Finite Group Databasebeta

← #23945

Subgroups — #23945

Order 256.

691 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 691 conjugacy classes of subgroups — too many to draw. Only its 131 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×#542C211yesmaximal, characteristic, complemented
2C2×#450C211yesmaximal, characteristic, complemented
3C2×#447C211yesmaximal, characteristic
4#2309C211yesmaximal, characteristic, complemented
5#2023C212yesmaximal, complemented
6#2012C212yesmaximal, complemented
7#1880C212yesmaximal, complemented
8#1879C212yesmaximal, complemented
9#1875C211yesmaximal, characteristic, complemented
10#1873C211yesmaximal, characteristic, complemented
11#1818C211yesmaximal, characteristic, complemented
12C23×D4C2211yescharacteristic
13C22×SD16C2212yes—
14C22×D8C2212yes—
15#542—24no—
16#542—24no—
17C2×C4⋊Q8C2211yescharacteristic, complemented
18C2×#126—21no—
19C2×#124—21no—
20C2×#124C2211yescharacteristic, complemented
21C2×#123—21no—
22C2×C22⋊Q8—21no—
23C2×C22⋊Q8C2211yescharacteristic
24C2×#121C2211yescharacteristic
25C2×C4×Q8C2211yescharacteristic, complemented
26C2×C4×D4C2211yescharacteristic, complemented
27C2×#117C2211yescharacteristic
28C2×#117C2211yescharacteristic
29#450—24no—
30#447—24no—
31C2×#103C2212yescomplemented
32C2×#102C2212yes—
33C2×#102C2212yes—
34C2×#102C2212yescomplemented
35C2×C22⋊C8C2211yescharacteristic
36C2×C22⋊C8C2211yescharacteristic
37#390—21no—
38#390—21no—
39#390C2212yes—
40#390C2212yes—
41#379—21no—
42#379—21no—
43#340C2212yescomplemented
44#340C2212yescomplemented
45#336C2212yescomplemented
46#336C2212yescomplemented
47C25—21no—
48C22×Q8—21no—
49C22×Q8C2311yescharacteristic
50C22×D4—21no—
51C22×D4—24no—
52C22×D4—21no—
53C22×D4—22no—
54C22×D4—22no—
55C22×D4—42no—
56C22×D4C2311yescharacteristic
57C22×D4C2312yes—
58C23×C4—21no—
59C23×C4C2311yescharacteristic
60C2×SD16—24no—
61C2×SD16—24no—
62C2×SD16—42no—
63C2×D8—24no—
64C2×D8—24no—
65C2×D8—42no—
66C22×C8C2312yes—
67C22×C8C2312yes—
68C4⋊Q8—22no—
69C4⋊Q8—22no—
70#126—28no—
71#124—28no—
72#124—22no—
73#124—22no—
74#123—28no—
75C22⋊Q8—28no—
76C22⋊Q8—22no—
77C22⋊Q8D414yes—
78#121—22no—
79#121D414yes—
80C4×Q8—22no—
81C4×Q8—22no—
82C4×D4—22no—
83C4×D4—22no—
84#117—22no—
85#117—22no—
86#117—42no—
87C2×C4⋊C4—21no—
88C2×C4⋊C4—21no—
89C2×C4⋊C4—21no—
90C2×C4⋊C4D411yescharacteristic
91C2×C4⋊C4D411yescharacteristic
92C2×C4⋊C4C2312yes—
93C2×C4⋊C4C2311yescharacteristic
94C2×C4⋊C4C2311yescharacteristic
95C2×C22⋊C4—21no—
96C2×C22⋊C4—21no—
97C2×C22⋊C4—21no—
98C2×C22⋊C4—21no—
99C2×C22⋊C4—21no—
100C2×C22⋊C4—21no—
101C2×C22⋊C4—22no—
102C2×C22⋊C4—22no—
103C2×C22⋊C4—22no—
104C2×C22⋊C4—22no—
105C2×C22⋊C4D411yescharacteristic
106C2×C22⋊C4D411yescharacteristic
107C2×C42—21no—
108C2×C42—21no—
109C2×C42C2311yescharacteristic
110C2×C42C2311yescharacteristic
111#103—42no—
112#102—24no—
113#102—24no—
114#102—42no—
115C22⋊C8—22no—
116C22⋊C8—22no—
117C24—24no—
118C24—21no—
119C24—22no—
120C24—42no—
121C24—42no—
122C24—82no—
123C24—82no—
124C24C2×D411yescharacteristic
125C2×Q8—24no—
126C2×Q8—21no—
127C2×Q8—22no—
128C2×Q8—22no—
129C2×Q8—82no—
130C2×Q8C2×D412yes—
131C2×D4—24no—
132C2×D4—24no—
133C2×D4—22no—
134C2×D4—22no—
135C2×D4—21no—
136C2×D4—22no—
137C2×D4—22no—
138C2×D4—22no—
139C2×D4—22no—
140C2×D4—22no—
141C2×D4—48no—
142C2×D4—48no—
143C2×D4—42no—
144C2×D4—42no—
145C2×D4—82no—
146C2×D4C2×D412yes—
