Finite Group Databasebeta

← #49122166

Subgroups — #49122166

Order 36224.

296 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 296 conjugacy classes of subgroups — too many to draw. Only its 97 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×#22583546C211yesmaximal, characteristic, complemented
2#31826474C211yesmaximal, characteristic, complemented
3#31826470C211yesmaximal, characteristic
4C2×#22583529C211yesmaximal, characteristic, complemented
5C2×#22583527C211yesmaximal, characteristic
6#31826420C211yesmaximal, characteristic, complemented
7C22×SD16×C283C211yesmaximal, characteristic
8C2×#17985376C2211yescharacteristic
9C2×#17985376C2211yescharacteristic
10C2×#17985376C2211yescharacteristic
11#22583569C2211yescharacteristic
12#22583546—22no—
13#22583534—22no—
14#22583531—22no—
15#22583529—22no—
16#22583527—22no—
17#22583520—21no—
18C8×Dic283—21no—
19C22×Q8×C283C2211yescharacteristic
20C22×D4×C283C2211yescharacteristic
21C2×SD16×C283—21no—
22C2×SD16×C283—21no—
23C2×SD16×C283—22no—
24C2×SD16×C283C414yescomplemented
25C22×C8×C283C2211yescharacteristic
26#17985376—21no—
27#17985376—21no—
28#17985376—21no—
29#17985376—21no—
30#17985376—21no—
31#17985376D412yes—
32C2×C283⋊C8—21no—
33C2×C283⋊C8—21no—
34C2×C283⋊C8—21no—
35C4⋊Dic283D412yes—
36C22⋊Dic283—21no—
37C4×Dic283—21no—
38C24×C283—21no—
39C2×Q8×C283—21no—
40C2×Q8×C283—21no—
41C2×Q8×C283—21no—
42C2×Q8×C283—21no—
43C2×Q8×C283—21no—
44C2×Q8×C283C2×C412yes—
45C2×D4×C283—21no—
46C2×D4×C283—21no—
47C2×D4×C283—21no—
48C2×D4×C283—21no—
49C2×D4×C283—21no—
50C2×D4×C283C2×C412yes—
51C22×C4×C283—21no—
52C22×C4×C283C2311yescharacteristic
53SD16×C283—24no—
54SD16×C283—24no—
55C2×C8×C283C2×C412yes—
56C2×C8×C283D411yescharacteristic
57C2×C8×C283D411yescharacteristic
58C2×C8×C283D412yes—
59C2×Dic283—22no—
60C283⋊C8—22no—
61C283⋊C8—41no—
62C283⋊C8—41no—
63C23×C283—22no—
64C23×C283—41no—
65C23×C283—41no—
66C23×C283—82no—
67C23×C283#4111yescharacteristic
68Q8×C283—22no—
69Q8×C283—22no—
70Q8×C283—41no—
71Q8×C283—41no—
72D4×C283—22no—
73D4×C283—22no—
74D4×C283—41no—
75D4×C283—41no—
76C2×C4×C283—22no—
77C2×C4×C283—41no—
78C2×C4×C283—41no—
79C2×C4×C283C22×C411yescharacteristic
80C2×C4×C283C2×D411yescharacteristic
81C2×C4×C283C2×D411yescharacteristic
82C2×C4×C283C2×D411yescharacteristic
83C2×C4×C283C2×D411yescharacteristic
84C2×C4×C283#4111yescharacteristic
85C8×C283C22⋊C412yes—
86C8×C283C22⋊C412yes—
87C22×C283—22no—
88C22×C283—42no—
89C22×C283—82no—
90C22×C283—82no—
91C22×C283—82no—
92C22×C283C4×D411yescharacteristic
93C22×C283#12111yescharacteristic
94C22×C283#12411yescharacteristic
95C4×C283—42no—
96C4×C283C2×C22⋊C411yescharacteristic
97C4×C283C4×D411yescharacteristic
98C4×C283#12111yescharacteristic
99C4×C283#12711yescharacteristic
100Dic283—42no—
101C2×C283—21no—
102C2×C283—21no—
103C2×C283—82no—
104C2×C283#39011yescharacteristic
105C2×C283#47112yes—
106C283#197711yescharacteristic, complemented, Sylow
