Finite Group Databasebeta

← #19451

Subgroups — #19451

Order 256.

309 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 309 conjugacy classes of subgroups — too many to draw. Only its 113 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#3229C211yesmaximal, characteristic, complemented
2#3076C211yesmaximal, characteristic
3#3039C211yesmaximal, characteristic
4#2989C211yesmaximal, characteristic
5#1700C212yesmaximal, complemented
6#1699C212yesmaximal, complemented
7#1692C212yesmaximal, complemented
8#1691C212yesmaximal, complemented
9#1571C211yesmaximal, characteristic, complemented
10#1550C211yesmaximal, characteristic, complemented
11C2×#332C211yesmaximal, characteristic, complemented
12C2×#137—21no—
13#533C2211yescharacteristic
14C2×C4⋊Q8C2211yescharacteristic
15C4×#41C2211yescharacteristic
16#498C2212yes—
17#487C2212yes—
18#483—22no—
19#479—21no—
20#478—21no—
21#477C2211yescharacteristic
22#476C2211yescharacteristic
23#475—21no—
24#475—21no—
25#475C2211yescharacteristic
26#474C2211yescharacteristic
27#462—21no—
28#462C2212yes—
29#461—21no—
30#461C2212yes—
31#446C2212yes—
32#436C2212yes—
33#435—22no—
34#432—21no—
35#431C2211yescharacteristic
36#427—21no—
37#423—21no—
38#423C2211yescharacteristic
39C2×C4⋊C8C2211yescharacteristic
40C2×C4⋊C8C2211yescharacteristic
41#419—21no—
42#416—21no—
43#335C2212yes—
44#335C2212yes—
45#333C2212yes—
46#332C2212yes—
47#332C2212yes—
48#332C2212yes—
49C2×#41D411yescharacteristic
50C22×Q8—21no—
51#137—44no—
52C2×Q16—22no—
53C2×SD16—22no—
54C2×M16—21no—
55C2×M16—22no—
56C22×C8—21no—
57C22×C8—21no—
58C4⋊Q8—21no—
59C4⋊Q8—21no—
60C4⋊Q8—21no—
61C4⋊Q8C2312yes—
62C4⋊Q8C2311yescharacteristic
63C4⋊Q8C2311yescharacteristic
64#125—21no—
65C22⋊Q8—21no—
66C22⋊Q8—21no—
67C4×Q8—21no—
68C4×Q8—21no—
69C4×Q8D411yescharacteristic
70C4×Q8C2311yescharacteristic
71C4×D4—21no—
72C4×D4—21no—
73C4×D4D411yescharacteristic
74C4×D4C2311yescharacteristic
75#117—21no—
76#117—21no—
77#117D411yescharacteristic
78C2×C4⋊C4—21no—
79C2×C4⋊C4—21no—
80C2×C42—21no—
81C2×C42C2311yescharacteristic
82#107—21no—
83#107—21no—
84#106—22no—
85#106—22no—
86#106—21no—
87#106—21no—
88C4⋊C8—22no—
89C4⋊C8—21no—
90C4⋊C8—21no—
91C4⋊C8—21no—
92C4⋊C8D411yescharacteristic
93C4⋊C8D411yescharacteristic
94C4⋊C8C2312yes—
95C4⋊C8C2311yescharacteristic
96C4⋊C8C2311yescharacteristic
97C4⋊C8C2312yes—
98#103—21no—
99#103—21no—
100#103—22no—
101#103—22no—
102#103—22no—
103#103—22no—
104#102—21no—
105#102—21no—
106C22⋊C8—22no—
107C22⋊C8D412yes—
108#97—22no—
109C4×C8C2312yes—
110#41—22no—
111#41—22no—
112C2×Q8—21no—
113C2×Q8—22no—
114C2×Q8—21no—
115C2×Q8—41no—
116C2×Q8—42no—
