Finite Group Databasebeta

← #10023

Subgroups — #10023

Order 256.

259 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 259 conjugacy classes of subgroups — too many to draw. Only its 49 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#3166C211yesmaximal, characteristic, complemented
2#3110C211yesmaximal, characteristic
3#2633C211yesmaximal, characteristic
4#1534C211yesmaximal, characteristic, complemented
5#1515C211yesmaximal, characteristic, complemented
6#1490C211yesmaximal, characteristic
7#1288C211yesmaximal, characteristic, complemented
8#533C2211yescharacteristic
9C2×#125C2211yescharacteristic
10C2×C4×D4C2211yescharacteristic
11#501—21no—
12#501—21no—
13#499—22no—
14#484—21no—
15#474—21no—
16#461—21no—
17#450—21no—
18#440—22no—
19#432C2211yescharacteristic
20#432C2211yescharacteristic
21#428—21no—
22#428—21no—
23C4×M16C2211yescharacteristic
24#399—21no—
25#398—21no—
26#397—21no—
27#394—21no—
28#385—22no—
29#383C2211yescharacteristic
30#330—22no—
31#328—22no—
32D4⋊C8—22no—
33C22×D4D411yescharacteristic
34C23×C4—21no—
35C2×SD16—22no—
36C2×M16—21no—
37C4⋊Q8D411yescharacteristic
38C4⋊Q8D411yescharacteristic
39#125—21no—
40#125—21no—
41#125—21no—
42#125—21no—
43#125—21no—
44C22⋊Q8—21no—
45C22⋊Q8—21no—
46C4×Q8—21no—
47C4×Q8—21no—
48C4×D4—21no—
49C4×D4—21no—
50C4×D4—21no—
51C4×D4—21no—
52#117—21no—
53C2×C4⋊C4—21no—
54C2×C4⋊C4—21no—
55C2×C4⋊C4—22no—
56C2×C4⋊C4D411yescharacteristic
57C2×C4⋊C4D411yescharacteristic
58C2×C4⋊C4D411yescharacteristic
59C2×C22⋊C4—22no—
60C2×C22⋊C4—21no—
61C2×C22⋊C4—22no—
62C2×C42C2311yescharacteristic
63#107—22no—
64#107—22no—
65#107—22no—
66#106—22no—
67#106—22no—
68C4⋊C8—21no—
69C4⋊C8—21no—
70#103—22no—
71#102—22no—
72C22⋊C8—21no—
73C22⋊C8—21no—
74#97—21no—
75#97—21no—
76#97D412yes—
77C4×C8D412yes—
78#95—22no—
79#95—21no—
80#95—21no—
81#95D412yes—
82C24—21no—
83C2×Q8—21no—
84C2×Q8—21no—
85C2×D4—21no—
86C2×D4—21no—
87C2×D4—42no—
88C22×C4—22no—
89C22×C4—21no—
90C22×C4—21no—
91C22×C4—21no—
92C22×C4—41no—
93C22×C4—41no—
94C22×C4—82no—
95C22×C4C2×D411yescharacteristic
96C22×C4C2×D411yescharacteristic
97C22×C4C2×D411yescharacteristic
98SD16—44no—
99M16—24no—
100C2×C8—22no—
101C2×C8—22no—
102C2×C8—41no—
103C4⋊C4—21no—
104C4⋊C4—21no—
105C4⋊C4—21no—
106C4⋊C4—21no—
107C4⋊C4—21no—
108C4⋊C4—21no—
109C4⋊C4—21no—
110C4⋊C4—21no—
111C4⋊C4—21no—
112C4⋊C4—21no—
113C4⋊C4—41no—
114C4⋊C4—41no—
115C4⋊C4—41no—
116C4⋊C4—41no—
117C4⋊C4—41no—
118C4⋊C4—41no—
119C4⋊C4—82no—
120C22⋊C4—44no—
121C22⋊C4—41no—
122C22⋊C4—41no—
123C22⋊C4—41no—
124C22⋊C4—82no—
125C42—41no—
126C42C2×D411yescharacteristic
127C42C2×D411yescharacteristic
128C42C2×D411yescharacteristic
129C42#4111yescharacteristic
130C23—22no—
131C23—42no—
132C23—41no—
133C23—41no—
134C23—42no—
135C23#12011yescharacteristic
136Q8—42no—
137Q8—81no—
138D4—22no—
139D4—22no—
140D4—41no—
141D4—41no—
142C2×C4—22no—
143C2×C4—41no—
144C2×C4—41no—
145C2×C4—42no—
146C2×C4—42no—
147C2×C4—41no—
148C2×C4—42no—
149C2×C4—41no—
150C2×C4—42no—
151C2×C4—41no—
152C2×C4—41no—
153C2×C4—41no—
154C2×C4—41no—
155C2×C4—41no—
156C2×C4—42no—
157C2×C4—82no—
158C2×C4—82no—
159C2×C4#12011yescharacteristic
160C2×C4#12011yescharacteristic
161C2×C4#12111yescharacteristic
162C2×C4#12111yescharacteristic
163C2×C4#12111yescharacteristic
164C2×C4#12711yescharacteristic
165C8—24no—
166C8—42no—
167C8—81no—
168C22—21no—
169C22—21no—
170C22—44no—
171C22—42no—
172C22—82no—
173C22—81no—
174C22—81no—
175C22#39211yescharacteristic
176C22#45612yes—
177C4—21no—
178C4—21no—
179C4—21no—
180C4—21no—
181C4—21no—
182C4—21no—
183C4—82no—
184C4—81no—
185C4—81no—
186C4—81no—
187C4—81no—
188C4—81no—
189C2—22no—
190C2—82no—
191C2#200511yescharacteristic
192C2#200811yescharacteristic
193C2#201011yescharacteristic
subgroupG/H
center, socleC22#392
commutator, Frattini, cosocleC2×C42C23
prime radicalC22×D4D4
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #3166C2semidirect
4. #1534C2semidirect
5. #1515C2semidirect
7. #1288C2semidirect

Maximal subgroups

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

subgroupindex
#31662
#31102
#26332
#15342
#15152
#14902
#12882