Finite Group Databasebeta

← #19513

Subgroups — #19513

Order 256.

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

normal subgroups only

This group has 305 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#3227C211yesmaximal, characteristic, complemented
2#3227C211yesmaximal, characteristic, complemented
3#3214C211yesmaximal, characteristic
4#2981C211yesmaximal, characteristic, complemented
5#1697C212yesmaximal, complemented
6#1691C212yesmaximal, complemented
7#1677C212yesmaximal, complemented
8#1671C212yesmaximal, complemented
9#1572C211yesmaximal, characteristic, complemented
10#1558C211yesmaximal, characteristic, complemented
11#1553C211yesmaximal, characteristic, complemented
12#532C2211yescharacteristic
13#532C2211yescharacteristic
14C2×C4⋊Q8C2211yescharacteristic
15#498—21no—
16#498C2211yescharacteristic
17#498C2211yescharacteristic
18#496—21no—
19#489—21no—
20#486—21no—
21#486C2211yescharacteristic
22#486C2211yescharacteristic
23#481—22no—
24#481—22no—
25#476—22no—
26#476C2212yes—
27#476C2212yes—
28#461C2212yes—
29#459C2212yes—
30#449—22no—
31#436C2212yes—
32#435—22no—
33C8×D4C2212yes—
34#432C2211yescharacteristic
35#432C2211yescharacteristic
36#431C2211yescharacteristic
37#431C2211yescharacteristic
38#408—22no—
39#335C2212yes—
40#334C2212yes—
41#333C2212yes—
42#333C2212yes—
43#332C2212yes—
44#331C2212yes—
45C2×#41—21no—
46C2×#41—21no—
47C22×Q8—21no—
48C2×SD16—22no—
49C2×D8—22no—
50C2×M16—22no—
51C22×C8—22no—
52C4⋊Q8—21no—
53C4⋊Q8—21no—
54C4⋊Q8—21no—
55C4⋊Q8D411yescharacteristic
56C4⋊Q8C2311yescharacteristic
57C4⋊Q8C2311yescharacteristic
58C4⋊Q8C2311yescharacteristic
59#127D411yescharacteristic
60#127C2311yescharacteristic
61#124—21no—
62#124—21no—
63#121—21no—
64#121—21no—
65#121—21no—
66#121D412yes—
67C4×D4—21no—
68C4×D4—21no—
69C4×D4—21no—
70C4×D4C2312yes—
71C2×C4⋊C4—21no—
72C2×C4⋊C4—21no—
73C2×C42C2311yescharacteristic
74#106—22no—
75#106—22no—
76#106—22no—
77#106—22no—
78C4⋊C8D412yes—
79C4⋊C8C2312yes—
80C4⋊C8C2312yes—
81C4⋊C8C2312yes—
82#103—22no—
83#102—22no—
84#102—22no—
85#102—22no—
86#102—22no—
87#102—22no—
88C22⋊C8—21no—
89C22⋊C8—21no—
90C22⋊C8—21no—
91C22⋊C8—22no—
92C22⋊C8—22no—
93C22⋊C8D412yes—
94#97—21no—
95#97—21no—
96C4×C8—21no—
97C4×C8C2311yescharacteristic
98C4×C8C2311yescharacteristic
99#41—82no—
100#41—82no—
101C2×Q8—21no—
102C2×Q8—21no—
103C2×Q8—21no—
104C2×Q8—21no—
105C2×Q8—41no—
106C2×Q8—82no—
107C2×D4—22no—
108C2×D4—21no—
109C2×D4—21no—
110C2×D4—21no—
111C2×D4—21no—
112C2×D4—21no—
113C2×D4—41no—
114C2×D4C2×D412yes—
115C22×C4—22no—
116C22×C4—41no—
117C22×C4—41no—
118C22×C4C2×D411yescharacteristic
119C22×C4C2×D411yescharacteristic
120C22×C4#4111yescharacteristic
121SD16—82no—
122D8—82no—
123M16—82no—
124C2×C8—22no—
125C2×C8—22no—
126C2×C8—22no—
127C2×C8—82no—
128C2×C8C2×D412yes—
129C2×C8C2×D412yes—
130C4⋊C4—21no—
131C4⋊C4—21no—
132C4⋊C4—21no—
133C4⋊C4—21no—
134C4⋊C4—21no—
135C4⋊C4—21no—
136C4⋊C4—21no—
137C4⋊C4—41no—
138C4⋊C4—41no—
139C4⋊C4C2×D412yes—
140C22⋊C4—22no—
141C22⋊C4—22no—
142C22⋊C4—41no—
143C22⋊C4—41no—
144C42C2×D411yescharacteristic
145C42C2×D411yescharacteristic
146C42#4111yescharacteristic
147C42C2411yescharacteristic
148C23—22no—
149C23—41no—
150C23#12111yescharacteristic
151Q8—42no—
152Q8—81no—
153Q8—81no—
154Q8—81no—
155Q8—82no—
156D4—22no—
157D4—22no—
158D4—42no—
159D4—42no—
160D4—81no—
161D4—81no—
162D4—81no—
163D4—82no—
164D4—82no—
165C2×C4—22no—
166C2×C4—22no—
167C2×C4—22no—
168C2×C4—41no—
169C2×C4—42no—
170C2×C4—42no—
171C2×C4—41no—
172C2×C4—41no—
173C2×C4—41no—
174C2×C4—82no—
175C2×C4—82no—
176C2×C4#12111yescharacteristic
177C2×C4#12111yescharacteristic
178C2×C4#12111yescharacteristic
179C2×C4C22×D411yescharacteristic
180C2×C4C22×D411yescharacteristic
181C2×C4C2×#4111yescharacteristic
182C8—22no—
183C8—22no—
184C8—42no—
185C8—42no—
186C8—42no—
187C22—21no—
188C22—21no—
189C22—21no—
190C22—42no—
191C22—42no—
192C22—42no—
193C22—81no—
194C22—81no—
195C22—81no—
196C22C2×#12111yescharacteristic
197C4—21no—
198C4—21no—
199C4—21no—
200C4—21no—
201C4—42no—
202C4—81no—
203C4—81no—
204C4—81no—
205C4#57712yes—
206C4#57712yes—
207C2—41no—
208C2—82no—
209C2—161no—
210C2#304211yescharacteristic
211C2#305711yescharacteristic
212C2#306811yescharacteristic
subgroupG/H
center, socleC22C2×#121
commutator, Frattini, cosocleC42C24
prime radical#3214C2
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. #3227C2semidirect
2. #3227C2semidirect
4. #2981C2semidirect
5. #1697C2semidirect
6. #1691C2semidirect
7. #1677C2semidirect
8. #1671C2semidirect
9. #1572C2semidirect
10. #1558C2semidirect
11. #1553C2semidirect

Maximal subgroups

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

subgroupindex
#32272
#32142
#29812
#16972
#16912
#16772
#16712
#15722
#15582
#15532