Finite Group Databasebeta

← #49122303

Subgroups — #49122303

Order 36224.

406 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 406 conjugacy classes of subgroups — too many to draw. Only its 75 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
1C2×#22583635C212yesmaximal, complemented
2C2×#22583537C212yesmaximal, complemented
3#31826366C211yesmaximal, characteristic, complemented
4#31826234C211yesmaximal, characteristic, complemented
5#537×C283C211yesmaximal, characteristic, complemented
6C22×D1132C2211yescharacteristic
7#22583635—24no—
8#22583635—24no—
9C2×#17985379—22no—
10C2×#17985377—22no—
11C2×#17985376C2212yes—
12C2×(C2×D283)⋊C4—22no—
13C2×C4×Dic283C2211yescharacteristic
14#22583537—24no—
15C2×#41×C283C2212yes—
16C22⋊Q8×C283—22no—
17#121×C283—22no—
18#121×C283—22no—
19C2×C4⋊C4×C283C2211yescharacteristic
20C23×D283—21no—
21C2×D1132—22no—
22C2×D1132—22no—
23C2×D1132—21no—
24C2×D1132D412yescomplemented
25C22×Dic283—21no—
26#17985379—28no—
27#17985377—28no—
28#17985376—22no—
29#17985376D414yescomplemented
30C2×C283⋊C8—22no—
31(C2×D283)⋊C4—44no—
32C4×Dic283—22no—
33#41×C283—22no—
34#41×C283—22no—
35#41×C283—22no—
36#41×C283—22no—
37C2×Q8×C283D412yes—
38C2×D4×C283—22no—
39C2×D4×C283—22no—
40C2×D4×C283D412yes—
41C22×C4×C283—22no—
42C22×C4×C283D412yes—
43C22×C4×C283C2311yescharacteristic
44C4⋊C4×C283—22no—
45C4⋊C4×C283—22no—
46C22⋊C4×C283—24no—
47C22×D283—22no—
48C22×D283—42no—
49C22×D283—42no—
50C22×D283—41no—
51C22×D283—41no—
52D1132—24no—
53D1132—42no—
54C2×Dic283—22no—
55C2×Dic283—41no—
56C2×Dic283—41no—
57C283⋊C8—24no—
58C23×C283—22no—
59C23×C283C2×D411yescharacteristic
60Q8×C283—22no—
61Q8×C283—22no—
62D4×C283—22no—
63D4×C283—22no—
64D4×C283—42no—
65D4×C283—42no—
66D4×C283—82no—
67C2×C4×C283—22no—
68C2×C4×C283—22no—
69C2×C4×C283—42no—
70C2×C4×C283—42no—
71C2×C4×C283—42no—
72C2×C4×C283C2×D411yescharacteristic
73C2×C4×C283C2×D412yes—
74C2×C4×C283C2×D412yes—
75C2×C4×C283#4111yescharacteristic
76C22×C283—21no—
77C22×C283—21no—
78C22×C283—42no—
79C22×C283—42no—
80C22×C283—42no—
81C22×C283#12011yescharacteristic
82C22×C283#12111yescharacteristic
83C22×C283#12711yescharacteristic
84C2×D283—44no—
85C2×D283—42no—
86C2×D283—82no—
87C2×D283—81no—
88C2×D283—81no—
89C4×C283—42no—
90C4×C283—42no—
91C4×C283#12012yes—
92C4×C283#12112yes—
93Dic283—42no—
94C2×C283—21no—
95C2×C283—21no—
96C2×C283—82no—
97C2×C283#39211yescharacteristic
98C2×C283#45311yescharacteristic
99C2×C283#45411yescharacteristic
100D283—82no—
101C283#200611yescharacteristic, complemented, Sylow
102#2006—2831nomaximal, Sylow
103C2×AGL(1,ℤ/8)—2832no—
104#537D28311yescharacteristic, complemented
105C2×#104—2832no—
106#390—2831no—
107#343—2831no—
108C2×#41C2×D28312yes—
109C22×D4—2831no—
110AGL(1,ℤ/8)—5664no—
111AGL(1,ℤ/8)—5664no—
112C2×SD16—5662no—
113C2×D8—5662no—
114C2×M16—2832no—
115C22⋊Q8—22no—
116#121—22no—
117#121—22no—
118C2×C4⋊C4C2×D28311yescharacteristic
119C2×C22⋊C4—5662no—
120C2×C42—2831no—
121#104—5664no—
122C24—5661no—
123#41—22no—
124#41—22no—
125#41—22no—
126#41—22no—
127C2×Q8C22⋊D28312yes—
128C2×D4—22no—
129C2×D4—22no—
130C2×D4—2832no—
131C2×D4—5662no—
132C2×D4—5662no—
133C2×D4—5661no—
134C2×D4C22⋊D28312yes—
135C22×C4—22no—
136C22×C4—5661no—
137C22×C4C22⋊D28312yes—
138C22×C4C22×D28311yescharacteristic
139SD16—5668no—
140D8—5668no—
141M16—2834no—
142M16—5662no—
143C2×C8—5662no—
144C4⋊C4—22no—
145C4⋊C4—22no—
146C22⋊C4—24no—
147C22⋊C4—11324no—
148C42—5662no—
149C23—22no—
150C23—5662no—
151C23—11322no—
152C23—11322no—
153C23—11321no—
154C23—11321no—
155C23C2×C22⋊D28311yescharacteristic
156Q8—22no—
157Q8—22no—
158D4—22no—
159D4—22no—
160D4—42no—
161D4—42no—
162D4—82no—
163D4—5664no—
164D4—11322no—
165C2×C4—22no—
166C2×C4—22no—
167C2×C4—42no—
168C2×C4—42no—
169C2×C4—42no—
170C2×C4—5662no—
171C2×C4—11321no—
172C2×C4—11321no—
173C2×C4D4×D28311yescharacteristic
174C2×C4D4×D28312yes—
175C2×C4C2×C22⋊D28312yes—
176C2×C4#1798539111yescharacteristic
177C8—5664no—
178C22—21no—
179C22—21no—
180C22—42no—
181C22—42no—
182C22—42no—
183C22—11324no—
184C22—11322no—
185C22—22642no—
186C22—22641no—
187C22—22641no—
188C22#2258357911yescharacteristic
189C22#2258358111yescharacteristic
190C22#2258360611yescharacteristic
191C4—42no—
192C4—42no—
193C4—11322no—
194C4#2258357812yes—
195C4#2258358212yes—
196C2—21no—
197C2—21no—
198C2—82no—
199C2—22642no—
200C2#3182637311yescharacteristic
201C2#3182663311yescharacteristic
202C2#3182663811yescharacteristic
subgroupG/H
centerC22#22583579
commutator, cosocleC22×C4×C283C23
FrattiniC22×C4C22×D283
Fitting#537×C283C2
socleC22×C283#120
hypercenter#537D283
hypocenterC283#2006
LCSG ▷ C22×C4×C283 ▷ C2×C283 ▷ C283
UCSC1 ◁ C22 ◁ C22×C4 ◁ #537

Complements

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

NH ≅ G/Ntype
1. C2×#22583635C2semidirect
2. C2×#22583537C2semidirect
3. #31826366C2semidirect
4. #31826234C2semidirect
5. #537×C283C2semidirect
24. C2×D1132D4semidirect
29. #17985376D4semidirect
101. C283#2006semidirect
104. #537D283semidirect

Maximal subgroups

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

subgroupindex
C2×#225836352
C2×#225835372
#318263662
#318262342
#537×C2832
#2006283