Finite Group Databasebeta

← #25892

Subgroups — #25892

Order 256.

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

normal subgroups only

This group has 646 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×#547C211yesmaximal, characteristic
2C2×#491C211yesmaximal, characteristic, complemented
3C2×#478C211yesmaximal, characteristic, complemented
4#2562C211yesmaximal, characteristic, complemented
5#2036C212yesmaximal, complemented
6#2035C212yesmaximal, complemented
7#1981C211yesmaximal, characteristic, complemented
8#1955C211yesmaximal, characteristic, complemented
9#1788C212yesmaximal, complemented
10#1788C212yesmaximal, complemented
11#1762C211yesmaximal, characteristic, complemented
12C22×#41C2212yes—
13#547—24no—
14#547—24no—
15C2×C4⋊Q8C2211yescharacteristic
16C2×#125C2211yescharacteristic
17C2×#123—22no—
18C2×C22⋊Q8C2212yes—
19C2×#121—21no—
20C2×C4×D4C2211yescharacteristic
21C2×C4×D4C2211yescharacteristic
22C22×C4⋊C4—21no—
23#491—24no—
24#478—24no—
25C2×#107C2211yescharacteristic
26C2×#106C2211yescharacteristic
27C2×C4⋊C8C2211yescharacteristic
28C2×#103C2212yes—
29C2×#103C2212yes—
30C2×#103C2212yes—
31C2×#103C2212yes—
32C2×#102C2211yescharacteristic
33C2×#102C2211yescharacteristic
34C2×#102C2212yes—
35C2×C22⋊C8C2212yes—
36C2×C22⋊C8C2212yes—
37C2×#97C2211yescharacteristic
38#400—21no—
39#394—21no—
40#388—22no—
41#380C2212yes—
42#380C2212yes—
43C4×C4⋊C4C2211yescharacteristic
44#340C2212yes—
45C2×#41—24no—
46C2×#41—24no—
47C2×#41—24no—
48C2×#41—24no—
49C2×#41—42no—
50C2×#41—42no—
51C22×Q8C2312yes—
52C22×D4—22no—
53C22×D4D411yescharacteristic
54C22×D4C2311yescharacteristic
55C23×C4—21no—
56C23×C4—22no—
57C23×C4C2312yes—
58C22×C8C2312yes—
59C22×C8C2311yescharacteristic
60C22×C8C2311yescharacteristic
61C4⋊Q8—22no—
62C4⋊Q8D414yes—
63#125—22no—
64#125—22no—
65#123—216no—
66C22⋊Q8—24no—
67C22⋊Q8—24no—
68#121—28no—
69C4×D4—22no—
70C4×D4—22no—
71C4×D4—22no—
72C4×D4D414yes—
73C2×C4⋊C4—28no—
74C2×C4⋊C4—21no—
75C2×C4⋊C4—22no—
76C2×C4⋊C4—21no—
77C2×C4⋊C4—21no—
78C2×C4⋊C4—21no—
79C2×C4⋊C4D411yescharacteristic
80C2×C4⋊C4C2311yescharacteristic
81C2×C4⋊C4C2311yescharacteristic
82C2×C4⋊C4C2312yes—
83C2×C4⋊C4C2311yescharacteristic
84C2×C22⋊C4—22no—
85C2×C22⋊C4—21no—
86C2×C22⋊C4—22no—
87C2×C22⋊C4—22no—
88C2×C22⋊C4D412yes—
89C2×C42—21no—
90C2×C42C2311yescharacteristic
91#107—22no—
92#106—22no—
93C4⋊C8—22no—
94#103—24no—
95#103—24no—
96#103—42no—
97#103—42no—
98#102—22no—
99#102—22no—
100#102—24no—
101C22⋊C8—42no—
102C22⋊C8—42no—
103#97—22no—
104#95—21no—
105#95—22no—
106#95—22no—
107C24—21no—
108C24C2×D412yes—
109#41—48no—
110#41—48no—
111#41—48no—
112#41—48no—
113C2×Q8—22no—
114C2×Q8—22no—
115C2×Q8—22no—
116C2×Q8—22no—
117C2×Q8—22no—
118C2×Q8C2×D414yes—
119C2×D4—28no—
120C2×D4—22no—
121C2×D4—21no—
122C2×D4—22no—
123C2×D4—42no—
124C2×D4—44no—
125C2×D4—44no—
126C2×D4C2×D412yes—
