Finite Group Databasebeta

← #24266

Subgroups — #24266

Order 256.

739 conjugacy classes of subgroups; only the 147 normal subgroups are shown.

normal subgroups only

This group has 739 conjugacy classes of subgroups — too many to draw. Only its 147 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
1C22×#135C211yesmaximal, characteristic, complemented
2C2×#467C211yesmaximal, characteristic, complemented
3C2×#466C211yesmaximal, characteristic, complemented
4C2×C8×D4C211yesmaximal, characteristic, complemented
5#2407C212yesmaximal, complemented
6#2055C212yesmaximal, complemented
7#2000C212yesmaximal, complemented
8#1970C211yesmaximal, characteristic, complemented
9#1969C211yesmaximal, characteristic, complemented
10#1808C211yesmaximal, characteristic, complemented
11#1788C212yesmaximal, complemented
12C22×#41C2212yes—
13C2×#135—24no—
14C2×#135—24no—
15C2×#135—24no—
16C22×Q16C2211yescharacteristic
17C22×SD16C2212yes—
18C22×D8C2211yescharacteristic
19C23×C8C2211yescharacteristic
20C23×C8C2211yescharacteristic
21C2×C22⋊Q8C2212yes—
22C2×#121C2212yes—
23C2×C4×D4C2211yescharacteristic
24#467—24no—
25#466—24no—
26C8×D4—24no—
27C2×#107C2211yescharacteristic
28C2×#107C2211yescharacteristic
29C2×C4⋊C8C2211yescharacteristic
30C2×#103C2212yes—
31C2×#103C2212yes—
32C2×#102C2212yes—
33C2×#102C2212yes—
34C2×C22⋊C8C2211yescharacteristic
35C2×C22⋊C8C2211yescharacteristic
36C2×C4×C8C2211yescharacteristic
37#394—22no—
38#392—22no—
39#391C2212yes—
40#390C2212yes—
41#386—22no—
42#385—22no—
43#380C2212yes—
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×D4—22no—
54C22×D4D411yescharacteristic
55C22×D4C2312yes—
56C23×C4—22no—
57C23×C4C2311yescharacteristic
58C23×C4C2311yescharacteristic
59#135—232no—
60C2×Q16—22no—
61C2×Q16—22no—
62C2×Q16D414yescomplemented
63C2×SD16—24no—
64C2×SD16—24no—
65C2×SD16—24no—
66C2×D8—22no—
67C2×D8—22no—
68C2×D8D414yescomplemented
69C22×C8—22no—
70C22×C8—22no—
71C22×C8—22no—
72C22×C8—22no—
73C22×C8—22no—
74C22×C8D414yescomplemented
75C22×C8C2311yescharacteristic
76C22×C8C2311yescharacteristic
77C22×C8C2311yescharacteristic
78C22×C8C2311yescharacteristic
79C22⋊Q8—24no—
80C22⋊Q8—42no—
81#121—24no—
82#121—42no—
83C4×D4—22no—
84C4×D4—22no—
85C2×C4⋊C4—22no—
86C2×C4⋊C4D411yescharacteristic
87C2×C4⋊C4C2312yes—
88C2×C4⋊C4C2312yes—
89C2×C22⋊C4—22no—
90C2×C22⋊C4—22no—
91C2×C22⋊C4—22no—
92C2×C22⋊C4—22no—
93C2×C22⋊C4—22no—
94C2×C22⋊C4D411yescharacteristic
95C2×C22⋊C4D411yescharacteristic
96C2×C42C2311yescharacteristic
97#107—22no—
98#107—22no—
99C4⋊C8—22no—
100#103—24no—
101#103—42no—
102#102—24no—
103#102—42no—
104C22⋊C8—22no—
105C22⋊C8—22no—
106C4×C8—22no—
107#95—22no—
108#95—22no—
109#95—22no—
110C24—22no—
111C24C2×D411yescharacteristic
112C24C2×D411yescharacteristic
113#41—216no—
114#41—48no—
115#41—48no—
116#41—48no—
117C2×Q8—22no—
118C2×Q8—22no—
119C2×Q8—22no—
120C2×Q8—22no—
121C2×Q8—22no—
122C2×Q8C2×D414yes—
123C2×D4—22no—
124C2×D4—22no—
125C2×D4—22no—
126C2×D4—22no—
127C2×D4—22no—
128C2×D4—22no—
129C2×D4—22no—
130C2×D4—22no—
131C2×D4—44no—
132C2×D4—44no—
133C2×D4—44no—
134C2×D4—44no—
135C2×D4—44no—
136C2×D4—82no—
137C2×D4C2×D414yes—
138C22×C4—21no—
139C22×C4—21no—
140C22×C4—21no—
141C22×C4—21no—
142C22×C4—22no—
143C22×C4—21no—
144C22×C4—21no—
145C22×C4—21no—
146C22×C4—21no—
147C22×C4—21no—
148C22×C4—21no—
149C22×C4—22no—
150C22×C4—22no—
151C22×C4—44no—
152C22×C4—44no—
153C22×C4—44no—
154C22×C4C2×D411yescharacteristic