117C2×Q8—42no—
118C2×Q8C2×D411yescharacteristic
119C2×D4—21no—
120C2×D4C2×D411yescharacteristic
121C22×C4—21no—
122C22×C4—21no—
123C22×C4—21no—
124C22×C4—41no—
125C22×C4C2×D411yescharacteristic
126C22×C4C2×D411yescharacteristic
127C22×C4#4111yescharacteristic
128Q16—44no—
129SD16—44no—
130M16—42no—
131M16—82no—
132C2×C8—22no—
133C2×C8—21no—
134C2×C8—21no—
135C2×C8—22no—
136C2×C8—42no—
137C2×C8—81no—
138C2×C8C2×D412yes—
139C2×C8C2×D412yes—
140C4⋊C4—21no—
141C4⋊C4—21no—
142C4⋊C4—21no—
143C4⋊C4—21no—
144C4⋊C4—22no—
145C4⋊C4—21no—
146C4⋊C4—22no—
147C4⋊C4—21no—
148C4⋊C4—41no—
149C4⋊C4—41no—
150C4⋊C4—41no—
151C4⋊C4—41no—
152C4⋊C4C2×D411yescharacteristic
153C4⋊C4C2×D411yescharacteristic
154C22⋊C4—21no—
155C22⋊C4—21no—
156C22⋊C4—21no—
157C22⋊C4—41no—
158C42—21no—
159C42—21no—
160C42—21no—
161C42—41no—
162C42C2×D411yescharacteristic
163C42C2×D411yescharacteristic
164C42#4111yescharacteristic
165C42C2411yescharacteristic
166C23—21no—
167C23#12111yescharacteristic
168Q8—21no—
169Q8—21no—
170Q8—42no—
171Q8—42no—
172Q8—41no—
173Q8—41no—
174Q8—81no—
175Q8—82no—
176D4—21no—
177D4—21no—
178D4—42no—
179C2×C4—21no—
180C2×C4—21no—
181C2×C4—21no—
182C2×C4—21no—
183C2×C4—21no—
184C2×C4—41no—
185C2×C4—41no—
186C2×C4—41no—
187C2×C4—42no—
188C2×C4—41no—
189C2×C4—41no—
190C2×C4—42no—
191C2×C4—81no—
192C2×C4—82no—
193C2×C4#12111yescharacteristic
194C2×C4#12111yescharacteristic
195C2×C4#12111yescharacteristic
196C2×C4AGL(1,ℤ/8)12yes—
197C2×C4#13712yes—
198C2×C4C22×D411yescharacteristic
199C2×C4C22×D411yescharacteristic
200C2×C4C2×#4111yescharacteristic
201C8—22no—
202C8—22no—
203C8—41no—
204C8—41no—
205C8—42no—
206C8—42no—
207C22—21no—
208C22—21no—
209C22—41no—
210C22—41no—
211C22—41no—
212C22#46812yes—
213C22C2×#12111yescharacteristic
214C4—21no—
215C4—21no—
216C4—21no—
217C4—41no—
218C4—41no—
219C4—41no—
220C4—81no—
221C4—82no—
222C4—81no—
223C4#46812yes—
224C4C2×AGL(1,ℤ/8)11yescharacteristic
225C4C2×#13711yescharacteristic
226C4#57711yescharacteristic
227C4#57811yescharacteristic
228C2—22no—
229C2—81no—
230C2#304611yescharacteristic
231C2C2×#46811yescharacteristic
232C2#306911yescharacteristic
subgroupG/H
center, socleC22C2×#121
commutator, Frattini, cosocleC42C24
prime radicalC2×#41D4
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. #3229C2semidirect
5. #1700C2semidirect
6. #1699C2semidirect
7. #1692C2semidirect
8. #1691C2semidirect
9. #1571C2semidirect
10. #1550C2semidirect
11. C2×#332C2semidirect

Maximal subgroups

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

subgroupindex
#32292
#30762
#30392
#29892
#17002
#16992
#16922
#16912
#15712
#15502
C2×#3322