127C2×D4#4114yes—
128C22×C4—28no—
129C22×C4—21no—
130C22×C4—22no—
131C22×C4—24no—
132C22×C4—22no—
133C22×C4—22no—
134C22×C4—22no—
135C22×C4—22no—
136C22×C4—22no—
137C22×C4—22no—
138C22×C4—22no—
139C22×C4—21no—
140C22×C4—21no—
141C22×C4—44no—
142C22×C4—44no—
143C22×C4—44no—
144C22×C4C2×D412yes—
145C22×C4C2×D411yescharacteristic
146C22×C4C2×D411yescharacteristic
147C22×C4C2411yescharacteristic
148C2×C8—22no—
149C2×C8—21no—
150C2×C8—21no—
151C2×C8—22no—
152C2×C8—22no—
153C2×C8—22no—
154C2×C8—22no—
155C4⋊C4—28no—
156C4⋊C4—22no—
157C4⋊C4—24no—
158C4⋊C4—24no—
159C4⋊C4—21no—
160C4⋊C4—21no—
161C4⋊C4—21no—
162C4⋊C4—21no—
163C4⋊C4—21no—
164C4⋊C4—22no—
165C4⋊C4—22no—
166C4⋊C4—42no—
167C4⋊C4—42no—
168C4⋊C4C2×D412yes—
169C22⋊C4—28no—
170C22⋊C4—24no—
171C22⋊C4—24no—
172C22⋊C4—44no—
173C22⋊C4—82no—
174C42—81no—
175C42C2×D412yes—
176C42C2×D412yes—
177C23—22no—
178C23—24no—
179C23—22no—
180C23—22no—
181C23—22no—
182C23—22no—
183C23—22no—
184C23—22no—
185C23—22no—
186C23—42no—
187C23—41no—
188C23—41no—
189C23C22×D411yescharacteristic
190Q8—24no—
191Q8—24no—
192Q8—42no—
193Q8—42no—
194D4—22no—
195D4—24no—
196D4—44no—
197D4—416no—
198D4#12414yes—
199C2×C4—22no—
200C2×C4—21no—
201C2×C4—21no—
202C2×C4—22no—
203C2×C4—22no—
204C2×C4—24no—
205C2×C4—22no—
206C2×C4—22no—
207C2×C4—48no—
208C2×C4—416no—
209C2×C4—42no—
210C2×C4—42no—
211C2×C4—42no—
212C2×C4—42no—
213C2×C4—42no—
214C2×C4—42no—
215C2×C4—42no—
216C2×C4—41no—
217C2×C4—41no—
218C2×C4—42no—
219C2×C4—42no—
220C2×C4—42no—
221C2×C4—42no—
222C2×C4—41no—
223C2×C4—41no—
224C2×C4#12012yes—
225C2×C4#12012yes—
226C2×C4C22×D411yescharacteristic
227C2×C4C22×D411yescharacteristic
228C2×C4C2×#4112yes—
229C2×C4H(5,𝔽2)12yes—
230C8—41no—
231C8—41no—
232C8—42no—
233C22—24no—
234C22—42no—
235C22—42no—
236C22—44no—
237C22—44no—
238C22—42no—
239C22—42no—
240C22—42no—
241C22—42no—
242C22—42no—
243C22—42no—
244C22—42no—
245C22C2×#12011yescharacteristic
246C22#54612yes—
247C22#57512yes—
248C22#57711yescharacteristic
249C22#57811yescharacteristic
250C4—22no—
251C4—22no—
252C4—42no—
253C4—44no—
254C4—42no—
255C4—81no—
256C4—81no—
257C4—82no—
258C4C2×#12411yescharacteristic
259C4#53511yescharacteristic
260C4#54612yes—
261C2—44no—
262C2—82no—
263C2#243511yescharacteristic
264C2#300511yescharacteristic
265C2#301311yescharacteristic
266C2#330712yes—
267C2#331512yes—
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC2×#547C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. C2×#491C2semidirect
3. C2×#478C2semidirect
4. #2562C2semidirect
5. #2036C2semidirect
6. #2035C2semidirect
7. #1981C2semidirect
8. #1955C2semidirect
9. #1788C2semidirect
10. #1788C2semidirect
11. #1762C2semidirect

Maximal subgroups

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

subgroupindex
C2×#5472
C2×#4912
C2×#4782
#25622
#20362
#20352
#19812
#19552
#17882
#17622