Finite Group Databasebeta

← #30339

Subgroups — #30339

Order 256.

371 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 371 conjugacy classes of subgroups — too many to draw. Only its 121 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
1#2862C211yesmaximal, characteristic, complemented
2#2860C211yesmaximal, characteristic
3#2066C212yesmaximal, complemented
4#2058C211yesmaximal, characteristic, complemented
5#2055C212yesmaximal
6#2043C211yesmaximal, characteristic, complemented
7#2023C211yesmaximal, characteristic
8#2013C211yesmaximal, characteristic
9#1968C212yesmaximal
10#1933C212yesmaximal, complemented
11C4×C4⋊C8C211yesmaximal, characteristic, complemented
12C22×SD16C2212yes—
13C2×C4⋊Q8C2211yescharacteristic
14C2×C4⋊Q8C2211yescharacteristic
15C2×#126—21no—
16C2×#106C2212yes—
17C2×C4⋊C8C2212yes—
18C2×#103C2212yes—
19C2×#103C2212yes—
20C2×#103C2212yes—
21C2×#102C2212yes—
22C2×C4×C8C2212yes—
23#391—21no—
24#391—21no—
25#391C2212yes—
26#391C2212yes—
27#390C2211yescharacteristic
28#390C2211yescharacteristic
29#390C2211yescharacteristic
30#389C2211yescharacteristic
31#389C2211yescharacteristic
32#389C2211yescharacteristic
33#388—21no—
34#388—21no—
35#388—21no—
36#382—21no—
37#382—21no—
38C43C2211yescharacteristic
39#340C2212yes—
40#340C2212yes—
41#336C2211yescharacteristic
42#336C2211yescharacteristic
43C22×Q8—21no—
44C22×Q8C2312yes—
45C22×D4C2311yescharacteristic
46C2×SD16—24no—
47C2×SD16—42no—
48C2×SD16—42no—
49C22×C8C2312yes—
50C22×C8C2312yes—
51C4⋊Q8—42no—
52C4⋊Q8—42no—
53#126—44no—
54C2×C4⋊C4—21no—
55C2×C4⋊C4—21no—
56C2×C4⋊C4—21no—
57C2×C4⋊C4—21no—
58C2×C4⋊C4—21no—
59C2×C4⋊C4—21no—
60C2×C4⋊C4C2312yes—
61C2×C4⋊C4C2311yescharacteristic
62C2×C4⋊C4C2311yescharacteristic
63C2×C4⋊C4C2311yescharacteristic
64C2×C22⋊C4—21no—
65C2×C22⋊C4—21no—
66C2×C22⋊C4—21no—
67C2×C22⋊C4—21no—
68C2×C22⋊C4—21no—
69C2×C22⋊C4—21no—
70C2×C42D411yescharacteristic
71C2×C42D411yescharacteristic
72C2×C42D411yescharacteristic
73C2×C42D411yescharacteristic
74C2×C42C2311yescharacteristic
75C2×C42C2311yescharacteristic
76C2×C42C2311yescharacteristic
77#106—42no—
78C4⋊C8—24no—
79#103—24no—
80#103—42no—
81#103—42no—
82#102—42no—
83C4×C8—42no—
84#95—22no—
85#95—22no—
86#95—22no—
87#95—22no—
88#95—21no—
89#95—21no—
90#95—21no—
91C24—21no—
92C2×Q8—22no—
93C2×Q8—22no—
94C2×Q8—22no—
95C2×Q8—22no—
96C2×Q8—22no—
97C2×Q8—22no—
98C2×Q8—42no—
99C2×Q8—42no—
100C2×Q8—42no—
101C2×D4—22no—
102C2×D4—22no—
103C2×D4—21no—
104C2×D4—21no—
105C22×C4—22no—
106C22×C4—22no—
107C22×C4—21no—
108C22×C4—21no—
109C22×C4—21no—
110C22×C4C2×D411yescharacteristic
111C22×C4C2×D411yescharacteristic
112C22×C4C2×D411yescharacteristic