107#1977—2831nomaximal, Sylow
108C22×SD16D28311yescharacteristic
109C2×#108—2831no—
110#419—2831no—
111#418—2831no—
112C2×#101—2831no—
113C2×#100—2831no—
114#405—2831no—
115C22×Q8C2×D28311yescharacteristic
116C22×D4C2×D28311yescharacteristic
117C2×SD16—21no—
118C2×SD16—21no—
119C2×SD16—22no—
120C2×SD16Dic28314yescomplemented
121C2×M16—2831no—
122C2×M16—2831no—
123C2×M16—2831no—
124C22×C8C2×D28311yescharacteristic
125#117—2831no—
126#108—5662no—
127#103—5662no—
128#102—5662no—
129#101—5662no—
130#100—5662no—
131#97—5661no—
132C4×C8—5661no—
133C24—21no—
134C2×Q8—21no—
135C2×Q8—21no—
136C2×Q8—21no—
137C2×Q8—21no—
138C2×Q8—21no—
139C2×Q8C2×Dic28312yes—
140C2×D4—21no—
141C2×D4—21no—
142C2×D4—21no—
143C2×D4—21no—
144C2×D4—21no—
145C2×D4C2×Dic28312yes—
146C22×C4—21no—
147C22×C4C22×D28311yescharacteristic
148SD16—24no—
149SD16—24no—
150M16—2832no—
151M16—5661no—
152M16—5661no—
153M16—5661no—
154M16—5661no—
155M16—5661no—
156C2×C8—5661no—
157C2×C8—5661no—
158C2×C8—5661no—
159C2×C8C2×Dic28312yes—
160C2×C8C22⋊D28311yescharacteristic
161C2×C8C22⋊D28311yescharacteristic
162C2×C8C22⋊D28312yes—
163C4⋊C4—2832no—
164C22⋊C4—5661no—
165C42—5661no—
166C23—22no—
167C23—41no—
168C23—41no—
169C23—82no—
170C23#1798539111yescharacteristic
171Q8—22no—
172Q8—22no—
173Q8—41no—
174Q8—41no—
175D4—22no—
176D4—22no—
177D4—41no—
178D4—41no—
179C2×C4—22no—
180C2×C4—41no—
181C2×C4—41no—
182C2×C4—5662no—
183C2×C4C22×Dic28311yescharacteristic
184C2×C4D4×D28311yescharacteristic
185C2×C4D4×D28311yescharacteristic
186C2×C4C2×C22⋊D28311yescharacteristic
187C2×C4C2×C22⋊D28311yescharacteristic
188C2×C4#1798539111yescharacteristic
189C8—5662no—
190C8—11321no—
191C8—11321no—
192C8C22⋊Dic28312yes—
193C8C22⋊Dic28312yes—
194C22—22no—
195C22—42no—
196C22—82no—
197C22—82no—
198C22—82no—
199C22D4×Dic28311yescharacteristic
200C22#2258358111yescharacteristic
201C22#2258359711yescharacteristic
202C4—42no—
203C4—11322no—
204C4C2×C22⋊Dic28311yescharacteristic
205C4D4×Dic28311yescharacteristic
206C4#2258358111yescharacteristic
207C4#2258360611yescharacteristic
208C2—21no—
209C2—21no—
210C2—82no—
211C2#3182636811yescharacteristic
212C2#3182672512yes—
subgroupG/H
centerC22#22583581
commutatorC2×C4×C283C22×C4
FrattiniC22×C4C22×D283
FittingC22×SD16×C283C2
socleC22×C283#121
prime radicalC22×D4×C283C22
cosocleC22×C4×C283C23
hypercenterC22×SD16D283
hypocenterC283#1977
LCSG ▷ C2×C4×C283 ▷ C2×C283 ▷ C283
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×SD16

Complements

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

NH ≅ G/Ntype
1. C2×#22583546C2semidirect
2. #31826474C2semidirect
4. C2×#22583529C2semidirect
6. #31826420C2semidirect
24. C2×SD16×C283C4semidirect
106. C283#1977semidirect
120. C2×SD16Dic283semidirect

Maximal subgroups

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

subgroupindex
C2×#225835462
#318264742
#318264702
C2×#225835292
C2×#225835272
#318264202
C22×SD16×C2832
#1977283