Finite Group Databasebeta

← #25733

Subgroups — #25733

Order 256.

783 conjugacy classes of subgroups; only the 151 normal subgroups are shown.

normal subgroups only

This group has 783 conjugacy classes of subgroups — too many to draw. Only its 151 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×#536C211yesmaximal, characteristic, complemented
2C2×#496C211yesmaximal, characteristic, complemented
3C2×#492C211yesmaximal, characteristic, complemented
4C2×#449C212yesmaximal, complemented
5C2×#448C212yesmaximal, complemented
6#2619C211yesmaximal, characteristic, complemented
7#2013C212yesmaximal, complemented
8#2012C212yesmaximal, complemented
9#1960C211yesmaximal, characteristic, complemented
10#1958C211yesmaximal, characteristic, complemented
11#1750C211yesmaximal, characteristic, complemented
12C22×#41C2211yescharacteristic
13C23×Q8C2211yescharacteristic
14C22×SD16C2212yes—
15C22×SD16C2212yes—
16C22×SD16C2212yes—
17C22×D8C2212yes—
18#536—24no—
19#536—24no—
20C2×C4⋊Q8C2211yescharacteristic
21C2×C4⋊Q8C2211yescharacteristic
22C2×#127C2211yescharacteristic
23C2×#124C2211yescharacteristic
24C2×#124C2211yescharacteristic
25C2×#123—22no—
26C2×C22⋊Q8—22no—
27C2×#121C2212yes—
28C2×#117C2211yescharacteristic
29#496—24no—
30#492—24no—
31#449—28no—
32#448—28no—
33C2×#103C2212yes—
34C2×#103C2212yes—
35C2×#103C2212yes—
36C2×#102C2212yes—
37C2×C22⋊C8C2212yes—
38C2×#97C2211yescharacteristic
39C2×C4×C8C2211yescharacteristic
40#394—22no—
41#392—22no—
42#390C2212yes—
43#382C2211yescharacteristic
44#336C2212yes—
45C2×#41—22no—
46C2×#41—22no—
47C2×#41—24no—
48C2×#41—42no—
49C22×Q8—22no—
50C22×Q8—22no—
51C22×Q8—21no—
52C22×Q8—24no—
53C22×Q8—42no—
54C22×Q8C2311yescharacteristic
55C22×Q8C2311yescharacteristic
56C22×Q8C2311yescharacteristic
57C22×D4—21no—
58C22×D4—22no—
59C22×D4—22no—
60C22×D4C2312yes—
61C22×D4C2311yescharacteristic
62C23×C4—21no—
63C23×C4—21no—
64C23×C4C2311yescharacteristic
65C2×SD16—24no—
66C2×SD16—24no—
67C2×SD16—24no—
68C2×SD16—24no—
69C2×SD16—24no—
70C2×SD16—24no—
71C2×SD16—42no—
72C2×SD16—42no—
73C2×SD16—42no—
74C2×D8—24no—
75C2×D8—24no—
76C2×D8—42no—
77C22×C8C2312yes—
78C22×C8C2312yes—
79C4⋊Q8—22no—
80C4⋊Q8—22no—
81C4⋊Q8—22no—
82C4⋊Q8D414yescomplemented
83#127—22no—
84#127—22no—
85#124—22no—
86#124—22no—
87#124—22no—
88#124D414yescomplemented
89#123—216no—
90C22⋊Q8—216no—
91#121—24no—
92#121—24no—
93#117—22no—
94#117—22no—
95C2×C4⋊C4—22no—
96C2×C4⋊C4—22no—
97C2×C4⋊C4D412yes—
98C2×C4⋊C4C2312yes—
99C2×C22⋊C4—22no—
100C2×C22⋊C4—22no—
101C2×C22⋊C4—22no—
102C2×C22⋊C4—24no—
103C2×C22⋊C4D412yes—
104C2×C42C2311yescharacteristic
105C2×C42C2311yescharacteristic
106#103—24no—
107#103—24no—
108#103—24no—
109#102—24no—
110C22⋊C8D418yescomplemented
111#97—22no—
112C4×C8—22no—
113#95—22no—
114C24—21no—
115C24—22no—
116C24C2×D411yescharacteristic
117#41—48no—
118#41—48no—
119C2×Q8—24no—
120C2×Q8—21no—
121C2×Q8—21no—
122C2×Q8—21no—
123C2×Q8—22no—
124C2×Q8—22no—
125C2×Q8—22no—
126C2×Q8—22no—
127C2×Q8—22no—
128C2×Q8—22no—
129C2×Q8—48no—
