Finite Group Databasebeta

← #49122165

Subgroups — #49122165

Order 36224.

296 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 296 conjugacy classes of subgroups — too many to draw. Only its 83 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
1C22×#17985379C211yesmaximal, characteristic
2C2×#22583545C211yesmaximal, characteristic, complemented
3#31826474C211yesmaximal, characteristic, complemented
4#31826471C211yesmaximal, characteristic
5C2×#22583526C211yesmaximal, characteristic
6#31826423C211yesmaximal, characteristic, complemented
7C2×#101×C283C211yesmaximal, characteristic, complemented
8C22×D1132C2211yescharacteristic
9C2×#17985379—21no—
10C2×#17985379—21no—
11C2×#17985379—22no—
12C2×#17985379C414yescomplemented
13C2×#17985376C2211yescharacteristic
14C22×C283⋊C8C2211yescharacteristic
15#22583569C2211yescharacteristic
16#22583545—22no—
17#22583534—22no—
18#22583532—22no—
19#22583526—22no—
20#22583520—21no—
21C8×Dic283—21no—
22C22×Q8×C283C2211yescharacteristic
23C2×M16×C283C2211yescharacteristic
24C2×M16×C283C2211yescharacteristic
25#101×C283—22no—
26C23×D283—21no—
27C2×D1132—21no—
28C2×D1132—21no—
29C2×D1132—21no—
30C2×D1132—21no—
31C2×D1132—21no—
32C2×D1132C2×C412yes—
33#17985379—24no—
34#17985379—24no—
35#17985376—21no—
36#17985376—21no—
37C2×C283⋊C8—21no—
38C2×C283⋊C8C2×C412yes—
39C2×C283⋊C8D411yescharacteristic
40C2×C283⋊C8D411yescharacteristic
41C2×C283⋊C8D412yes—
42C4⋊Dic283D412yes—
43C22⋊Dic283—21no—
44C4×Dic283—21no—
45C2×Q8×C283—21no—
46C2×Q8×C283—21no—
47C2×Q8×C283—21no—
48C2×Q8×C283—21no—
49C2×Q8×C283—21no—
50C2×Q8×C283C2×C412yes—
51C22×C4×C283—21no—
52C22×C4×C283C2311yescharacteristic
53M16×C283—21no—
54M16×C283—21no—
55M16×C283—21no—
56M16×C283D412yes—
57C2×C8×C283—21no—
58C2×C8×C283—21no—
59C22×D283—22no—
60C22×D283—41no—
61C22×D283—41no—
62C22×D283—82no—
63D1132—22no—
64D1132—22no—
65D1132—41no—
66D1132—41no—
67C2×Dic283—22no—
68C283⋊C8—41no—
69C283⋊C8C22⋊C412yes—
70C283⋊C8C22⋊C412yes—
71C23×C283#4111yescharacteristic
72Q8×C283—22no—
73Q8×C283—22no—
74Q8×C283—41no—
75Q8×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×C283—22no—
86C8×C283—41no—
87C22×C283—22no—
88C22×C283C4×D411yescharacteristic
89C22×C283#12111yescharacteristic
90C22×C283#12411yescharacteristic
91C2×D283—42no—
92C2×D283—82no—
93C2×D283—82no—
94C2×D283—82no—
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#39011yescharacteristic
104C2×C283#47112yes—
105D283—82no—
106C283#197711yescharacteristic, complemented, Sylow
107#1977—2831nomaximal, Sylow
108C22×SD16—2831no—
109C2×#108—2831no—
110#419—2831no—
111#418—2831no—
112C2×#101D28311yescharacteristic, complemented
113C2×#100—2831no—
114#405—2831no—
115C22×Q8C2×D28311yescharacteristic
116C22×D4—2831no—
117C2×SD16—2834no—
118C2×SD16—5661no—
119C2×SD16—5661no—
120C2×SD16—5662no—
121C2×M16—2831no—
122C2×M16C2×D28311yescharacteristic
123C2×M16C2×D28311yescharacteristic
124C22×C8—2831no—
125#117—2831no—
126#108—5662no—
127#103—5662no—
128#102—5662no—
129#101—22no—
130#100—5662no—
131#97—5661no—
132C4×C8—5661no—
133C24—5661no—
134C2×Q8—21no—
135C2×Q8—21no—
136C2×Q8—21no—
137C2×Q8—21no—
138C2×Q8—21no—
139C2×Q8C4×D28312yes—
140C2×D4—2832no—
141C2×D4—5661no—
142C2×D4—5661no—
143C2×D4—5661no—
144C2×D4—5661no—
145C2×D4—5661no—
146C22×C4—21no—
147C22×C4C22×D28311yescharacteristic
148SD16—5664no—
149SD16—5664no—
150M16—21no—
151M16—21no—
152M16—21no—
153M16—5661no—
154M16—5661no—
155M16C22⋊D28312yes—
156C2×C8—21no—
157C2×C8—21no—
158C2×C8—2832no—
159C2×C8—2831no—
160C2×C8—2831no—
161C2×C8—2832no—
162C2×C8—5661no—
163C4⋊C4—2832no—
164C22⋊C4—5661no—
165C42—5661no—
166C23—5662no—
167C23—11321no—
168C23—11321no—
169C23—22642no—
170C23#1798539211yescharacteristic
171Q8—22no—
172Q8—22no—
173Q8—41no—
174Q8—41no—
175D4—5662no—
176D4—5662no—
177D4—11321no—
178D4—11321no—
179C2×C4—22no—
180C2×C4—41no—
181C2×C4—41no—
182C2×C4—5662no—
183C2×C4C2×C4×D28311yescharacteristic
184C2×C4D4×D28311yescharacteristic
185C2×C4D4×D28311yescharacteristic
186C2×C4D4×D28311yescharacteristic
187C2×C4C2×C22⋊D28311yescharacteristic
188C2×C4#1798539111yescharacteristic
189C8—22no—
190C8—41no—
191C8—2832no—
192C8—2832no—
193C8—11321no—
194C22—22no—
195C22—11322no—
196C22—22642no—
197C22—22642no—
198C22—22642no—
199C22C4×C22⋊D28311yescharacteristic
200C22#2258358311yescharacteristic
201C22#2258359611yescharacteristic
202C4—42no—
203C4—11322no—
204C4C22⋊C4×D28311yescharacteristic
205C4(C22⋊D283)⋊C411yescharacteristic
206C4#2258358211yescharacteristic
207C4#2258360611yescharacteristic
208C2—21no—
209C2—21no—
210C2—22642no—
211C2#3182636511yescharacteristic
212C2#3182673112yes—
subgroupG/H
centerC22#22583583
commutatorC2×C4×C283C22×C4
FrattiniC22×C4C22×D283
FittingC2×#101×C283C2
socleC22×C283#121
prime radicalC22×D1132C22
cosocleC22×C4×C283C23
hypercenterC2×#101D283
hypocenterC283#1977
LCSG ▷ C2×C4×C283 ▷ C2×C283 ▷ C283
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#101

Complements

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

NH ≅ G/Ntype
2. C2×#22583545C2semidirect
3. #31826474C2semidirect
6. #31826423C2semidirect
7. C2×#101×C283C2semidirect
12. C2×#17985379C4semidirect
106. C283#1977semidirect
112. C2×#101D283semidirect

Maximal subgroups

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

subgroupindex
C22×#179853792
C2×#225835452
#318264742
#318264712
C2×#225835262
#318264232
C2×#101×C2832
#1977283