147C2×D4#4114yes—
148C22×C4—22no—
149C22×C4—22no—
150C22×C4—21no—
151C22×C4—21no—
152C22×C4—21no—
153C22×C4—24no—
154C22×C4—22no—
155C22×C4—82no—
156C22×C4C2×D411yescharacteristic
157C22×C4C2×D411yescharacteristic
158C22×C4C2×D411yescharacteristic
159C22×C4C2×D411yescharacteristic
160C22×C4C2×D411yescharacteristic
161C22×C4C2×D412yes—
162C22×C4C2×D412yes—
163C22×C4C2411yescharacteristic
164SD16—48no—
165D8—48no—
166C2×C8—22no—
167C2×C8—22no—
168C2×C8—22no—
169C2×C8—22no—
170C2×C8—22no—
171C2×C8—22no—
172C4⋊C4—24no—
173C4⋊C4—24no—
174C4⋊C4—24no—
175C4⋊C4—22no—
176C4⋊C4—22no—
177C4⋊C4—21no—
178C4⋊C4—21no—
179C4⋊C4—22no—
180C4⋊C4—22no—
181C4⋊C4C2×D412yes—
182C4⋊C4C2×D412yes—
183C22⋊C4—24no—
184C22⋊C4—24no—
185C22⋊C4—24no—
186C22⋊C4—24no—
187C22⋊C4—22no—
188C22⋊C4—22no—
189C22⋊C4—24no—
190C22⋊C4—42no—
191C22⋊C4—82no—
192C22⋊C4—82no—
193C22⋊C4—82no—
194C22⋊C4—82no—
195C42—24no—
196C42—21no—
197C42—21no—
198C42—21no—
199C42—21no—
200C42—42no—
201C23—21no—
202C23—21no—
203C23—22no—
204C23—22no—
205C23—22no—
206C23—24no—
207C23—48no—
208C23—42no—
209C23—41no—
210C23—42no—
211C23—41no—
212C23—42no—
213C23—42no—
214C23—42no—
215C23—42no—
216C23—42no—
217C23—42no—
218C23—84no—
219C23—84no—
220C23—88no—
221C23—88no—
222C23#12012yes—
223C23C22×D411yescharacteristic
224Q8—24no—
225Q8—42no—
226Q8—84no—
227D4—24no—
228D4—24no—
229D4—24no—
230D4—48no—
231D4—42no—
232D4—42no—
233D4—42no—
234D4—84no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—24no—
238C2×C4—24no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—22no—
242C2×C4—21no—
243C2×C4—21no—
244C2×C4—22no—
245C2×C4—22no—
246C2×C4—22no—
247C2×C4—22no—
248C2×C4—22no—
249C2×C4—22no—
250C2×C4—41no—
251C2×C4—41no—
252C2×C4—42no—
253C2×C4—42no—
254C2×C4—42no—
255C2×C4—42no—
256C2×C4—42no—
257C2×C4—84no—
258C2×C4#12012yes—
259C2×C4C22×D411yescharacteristic
260C2×C4C22×D411yescharacteristic
261C2×C4C2×#4112yes—
262C2×C4H(5,𝔽2)12yes—
263C8—42no—
264C8—42no—
265C22—22no—
266C22—22no—
267C22—22no—
268C22—42no—
269C22—42no—
270C22—44no—
271C22—44no—
272C22—44no—
273C22—42no—
274C22—42no—
275C22—42no—
276C22—816no—
277C22—82no—
278C22—82no—
279C22—82no—
280C22—82no—
281C22—82no—
282C22—82no—
283C22C2×#12011yescharacteristic
284C22#52511yescharacteristic
285C22#53511yescharacteristic
286C22#57612yes—
287C22#57712yes—
288C4—22no—
289C4—42no—
290C4—41no—
291C4—41no—
292C4—42no—
293C4—42no—
294C4—41no—
295C4—41no—
296C4—82no—
297C4#54612yes—
298C4#54612yes—
299C2—42no—
300C2—82no—
301C2—84no—
302C2#240411yescharacteristic
303C2#300811yescharacteristic
304C2#301011yescharacteristic
305C2#311812yes—
306C2#322412yes—
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC2×#447C2
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×#542C2semidirect
2. C2×#450C2semidirect
4. #2309C2semidirect
5. #2023C2semidirect
6. #2012C2semidirect
7. #1880C2semidirect
8. #1879C2semidirect
9. #1875C2semidirect
10. #1873C2semidirect
11. #1818C2semidirect
17. C2×C4⋊Q8C22semidirect
20. C2×#124C22semidirect
25. C2×C4×Q8C22semidirect
26. C2×C4×D4C22semidirect
31. C2×#103C22semidirect
34. C2×#102C22semidirect
43. #340C22semidirect
44. #340C22semidirect
45. #336C22semidirect
46. #336C22semidirect

Maximal subgroups

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

subgroupindex
C2×#5422
C2×#4502
C2×#4472
#23092
#20232
#20122
#18802
#18792
#18752
#18732
#18182