Finite Group Databasebeta

← #24659

Subgroups — #24659

Order 256.

480 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 480 conjugacy classes of subgroups — too many to draw. Only its 133 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×#461C211yesmaximal, characteristic, complemented
2C2×#432C211yesmaximal, characteristic, complemented
3#2857C211yesmaximal, characteristic, complemented
4C4×#121C211yesmaximal, characteristic
5#2036C212yesmaximal, complemented
6#2000C212yesmaximal
7#1981C211yesmaximal, characteristic, complemented
8#1973C212yesmaximal, complemented
9#1947C211yesmaximal, characteristic, complemented
10#1935C212yesmaximal, complemented
11#1802C211yesmaximal, characteristic, complemented
12C22×SD16C2212yes—
13C2×C4⋊Q8C2211yescharacteristic
14C2×C22⋊Q8C2212yes—
15C2×#121C2211yescharacteristic
16C2×C4×D4—21no—
17C2×C4×D4C2211yescharacteristic
18C22×C42C2211yescharacteristic
19#461—24no—
20#432—24no—
21C2×#107C2212yes—
22C2×C4⋊C8C2211yescharacteristic
23C2×C4⋊C8C2211yescharacteristic
24C2×#103C2212yes—
25C2×#103C2212yes—
26C2×#103C2212yes—
27C2×#102C2212yes—
28C2×C22⋊C8C2212yes—
29C2×#97C2212yes—
30#391C2212yes—
31#390C2211yescharacteristic
32#389—21no—
33#389—21no—
34#389C2211yescharacteristic
35#388—21no—
36#386—21no—
37#384C2211yescharacteristic
38#380—21no—
39#380—21no—
40#380C2211yescharacteristic
41C4×C4⋊C4C2211yescharacteristic
42C4×C22⋊C4—21no—
43#340C2212yes—
44#340C2212yes—
45C22×Q8C2312yes—
46C22×D4—21no—
47C22×D4C2311yescharacteristic
48C23×C4—21no—
49C23×C4D411yescharacteristic
50C23×C4D411yescharacteristic
51C23×C4C2311yescharacteristic
52C2×SD16—24no—
53C2×SD16—24no—
54C2×SD16—42no—
55C22×C8C2312yes—
56C22×C8C2312yes—
57C4⋊Q8—22no—
58C4⋊Q8—22no—
59C22⋊Q8—42no—
60C22⋊Q8—42no—
61#121—22no—
62#121—22no—
63C4×D4—22no—
64C4×D4—22no—
65C4×D4—44no—
66C2×C4⋊C4—21no—
67C2×C4⋊C4—22no—
68C2×C4⋊C4—21no—
69C2×C4⋊C4—21no—
70C2×C4⋊C4—21no—
71C2×C4⋊C4—21no—
72C2×C4⋊C4C2311yescharacteristic
73C2×C4⋊C4C2312yes—
74C2×C4⋊C4C2311yescharacteristic
75C2×C4⋊C4C2311yescharacteristic
76C2×C22⋊C4—21no—
77C2×C22⋊C4—21no—
78C2×C22⋊C4—22no—
79C2×C22⋊C4—21no—
80C2×C22⋊C4—21no—
81C2×C22⋊C4—21no—
82C2×C42—22no—
83C2×C42—22no—
84C2×C42—21no—
85C2×C42D411yescharacteristic
86C2×C42D411yescharacteristic
87C2×C42C2311yescharacteristic
88C2×C42C2311yescharacteristic
89#107—42no—
90C4⋊C8—22no—
91C4⋊C8D414yescomplemented
92#103—24no—
93#103—24no—
94#103—42no—
95#102—42no—
96C22⋊C8—24no—
97#97—24no—
98#95—21no—
99#95—21no—
100#95—21no—
101#95—22no—
102#95—22no—
103C24—21no—
104C24C2×D411yescharacteristic
105C2×Q8—22no—
106C2×Q8—22no—
107C2×Q8—22no—
108C2×Q8—22no—
109C2×Q8—22no—
110C2×Q8—22no—
111C2×D4—24no—
112C2×D4—21no—
113C2×D4—21no—
114C2×D4—21no—
115C2×D4—21no—
116C2×D4—21no—
117C2×D4—42no—
118C2×D4—42no—
119C2×D4#4112yes—
120C22×C4—22no—
121C22×C4—21no—
122C22×C4—22no—
123C22×C4—22no—
124C22×C4—22no—
125C22×C4—22no—
126C22×C4—21no—
127C22×C4—21no—
128C22×C4—21no—
129C22×C4—21no—
130C22×C4—21no—
131C22×C4—21no—
132C22×C4—21no—
133C22×C4—21no—
134C22×C4—22no—
135C22×C4—22no—
136C22×C4—41no—
137C22×C4—42no—
138C22×C4—42no—
139C22×C4—42no—
140C22×C4C2×D411yescharacteristic
141C22×C4C2×D411yescharacteristic
