Finite Group Databasebeta

← #19519

Subgroups — #19519

Order 256.

301 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 301 conjugacy classes of subgroups — too many to draw. Only its 105 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#3253C211yesmaximal, characteristic
2#3246C211yesmaximal, characteristic, complemented
3#3228C211yesmaximal, characteristic, complemented
4#2981C211yesmaximal, characteristic, complemented
5#1697C212yesmaximal, complemented
6#1691C212yesmaximal, complemented
7#1681C212yesmaximal, complemented
8#1675C212yesmaximal, complemented
9#1571C211yesmaximal, characteristic, complemented
10#1559C211yesmaximal, characteristic, complemented
11#1551C211yesmaximal, characteristic, complemented
12#533C2211yescharacteristic
13#532C2211yescharacteristic
14#532C2211yescharacteristic
15#500C2211yescharacteristic
16#498C2211yescharacteristic
17#486C2211yescharacteristic
18#486C2211yescharacteristic
19#482—22no—
20#478—22no—
21#476—21no—
22#476C2212yes—
23#476C2212yes—
24#475—21no—
25#469—22no—
26#465—22no—
27#461C2212yes—
28#460—21no—
29#459—21no—
30#459C2212yes—
31#436C2212yes—
32#435—22no—
33C8×D4C2212yes—
34#432C2211yescharacteristic
35#431C2211yescharacteristic
36#423C2211yescharacteristic
37C2×C4⋊C8C2211yescharacteristic
38#408—22no—
39#335C2212yes—
40#335C2212yes—
41#333C2212yes—
42#333C2212yes—
43#332C2212yes—
44#331C2212yes—
45C2×#41—21no—
46C2×#41—21no—
47C2×SD16—22no—
48C2×D8—22no—
49C2×M16—21no—
50C2×M16—22no—
51C22×C8—21no—
52C22×C8—22no—
53C4⋊Q8D411yescharacteristic
54C4⋊Q8C2311yescharacteristic
55C4⋊Q8C2311yescharacteristic
56C4⋊Q8C2311yescharacteristic
57#127D411yescharacteristic
58#127C2311yescharacteristic
59#125—21no—
60#124—21no—
61#124—21no—
62C22⋊Q8—21no—
63C22⋊Q8—21no—
64#121—21no—
65#121—21no—
66#121—21no—
67#121D412yes—
68C4×Q8—21no—
69C4×Q8—21no—
70C4×D4—21no—
71C4×D4—21no—
72C4×D4—21no—
73C4×D4C2312yes—
74#117—21no—
75C2×C42C2311yescharacteristic
76#107—22no—
77#106—22no—
78#106—22no—
79#106—22no—
80C4⋊C8—21no—
81C4⋊C8—21no—
82C4⋊C8D412yes—
83C4⋊C8C2312yes—
84C4⋊C8C2312yes—
85C4⋊C8C2312yes—
86#103—22no—
87#102—22no—
88#102—22no—
89#102—22no—
90#102—22no—
91#102—22no—
92C22⋊C8—21no—
93C22⋊C8—22no—
94C22⋊C8—22no—
95C22⋊C8D412yes—
96#97—21no—
97C4×C8C2311yescharacteristic
98C4×C8C2311yescharacteristic
99#41—82no—
100#41—82no—
101C2×Q8—21no—
102C2×Q8—21no—
103C2×Q8—21no—
104C2×Q8—21no—
105C2×D4—22no—
106C2×D4—21no—
107C2×D4—21no—
108C2×D4—21no—
109C2×D4—21no—
110C2×D4—21no—
111C2×D4—41no—
112C2×D4C2×D412yes—
113C22×C4—22no—
114C22×C4—41no—
115C22×C4C2×D411yescharacteristic
116C22×C4C2×D411yescharacteristic
117C22×C4#4111yescharacteristic
118SD16—82no—
119D8—82no—
120M16—82no—
121M16—81no—
122C2×C8—22no—
123C2×C8—22no—
124C2×C8—22no—
125C2×C8—82no—
126C2×C8—81no—
127C2×C8C2×D412yes—
128C2×C8C2×D412yes—
129C4⋊C4—21no—
130C4⋊C4—21no—
131C4⋊C4—21no—
132C4⋊C4—21no—
133C4⋊C4—21no—
134C4⋊C4—21no—
135C4⋊C4—21no—
136C4⋊C4—41no—
137C4⋊C4—41no—
138C4⋊C4C2×D412yes—
139C22⋊C4—22no—
140C22⋊C4—22no—
141C22⋊C4—41no—
142C22⋊C4—41no—
143C22⋊C4—41no—
144C42—41no—
145C42C2×D411yescharacteristic
146C42C2×D411yescharacteristic
147C42#4111yescharacteristic
148C42C2411yescharacteristic
149C23—22no—
150C23—41no—
151C23#12111yescharacteristic
152Q8—41no—
153Q8—41no—
154Q8—81no—
155Q8—81no—
156Q8—81no—
157D4—22no—
158D4—22no—
159D4—41no—
160D4—42no—
161D4—41no—
162D4—81no—
163D4—81no—
164D4—81no—
165D4—82no—
166D4—82no—
167C2×C4—21no—
168C2×C4—22no—
169C2×C4—21no—
170C2×C4—22no—
171C2×C4—41no—
172C2×C4—42no—
173C2×C4—42no—
174C2×C4—41no—
175C2×C4—41no—
176C2×C4—41no—
177C2×C4—82no—
178C2×C4#12111yescharacteristic
179C2×C4#12111yescharacteristic
180C2×C4#12111yescharacteristic
181C2×C4C22×D411yescharacteristic
182C2×C4C22×D411yescharacteristic
183C2×C4C2×#4111yescharacteristic
184C8—22no—
185C8—22no—
186C8—42no—
187C8—42no—
188C8—42no—
189C22—21no—
190C22—21no—
191C22—21no—
192C22—42no—
193C22—42no—
194C22—42no—
195C22—81no—
196C22—81no—
197C22—81no—
198C22C2×#12111yescharacteristic
199C4—21no—
200C4—21no—
201C4—21no—
202C4—21no—
203C4—42no—
204C4—81no—
205C4—81no—
206C4—81no—
207C4#57611yescharacteristic
208C4#57712yes—
209C4#57811yescharacteristic
210C2—41no—
211C2—82no—
212C2—161no—
213C2#304211yescharacteristic
214C2#306111yescharacteristic
215C2#306811yescharacteristic
subgroupG/H
center, socleC22C2×#121
commutator, Frattini, cosocleC42C24
prime radical#3253C2
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
2. #3246C2semidirect
3. #3228C2semidirect
4. #2981C2semidirect
5. #1697C2semidirect
6. #1691C2semidirect
7. #1681C2semidirect
8. #1675C2semidirect
9. #1571C2semidirect
10. #1559C2semidirect
11. #1551C2semidirect

Maximal subgroups

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

subgroupindex
#32532
#32462
#32282
#29812
#16972
#16912
#16812
#16752
#15712
#15592
#15512