Finite Group Databasebeta

← #29631

Subgroups — #29631

Order 256.

651 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 651 conjugacy classes of subgroups — too many to draw. Only its 127 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×#449C211yesmaximal, characteristic
3C2×#448C211yesmaximal, characteristic, complemented
4#2624C211yesmaximal, characteristic, complemented
5#2043C212yesmaximal, complemented
6#2042C212yesmaximal, complemented
7#2013C212yesmaximal, complemented
8#2012C212yesmaximal, complemented
9#1818C211yesmaximal, characteristic, complemented
10#1788C211yesmaximal, characteristic, complemented
11#1787C211yesmaximal, characteristic, complemented
12C22×#41C2211yescharacteristic
13C23×Q8C2211yescharacteristic
14C22×SD16C2212yes—
15C22×SD16C2211yescharacteristic
16C22×D8C2211yescharacteristic
17#536—24no—
18#536—24no—
19C2×C4⋊Q8C2212yes—
20C2×#124C2212yes—
21C2×#123—22no—
22C2×C22⋊Q8—21no—
23C2×C22⋊Q8—21no—
24C2×#121C2211yescharacteristic
25C2×#117C2211yescharacteristic
26#449—24no—
27#448—24no—
28C2×#103C2212yes—
29C2×#103C2211yescharacteristic
30C2×#103C2212yes—
31C2×#103C2212yes—
32C2×#103C2212yes—
33C2×#102C2211yescharacteristic
34C2×#102C2212yes—
35C2×C22⋊C8C2211yescharacteristic
36C2×C22⋊C8C2211yescharacteristic
37#399—21no—
38#397—21no—
39#394—22no—
40#390C2212yes—
41#389C2212yes—
42#380C2211yescharacteristic
43#336C2212yes—
44#336C2212yes—
45C2×#41—22no—
46C2×#41—42no—
47C2×#41—42no—
48C2×#41D414yes—
49C22×Q8—21no—
50C22×Q8—22no—
51C22×Q8—42no—
52C22×Q8—42no—
53C22×Q8D414yes—
54C22×Q8C2312yes—
55C22×Q8C2311yescharacteristic
56C22×D4—21no—
57C22×D4—21no—
58C22×D4C2311yescharacteristic
59C22×D4C2311yescharacteristic
60C23×C4—21no—
61C23×C4—21no—
62C23×C4C2311yescharacteristic
63C2×SD16—24no—
64C2×SD16—22no—
65C2×SD16—42no—
66C2×SD16—42no—
67C2×SD16—42no—
68C2×D8—22no—
69C2×D8—42no—
70C22×C8C2311yescharacteristic
71C22×C8C2311yescharacteristic
72C22×C8C2312yes—
73C4⋊Q8—24no—
74C4⋊Q8—24no—
75#124—24no—
76#124—24no—
77#123—216no—
78C22⋊Q8—28no—
79C22⋊Q8—28no—
80#121—22no—
81#121—22no—
82#117—22no—
83#117—22no—
84C2×C4⋊C4—22no—
85C2×C4⋊C4—22no—
86C2×C4⋊C4—22no—
87C2×C4⋊C4D411yescharacteristic
88C2×C4⋊C4D411yescharacteristic
89C2×C4⋊C4C2311yescharacteristic
90C2×C4⋊C4C2312yes—
91C2×C22⋊C4—22no—
92C2×C22⋊C4—22no—
93C2×C22⋊C4—21no—
94C2×C22⋊C4—21no—
95C2×C22⋊C4—22no—
96C2×C22⋊C4—22no—
97C2×C22⋊C4D412yes—
98C2×C42C2312yes—
99#103—24no—
100#103—22no—
101#103—42no—
102#103—42no—
103#103—42no—
104#102—22no—
105#102—42no—
106C22⋊C8—22no—
107C22⋊C8—22no—
108#95—21no—
109#95—22no—
110C24—21no—
111C24—21no—
112C24C2×D411yescharacteristic
113#41—44no—
114#41—44no—
115#41—48no—
116C2×Q8—24no—
117C2×Q8—22no—
118C2×Q8—21no—
119C2×Q8—22no—
120C2×Q8—22no—
121C2×Q8—22no—
122C2×Q8—22no—
123C2×Q8—22no—
124C2×Q8—22no—
125C2×Q8—48no—
126C2×Q8—44no—
127C2×Q8—44no—
128C2×Q8—44no—
129C2×Q8C2×D414yes—
130C2×Q8C2×D412yes—
131C2×D4—24no—
132C2×D4—21no—
133C2×D4—21no—