142C22×C4C2×D411yescharacteristic
143C22×C4C2×D411yescharacteristic
144C22×C4C2×D411yescharacteristic
145C22×C4C2411yescharacteristic
146SD16—48no—
147C2×C8—22no—
148C2×C8—22no—
149C2×C8—22no—
150C2×C8—22no—
151C2×C8—22no—
152C2×C8C2×D414yes—
153C4⋊C4—21no—
154C4⋊C4—21no—
155C4⋊C4—21no—
156C4⋊C4—22no—
157C4⋊C4—21no—
158C4⋊C4—21no—
159C4⋊C4—22no—
160C4⋊C4—42no—
161C4⋊C4—42no—
162C4⋊C4—44no—
163C4⋊C4—42no—
164C4⋊C4—42no—
165C4⋊C4—42no—
166C4⋊C4#4112yes—
167C22⋊C4—24no—
168C22⋊C4—42no—
169C22⋊C4—42no—
170C22⋊C4—42no—
171C22⋊C4—44no—
172C22⋊C4—42no—
173C42—22no—
174C42—22no—
175C42—21no—
176C42—21no—
177C42—42no—
178C42C2×D412yes—
179C42#4112yes—
180C23—21no—
181C23—21no—
182C23—21no—
183C23—21no—
184C23—21no—
185C23—21no—
186C23—22no—
187C23—41no—
188C23—41no—
189C23—41no—
190C23—41no—
191C23—41no—
192C23—41no—
193C23#13712yes—
194C23C22×D411yescharacteristic
195Q8—42no—
196Q8—42no—
197Q8—42no—
198Q8—42no—
199D4—22no—
200D4—22no—
201D4—48no—
202D4—41no—
203D4—41no—
204C2×C4—21no—
205C2×C4—21no—
206C2×C4—21no—
207C2×C4—21no—
208C2×C4—21no—
209C2×C4—21no—
210C2×C4—22no—
211C2×C4—21no—
212C2×C4—21no—
213C2×C4—21no—
214C2×C4—21no—
215C2×C4—21no—
216C2×C4—44no—
217C2×C4—41no—
218C2×C4—41no—
219C2×C4—41no—
220C2×C4—41no—
221C2×C4—41no—
222C2×C4—41no—
223C2×C4—44no—
224C2×C4—41no—
225C2×C4—42no—
226C2×C4—42no—
227C2×C4—42no—
228C2×C4—42no—
229C2×C4—41no—
230C2×C4—41no—
231C2×C4—41no—
232C2×C4—42no—
233C2×C4—41no—
234C2×C4—41no—
235C2×C4—42no—
236C2×C4—84no—
237C2×C4#12112yes—
238C2×C4#12112yes—
239C2×C4AGL(1,ℤ/8)12yes—
240C2×C4#13712yes—
241C2×C4C22×D411yescharacteristic
242C2×C4C2×#4111yescharacteristic
243C2×C4C2×#4111yescharacteristic
244C2×C4C2×#4111yescharacteristic
245C2×C4H(5,𝔽2)11yescharacteristic
246C2×C4H(5,𝔽2)11yescharacteristic
247C8—24no—
248C8—42no—
249C22—22no—
250C22—22no—
251C22—22no—
252C22—41no—
253C22—41no—
254C22—41no—
255C22—41no—
256C22—42no—
257C22—42no—
258C22—42no—
259C22—81no—
260C22—81no—
261C22—81no—
262C22—81no—
263C22C2×#12111yescharacteristic
264C22#52511yescharacteristic
265C22#53511yescharacteristic
266C22C2×AGL(1,ℤ/8)11yescharacteristic
267C22C2×#13711yescharacteristic
268C22C2×#13711yescharacteristic
269C22#57511yescharacteristic
270C4—22no—
271C4—42no—
272C4—41no—
273C4—41no—
274C4—41no—
275C4—82no—
276C4—82no—
277C4—81no—
278C4—81no—
279C4#46814yes—
280C4#54611yescharacteristic
281C4#55011yescharacteristic
282C4#55311yescharacteristic
283C4#55511yescharacteristic
284C2—42no—
285C2—82no—
286C2#241411yescharacteristic
287C2C2×#46811yescharacteristic
288C2#305611yescharacteristic
289C2#311011yescharacteristic
290C2#311611yescharacteristic
291C2#321411yescharacteristic
292C2#321811yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC2×#121C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#461C2semidirect
2. C2×#432C2semidirect
3. #2857C2semidirect
5. #2036C2semidirect
7. #1981C2semidirect
8. #1973C2semidirect
9. #1947C2semidirect
10. #1935C2semidirect
11. #1802C2semidirect
91. C4⋊C8D4semidirect

Maximal subgroups

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

subgroupindex
C2×#4612
C2×#4322
#28572
C4×#1212
#20362
#20002
#19812
#19732
#19472
#19352
#18022