Finite Group Databasebeta

← #33040

Subgroups — #33040

Order 256.

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

normal subgroups only

This group has 438 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×#478C211yesmaximal, characteristic
2C2×#461C211yesmaximal, characteristic
3#2642C211yesmaximal, characteristic, complemented
4#2275C211yesmaximal, characteristic
5#1989C211yesmaximal, characteristic, complemented
6#1981C211yesmaximal, characteristic, complemented
7#1928C212yesmaximal, complemented
8#1927C212yesmaximal, complemented
9#1877C212yesmaximal
10#1877C212yesmaximal
11#1832C211yesmaximal, characteristic, complemented
12C22×SD16C2212yes—
13C2×C4⋊Q8C2211yescharacteristic
14C2×#125C2211yescharacteristic
15C2×C4×D4C2212yes—
16C2×C4×D4C2211yescharacteristic
17#478—24no—
18#461—24no—
19C2×#107C2211yescharacteristic
20C2×#106C2211yescharacteristic
21C2×C4⋊C8C2211yescharacteristic
22C2×C4⋊C8C2211yescharacteristic
23C2×#103C2212yes—
24C2×#103C2212yes—
25C2×#103C2212yes—
26C2×#102C2211yescharacteristic
27C2×#102C2211yescharacteristic
28#400—21no—
29#400—22no—
30#400—21no—
31#391C2212yes—
32#391C2212yes—
33C4×C4⋊C4C2212yes—
34C4×C4⋊C4C2211yescharacteristic
35C4×C22⋊C4—22no—
36C4×C22⋊C4—21no—
37#376C2211yescharacteristic
38C2×#95—21no—
39#340C2212yes—
40#340C2212yes—
41#336C2212yes—
42#336C2212yes—
43C22×Q8C2312yes—
44C22×D4C2311yescharacteristic
45C23×C4—22no—
46C23×C4—21no—
47C2×SD16—24no—
48C2×SD16—24no—
49C2×SD16—24no—
50C22×C8C2312yes—
51C22×C8C2311yescharacteristic
52C22×C8C2311yescharacteristic
53C4⋊Q8—22no—
54C4⋊Q8—22no—
55#125—22no—
56#125—22no—
57C4×D4—24no—
58C4×D4—24no—
59C4×D4—22no—
60C4×D4D414yes—
61C2×C4⋊C4—21no—
62C2×C4⋊C4—21no—
63C2×C4⋊C4—21no—
64C2×C4⋊C4—21no—
65C2×C4⋊C4C2311yescharacteristic
66C2×C4⋊C4C2311yescharacteristic
67C2×C4⋊C4C2311yescharacteristic
68C2×C4⋊C4C2312yes—
69C2×C22⋊C4—22no—
70C2×C22⋊C4—21no—
71C2×C42—22no—
72C2×C42—21no—
73C2×C42C2312yes—
74C2×C42C2311yescharacteristic
75#107—22no—
76#106—22no—
77C4⋊C8—22no—
78C4⋊C8—22no—
79#103—24no—
80#103—24no—
81#103—24no—
82#102—22no—
83#102—22no—
84#95—22no—
85#95—21no—
86#95—22no—
87#95—22no—
88#95—22no—
89#95—22no—
90#95D411yescharacteristic
91#95D412yes—
92#95D411yescharacteristic
93C24—21no—
94C2×Q8—22no—
95C2×Q8—22no—
96C2×Q8—22no—
97C2×Q8—22no—
98C2×Q8—22no—
99C2×Q8—22no—
100C2×D4—21no—
101C2×D4—22no—
102C2×D4C2×D412yes—
103C2×D4#4114yes—
104C22×C4—21no—
105C22×C4—22no—
106C22×C4—28no—
107C22×C4—24no—
108C22×C4—21no—
109C22×C4—21no—
110C22×C4—22no—
111C22×C4—82no—
112C22×C4—82no—
113C22×C4—82no—
114C22×C4C2×D412yes—
115C22×C4C2×D412yes—
116C22×C4C2×D411yescharacteristic
117C22×C4C2×D411yescharacteristic
118C22×C4C2411yescharacteristic
119SD16—48no—
120C2×C8—21no—
121C2×C8—21no—
122C2×C8—22no—
123C2×C8—22no—
124C2×C8—22no—
125C2×C8—22no—
126C2×C8—22no—
127C4⋊C4—21no—
128C4⋊C4—22no—
129C4⋊C4—21no—
130C4⋊C4—21no—
131C4⋊C4—21no—
132C4⋊C4—21no—
133C4⋊C4—42no—
134C4⋊C4—42no—
135C4⋊C4—42no—
136C4⋊C4—42no—
137C4⋊C4C2×D412yes—
138C4⋊C4#4114yes—
139C22⋊C4—28no—
140C22⋊C4—24no—
141C42—22no—
142C42—22no—
143C42—81no—
144C42—82no—
145C42C2×D412yes—
146C42#4112yes—
147C23—22no—
148C23—24no—
149C23—42no—
150C23—41no—
151C23—41no—
152C23C22×D411yescharacteristic
153Q8—42no—
154Q8—42no—
155Q8—42no—
156Q8—42no—
157D4—24no—
158D4—22no—
159D4#12414yes—
160C2×C4—21no—
161C2×C4—22no—
162C2×C4—21no—
163C2×C4—22no—
164C2×C4—22no—
165C2×C4—22no—
166C2×C4—24no—
167C2×C4—22no—
168C2×C4—22no—
169C2×C4—22no—
170C2×C4—22no—
171C2×C4—42no—
172C2×C4—42no—
173C2×C4—42no—
174C2×C4—41no—
175C2×C4—41no—
176C2×C4—42no—
177C2×C4—42no—
178C2×C4—42no—
179C2×C4—42no—
180C2×C4—41no—
181C2×C4—41no—
182C2×C4—84no—
183C2×C4—88no—
184C2×C4#12112yes—
185C2×C4#12112yes—
186C2×C4C22×D411yescharacteristic
187C2×C4C2×#4112yes—
188C2×C4C2×#4111yescharacteristic
189C2×C4C2×#4112yes—
190C8—42no—
191C8—41no—
192C8—41no—
193C22—24no—
194C22—44no—
195C22—44no—
196C22—42no—
197C22—42no—
198C22C2×#12111yescharacteristic
199C22#52512yes—
200C22#57712yes—
201C22#57711yescharacteristic
202C22#57811yescharacteristic
203C4—22no—
204C4—22no—
205C4—42no—
206C4—42no—
207C4—44no—
208C4—42no—
209C4—82no—
210C4—81no—
211C4—81no—
212C4C2×#12411yescharacteristic
213C4#52912yes—
214C4#53211yescharacteristic
215C2—44no—
216C2#285011yescharacteristic
217C2#304211yescharacteristic
218C2#304611yescharacteristic
219C2#312012yes—
220C2#312312yes—
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC22×D4C23
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
3. #2642C2semidirect
5. #1989C2semidirect
6. #1981C2semidirect
7. #1928C2semidirect
8. #1927C2semidirect
11. #1832C2semidirect

Maximal subgroups

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

subgroupindex
C2×#4782
C2×#4612
#26422
#22752
#19892
#19812
#19282
#19272
#18772
#18322