Finite Group Databasebeta

← #19497

Subgroups — #19497

Order 256.

301 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 301 conjugacy classes of subgroups — too many to draw. Only its 109 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#3209C211yesmaximal, characteristic, complemented
2#3107C211yesmaximal, characteristic
3#3079C211yesmaximal, characteristic
4#2991C211yesmaximal, characteristic
5#1699C212yesmaximal, complemented
6#1689C212yesmaximal, complemented
7#1680C212yesmaximal, complemented
8#1670C212yesmaximal, complemented
9#1574C211yesmaximal, characteristic, complemented
10#1561C211yesmaximal, characteristic, complemented
11#1546C211yesmaximal, characteristic, complemented
12#533C2211yescharacteristic
13C2×C4⋊Q8C2211yescharacteristic
14C4×#41C2211yescharacteristic
15#494C2212yes—
16#487C2212yes—
17#485—21no—
18#477—21no—
19#476—21no—
20#475C2212yes—
21#475C2211yescharacteristic
22#474C2212yes—
23#474C2211yescharacteristic
24#462—21no—
25#462—21no—
26#462C2211yescharacteristic
27#461C2211yescharacteristic
28#452—21no—
29#451—22no—
30C8×Q8C2212yes—
31C4×Q16—21no—
32C4×SD16—21no—
33#436—22no—
34#436C2212yes—
35#431—21no—
36#431C2211yescharacteristic
37#431C2211yescharacteristic
38#428—21no—
39#423—21no—
40#423C2211yescharacteristic
41#423C2211yescharacteristic
42#417—21no—
43#335C2212yes—
44#334C2212yes—
45#332C2212yes—
46#332C2212yes—
47#332C2212yes—
48#332C2212yes—
49C2×#41D411yescharacteristic
50C22×Q8—21no—
51C2×Q16—21no—
52C2×Q16—22no—
53C2×Q16—22no—
54C2×SD16—21no—
55C2×M16—21no—
56C2×M16—21no—
57C2×M16—22no—
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—22no—
83#107—21no—
84#106—22no—
85#106—21no—
86C4⋊C8—22no—
87C4⋊C8—21no—
88C4⋊C8—21no—
89C4⋊C8D411yescharacteristic
90C4⋊C8D411yescharacteristic
91C4⋊C8C2312yes—
92C4⋊C8C2311yescharacteristic
93C4⋊C8C2312yes—
94C4⋊C8C2311yescharacteristic
95#103—22no—
96#103—21no—
97#103—22no—
98#103—22no—
99#103—21no—
100#103—21no—
101#102—22no—
102#102—21no—
103C22⋊C8—21no—
104C22⋊C8—22no—
105C22⋊C8D412yes—
106C4×C8—22no—
107C4×C8—21no—
108C4×C8C2312yes—
109#41—22no—
110#41—41no—
111C2×Q8—21no—
112C2×Q8—22no—
113C2×Q8—21no—
114C2×Q8—41no—
115C2×Q8—81no—
116C2×Q8—81no—
117C2×Q8C2×D411yescharacteristic
118C2×D4—21no—
119C2×D4C2×D411yescharacteristic
120C22×C4—21no—
121C22×C4—21no—
122C22×C4—21no—
123C22×C4—41no—
124C22×C4C2×D411yescharacteristic
125C22×C4C2×D411yescharacteristic
126C22×C4#4111yescharacteristic
127Q16—42no—
128Q16—82no—
129Q16—82no—
130SD16—42no—
131M16—42no—
132M16—82no—
133M16—81no—
134C2×C8—21no—
135C2×C8—21no—
136C2×C8—22no—
137C2×C8—22no—
138C2×C8C2×D412yes—
139C2×C8C2×D412yes—
140C4⋊C4—21no—
141C4⋊C4—21no—
142C4⋊C4—22no—
143C4⋊C4—21no—
144C4⋊C4—21no—
145C4⋊C4—22no—
146C4⋊C4—21no—
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—41no—
170Q8—41no—
171Q8—41no—
172Q8—42no—
173Q8—41no—
174Q8—42no—
175Q8—82no—
176Q8#13712yes—
177D4—21no—
178D4—21no—
179D4—42no—
180C2×C4—21no—
181C2×C4—21no—
182C2×C4—21no—
183C2×C4—21no—
184C2×C4—21no—
185C2×C4—22no—
186C2×C4—21no—
187C2×C4—21no—
188C2×C4—42no—
189C2×C4—41no—
190C2×C4—41no—
191C2×C4—42no—
192C2×C4—41no—
193C2×C4—41no—
194C2×C4—42no—
195C2×C4—82no—
196C2×C4#12111yescharacteristic
197C2×C4#12111yescharacteristic
198C2×C4#12111yescharacteristic
199C2×C4C22×D411yescharacteristic
200C2×C4C22×D411yescharacteristic
201C2×C4C2×#4111yescharacteristic
202C8—22no—
203C8—22no—
204C8—41no—
205C8—42no—
206C8—41no—
207C8—42no—
208C22—21no—
209C22—21no—
210C22—21no—
211C22—41no—
212C22—41no—
213C22—41no—
214C22C2×#12111yescharacteristic
215C4—21no—
216C4—21no—
217C4—22no—
218C4—21no—
219C4—41no—
220C4—41no—
221C4—82no—
222C4—81no—
223C4—81no—
224C4#47312yes—
225C4C2×#13711yescharacteristic
226C4#57511yescharacteristic
227C4#57611yescharacteristic
228C4#57711yescharacteristic
229C2—41no—
230C2—81no—
231C2#304511yescharacteristic
232C2#305611yescharacteristic
233C2C2×#47311yescharacteristic
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. #3209C2semidirect
5. #1699C2semidirect
6. #1689C2semidirect
7. #1680C2semidirect
8. #1670C2semidirect
9. #1574C2semidirect
10. #1561C2semidirect
11. #1546C2semidirect

Maximal subgroups

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

subgroupindex
#32092
#31072
#30792
#29912
#16992
#16892
#16802
#16702
#15742
#15612
#15462