Finite Group Databasebeta

← #19408

Subgroups — #19408

Order 256.

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

normal subgroups only

This group has 337 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#3206C211yesmaximal, characteristic, complemented
2#3079C211yesmaximal, characteristic, complemented
3#3035C211yesmaximal, characteristic
4#2988C211yesmaximal, characteristic, complemented
5#1680C212yesmaximal, complemented
6#1679C212yesmaximal, complemented
7#1666C212yesmaximal, complemented
8#1665C212yesmaximal, complemented
9#1570C211yesmaximal, characteristic, complemented
10#1547C211yesmaximal, characteristic, complemented
11C2×#332C211yesmaximal, characteristic, complemented
12C2×#135—21no—
13#532C2211yescharacteristic
14C2×C4⋊Q8C2211yescharacteristic
15C4×#41C2211yescharacteristic
16#492C2212yes—
17#487C2212yes—
18#477—21no—
19#476—21no—
20#475C2212yes—
21#474C2212yes—
22#464—21no—
23#463—21no—
24#462C2211yescharacteristic
25#461—21no—
26#461—21no—
27#461C2211yescharacteristic
28#460C2211yescharacteristic
29#459C2211yescharacteristic
30#448—22no—
31C8×Q8C2212yes—
32C8×D4—22no—
33C8×D4C2212yes—
34#431—21no—
35#431C2211yescharacteristic
36#428—21no—
37#423C2211yescharacteristic
38C2×C4⋊C8—21no—
39C2×C4⋊C8C2211yescharacteristic
40C2×C4⋊C8C2211yescharacteristic
41#417—21no—
42#416—21no—
43#334C2212yes—
44#334C2212yes—
45#332C2212yes—
46#332C2212yes—
47#332C2212yes—
48#331C2212yes—
49C2×#41—21no—
50C2×#41D411yescharacteristic
51C22×Q8—21no—
52#135—44no—
53C2×Q16—21no—
54C2×SD16—21no—
55C2×SD16—21no—
56C2×SD16—22no—
57C2×SD16—22no—
58C2×D8—21no—
59C2×M16—21no—
60C22×C8—21no—
61C22×C8—21no—
62C22×C8—22no—
63C4⋊Q8—21no—
64C4⋊Q8—21no—
65C4⋊Q8—21no—
66C4⋊Q8C2312yes—
67C4⋊Q8C2311yescharacteristic
68#127C2311yescharacteristic
69#124—21no—
70#121—21no—
71#121—21no—
72C4×Q8D411yescharacteristic
73C4×Q8C2311yescharacteristic
74C4×D4—21no—
75C4×D4—21no—
76C4×D4—21no—
77C4×D4—21no—
78C4×D4D411yescharacteristic
79C4×D4C2311yescharacteristic
80#117—21no—
81#117D411yescharacteristic
82C2×C4⋊C4—21no—
83C2×C4⋊C4—21no—
84C2×C42—21no—
85C2×C42C2311yescharacteristic
86#107—22no—
87#106—22no—
88C4⋊C8—22no—
89C4⋊C8—21no—
90C4⋊C8—21no—
91C4⋊C8—21no—
92C4⋊C8D411yescharacteristic
93C4⋊C8D411yescharacteristic
94C4⋊C8C2312yes—
95C4⋊C8C2312yes—
96C4⋊C8C2311yescharacteristic
97C4⋊C8C2311yescharacteristic
98#103—22no—
99#103—22no—
100#103—22no—
101#103—21no—
102#103—21no—
103#102—22no—
104#102—21no—
105#102—21no—
106C22⋊C8—22no—
107C22⋊C8D412yes—
108C4×C8—22no—
109C4×C8C2312yes—
110#41—22no—
111#41—42no—
112#41—42no—
113#41SD1614yescomplemented
114C2×Q8—21no—
115C2×Q8—22no—
116C2×Q8—41no—
117C2×Q8—81no—
118C2×Q8—81no—
119C2×Q8C2×D411yescharacteristic
120C2×D4—21no—
121C2×D4—21no—
122C2×D4—21no—
123C2×D4—21no—
124C2×D4—41no—
