Finite Group Databasebeta

← #49122307

Subgroups — #49122307

Order 36224.

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

normal subgroups only

This group has 370 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×#22583641C212yesmaximal, complemented
2#31827046C211yesmaximal, characteristic
3C2×#22583538C212yesmaximal, complemented
4#31826234C211yesmaximal, characteristic, complemented
5#391×C283C211yesmaximal, characteristic, complemented
6C2×#17985392C2212yes—
7#22583641—24no—
8#22583641—24no—
9C2×#17985382—22no—
10C2×#17985379—22no—
11C2×#17985376C2212yes—
12#22583590—22no—
13#22583585—22no—
14#22583584—22no—
15#22583538—24no—
16C22×Q8×C283C2211yescharacteristic
17C2×C4⋊C4×C283C2211yescharacteristic
18C2×C42×C283C2211yescharacteristic
19#95×C283—22no—
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#17985382—28no—
30#17985379—28no—
31#17985376—22no—
32#17985376D414yescomplemented
33C2×C283⋊C8—22no—
34C4⋊Dic283—22no—
35(C2×D283)⋊C4—24no—
36C2×Q8×C283—21no—
37C2×Q8×C283—22no—
38C2×Q8×C283—22no—
39C2×Q8×C283D412yescomplemented
40C22×C4×C283—21no—
41C22×C4×C283—21no—
42C22×C4×C283D412yes—
43C22×C4×C283C2311yescharacteristic
44C4⋊C4×C283—22no—
45C42×C283—22no—
46C22×D283—22no—
47Dic566—22no—
48Dic566—22no—
49D1132—22no—
50D1132—22no—
51D1132—82no—
52C22⋊D283—42no—
53C22⋊D283—42no—
54C2×Dic283—22no—
55C4×D283—42no—
56C4×D283—42no—
57C283⋊C8—24no—
58C23×C283C2×D411yescharacteristic
59Q8×C283—24no—
60Q8×C283—42no—
61C2×C4×C283—22no—
62C2×C4×C283—22no—
63C2×C4×C283—22no—
64C2×C4×C283—41no—
65C2×C4×C283—41no—
66C2×C4×C283—41no—
67C2×C4×C283—41no—
68C2×C4×C283—42no—
69C2×C4×C283C2×D412yes—
70C2×C4×C283C2×D412yes—
71C2×C4×C283C2×D411yescharacteristic
72C2×C4×C283#4111yescharacteristic
73C22×C283—21no—
74C22×C283—21no—
75C22×C283#12011yescharacteristic
76C22×C283#12111yescharacteristic
77C22×C283#12711yescharacteristic
78C2×D283—42no—
79C2×D283—42no—
80C2×D283—42no—
81C4×C283—42no—
82C4×C283—42no—
83C4×C283—42no—
84C4×C283#12012yes—
85C4×C283#12112yes—
86Dic283—42no—
87C2×C283—21no—
88C2×C283—21no—
89C2×C283#39211yescharacteristic
90C2×C283#45511yescharacteristic
91C2×C283#45611yescharacteristic
92D283—82no—
93C283#200711yescharacteristic, complemented, Sylow
94#2007—2831nomaximal, Sylow
95C2×#137—2832no—
96#537—2831no—
97C2×#104—2832no—
98#391D28311yescharacteristic, complemented
99#343—2831no—
100C2×#41—2832no—
101C22×Q8C2×D28311yescharacteristic
102#137—5664no—
103#137—5664no—
104C2×Q16—5662no—
105C2×SD16—5662no—
106C2×M16—2832no—
107C22⋊Q8—5662no—
108#121—5662no—
109#121—5662no—
110C2×C4⋊C4C2×D28311yescharacteristic
111C2×C42C2×D28311yescharacteristic
112#104—5664no—
113#95—22no—
114#41—5662no—
115#41—5662no—
116#41—5662no—
117#41—5662no—
118C2×Q8—21no—
119C2×Q8—22no—
120C2×Q8—22no—
121C2×Q8—2832no—
122C2×Q8D113212yescomplemented
123C2×D4—2832no—
124C2×D4—5662no—
125C2×D4—5662no—
126C22×C4—21no—
127C22×C4—21no—
128C22×C4—5662no—
129C22×C4C22⋊D28312yes—
130C22×C4C22×D28311yescharacteristic
131Q16—5668no—
132SD16—5668no—
133M16—2834no—
134M16—5662no—
135C2×C8—5662no—
136C4⋊C4—22no—
137C4⋊C4—5662no—
138C22⋊C4—5664no—
139C42—22no—
140C23—5662no—
141C23C2×C22⋊D28311yescharacteristic
142Q8—24no—
143Q8—42no—
144Q8—5662no—
145Q8—5662no—
146D4—5662no—
147D4—5662no—
148D4—11322no—
149D4—11322no—
150D4—22642no—
151C2×C4—22no—
152C2×C4—22no—
153C2×C4—22no—
154C2×C4—41no—
155C2×C4—41no—
156C2×C4—41no—
157C2×C4—41no—
158C2×C4—42no—
159C2×C4—5662no—
160C2×C4—11322no—
161C2×C4—11322no—
162C2×C4D4×D28312yes—
163C2×C4D4×D28312yes—
164C2×C4C2×D113211yescharacteristic
165C2×C4#1798539211yescharacteristic
166C8—5664no—
167C22—21no—
168C22—21no—
169C22—11322no—
170C22—11322no—
171C22—11322no—
172C22#2258357811yescharacteristic
173C22#2258358411yescharacteristic
174C22#2258360611yescharacteristic
175C4—42no—
176C4—42no—
177C4—42no—
178C4—11322no—
179C4#2258358012yes—
180C4#2258358312yes—
181C2—21no—
182C2—21no—
183C2—22642no—
184C2#3182637411yescharacteristic
185C2#3182664211yescharacteristic
186C2#3182664711yescharacteristic
subgroupG/H
centerC22#22583578
commutator, cosocleC22×C4×C283C23
FrattiniC22×C4C22×D283
Fitting#391×C283C2
socleC22×C283#120
prime radical#31827046C2
hypercenter#391D283
hypocenterC283#2007
LCSG ▷ C22×C4×C283 ▷ C2×C283 ▷ C283
UCSC1 ◁ C22 ◁ C22×C4 ◁ #391

Complements

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

NH ≅ G/Ntype
1. C2×#22583641C2semidirect
3. C2×#22583538C2semidirect
4. #31826234C2semidirect
5. #391×C283C2semidirect
32. #17985376D4semidirect
39. C2×Q8×C283D4semidirect
93. C283#2007semidirect
98. #391D283semidirect
122. C2×Q8D1132semidirect

Maximal subgroups

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

subgroupindex
C2×#225836412
#318270462
C2×#225835382
#318262342
#391×C2832
#2007283