Finite Group Databasebeta

← #49123235

Subgroups — #49123235

Order 36224.

216 conjugacy classes of subgroups; only the 47 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 47 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#31827266C211yesmaximal, characteristic
2#31826908C211yesmaximal, characteristic, complemented
3C2×#22583554C211yesmaximal, characteristic, complemented
4#31826741C211yesmaximal, characteristic, complemented
5#31826271C211yesmaximal, characteristic, complemented
6#31826250C211yesmaximal, characteristic, complemented
7#361×C283C211yesmaximal, characteristic, complemented
8#22583655—21no—
9#22583643—21no—
10#22583636—21no—
11#22583634—22no—
12#22583632C2211yescharacteristic
13#22583629—21no—
14C2×Dic1132C2211yescharacteristic, complemented
15D8×D283—21no—
16#22583616C2211yescharacteristic
17Dic2264—21no—
18#22583555—21no—
19#22583554—22no—
20C2×C283⋊C16C2211yescharacteristic, complemented
21#22583528—21no—
22C2×D8×C283C2211yescharacteristic
23M32×C283C2211yescharacteristic, complemented
24#108×C283C2211yescharacteristic, complemented
25#17985392—21no—
26#17985392D411yescharacteristic
27#17985391—22no—
28#17985391—41no—
29C2×Dic566—21no—
30C2×C22⋊D283—42no—
31D4×D283—22no—
32D4×D283—41no—
33Dic1132—21no—
34Dic1132—21no—
35Dic1132—41no—
36Dic1132D411yescharacteristic
37#17985381—21no—
38#17985381—41no—
39#17985380—22no—
40D2264D411yescharacteristic
41#17985377—22no—
42#17985376D411yescharacteristic
43#17985375—21no—
44C8×D283—21no—
45C283⋊C16—21no—
46C2×D4×C283—21no—
47D8×C283—21no—
48D8×C283D412yes—
49M16×C283—21no—
50C2×C8×C283C2311yescharacteristic
51C16×C283—21no—
52C22×D283—42no—
53Dic566—21no—
54Dic566—41no—
55Dic566—41no—
56Dic566D811yescharacteristic
57D1132—21no—
58D1132D811yescharacteristic
59C22⋊D283—21no—
60C22⋊D283—42no—
61C22⋊D283—42no—
62C22⋊D283—41no—
63C2×Dic283—42no—
64C2×Dic283—41no—
65C4×D283—21no—
66C4×D283—41no—
67C283⋊C8—21no—
68C23×C283—41no—
69D4×C283—22no—
70D4×C283—41no—
71C2×C4×C283C2×D411yescharacteristic
72C8×C283—41no—
73C8×C283C2×D411yescharacteristic
74C8×C283C2×D411yescharacteristic
75C22×C283—42no—
76C22×C283—161no—
77C22×C283AGL(1,ℤ/8)11yescharacteristic
78C2×D283—21no—
79C2×D283—41no—
80C2×D283—162no—
81C4×C283#12011yescharacteristic
82C4×C283C2×D811yescharacteristic
83Dic283—21no—
84Dic283—41no—
85Dic283—81no—
86C2×C283—21no—
87C2×C283—82no—
88C2×C283#44711yescharacteristic
89D283—41no—
90D283—81no—
91C283#219411yescharacteristic, complemented, Sylow
92#2194—2831nomaximal, Sylow
93#576—2831no—
94#510—2831no—
95C2×SD32—2831no—
96#473—2831no—
97#361D28311yescharacteristic, complemented
98#359—2831no—
99#350—2831no—
100H(5,𝔽2)—5661no—
101#137—5661no—
102AGL(1,ℤ/8)—5661no—
103AGL(1,ℤ/8)—5662no—
104#135—2831no—
105#135—5661no—
106C2×Q16—2831no—
107C2×D8—5661no—
108C2×D8C2×D28311yescharacteristic
109#131—2831no—
110Q32—5661no—
111SD32—5661no—
112SD32—5662no—
113M32C2×D28311yescharacteristic, complemented
114C2×C16—2831no—
115#108C2×D28311yescharacteristic, complemented
116#101—5661no—
117#41—2831no—
118#41—5662no—
119#41—5661no—
120#41—11321no—
121C2×Q8—5661no—
122C2×D4—21no—
123C2×D4—5662no—
124C2×D4—11321no—
125C2×D4—11322no—
126Q16—2831no—
127Q16—5661no—
128Q16—5661no—
129Q16—11321no—
130SD16—5662no—
131SD16—5661no—
132SD16—11321no—
133D8—21no—
134D8—2831no—
135D8—5662no—
136D8D113212yes—
137M16—21no—
138M16—2831no—
139M16—5661no—
140C2×C8—5661no—
141C2×C8C22×D28311yescharacteristic
142C16—21no—
143C16—5661no—
144C23—41no—
145C23—11322no—
146Q8—2831no—
147Q8—5661no—
148Q8—11321no—
149Q8—11321no—
150D4—22no—
151D4—41no—
152D4—2831no—
153D4—5661no—
154D4—5661no—
155D4—11322no—
156D4—11322no—
157D4—11321no—
158C2×C4—5661no—
159C2×C4—11322no—
160C2×C4—11321no—
161C2×C4—11321no—
162C2×C4D4×D28311yescharacteristic
163C8—41no—
164C8—5661no—
165C8D4×D28311yescharacteristic
166C8C2×D113211yescharacteristic
167C22—42no—
168C22—161no—
169C22—5661no—
170C22—11321no—
171C22—45282no—
172C22#2258363711yescharacteristic
173C4—5661no—
174C4—11321no—
175C4—22641no—
176C4#2258358011yescharacteristic
177C4D8×D28311yescharacteristic
178C2—21no—
179C2—82no—
180C2—11321no—
181C2—22641no—
182C2#3182660011yescharacteristic
subgroupG/H
centerC2#31826600
commutator, cosocleC2×C8×C283C23
FrattiniC2×C8C22×D283
Fitting#361×C283C2
socleC2×C283#447
prime radical#31827266C2
hypercenter#361D283
hypocenterC283#2194
LCSG ▷ C2×C8×C283 ▷ C4×C283 ▷ C2×C283 ▷ C283
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ #361

Complements

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

NH ≅ G/Ntype
2. #31826908C2semidirect
3. C2×#22583554C2semidirect
4. #31826741C2semidirect
5. #31826271C2semidirect
6. #31826250C2semidirect
7. #361×C283C2semidirect
14. C2×Dic1132C22semidirect
20. C2×C283⋊C16C22semidirect
23. M32×C283C22semidirect
24. #108×C283C22semidirect
91. C283#2194semidirect
97. #361D283semidirect
113. M32C2×D283semidirect
115. #108C2×D283semidirect

Maximal subgroups

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

subgroupindex
#318272662
#318269082
C2×#225835542
#318267412
#318262712
#318262502
#361×C2832
#2194283