125C2×D4C2×D411yescharacteristic
126C22×C4—21no—
127C22×C4—21no—
128C22×C4—21no—
129C22×C4—41no—
130C22×C4—41no—
131C22×C4C2×D411yescharacteristic
132C22×C4C2×D411yescharacteristic
133C22×C4#4111yescharacteristic
134Q16—42no—
135SD16—42no—
136SD16—42no—
137SD16—82no—
138SD16—82no—
139D8—42no—
140M16—42no—
141C2×C8—22no—
142C2×C8—22no—
143C2×C8—21no—
144C2×C8—21no—
145C2×C8—42no—
146C2×C8—82no—
147C2×C8—81no—
148C2×C8C2×D412yes—
149C2×C8C2×D412yes—
150C4⋊C4—21no—
151C4⋊C4—21no—
152C4⋊C4—22no—
153C4⋊C4—21no—
154C4⋊C4—22no—
155C4⋊C4—21no—
156C4⋊C4—41no—
157C4⋊C4—41no—
158C4⋊C4C2×D411yescharacteristic
159C4⋊C4C2×D411yescharacteristic
160C22⋊C4—21no—
161C22⋊C4—21no—
162C22⋊C4—21no—
163C22⋊C4—41no—
164C22⋊C4—41no—
165C42—21no—
166C42—21no—
167C42—21no—
168C42C2×D411yescharacteristic
169C42C2×D411yescharacteristic
170C42#4111yescharacteristic
171C42C2411yescharacteristic
172C23—21no—
173C23—41no—
174C23#12111yescharacteristic
175Q8—21no—
176Q8—41no—
177Q8—41no—
178Q8—42no—
179Q8—42no—
180Q8—82no—
181Q8C2×SD1612yes—
182D4—24no—
183D4—21no—
184D4—41no—
185D4—41no—
186D4—41no—
187D4—41no—
188D4—81no—
189D4—82no—
190D4C2×SD1612yes—
191C2×C4—21no—
192C2×C4—21no—
193C2×C4—21no—
194C2×C4—24no—
195C2×C4—21no—
196C2×C4—21no—
197C2×C4—41no—
198C2×C4—41no—
199C2×C4—41no—
200C2×C4—41no—
201C2×C4—42no—
202C2×C4—81no—
203C2×C4—82no—
204C2×C4—82no—
205C2×C4#12111yescharacteristic
206C2×C4#12111yescharacteristic
207C2×C4#12111yescharacteristic
208C2×C4C2×SD1612yes—
209C2×C4#13512yes—
210C2×C4C22×D411yescharacteristic
211C2×C4C22×D411yescharacteristic
212C2×C4C2×#4111yescharacteristic
213C8—22no—
214C8—22no—
215C8—41no—
216C8—41no—
217C8—42no—
218C8—42no—
219C22—21no—
220C22—21no—
221C22—22no—
222C22—41no—
223C22—41no—
224C22—81no—
225C22—81no—
226C22—81no—
227C22#46512yes—
228C22C2×#12111yescharacteristic
229C4—21no—
230C4—22no—
231C4—21no—
232C4—21no—
233C4—41no—
234C4—41no—
235C4—82no—
236C4—81no—
237C4#46512yes—
238C4C22×SD1611yescharacteristic
239C4C2×#13511yescharacteristic
240C4#57611yescharacteristic
241C4#57711yescharacteristic
242C2—22no—
243C2—42no—
244C2—161no—
245C2#304511yescharacteristic
246C2C2×#46511yescharacteristic
247C2#306911yescharacteristic
subgroupG/H
center, socleC22C2×#121
commutator, Frattini, cosocleC42C24
prime radical#3035C2
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ G
LECSG ▷ C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #3206C2semidirect
2. #3079C2semidirect
4. #2988C2semidirect
5. #1680C2semidirect
6. #1679C2semidirect
7. #1666C2semidirect
8. #1665C2semidirect
9. #1570C2semidirect
10. #1547C2semidirect
11. C2×#332C2semidirect
113. #41SD16semidirect

Maximal subgroups

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

subgroupindex
#32062
#30792
#30352
#29882
#16802
#16792
#16662
#16652
#15702
#15472
C2×#3322