Finite Group Databasebeta

← #24249

Subgroups — #24249

Order 256.

744 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 744 conjugacy classes of subgroups — too many to draw. Only its 143 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, complemented
2C2×#461C211yesmaximal, characteristic, complemented
3C2×#449C212yesmaximal, complemented
4C2×#442C211yesmaximal, characteristic, complemented
5#2535C211yesmaximal, characteristic, complemented
6#2035C212yesmaximal, complemented
7#2000C212yesmaximal, complemented
8#1960C211yesmaximal, characteristic, complemented
9#1955C211yesmaximal, characteristic, complemented
10#1804C211yesmaximal, characteristic, complemented
11#1788C212yesmaximal, complemented
12C22×#41C2212yes—
13C22×SD16C2212yes—
14C22×D8C2211yescharacteristic
15#547—24no—
16#547—24no—
17C2×C4⋊Q8C2211yescharacteristic
18C2×#123—22no—
19C2×C22⋊Q8C2212yes—
20C2×#121—21no—
21C2×#121C2212yes—
22C2×C4×D4C2211yescharacteristic
23C2×C4×D4C2211yescharacteristic
24C2×C4×D4C2211yescharacteristic
25C22×C4⋊C4—21no—
26#461—24no—
27#449—28no—
28#442—24no—
29C2×#106C2211yescharacteristic
30C2×C4⋊C8C2211yescharacteristic
31C2×#103C2212yes—
32C2×#103C2212yes—
33C2×#103C2212yes—
34C2×#102C2212yes—
35C2×#102C2211yescharacteristic
36C2×#102C2211yescharacteristic
37C2×C22⋊C8C2212yes—
38C2×C22⋊C8C2212yes—
39C2×#97C2211yescharacteristic
40#397—21no—
41#392—21no—
42#386—22no—
43#382C2211yescharacteristic
44#380C2212yes—
45#340C2212yes—
46C2×#41—24no—
47C2×#41—24no—
48C2×#41—24no—
49C2×#41—24no—
50C2×#41—42no—
51C2×#41—42no—
52C22×Q8C2312yes—
53C22×D4—22no—
54C22×D4—22no—
55C22×D4D411yescharacteristic
56C22×D4C2311yescharacteristic
57C22×D4C2311yescharacteristic
58C23×C4—21no—
59C23×C4—22no—
60C23×C4—21no—
61C23×C4C2312yes—
62C2×SD16—24no—
63C2×SD16—24no—
64C2×SD16—42no—
65C2×D8—22no—
66C2×D8—22no—
67C2×D8D414yescomplemented
68C22×C8C2312yes—
69C22×C8C2311yescharacteristic
70C22×C8C2311yescharacteristic
71C4⋊Q8—22no—
72C4⋊Q8D414yescomplemented
73#123—216no—
74C22⋊Q8—24no—
75C22⋊Q8—24no—
76#121—28no—
77#121—24no—
78#121—42no—
79C4×D4—22no—
80C4×D4—22no—
81C4×D4—22no—
82C4×D4—22no—
83C4×D4—22no—
84C4×D4D414yescomplemented
85C2×C4⋊C4—28no—
86C2×C4⋊C4—21no—
87C2×C4⋊C4—22no—
88C2×C4⋊C4—21no—
89C2×C4⋊C4D411yescharacteristic
90C2×C4⋊C4C2311yescharacteristic
91C2×C4⋊C4C2312yes—
92C2×C4⋊C4C2311yescharacteristic
93C2×C22⋊C4—22no—
94C2×C22⋊C4—21no—
95C2×C22⋊C4—21no—
96C2×C22⋊C4—22no—
97C2×C22⋊C4—22no—
98C2×C22⋊C4D412yes—
99C2×C42C2311yescharacteristic
100#106—22no—
101C4⋊C8—22no—
102#103—24no—
103#103—24no—
104#103—42no—
105#102—22no—
106#102—24no—
107#102—22no—
108C22⋊C8—24no—
109C22⋊C8—42no—
110#97—22no—
111#95—22no—
112#95—21no—
113#95—21no—
114C24—21no—
115C24—21no—
116C24C2×D412yes—
117#41—48no—
118#41—48no—
119#41—48no—
120#41—48no—
121C2×Q8—22no—
122C2×Q8—22no—
123C2×Q8—22no—
124C2×Q8—22no—
125C2×Q8—22no—
126C2×Q8C2×D414yes—
127C2×D4—28no—
128C2×D4—22no—
129C2×D4—21no—
130C2×D4—21no—
131C2×D4—21no—
132C2×D4—21no—
133C2×D4—21no—
134C2×D4—21no—
135C2×D4—21no—
136C2×D4—21no—
137C2×D4—42no—
138C2×D4—44no—
139C2×D4—44no—
140C2×D4—44no—
141C2×D4—44no—
142C2×D4—82no—
143C2×D4C2×D412yes—
144C2×D4C2×D412yes—
145C2×D4C2×D412yes—
146C2×D4#4112yes—
