Finite Group Databasebeta

← #49120566

Subgroups — #49120566

Order 36224.

228 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 45 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
1#31827207C211yesmaximal, characteristic
2#31826857C211yesmaximal, characteristic, complemented
3#31826827C211yesmaximal, characteristic, complemented
4#31826205C211yesmaximal, characteristic, complemented
5#31826204C211yesmaximal, characteristic, complemented
6#31826204C211yesmaximal, characteristic, complemented
7#489×C283C211yesmaximal, characteristic, complemented
8C2×#17985382—21no—
9C2×#17985379—21no—
10#22583599—21no—
11#22583598C2211yescharacteristic
12#22583598C2211yescharacteristic
13#22583596—21no—
14#22583596—21no—
15#22583580—21no—
16#22583580—21no—
17#22583578—21no—
18#22583533—21no—
19#22583530—21no—
20#22583521C2211yescharacteristic, complemented
21#22583521C2211yescharacteristic, complemented
22C4⋊Q8×C283C2211yescharacteristic, complemented
23#124×C283C2211yescharacteristic
24#103×C283—21no—
25#102×C283—21no—
26#97×C283C2211yescharacteristic, complemented
27C23×D283—21no—
28C2×Dic566D411yescharacteristic
29C2×Dic566D411yescharacteristic
30C2×D1132—21no—
31C2×D1132D411yescharacteristic
32C2×D1132D411yescharacteristic
33C2×C22⋊D283—21no—
34C2×C22⋊D283—21no—
35#17985382—42no—
36#17985379—42no—
37C2×C283⋊C8—21no—
38C2×C283⋊C8—21no—
39C22⋊Dic283—21no—
40(C2×D283)⋊C4—21no—
41(C2×D283)⋊C4—21no—
42(C2×D283)⋊C4—21no—
43(C2×D283)⋊C4—21no—
44(C2×D283)⋊C4—21no—
45(C2×D283)⋊C4—21no—
46C4×Dic283—21no—
47C2×Q8×C283—21no—
48C2×Q8×C283D411yescharacteristic
49C2×D4×C283D411yescharacteristic
50C2×C8×C283—21no—
51C4⋊C4×C283—21no—
52C4⋊C4×C283—21no—
53C22⋊C4×C283—21no—
54C22⋊C4×C283—21no—
55C42×C283C2311yescharacteristic
56C22×D283—21no—
57C22×D283—21no—
58C22×D283—81no—
59C22×D283—81no—
60C22×D283—81no—
61Dic566—41no—
62Dic566—41no—
63D1132—41no—
64D1132—41no—
65D1132—81no—
66C22⋊D283—81no—
67C22⋊D283—81no—
68C2×Dic283—21no—
69C2×Dic283—21no—
70C283⋊C8—22no—
71C283⋊C8—41no—
72C23×C283—21no—
73Q8×C283—41no—
74Q8×C283—41no—
75Q8×C283—41no—
76D4×C283—41no—
77C2×C4×C283—21no—
78C2×C4×C283—41no—
79C2×C4×C283C2×D411yescharacteristic
80C2×C4×C283C2×D411yescharacteristic
81C2×C4×C283C2×D411yescharacteristic
82C8×C283—41no—
83C22×C283—41no—
84C22×C283—41no—
85C22×C283—41no—
86C22×C283#12011yescharacteristic
87C2×D283—41no—
88C2×D283—41no—
89C2×D283—41no—
90C2×D283—41no—
91C2×D283—41no—
92C2×D283—41no—
93C2×D283—84no—
94C4×C283—21no—
95C4×C283—21no—
96C4×C283—21no—
97C4×C283—41no—
98C4×C283—81no—
99Dic283—41no—
100Dic283—41no—
101C2×C283—81no—
102C2×C283#45311yescharacteristic
103C2×C283#45511yescharacteristic
104C2×C283#45511yescharacteristic
105D283—81no—
106D283—81no—
107C283#165911yescharacteristic, complemented, Sylow
108#1659—2831nomaximal, Sylow