130C2×Q8—48no—
131C2×Q8—42no—
132C2×Q8—42no—
133C2×Q8C2×D412yes—
134C2×Q8C2×D412yes—
135C2×Q8C2×D412yes—
136C2×D4—24no—
137C2×D4—22no—
138C2×D4—21no—
139C2×D4—22no—
140C2×D4—22no—
141C2×D4—22no—
142C2×D4—22no—
143C2×D4—22no—
144C2×D4—22no—
145C2×D4—22no—
146C2×D4—42no—
147C2×D4—42no—
148C2×D4—44no—
149C2×D4—44no—
150C2×D4—84no—
151C2×D4—84no—
152C2×D4C2×D412yes—
153C22×C4—21no—
154C22×C4—21no—
155C22×C4—22no—
156C22×C4—24no—
157C22×C4—24no—
158C22×C4—21no—
159C22×C4—21no—
160C22×C4—21no—
161C22×C4—22no—
162C22×C4—42no—
163C22×C4—42no—
164C22×C4—42no—
165C22×C4—42no—
166C22×C4C2×D412yes—
167C22×C4C2×D412yes—
168C22×C4C2×D411yescharacteristic
169C22×C4C2×D414yes—
170C22×C4C2411yescharacteristic
171SD16—48no—
172SD16—48no—
173SD16—48no—
174D8—48no—
175C2×C8—22no—
176C2×C8—22no—
177C2×C8—22no—
178C2×C8—22no—
179C2×C8C2×D414yes—
180C2×C8C2×D414yes—
181C4⋊C4—28no—
182C4⋊C4—28no—
183C4⋊C4—24no—
184C4⋊C4—22no—
185C4⋊C4—22no—
186C22⋊C4—28no—
187C22⋊C4—28no—
188C22⋊C4—24no—
189C22⋊C4—44no—
190C22⋊C4—84no—
191C42—21no—
192C42—21no—
193C42C2×D412yes—
194C42C2×D412yes—
195C23—21no—
196C23—21no—
197C23—24no—
198C23—22no—
199C23—41no—
200C23—41no—
201C23—44no—
202C23—42no—
203C23—42no—
204C23—42no—
205C23—44no—
206C23—44no—
207C23#12712yes—
208C23C22×D411yescharacteristic
209Q8—24no—
210Q8—24no—
211Q8—24no—
212Q8—48no—
213Q8—42no—
214Q8—42no—
215Q8—42no—
216D4—24no—
217D4—48no—
218D4—42no—
219D4—42no—
220D4—42no—
221D4—42no—
222D4—42no—
223D4—88no—
224D4—88no—
225C2×C4—21no—
226C2×C4—21no—
227C2×C4—24no—
228C2×C4—24no—
229C2×C4—24no—
230C2×C4—22no—
231C2×C4—22no—
232C2×C4—22no—
233C2×C4—22no—
234C2×C4—24no—
235C2×C4—48no—
236C2×C4—48no—
237C2×C4—41no—
238C2×C4—41no—
239C2×C4—41no—
240C2×C4—41no—
241C2×C4—41no—
242C2×C4—41no—
243C2×C4—42no—
244C2×C4—44no—
245C2×C4#12014yes—
246C2×C4#12712yes—
247C2×C4C22×D411yescharacteristic
248C2×C4C22×D411yescharacteristic
249C2×C4C22×D412yes—
250C2×C4H(5,𝔽2)12yes—
251C8—24no—
252C8—24no—
253C22—24no—
254C22—22no—
255C22—42no—
256C22—42no—
257C22—42no—
258C22—44no—
259C22—82no—
260C22—82no—
261C22—82no—
262C22—82no—
263C22—82no—
264C22—84no—
265C22—84no—
266C22C2×#12011yescharacteristic
267C22C2×#12711yescharacteristic
268C22#53511yescharacteristic
269C22#57714yes—
270C4—22no—
271C4—24no—
272C4—42no—
273C4—42no—
274C4—42no—
275C4—42no—
276C4—82no—
277C4D4212yes—
278C4D4212yes—
279C2—42no—
280C2—82no—
281C2—162no—
282C2#243311yescharacteristic
283C2#301012yes—
284C2#315412yes—
285C2#322112yes—
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#536C2semidirect
2. C2×#496C2semidirect
3. C2×#492C2semidirect
4. C2×#449C2semidirect
5. C2×#448C2semidirect
6. #2619C2semidirect
7. #2013C2semidirect
8. #2012C2semidirect
9. #1960C2semidirect
10. #1958C2semidirect
11. #1750C2semidirect
82. C4⋊Q8D4semidirect
88. #124D4semidirect
110. C22⋊C8D4semidirect

Maximal subgroups

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

subgroupindex
C2×#5362
C2×#4962
C2×#4922
C2×#4492
C2×#4482
#26192
#20132
#20122
#19602
#19582
#17502