147C22×C4—28no—
148C22×C4—21no—
149C22×C4—22no—
150C22×C4—24no—
151C22×C4—22no—
152C22×C4—22no—
153C22×C4—22no—
154C22×C4—22no—
155C22×C4—22no—
156C22×C4—22no—
157C22×C4—22no—
158C22×C4—21no—
159C22×C4—44no—
160C22×C4—44no—
161C22×C4—44no—
162C22×C4—42no—
163C22×C4—82no—
164C22×C4C2×D412yes—
165C22×C4C2×D411yescharacteristic
166C22×C4C2×D411yescharacteristic
167C22×C4C2411yescharacteristic
168SD16—48no—
169D8—28no—
170C2×C8—21no—
171C2×C8—21no—
172C2×C8—21no—
173C2×C8—22no—
174C2×C8—22no—
175C2×C8—21no—
176C2×C8—22no—
177C2×C8—21no—
178C2×C8C2×D412yes—
179C4⋊C4—28no—
180C4⋊C4—22no—
181C4⋊C4—24no—
182C4⋊C4—24no—
183C4⋊C4—21no—
184C4⋊C4—22no—
185C4⋊C4—21no—
186C4⋊C4—21no—
187C4⋊C4—22no—
188C4⋊C4C2×D412yes—
189C22⋊C4—28no—
190C22⋊C4—24no—
191C22⋊C4—24no—
192C22⋊C4—44no—
193C22⋊C4—42no—
194C22⋊C4—82no—
195C42C2×D412yes—
196C42C2×D412yes—
197C23—22no—
198C23—22no—
199C23—22no—
200C23—22no—
201C23—22no—
202C23—22no—
203C23—22no—
204C23—22no—
205C23—22no—
206C23—22no—
207C23—22no—
208C23—42no—
209C23—42no—
210C23—41no—
211C23—41no—
212C23—42no—
213C23—42no—
214C23C22×D411yescharacteristic
215Q8—24no—
216Q8—24no—
217Q8—42no—
218Q8—42no—
219D4—22no—
220D4—22no—
221D4—22no—
222D4—22no—
223D4—22no—
224D4—44no—
225D4—416no—
226D4—41no—
227D4—41no—
228D4—88no—
229D4#12114yes—
230C2×C4—22no—
231C2×C4—22no—
232C2×C4—22no—
233C2×C4—22no—
234C2×C4—24no—
235C2×C4—22no—
236C2×C4—22no—
237C2×C4—48no—
238C2×C4—416no—
239C2×C4—42no—
240C2×C4—42no—
241C2×C4—42no—
242C2×C4—42no—
243C2×C4—42no—
244C2×C4—42no—
245C2×C4—42no—
246C2×C4—41no—
247C2×C4—42no—
248C2×C4—42no—
249C2×C4—42no—
250C2×C4—42no—
251C2×C4—84no—
252C2×C4#12012yes—
253C2×C4#12012yes—
254C2×C4C22×D411yescharacteristic
255C2×C4C22×D411yescharacteristic
256C2×C4C22×D411yescharacteristic
257C2×C4C2×#4111yescharacteristic
258C2×C4H(5,𝔽2)11yescharacteristic
259C2×C4H(5,𝔽2)11yescharacteristic
260C8—22no—
261C8—41no—
262C8—42no—
263C22—24no—
264C22—42no—
265C22—42no—
266C22—44no—
267C22—44no—
268C22—42no—
269C22—42no—
270C22—42no—
271C22—42no—
272C22—42no—
273C22—44no—
274C22—42no—
275C22—42no—
276C22—42no—
277C22—81no—
278C22—81no—
279C22—82no—
280C22C2×#12011yescharacteristic
281C22D4211yescharacteristic
282C22#54611yescharacteristic
283C22#57512yes—
284C22#57712yes—
285C4—22no—
286C4—22no—
287C4—42no—
288C4—44no—
289C4—42no—
290C4—81no—
291C4—82no—
292C4C2×#12111yescharacteristic
293C4#53511yescharacteristic
294C4D4211yescharacteristic
295C4#54611yescharacteristic
296C2—44no—
297C2—82no—
298C2—82no—
299C2#240811yescharacteristic
300C2#300511yescharacteristic
301C2#301011yescharacteristic
302C2#326712yes—
303C2#330712yes—
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
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
1. C2×#547C2semidirect
2. C2×#461C2semidirect
3. C2×#449C2semidirect
4. C2×#442C2semidirect
5. #2535C2semidirect
6. #2035C2semidirect
7. #2000C2semidirect
8. #1960C2semidirect
9. #1955C2semidirect
10. #1804C2semidirect
11. #1788C2semidirect
67. C2×D8D4semidirect
72. C4⋊Q8D4semidirect
84. C4×D4D4semidirect

Maximal subgroups

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

subgroupindex
C2×#5472
C2×#4612
C2×#4492
C2×#4422
#25352
#20352
#20002
#19602
#19552
#18042
#17882