Finite Group Databasebeta

← #28627

Subgroups — #28627

Order 256.

472 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 472 conjugacy classes of subgroups — too many to draw. Only its 123 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×#496C211yesmaximal, characteristic
2C2×#486C211yesmaximal, characteristic, complemented
3#2640C211yesmaximal, characteristic, complemented
4#2328C211yesmaximal, characteristic, complemented
5#2055C212yesmaximal, complemented
6#2054C212yesmaximal, complemented
7#1993C211yesmaximal, characteristic, complemented
8#1958C211yesmaximal, characteristic, complemented
9#1879C212yesmaximal, complemented
10#1877C212yesmaximal, complemented
11#1768C211yesmaximal, characteristic, complemented
12C22×SD16C2212yes—
13C22×D8C2212yes—
14C2×C4⋊Q8C2211yescharacteristic
15C2×#127C2211yescharacteristic
16C2×#124C2211yescharacteristic
17C2×C4×D4C2212yes—
18#496—24no—
19#486—24no—
20C2×C4⋊C8C2212yes—
21C2×#103C2212yes—
22C2×#102C2212yes—
23C2×#102C2212yes—
24C2×#102C2212yes—
25C2×#97C2211yescharacteristic
26C2×C4×C8C2211yescharacteristic
27#397—22no—
28#393—22no—
29#391C2211yescharacteristic
30#391C2211yescharacteristic
31#390C2211yescharacteristic
32#390C2212yes—
33#390C2211yescharacteristic
34#386—22no—
35#386—22no—
36#382C2211yescharacteristic
37#376C2211yescharacteristic
38#340C2212yes—
39#340C2212yes—
40#336C2212yes—
41C22×Q8—21no—
42C22×Q8C2311yescharacteristic
43C22×D4—21no—
44C22×D4—21no—
45C22×D4C2311yescharacteristic
46C22×D4C2312yes—
47C23×C4—22no—
48C2×SD16—24no—
49C2×SD16—24no—
50C2×SD16—42no—
51C2×D8—24no—
52C2×D8—24no—
53C2×D8—42no—
54C22×C8C2312yes—
55C22×C8C2312yes—
56C4⋊Q8—22no—
57C4⋊Q8—22no—
58#127—22no—
59#127D414yes—
60#124—22no—
61#124—22no—
62C4×D4—42no—
63C4×D4—42no—
64C2×C4⋊C4—21no—
65C2×C4⋊C4C2312yes—
66C2×C4⋊C4C2311yescharacteristic
67C2×C4⋊C4C2311yescharacteristic
68C2×C22⋊C4—22no—
69C2×C22⋊C4—21no—
70C2×C22⋊C4—21no—
71C2×C22⋊C4—22no—
72C2×C22⋊C4—22no—
73C2×C22⋊C4—22no—
74C2×C22⋊C4—22no—
75C2×C42C2311yescharacteristic
76C2×C42C2311yescharacteristic
77C2×C42C2311yescharacteristic
78C4⋊C8—24no—
79#103—42no—
80#102—24no—
81#102—24no—
82#102—42no—
83#97—22no—
84C4×C8—22no—
85#95—21no—
86#95—21no—
87#95—22no—
88#95—22no—
89#95D412yes—
90#95D412yes—
91C24—22no—
92C24—21no—
93C2×Q8—21no—
94C2×Q8—21no—
95C2×Q8—22no—
96C2×Q8—22no—
97C2×Q8—42no—
98C2×Q8—82no—
99C2×D4—24no—
100C2×D4—24no—
101C2×D4—21no—
102C2×D4—22no—
103C2×D4—22no—
104C2×D4—22no—
105C2×D4—21no—
106C2×D4—22no—
107C2×D4—22no—
108C2×D4—22no—
109C2×D4—22no—
110C2×D4—82no—
111C2×D4—82no—
112C2×D4C2×D414yes—
113C22×C4—21no—
114C22×C4—22no—
115C22×C4—21no—
116C22×C4—21no—
117C22×C4—44no—
118C22×C4—82no—
119C22×C4—82no—
120C22×C4C2×D412yes—
121C22×C4C2×D412yes—
122C22×C4C2×D411yescharacteristic
123C22×C4C2×D411yescharacteristic
124C22×C4C2411yescharacteristic
125SD16—48no—
126D8—48no—
127C2×C8—22no—
128C2×C8—22no—
129C2×C8—22no—
130C2×C8—22no—
131C2×C8—22no—
132C2×C8#4114yes—
133C4⋊C4—21no—
134C4⋊C4—21no—
135C4⋊C4—21no—
136C4⋊C4—21no—
137C4⋊C4—22no—
138C4⋊C4—22no—
139C4⋊C4—42no—
140C22⋊C4—44no—
141C22⋊C4—42no—
142C22⋊C4—42no—
143C22⋊C4—82no—
144C22⋊C4—82no—
145C22⋊C4—82no—
146C22⋊C4—82no—
147C42—22no—
148C42—22no—
149C42C2×D412yes—
150C42#4112yes—
151C23—24no—
152C23—42no—
153C23—42no—
154C23—42no—
155C23—42no—
156C23—42no—
157C23—42no—
158C23—41no—
159C23—41no—
160C23—41no—
161C23—42no—
162C23—42no—
163C23C22×D411yescharacteristic
164Q8—42no—
165Q8—42no—
166Q8—84no—
167D4—24no—
168D4—24no—
169D4—42no—
170D4—42no—
171D4—42no—
172D4—42no—
173D4—84no—
174D4—84no—
175C2×C4—22no—
176C2×C4—22no—
177C2×C4—22no—
178C2×C4—22no—
179C2×C4—22no—
180C2×C4—22no—
181C2×C4—22no—
182C2×C4—22no—
183C2×C4—41no—
184C2×C4—41no—
185C2×C4—41no—
186C2×C4—41no—
187C2×C4—42no—
188C2×C4—41no—
189C2×C4—42no—
190C2×C4—42no—
191C2×C4—42no—
192C2×C4—42no—
193C2×C4—88no—
194C2×C4#12112yes—
195C2×C4#12112yes—
196C2×C4C22×D411yescharacteristic
197C2×C4C2×#4112yes—
198C2×C4C2×#4111yescharacteristic
199C2×C4H(5,𝔽2)12yes—
200C8—24no—
201C8—42no—
202C22—44no—
203C22—44no—
204C22—44no—
205C22—82no—
206C22—82no—
207C22—82no—
208C22—82no—
209C22—82no—
210C22—82no—
211C22—81no—
212C22—81no—
213C22—81no—
214C22C2×#12111yescharacteristic
215C22#53211yescharacteristic
216C22#53511yescharacteristic
217C22#57612yes—
218C22#57712yes—
219C4—22no—
220C4—22no—
221C4—42no—
222C4—42no—
223C4—82no—
224C4—81no—
225C4—81no—
226C4—81no—
227C4#54612yes—
228C4#55912yes—
229C2—84no—
230C2—161no—
231C2#254411yescharacteristic
232C2#304111yescharacteristic
233C2#304211yescharacteristic
234C2#317912yes—
235C2#322412yes—
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC2×#496C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ #376 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#486C2semidirect
3. #2640C2semidirect
4. #2328C2semidirect
5. #2055C2semidirect
6. #2054C2semidirect
7. #1993C2semidirect
8. #1958C2semidirect
9. #1879C2semidirect
10. #1877C2semidirect
11. #1768C2semidirect

Maximal subgroups

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

subgroupindex
C2×#4962
C2×#4862
#26402
#23282
#20552
#20542
#19932
#19582
#18792
#18772
#17682