Finite Group Databasebeta

← #25685

Subgroups — #25685

Order 256.

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

normal subgroups only

This group has 548 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×#540C211yesmaximal, characteristic
2C2×#501C211yesmaximal, characteristic, complemented
3C2×#486C211yesmaximal, characteristic, complemented
4#2628C211yesmaximal, characteristic, complemented
5#2034C212yesmaximal, complemented
6#2033C212yesmaximal, complemented
7#1991C211yesmaximal, characteristic, complemented
8#1961C211yesmaximal, characteristic, complemented
9#1864C212yesmaximal, complemented
10#1862C212yesmaximal, complemented
11#1750C211yesmaximal, characteristic, complemented
12#540—24no—
13#540—24no—
14C2×C4⋊Q8C2211yescharacteristic
15C2×#127C2211yescharacteristic
16C2×#125C2211yescharacteristic
17C2×#123—21no—
18C2×#123—21no—
19C2×#121—21no—
20C2×#121—21no—
21C2×#121C2211yescharacteristic
22C2×#121C2211yescharacteristic
23C2×C4×D4C2211yescharacteristic
24C2×C4×D4C2211yescharacteristic
25C2×#117C2211yescharacteristic
26#501—24no—
27#486—24no—
28C2×#107C2212yes—
29C2×#106C2212yes—
30C2×#102C2212yes—
31C2×#102C2212yes—
32C2×#102C2212yes—
33C2×#102C2212yes—
34C2×C22⋊C8C2212yes—
35C2×#97C2211yescharacteristic
36C2×C4×C8C2211yescharacteristic
37#400—21no—
38#399—21no—
39#395—21no—
40#393—21no—
41#389C2212yes—
42#382C2211yescharacteristic
43#380C2212yes—
44#340C2212yes—
45#340C2212yes—
46#336C2212yes—
47C22×Q8—21no—
48C22×D4—21no—
49C22×D4—21no—
50C22×D4—21no—
51C22×D4—21no—
52C22×D4C2311yescharacteristic
53C22×D4C2311yescharacteristic
54C23×C4—21no—
55C23×C4—21no—
56C23×C4C2311yescharacteristic
57C22×C8C2312yes—
58C22×C8C2312yes—
59C4⋊Q8—22no—
60C4⋊Q8—22no—
61#127—22no—
62#127D414yes—
63#125—22no—
64#125D414yes—
65#123—28no—
66#123—28no—
67#121—28no—
68#121—28no—
69#121—22no—
70#121—22no—
71#121—22no—
72#121—22no—
73C4×D4—22no—
74C4×D4—22no—
75C4×D4—22no—
76C4×D4—22no—
77#117—22no—
78#117—22no—
79C2×C4⋊C4—21no—
80C2×C4⋊C4—21no—
81C2×C4⋊C4—21no—
82C2×C4⋊C4—22no—
83C2×C4⋊C4D411yescharacteristic
84C2×C4⋊C4D411yescharacteristic
85C2×C4⋊C4C2312yes—
86C2×C4⋊C4C2312yes—
87C2×C4⋊C4C2311yescharacteristic
88C2×C4⋊C4C2311yescharacteristic
89C2×C22⋊C4—21no—
90C2×C22⋊C4—21no—
91C2×C22⋊C4—21no—
92C2×C22⋊C4—21no—
93C2×C22⋊C4—22no—
94C2×C22⋊C4D411yescharacteristic
95C2×C22⋊C4D411yescharacteristic
96C2×C42C2311yescharacteristic
97C2×C42C2311yescharacteristic
98#107—24no—
99#106—24no—
100#102—24no—
101#102—24no—
102#102—24no—
103#102—24no—
104C22⋊C8—24no—
105#97—22no—
106C4×C8—22no—
107#95—22no—
108#95—21no—
109#95—21no—
110#95—22no—
111C24—21no—
112C24—21no—
113C24C2×D411yescharacteristic
114C2×Q8—42no—
115C2×Q8—82no—
116C2×D4—24no—
117C2×D4—24no—
118C2×D4—24no—
119C2×D4—24no—
120C2×D4—21no—
121C2×D4—21no—
122C2×D4—22no—
123C2×D4—22no—
124C2×D4—22no—
125C2×D4—22no—
126C2×D4—82no—
127C2×D4—82no—
128C2×D4—82no—
129C2×D4—82no—
130C2×D4C2×D412yes—
