Finite Group Databasebeta

← #49120437

Subgroups — #49120437

Order 36224.

308 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 308 conjugacy classes of subgroups — too many to draw. Only its 57 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#31827100C211yesmaximal, characteristic
2#31826742C211yesmaximal, characteristic, complemented
3#31826667C211yesmaximal, characteristic, complemented
4#31826209C211yesmaximal, characteristic, complemented
5#31826194C211yesmaximal, characteristic, complemented
6#31826194C211yesmaximal, characteristic, complemented
7#486×C283C211yesmaximal, characteristic, complemented
8C2×D4×D283—21no—
9C2×D4×D283—21no—
10C2×#17985379—21no—
11#22583581—21no—
12#22583581—21no—
13#22583578—22no—
14C4×D1132C2211yescharacteristic
15C4×D1132C2211yescharacteristic
16#22583543—21no—
17C4⋊(C283⋊C8)C2211yescharacteristic, complemented
18C4⋊(C283⋊C8)C2211yescharacteristic, complemented
19#22583534—21no—
20#22583530—21no—
21C4⋊Q8×C283C2211yescharacteristic, complemented
22#127×C283C2211yescharacteristic
23#102×C283—22no—
24C4×C8×C283C2211yescharacteristic, complemented
25C23×D283—22no—
26C2×D1132D411yescharacteristic
27C2×D1132D411yescharacteristic
28C2×C22⋊D283—22no—
29C2×C22⋊D283—22no—
30D4×D283—42no—
31D4×D283—42no—
32D4×D283—42no—
33D4×D283—42no—
34C2×C4×D283—21no—
35C2×C4×D283—21no—
36#17985379—42no—
37C2×C283⋊C8—21no—
38C2×C283⋊C8—21no—
39C4⋊Dic283D411yescharacteristic
40C4⋊Dic283D411yescharacteristic
41C22⋊Dic283—22no—
42(C2×D283)⋊C4—21no—
43(C2×D283)⋊C4—21no—
44C2×Q8×C283—21no—
45C2×D4×C283—21no—
46C2×D4×C283—21no—
47C2×D4×C283D412yes—
48C2×C8×C283—21no—
49C4⋊C4×C283—21no—
50C4⋊C4×C283—21no—
51C42×C283C2311yescharacteristic
52C22×D283—21no—
53C22×D283—21no—
54C22×D283—44no—
55C22×D283—44no—
56C22×D283—82no—
57D1132—21no—
58D1132—21no—
59D1132D812yes—
60D1132SD1612yescomplemented
61C22⋊D283—44no—
62C22⋊D283—44no—
63C2×Dic283—21no—
64C2×Dic283—21no—
65C4×D283—41no—
66C4×D283—41no—
67C4×D283—41no—
68C4×D283—41no—
69C283⋊C8—41no—
70C283⋊C8—41no—
71C23×C283—22no—
72Q8×C283—41no—
73Q8×C283—41no—
74D4×C283—42no—
75D4×C283—41no—
76D4×C283—41no—
77D4×C283—41no—
78D4×C283—41no—
79C2×C4×C283—41no—
80C2×C4×C283C2×D411yescharacteristic
81C2×C4×C283C2×D411yescharacteristic
82C2×C4×C283C2×D411yescharacteristic
83C8×C283—41no—
84C22×C283—42no—
85C22×C283—42no—
86C22×C283—42no—
87C22×C283#12011yescharacteristic
88C2×D283—22no—
89C2×D283—22no—
90C2×D283—41no—
91C2×D283—41no—
92C2×D283—41no—
93C2×D283—41no—
94C2×D283—88no—
95C4×C283—21no—
96C4×C283—81no—
97C4×C283C2×D811yescharacteristic
98C4×C283C2×SD1611yescharacteristic
99C4×C283AGL(1,ℤ/8)11yescharacteristic
100C4×C283AGL(1,ℤ/8)11yescharacteristic
101Dic283—41no—
102Dic283—41no—
103C2×C283—82no—
104C2×C283#44711yescharacteristic
105C2×C283#45011yescharacteristic
106C2×C283#45511yescharacteristic
107D283—42no—
108D283—42no—
109C283#163911yescharacteristic, complemented, Sylow
110#1639—2831nomaximal, Sylow
