C22⋊Q8—42no—
66C22⋊Q8—42no—
67#121—42no—
68C2×C4⋊C4—21no—
69C2×C4⋊C4—22no—
70C2×C4⋊C4—21no—
71C2×C4⋊C4—21no—
72C2×C4⋊C4—21no—
73C2×C4⋊C4—21no—
74C2×C4⋊C4C2311yescharacteristic
75C2×C4⋊C4C2311yescharacteristic
76C2×C4⋊C4C2312yes—
77C2×C22⋊C4—22no—
78C2×C22⋊C4—21no—
79C2×C22⋊C4—21no—
80C2×C22⋊C4—21no—
81C2×C22⋊C4—21no—
82C2×C22⋊C4—21no—
83C2×C22⋊C4—21no—
84C2×C42—22no—
85C2×C42—22no—
86C2×C42D411yescharacteristic
87C2×C42D411yescharacteristic
88C2×C42C2311yescharacteristic
89C2×C42C2311yescharacteristic
90C4⋊C8—24no—
91#103—42no—
92#103—42no—
93#103—42no—
94#102—42no—
95C22⋊C8—24no—
96#97—22no—
97#97D414yescomplemented
98#95—22no—
99#95—22no—
100#95—21no—
101#95—21no—
102#95—21no—
103#95—21no—
104C24—21no—
105C24C2×D411yescharacteristic
106C2×Q8—21no—
107C2×Q8—22no—
108C2×Q8—21no—
109C2×Q8—22no—
110C2×Q8—22no—
111C2×Q8—22no—
112C2×Q8—22no—
113C2×Q8—22no—
114C2×Q8—22no—
115C2×Q8—22no—
116C2×D4—21no—
117C2×D4—21no—
118C2×D4—22no—
119C2×D4—22no—
120C2×D4—42no—
121C2×D4—42no—
122C2×D4—42no—
123C22×C4—22no—
124C22×C4—22no—
125C22×C4—22no—
126C22×C4—22no—
127C22×C4—21no—
128C22×C4—21no—
129C22×C4—22no—
130C22×C4—22no—
131C22×C4—21no—
132C22×C4—21no—
133C22×C4—21no—
134C22×C4—22no—
135C22×C4—22no—
136C22×C4—41no—
137C22×C4—41no—
138C22×C4C2×D411yescharacteristic
139C22×C4C2×D411yescharacteristic
140C22×C4C2×D411yescharacteristic
141C22×C4C2×D411yescharacteristic
142C22×C4C2×D411yescharacteristic
143C22×C4C2411yescharacteristic
144Q16—48no—
145SD16—48no—
146C2×C8—22no—
147C2×C8—22no—
148C2×C8—22no—
149C2×C8—22no—
150C2×C8—22no—
151C2×C8C2×D414yes—
152C4⋊C4—22no—
153C4⋊C4—22no—
154C4⋊C4—22no—
155C4⋊C4—22no—
156C4⋊C4—42no—
157C4⋊C4—42no—
158C4⋊C4—42no—
159C4⋊C4—44no—
160C4⋊C4—42no—
161C4⋊C4—42no—
162C22⋊C4—42no—
163C22⋊C4—42no—
164C22⋊C4—44no—
165C22⋊C4—42no—
166C22⋊C4—42no—
167C22⋊C4—42no—
168C22⋊C4—42no—
169C42—22no—
170C42—22no—
171C42—21no—
172C42—21no—
173C42C2×D412yes—
174C42#4112yes—
175C23—21no—
176C23—21no—
177C23—21no—
178C23—21no—
179C23—21no—
180C23—21no—
181C23—41no—
182C23—41no—
183C23—41no—
184C23—41no—
185C23—41no—
186C23—41no—
187C23—41no—
188C23#13712yes—
189C23C22×D411yescharacteristic
190Q8—42no—
191Q8—42no—
192Q8—42no—
193Q8—42no—
194Q8—42no—
195Q8—42no—
196D4—42no—
197D4—42no—
198D4—84no—
199C2×C4—21no—
200C2×C4—21no—
201C2×C4—21no—
202C2×C4—21no—
203C2×C4—21no—
204C2×C4—21no—
205C2×C4—21no—
206C2×C4—21no—
207C2×C4—21no—
208C2×C4—21no—
209C2×C4—21no—
210C2×C4—21no—
211C2×C4—44no—
212C2×C4—44no—
213C2×C4—42no—
214C2×C4—42no—
215C2×C4—41no—
216C2×C4—42no—
217C2×C4—41no—
218C2×C4—42no—
219C2×C4—41no—
220C2×C4—41no—
221C2×C4—42no—
222C2×C4—42no—
223C2×C4—41no—
224C2×C4—41no—
225C2×C4—41no—
226C2×C4—41no—
227C2×C4—42no—
228C2×C4—42no—
229C2×C4—41no—
230C2×C4#12112yes—
231C2×C4#12112yes—
232C2×C4#13712yes—
233C2×C4C22×D411yescharacteristic
234C2×C4C2×#4111yescharacteristic
235C2×C4H(5,𝔽2)12yes—
236C2×C4H(5,𝔽2)12yes—
237C8—24no—
238C8—42no—
239C22—22no—
240C22—22no—
241C22—22no—
242C22—41no—
243C22—41no—
244C22—41no—
245C22—41no—
246C22—81no—
247C22—81no—
248C22—81no—
249C22—81no—
250C22—81no—
251C22—81no—
252C22—81no—
253C22C2×#12111yescharacteristic
254C22#53511yescharacteristic
255C22#53511yescharacteristic
256C22C2×#13711yescharacteristic
257C22C2×#13711yescharacteristic
258C22#57511yescharacteristic
259C22#57511yescharacteristic
260C4—22no—
261C4—22no—
262C4—22no—
263C4—41no—
264C4—41no—
265C4—81no—
266C4—82no—
267C4—81no—
268C4—81no—
269C4—82no—
270C4#53512yes—
271C4#54012yes—
272C2—42no—
273C2—161no—
274C2#261911yescharacteristic
275C2#305611yescharacteristic
276C2#305611yescharacteristic
277C2#321511yescharacteristic
278C2#321611yescharacteristic
279C2#321811yescharacteristic
280C2#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
1. C2×#497C2semidirect
2. C2×#432C2semidirect
3. #2857C2semidirect
5. #2056C2semidirect
6. #2055C2semidirect
7. #2001C2semidirect
9. #1960C2semidirect
10. #1959C2semidirect
11. #1836C2semidirect
97. #97D4semidirect

Maximal subgroups

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

subgroupindex
C2×#4972
C2×#4322
#28572
#28502
#20562
#20552
#20012
#20002
#19602
#19592
#18362