Finite Group Databasebeta

← #49122302

Subgroups — #49122302

Order 36224.

406 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 406 conjugacy classes of subgroups — too many to draw. Only its 73 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×#22583636C212yesmaximal, complemented
2#31827046C211yesmaximal, characteristic, complemented
3C2×#22583538C212yesmaximal, complemented
4#31826234C211yesmaximal, characteristic, complemented
5#390×C283C211yesmaximal, characteristic, complemented
6C2×#17985392C2212yes—
7#22583636—24no—
8#22583636—24no—
9C2×#17985380—22no—
10C2×#17985377—22no—
11C2×#17985376C2212yes—
12#22583590—22no—
13#22583585—22no—
14#22583584—22no—
15#22583538—24no—
16C22×D4×C283C2211yescharacteristic
17C2×C4⋊C4×C283C2211yescharacteristic
18C2×C22⋊C4×C283—22no—
19C2×C42×C283C2211yescharacteristic
20#17985392—22no—
21#17985392—22no—
22#17985392—22no—
23#17985392—22no—
24C2×Dic566D412yes—
25C2×D1132—22no—
26C2×D1132D412yes—
27C2×C22⋊D283—22no—
28C2×C4×D283—22no—
29#17985380—28no—
30#17985377—28no—
31#17985376—22no—
32#17985376D414yescomplemented
33C2×C283⋊C8—22no—
34C4⋊Dic283—22no—
35(C2×D283)⋊C4—24no—
36C24×C283—21no—
37C2×D4×C283—22no—
38C2×D4×C283—22no—
39C2×D4×C283—21no—
40C2×D4×C283D412yescomplemented
41C22×C4×C283—21no—
42C22×C4×C283D412yes—
43C22×C4×C283C2311yescharacteristic
44C4⋊C4×C283—22no—
45C22⋊C4×C283—44no—
46C42×C283—22no—
47C22×D283—22no—
48Dic566—22no—
49Dic566—22no—
50D1132—22no—
51D1132—22no—
52D1132—82no—
53C22⋊D283—42no—
54C22⋊D283—42no—
55C2×Dic283—22no—
56C4×D283—42no—
57C4×D283—42no—
58C283⋊C8—24no—
59C23×C283—22no—
60C23×C283—42no—
61C23×C283—42no—
62C23×C283—41no—
63C23×C283—41no—
64C23×C283C2×D411yescharacteristic
65D4×C283—24no—
66D4×C283—42no—
67C2×C4×C283—22no—
68C2×C4×C283—22no—
69C2×C4×C283—41no—
70C2×C4×C283—41no—
71C2×C4×C283—42no—
72C2×C4×C283C2×D412yes—
73C2×C4×C283C2×D412yes—
74C2×C4×C283C2×D411yescharacteristic
75C2×C4×C283#4111yescharacteristic
76C22×C283—21no—
77C22×C283—21no—
78C22×C283—44no—
79C22×C283—42no—
80C22×C283—82no—
81C22×C283—81no—
82C22×C283—81no—
83C22×C283#12011yescharacteristic
84C22×C283#12111yescharacteristic
85C22×C283#12711yescharacteristic
86C2×D283—42no—
87C2×D283—42no—
88C2×D283—42no—
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#537—2831no—
105C2×#104—2832no—
106#390D28311yescharacteristic, complemented
107#343—2831no—
108C2×#41—2832no—
109C22×D4C2×D28311yescharacteristic
110AGL(1,ℤ/8)—5664no—
111AGL(1,ℤ/8)—5664no—
112C2×SD16—5662no—
113C2×D8—5662no—
114C2×M16—2832no—
115C22⋊Q8—5662no—
116#121—5662no—
117#121—5662no—
118C2×C4⋊C4C2×D28311yescharacteristic
119C2×C22⋊C4—22no—
120C2×C42C2×D28311yescharacteristic
121#104—5664no—
122C24—21no—
123#41—5662no—
124#41—5662no—
125#41—5662no—
126#41—5662no—
127C2×Q8—2832no—
128C2×D4—22no—
129C2×D4—22no—
130C2×D4—21no—
131C2×D4—2832no—
132C2×D4—5662no—
133C2×D4—5662no—
134C2×D4D113212yescomplemented
135C22×C4—21no—
136C22×C4—5662no—
137C22×C4C22⋊D28312yes—
138C22×C4C22×D28311yescharacteristic
139SD16—5668no—
140D8—5668no—
141M16—2834no—
142M16—5662no—
143C2×C8—5662no—
144C4⋊C4—22no—
145C4⋊C4—5662no—
146C22⋊C4—44no—
147C22⋊C4—5664no—
148C42—22no—
149C23—22no—
150C23—42no—
151C23—42no—
152C23—41no—
153C23—41no—
154C23—5662no—
155C23C2×C22⋊D28311yescharacteristic
156Q8—5662no—
157Q8—5662no—
158D4—24no—
159D4—42no—
160D4—5662no—
161D4—5662no—
162D4—11322no—
163D4—11322no—
164D4—22642no—
165C2×C4—22no—
166C2×C4—22no—
167C2×C4—41no—
168C2×C4—41no—
169C2×C4—42no—
170C2×C4—5662no—
171C2×C4—11322no—
172C2×C4—11322no—
173C2×C4D4×D28312yes—
174C2×C4D4×D28312yes—
175C2×C4C2×D113211yescharacteristic
176C2×C4#1798539211yescharacteristic
177C8—5664no—
178C22—21no—
179C22—21no—
180C22—44no—
181C22—42no—
182C22—82no—
183C22—81no—
184C22—81no—
185C22—11322no—
186C22—11322no—
187C22—11322no—
188C22#2258357811yescharacteristic
189C22#2258358411yescharacteristic
190C22#2258360611yescharacteristic
191C4—42no—
192C4—42no—
193C4—11322no—
194C4#2258358012yes—
195C4#2258358312yes—
196C2—21no—
197C2—21no—
198C2—82no—
199C2—22642no—
200C2#3182637411yescharacteristic
201C2#3182663211yescharacteristic
202C2#3182663711yescharacteristic
subgroupG/H
centerC22#22583578
commutator, cosocleC22×C4×C283C23
FrattiniC22×C4C22×D283
Fitting#390×C283C2
socleC22×C283#120
hypercenter#390D283
hypocenterC283#2006
LCSG ▷ C22×C4×C283 ▷ C2×C283 ▷ C283
UCSC1 ◁ C22 ◁ C22×C4 ◁ #390

Complements

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

NH ≅ G/Ntype
1. C2×#22583636C2semidirect
2. #31827046C2semidirect
3. C2×#22583538C2semidirect
4. #31826234C2semidirect
5. #390×C283C2semidirect
32. #17985376D4semidirect
40. C2×D4×C283D4semidirect
101. C283#2006semidirect
106. #390D283semidirect
134. C2×D4D1132semidirect

Maximal subgroups

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

subgroupindex
C2×#225836362
#318270462
C2×#225835382
#318262342
#390×C2832
#2006283