155C22×C4C2×D411yescharacteristic
156C22×C4C2×D411yescharacteristic
157C22×C4C2×D412yes—
158C22×C4C2×D411yescharacteristic
159C22×C4#4112yes—
160C22×C4C2411yescharacteristic
161Q16—28no—
162SD16—216no—
163D8—28no—
164C2×C8—21no—
165C2×C8—21no—
166C2×C8—28no—
167C2×C8—21no—
168C2×C8—21no—
169C2×C8—21no—
170C2×C8—21no—
171C2×C8—21no—
172C2×C8—21no—
173C2×C8—44no—
174C2×C8C2×D412yes—
175C2×C8C2×D412yes—
176C2×C8C2×D412yes—
177C2×C8C2×D412yes—
178C4⋊C4—22no—
179C4⋊C4—22no—
180C4⋊C4—22no—
181C4⋊C4—22no—
182C4⋊C4—22no—
183C4⋊C4—44no—
184C22⋊C4—22no—
185C22⋊C4—22no—
186C22⋊C4—44no—
187C22⋊C4—44no—
188C22⋊C4—44no—
189C22⋊C4—82no—
190C22⋊C4—82no—
191C42—21no—
192C42—21no—
193C23—21no—
194C23—21no—
195C23—21no—
196C23—21no—
197C23—21no—
198C23—21no—
199C23—21no—
200C23—21no—
201C23—21no—
202C23—21no—
203C23—21no—
204C23—21no—
205C23—24no—
206C23—42no—
207C23—42no—
208C23—42no—
209C23—42no—
210C23—42no—
211C23—42no—
212C23#12112yes—
213C23#13512yescomplemented
214C23C22×D411yescharacteristic
215Q8—24no—
216Q8—24no—
217Q8—42no—
218Q8—42no—
219D4—28no—
220D4—24no—
221D4—24no—
222D4—416no—
223D4—42no—
224D4—42no—
225D4—88no—
226C2×C4—21no—
227C2×C4—21no—
228C2×C4—21no—
229C2×C4—21no—
230C2×C4—22no—
231C2×C4—22no—
232C2×C4—22no—
233C2×C4—22no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—24no—
238C2×C4—21no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—416no—
242C2×C4—42no—
243C2×C4—42no—
244C2×C4—41no—
245C2×C4—41no—
246C2×C4—41no—
247C2×C4—41no—
248C2×C4—42no—
249C2×C4—42no—
250C2×C4—42no—
251C2×C4—42no—
252C2×C4—42no—
253C2×C4—42no—
254C2×C4#12112yes—
255C2×C4#13512yescomplemented
256C2×C4C22×D411yescharacteristic
257C2×C4C22×D411yescharacteristic
258C2×C4C22×D411yescharacteristic
259C2×C4C2×#4111yescharacteristic
260C2×C4H(5,𝔽2)11yescharacteristic
261C2×C4H(5,𝔽2)11yescharacteristic
262C8—22no—
263C8—41no—
264C8—41no—
265C8#12014yes—
266C22—22no—
267C22—22no—
268C22—22no—
269C22—22no—
270C22—22no—
271C22—22no—
272C22—41no—
273C22—41no—
274C22—41no—
275C22—41no—
276C22—41no—
277C22—41no—
278C22—41no—
279C22—41no—
280C22—44no—
281C22—44no—
282C22—44no—
283C22—82no—
284C22—82no—
285C22—82no—
286C22—82no—
287C22C2×#12111yescharacteristic
288C22D4211yescharacteristic
289C22D4211yescharacteristic
290C22C2×#13511yescharacteristic
291C22C2×#13511yescharacteristic
292C22#57611yescharacteristic
293C22#57811yescharacteristic
294C4—22no—
295C4—22no—
296C4—41no—
297C4—44no—
298C4—41no—
299C4—82no—
300C4—82no—
301C4C2×#12011yescharacteristic
302C4#53511yescharacteristic
303C4#54611yescharacteristic
304C4#54611yescharacteristic
305C2—42no—
306C2—42no—
307C2—84no—
308C2#240811yescharacteristic
309C2#305011yescharacteristic
310C2#306111yescharacteristic
311C2#328011yescharacteristic
312C2#328311yescharacteristic
313C2#328511yescharacteristic
314C2#328711yescharacteristic
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. C22×#135C2semidirect
2. C2×#467C2semidirect
3. C2×#466C2semidirect
4. C2×C8×D4C2semidirect
5. #2407C2semidirect
6. #2055C2semidirect
7. #2000C2semidirect
8. #1970C2semidirect
9. #1969C2semidirect
10. #1808C2semidirect
11. #1788C2semidirect
62. C2×Q16D4semidirect
68. C2×D8D4semidirect
74. C22×C8D4semidirect
213. C23#135semidirect
255. C2×C4#135semidirect

Maximal subgroups

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

subgroupindex
C22×#1352
C2×#4672
C2×#4662
C2×C8×D42
#24072
#20552
#20002
#19702
#19692
#18082
#17882