109H(3,𝔽4)—2831no—
110#497—2831no—
111#489—2831no—
112#489D28311yescharacteristic, complemented
113#329—2831no—
114#329—2831no—
115#329—2831no—
116C2×Q16—5661no—
117C2×SD16—5661no—
118C4⋊Q8C2×D28311yescharacteristic, complemented
119#124—2831no—
120#124—2831no—
121#124—5661no—
122#124—5661no—
123#124—5661no—
124#124C2×D28311yescharacteristic
125#120—5661no—
126#120—5661no—
127#120—5661no—
128#103—21no—
129#103—5661no—
130#102—21no—
131#102—5661no—
132#97—2831no—
133#97—2831no—
134#97C2×D28311yescharacteristic, complemented
135C24—5661no—
136C2×Q8—21no—
137C2×Q8—2831no—
138C2×Q8—2831no—
139C2×Q8C22⋊D28311yescharacteristic
140C2×D4—2831no—
141C2×D4—2831no—
142C2×D4—5661no—
143C2×D4—5661no—
144C2×D4—5661no—
145C2×D4C22⋊D28311yescharacteristic
146Q16—11322no—
147SD16—11322no—
148C2×C8—21no—
149C2×C8—5661no—
150C2×C8—5661no—
151C4⋊C4—21no—
152C4⋊C4—21no—
153C22⋊C4—21no—
154C22⋊C4—21no—
155C22⋊C4—5661no—
156C22⋊C4—5661no—
157C22⋊C4—5661no—
158C22⋊C4—5661no—
159C22⋊C4—5661no—
160C22⋊C4—5661no—
161C22⋊C4—5661no—
162C42—5661no—
163C42C22×D28311yescharacteristic
164C23—21no—
165C23—5661no—
166C23—5661no—
167C23—22641no—
168C23—22641no—
169C23—22641no—
170Q8—41no—
171Q8—41no—
172Q8—41no—
173Q8—11321no—
174Q8—11321no—
175D4—41no—
176D4—11321no—
177D4—11321no—
178D4—22641no—
179D4—22641no—
180D4—22641no—
181C2×C4—21no—
182C2×C4—41no—
183C2×C4—5661no—
184C2×C4—5661no—
185C2×C4D4×D28311yescharacteristic
186C2×C4D4×D28311yescharacteristic
187C2×C4C2×C22⋊D28311yescharacteristic
188C8—41no—
189C8—5662no—
190C8—11321no—
191C22—41no—
192C22—41no—
193C22—41no—
194C22—11321no—
195C22—11321no—
196C22—11321no—
197C22—11321no—
198C22—11321no—
199C22—11321no—
200C22—22644no—
201C22#2258357811yescharacteristic
202C4—21no—
203C4—21no—
204C4—21no—
205C4—41no—
206C4—81no—
207C4—11321no—
208C4—11321no—
209C2—81no—
210C2—22641no—
211C2—22641no—
212C2#3182663111yescharacteristic
213C2#3182664111yescharacteristic
214C2#3182664211yescharacteristic
subgroupG/H
centerC22#22583578
commutator, cosocleC42×C283C23
FrattiniC42C22×D283
Fitting#489×C283C2
socleC22×C283#120
prime radical#31827207C2
hypercenter#489D283
hypocenterC283#1659
LCSG ▷ C42×C283 ▷ C22×C283 ▷ C283
UCSC1 ◁ C22 ◁ C42 ◁ #489

Complements

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

NH ≅ G/Ntype
2. #31826857C2semidirect
3. #31826827C2semidirect
4. #31826205C2semidirect
5. #31826204C2semidirect
6. #31826204C2semidirect
7. #489×C283C2semidirect
20. #22583521C22semidirect
21. #22583521C22semidirect
22. C4⋊Q8×C283C22semidirect
26. #97×C283C22semidirect
107. C283#1659semidirect
112. #489D283semidirect
118. C4⋊Q8C2×D283semidirect
134. #97C2×D283semidirect

Maximal subgroups

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

subgroupindex
#318272072
#318268572
#318268272
#318262052
#318262042
#489×C2832
#1659283