131C2×D4C2×D412yes—
132C22×C4—21no—
133C22×C4—21no—
134C22×C4—24no—
135C22×C4—24no—
136C22×C4—21no—
137C22×C4—21no—
138C22×C4—22no—
139C22×C4—22no—
140C22×C4—22no—
141C22×C4—22no—
142C22×C4—82no—
143C22×C4—82no—
144C22×C4C2×D411yescharacteristic
145C22×C4C2×D411yescharacteristic
146C22×C4C2×D411yescharacteristic
147C22×C4C2×D411yescharacteristic
148C22×C4C2×D411yescharacteristic
149C22×C4C2411yescharacteristic
150C2×C8—22no—
151C2×C8—22no—
152C2×C8—22no—
153C2×C8—22no—
154C2×C8—22no—
155C2×C8#4114yes—
156C4⋊C4—24no—
157C4⋊C4—24no—
158C4⋊C4—22no—
159C4⋊C4—22no—
160C4⋊C4—21no—
161C4⋊C4—21no—
162C4⋊C4—22no—
163C4⋊C4—22no—
164C4⋊C4—22no—
165C4⋊C4—22no—
166C4⋊C4—42no—
167C4⋊C4—82no—
168C4⋊C4C2×D412yes—
169C4⋊C4C2×D412yes—
170C22⋊C4—24no—
171C22⋊C4—24no—
172C22⋊C4—24no—
173C22⋊C4—24no—
174C22⋊C4—22no—
175C22⋊C4—22no—
176C22⋊C4—82no—
177C42—21no—
178C42—21no—
179C42C2×D412yes—
180C42C2×D412yes—
181C23—21no—
182C23—22no—
183C23—22no—
184C23—21no—
185C23—21no—
186C23—22no—
187C23—22no—
188C23—41no—
189C23—41no—
190C23—41no—
191C23—41no—
192C23—42no—
193C23—42no—
194C23—42no—
195C23—42no—
196C23C22×D411yescharacteristic
197Q8—84no—
198D4—24no—
199D4—24no—
200D4—42no—
201D4—42no—
202D4—84no—
203D4—84no—
204D4—84no—
205D4—84no—
206C2×C4—22no—
207C2×C4—22no—
208C2×C4—21no—
209C2×C4—21no—
210C2×C4—21no—
211C2×C4—22no—
212C2×C4—22no—
213C2×C4—21no—
214C2×C4—21no—
215C2×C4—22no—
216C2×C4—22no—
217C2×C4—42no—
218C2×C4—42no—
219C2×C4—42no—
220C2×C4—42no—
221C2×C4—42no—
222C2×C4—42no—
223C2×C4—42no—
224C2×C4—42no—
225C2×C4—42no—
226C2×C4—42no—
227C2×C4—84no—
228C2×C4—84no—
229C2×C4#12012yes—
230C2×C4#12012yes—
231C2×C4C22×D411yescharacteristic
232C2×C4C22×D411yescharacteristic
233C2×C4C2×#4112yes—
234C2×C42−1+412yes—
235C8—24no—
236C8—42no—
237C22—42no—
238C22—41no—
239C22—42no—
240C22—42no—
241C22—42no—
242C22—42no—
243C22—41no—
244C22—42no—
245C22—42no—
246C22—41no—
247C22—42no—
248C22—82no—
249C22—82no—
250C22—82no—
251C22—82no—
252C22C2×#12011yescharacteristic
253C22#53211yescharacteristic
254C22#53611yescharacteristic
255C22#57612yes—
256C22#57712yes—
257C4—22no—
258C4—22no—
259C4—41no—
260C4—41no—
261C4—42no—
262C4—41no—
263C4—42no—
264C4—82no—
265C4—82no—
266C4#54712yes—
267C4#54812yes—
268C2—81no—
269C2—82no—
270C2—82no—
271C2#243211yescharacteristic
272C2#300811yescharacteristic
273C2#300911yescharacteristic
274C2#317912yes—
275C2#324112yes—
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC2×#540C2
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
2. C2×#501C2semidirect
3. C2×#486C2semidirect
4. #2628C2semidirect
5. #2034C2semidirect
6. #2033C2semidirect
7. #1991C2semidirect
8. #1961C2semidirect
9. #1864C2semidirect
10. #1862C2semidirect
11. #1750C2semidirect

Maximal subgroups

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

subgroupindex
C2×#5402
C2×#5012
C2×#4862
#26282
#20342
#20332
#19912
#19612
#18642
#18622
#17502