134C2×D4—21no—
135C2×D4—22no—
136C2×D4—22no—
137C2×D4—22no—
138C2×D4—22no—
139C2×D4—44no—
140C2×D4—42no—
141C2×D4—82no—
142C2×D4C2×D412yes—
143C22×C4—22no—
144C22×C4—22no—
145C22×C4—24no—
146C22×C4—22no—
147C22×C4—21no—
148C22×C4—24no—
149C22×C4—22no—
150C22×C4—21no—
151C22×C4—44no—
152C22×C4—44no—
153C22×C4C2×D412yes—
154C22×C4C2×D412yes—
155C22×C4C2×D411yescharacteristic
156C22×C4C2×D412yes—
157C22×C4C2×D412yes—
158C22×C4C2411yescharacteristic
159SD16—48no—
160SD16—44no—
161D8—44no—
162C2×C8—21no—
163C2×C8—21no—
164C2×C8—22no—
165C2×C8—22no—
166C2×C8—22no—
167C2×C8—22no—
168C2×C8—22no—
169C4⋊C4—28no—
170C4⋊C4—28no—
171C4⋊C4—22no—
172C4⋊C4—22no—
173C4⋊C4—21no—
174C4⋊C4—21no—
175C4⋊C4—22no—
176C4⋊C4—22no—
177C4⋊C4—82no—
178C22⋊C4—28no—
179C22⋊C4—28no—
180C22⋊C4—24no—
181C22⋊C4—42no—
182C22⋊C4—82no—
183C22⋊C4—82no—
184C22⋊C4—81no—
185C42—22no—
186C42—22no—
187C23—22no—
188C23—21no—
189C23—22no—
190C23—21no—
191C23—22no—
192C23—42no—
193C23—42no—
194C23—41no—
195C23—41no—
196C23—41no—
197C23—42no—
198C23—41no—
199C23—42no—
200C23—41no—
201C23#12012yes—
202C23C22×D411yescharacteristic
203Q8—24no—
204Q8—22no—
205Q8—24no—
206Q8—22no—
207Q8—48no—
208Q8—42no—
209Q8—42no—
210Q8—42no—
211D4—22no—
212D4—22no—
213D4—48no—
214D4—42no—
215D4—41no—
216D4—41no—
217D4—42no—
218D4—84no—
219C2×C4—22no—
220C2×C4—22no—
221C2×C4—22no—
222C2×C4—24no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—24no—
226C2×C4—22no—
227C2×C4—22no—
228C2×C4—22no—
229C2×C4—22no—
230C2×C4—22no—
231C2×C4—48no—
232C2×C4—48no—
233C2×C4—42no—
234C2×C4—42no—
235C2×C4—42no—
236C2×C4—41no—
237C2×C4—42no—
238C2×C4—42no—
239C2×C4—42no—
240C2×C4—42no—
241C2×C4#12012yes—
242C2×C4C22×D411yescharacteristic
243C2×C4C22×D411yescharacteristic
244C2×C4H(5,𝔽2)12yes—
245C2×C4H(5,𝔽2)12yes—
246C8—41no—
247C8—41no—
248C8—42no—
249C22—22no—
250C22—22no—
251C22—22no—
252C22—42no—
253C22—42no—
254C22—42no—
255C22—42no—
256C22—42no—
257C22—82no—
258C22—82no—
259C22—81no—
260C22—82no—
261C22—82no—
262C22—81no—
263C22—81no—
264C22C2×#12011yescharacteristic
265C22#53512yes—
266C22#57711yescharacteristic
267C22#57712yes—
268C22#57811yescharacteristic
269C4—22no—
270C4—42no—
271C4—44no—
272C4—42no—
273C4—42no—
274C4—82no—
275C4—81no—
276C4#53411yescharacteristic
277C4#53411yescharacteristic
278C4#53412yes—
279C2—42no—
280C2—82no—
281C2—161no—
282C2#261311yescharacteristic
283C2#301011yescharacteristic
284C2#301311yescharacteristic
285C2#322112yes—
286C2#322512yes—
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC2×#449C2
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
3. C2×#448C2semidirect
4. #2624C2semidirect
5. #2043C2semidirect
6. #2042C2semidirect
7. #2013C2semidirect
8. #2012C2semidirect
9. #1818C2semidirect
10. #1788C2semidirect
11. #1787C2semidirect

Maximal subgroups

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

subgroupindex
C2×#5362
C2×#4492
C2×#4482
#26242
#20432
#20422
#20132
#20122
#18182
#17882
#17872