113C22×C4C2×D411yescharacteristic
114C22×C4C2×D411yescharacteristic
115C22×C4C2×D411yescharacteristic
116C22×C4C2411yescharacteristic
117SD16—48no—
118C2×C8—22no—
119C2×C8—22no—
120C2×C8—22no—
121C2×C8—22no—
122C2×C8—22no—
123C2×C8#4114yes—
124C4⋊C4—22no—
125C4⋊C4—22no—
126C4⋊C4—21no—
127C4⋊C4—22no—
128C4⋊C4—22no—
129C4⋊C4—21no—
130C4⋊C4—42no—
131C4⋊C4—42no—
132C4⋊C4—42no—
133C4⋊C4—42no—
134C4⋊C4—42no—
135C4⋊C4—42no—
136C22⋊C4—42no—
137C22⋊C4—42no—
138C22⋊C4—42no—
139C22⋊C4—42no—
140C22⋊C4—42no—
141C22⋊C4—42no—
142C42—22no—
143C42—22no—
144C42—22no—
145C42—21no—
146C42—22no—
147C42—21no—
148C42—41no—
149C42SD1614yescomplemented
150C23—41no—
151C23—41no—
152C23—41no—
153C23—41no—
154C23—41no—
155C23—41no—
156C23—41no—
157C23C22×D411yescharacteristic
158Q8—42no—
159Q8—42no—
160Q8—42no—
161Q8—42no—
162Q8—84no—
163D4—42no—
164D4—42no—
165C2×C4—21no—
166C2×C4—21no—
167C2×C4—21no—
168C2×C4—21no—
169C2×C4—21no—
170C2×C4—21no—
171C2×C4—21no—
172C2×C4—21no—
173C2×C4—21no—
174C2×C4—21no—
175C2×C4—21no—
176C2×C4—21no—
177C2×C4—21no—
178C2×C4—41no—
179C2×C4—42no—
180C2×C4—42no—
181C2×C4—41no—
182C2×C4—42no—
183C2×C4—41no—
184C2×C4—42no—
185C2×C4—41no—
186C2×C4—41no—
187C2×C4—41no—
188C2×C4—41no—
189C2×C4—42no—
190C2×C4—41no—
191C2×C4—41no—
192C2×C4—42no—
193C2×C4C2×SD1612yes—
194C2×C4C2×SD1612yes—
195C2×C4C2×SD1612yes—
196C2×C4#13512yes—
197C2×C4#13712yes—
198C2×C4C2×#4112yes—
199C2×C4H(5,𝔽2)11yescharacteristic
200C2×C4H(5,𝔽2)12yes—
201C2×C42−1+411yescharacteristic
202C8—24no—
203C8—42no—
204C22—81no—
205C22—81no—
206C22—81no—
207C22—81no—
208C22—81no—
209C22—81no—
210C22—81no—
211C22#53211yescharacteristic
212C22#53511yescharacteristic
213C22#53711yescharacteristic
214C22C22×SD1611yescharacteristic
215C22C2×#13511yescharacteristic
216C22C2×#13711yescharacteristic
217C22#57511yescharacteristic
218C4—22no—
219C4—22no—
220C4—22no—
221C4—22no—
222C4—22no—
223C4—41no—
224C4—81no—
225C4—82no—
226C4—81no—
227C4—82no—
228C4—81no—
229C4#53812yes—
230C4#54212yes—
231C2—161no—
232C2#266211yescharacteristic
233C2#316711yescharacteristic
234C2#317211yescharacteristic
235C2#320611yescharacteristic
236C2#321111yescharacteristic
237C2#324611yescharacteristic
238C2#324811yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC22×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #2862C2semidirect
3. #2066C2semidirect
4. #2058C2semidirect
6. #2043C2semidirect
10. #1933C2semidirect
11. C4×C4⋊C8C2semidirect
149. C42SD16semidirect

Maximal subgroups

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

subgroupindex
#28622
#28602
#20662
#20582
#20552
#20432
#20232
#20132
#19682
#19332
C4×C4⋊C82