111D42—2831no—
112#486D28311yescharacteristic, complemented
113#474—2831no—
114#461—2831no—
115#331—2831no—
116D4⋊C8—2831no—
117D4⋊C8—2831no—
118C22×D4—5661no—
119C22×D4—5661no—
120C2×SD16—5661no—
121C4⋊Q8C2×D28311yescharacteristic, complemented
122#127C2×D28311yescharacteristic
123#121—5661no—
124#121—5661no—
125#120—5662no—
126C4×D4—2831no—
127C4×D4—2831no—
128#107—5661no—
129C4⋊C8—2831no—
130C4⋊C8—2831no—
131#103—5661no—
132#102—22no—
133#102—5661no—
134C4×C8C2×D28311yescharacteristic, complemented
135C24—5662no—
136C2×Q8—21no—
137C2×D4—21no—
138C2×D4—21no—
139C2×D4—2831no—
140C2×D4—2831no—
141C2×D4—5662no—
142C2×D4—5662no—
143C2×D4—11322no—
144C2×D4—11322no—
145C2×D4—11322no—
146C2×D4—11322no—
147C2×D4C22⋊D28312yes—
148C22×C4—5661no—
149C22×C4—5661no—
150SD16—11322no—
151C2×C8—21no—
152C2×C8—5661no—
153C2×C8—5661no—
154C4⋊C4—21no—
155C4⋊C4—21no—
156C4⋊C4—2831no—
157C4⋊C4—2831no—
158C22⋊C4—5662no—
159C22⋊C4—5661no—
160C22⋊C4—5661no—
161C42C22×D28311yescharacteristic
162C23—22no—
163C23—5661no—
164C23—5661no—
165C23—11324no—
166C23—11324no—
167C23—22642no—
168Q8—41no—
169Q8—41no—
170D4—42no—
171D4—41no—
172D4—41no—
173D4—41no—
174D4—41no—
175D4—2832no—
176D4—2832no—
177D4—5661no—
178D4—5661no—
179D4—11324no—
180D4—11324no—
181C2×C4—41no—
182C2×C4—5661no—
183C2×C4—5661no—
184C2×C4—11321no—
185C2×C4—11321no—
186C2×C4—11321no—
187C2×C4—11321no—
188C2×C4D4×D28311yescharacteristic
189C2×C4D4×D28311yescharacteristic
190C2×C4C2×C22⋊D28311yescharacteristic
191C8—41no—
192C8—11321no—
193C8—11321no—
194C22—42no—
195C22—42no—
196C22—42no—
197C22—5662no—
198C22—5662no—
199C22—11321no—
200C22—11321no—
201C22—11321no—
202C22—11321no—
203C22—22648no—
204C22#2258357811yescharacteristic
205C4—21no—
206C4—81no—
207C4—11321no—
208C4—11321no—
209C4D8×D28311yescharacteristic
210C4SD16×D28311yescharacteristic
211C4#2258363411yescharacteristic
212C4#2258363711yescharacteristic
213C2—82no—
214C2—11322no—
215C2—11322no—
216C2#3182659711yescharacteristic
217C2#3182661411yescharacteristic
218C2#3182664211yescharacteristic
subgroupG/H
centerC22#22583578
commutator, cosocleC42×C283C23
FrattiniC42C22×D283
Fitting#486×C283C2
socleC22×C283#120
prime radical#31827100C2
hypercenter#486D283
hypocenterC283#1639
LCSG ▷ C42×C283 ▷ C22×C283 ▷ C283
UCSC1 ◁ C22 ◁ C42 ◁ #486

Complements

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

NH ≅ G/Ntype
2. #31826742C2semidirect
3. #31826667C2semidirect
4. #31826209C2semidirect
5. #31826194C2semidirect
6. #31826194C2semidirect
7. #486×C283C2semidirect
17. C4⋊(C283⋊C8)C22semidirect
18. C4⋊(C283⋊C8)C22semidirect
21. C4⋊Q8×C283C22semidirect
24. C4×C8×C283C22semidirect
60. D1132SD16semidirect
109. C283#1639semidirect
112. #486D283semidirect
121. C4⋊Q8C2×D283semidirect
134. C4×C8C2×D283semidirect

Maximal subgroups

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

subgroupindex
#318271002
#318267422
#318266672
#318262092
#318261942
#486×C2832
#1639283