Finite Group Databasebeta

← #49120493

Subgroups — #49120493

Order 36224.

240 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 240 conjugacy classes of subgroups — too many to draw. Only its 55 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#31827118C211yesmaximal, characteristic
2#31826742C211yesmaximal, characteristic, complemented
3#31826667C211yesmaximal, characteristic, complemented
4#31826211C211yesmaximal, characteristic, complemented
5#31826194C211yesmaximal, characteristic, complemented
6#31826194C211yesmaximal, characteristic, complemented
7#500×C283C211yesmaximal, characteristic, complemented
8C2×#17985393—21no—
9C2×#17985379—21no—
10#22583595—22no—
11#22583586—21no—
12#22583585—21no—
13C4×D1132C2211yescharacteristic
14C4×D1132C2211yescharacteristic
15C4⋊C4×D283—21no—
16#22583543—21no—
17C4⋊(C283⋊C8)C2211yescharacteristic
18C4⋊(C283⋊C8)C2211yescharacteristic
19#22583534—21no—
20#22583530—21no—
21C4⋊Q8×C283C2211yescharacteristic
22C4⋊Q8×C283C2211yescharacteristic
23#107×C283—22no—
24C4×C8×C283C2211yescharacteristic
25#17985393—42no—
26#17985393—42no—
27C2×D1132—22no—
28C2×D1132D411yescharacteristic
29C2×D1132D411yescharacteristic
30C2×C4×D283—22no—
31C2×C4×D283—21no—
32C2×C4×D283—21no—
33#17985379—42no—
34C2×C283⋊C8—21no—
35C2×C283⋊C8—21no—
36C4⋊Dic283D411yescharacteristic
37C4⋊Dic283D411yescharacteristic
38Dic283⋊C4—22no—
39(C2×D283)⋊C4—22no—
40(C2×D283)⋊C4—21no—
41(C2×D283)⋊C4—21no—
42C2×Q8×C283—21no—
43C2×Q8×C283—21no—
44C2×C8×C283—21no—
45C4⋊C4×C283—21no—
46C4⋊C4×C283—21no—
47C4⋊C4×C283—21no—
48C4⋊C4×C283D412yes—
49C42×C283C2311yescharacteristic
50C22×D283—21no—
51C22×D283—21no—
52D1132—21no—
53D1132—21no—
54D1132—21no—
55D1132—44no—
56D1132D812yes—
57C2×Dic283—21no—
58C2×Dic283—21no—
59C4×D283—44no—
60C4×D283—41no—
61C4×D283—41no—
62C4×D283—41no—
63C4×D283—41no—
64C283⋊C8—41no—
65C283⋊C8—41no—
66Q8×C283—41no—
67Q8×C283—41no—
68Q8×C283—41no—
69Q8×C283—41no—
70C2×C4×C283—22no—
71C2×C4×C283—41no—
72C2×C4×C283C2×D411yescharacteristic
73C2×C4×C283C2×D411yescharacteristic
74C2×C4×C283C2×D411yescharacteristic
75C8×C283—22no—
76C22×C283#12011yescharacteristic
77C2×D283—22no—
78C2×D283—41no—
79C2×D283—41no—
80C2×D283—41no—
81C2×D283—41no—
82C2×D283—41no—
83C4×C283—21no—
84C4×C283—42no—
85C4×C283—81no—
86C4×C283C2×D811yescharacteristic
87C4×C283#13511yescharacteristic
88C4×C283AGL(1,ℤ/8)11yescharacteristic
89C4×C283#13711yescharacteristic
90Dic283—41no—
91Dic283—41no—
92C2×C283#44711yescharacteristic
93C2×C283#45211yescharacteristic
94C2×C283#45611yescharacteristic
95D283—42no—
96D283—81no—
97C283#164711yescharacteristic, complemented, Sylow
98#1647—2831nomaximal, Sylow
99#547—2831no—
100#500D28311yescharacteristic, complemented
101#474—2831no—
102#461—2831no—
103#332—2831no—
104D4⋊C8—2831no—
105D4⋊C8—2831no—
106C2×#41—5661no—
107C2×SD16—5661no—
108C4⋊Q8C2×D28311yescharacteristic
109C4⋊Q8C2×D28311yescharacteristic
110#123—5662no—
111C22⋊Q8—5661no—
112#121—5661no—
113C4×D4—2831no—
114C4×D4—2831no—
115C2×C4⋊C4—5661no—
116#107—22no—
117#107—5661no—
118C4⋊C8—2831no—
119C4⋊C8—2831no—
120#103—5661no—
121#102—5661no—
122C4×C8C2×D28311yescharacteristic
123#41—11322no—
124#41—11322no—
125C2×Q8—21no—
126C2×Q8—21no—
127C2×D4—2831no—
128C2×D4—2831no—
129C2×D4—5662no—
130C22×C4—5662no—
131C22×C4—5661no—
132C22×C4—5661no—
133SD16—11322no—
134C2×C8—21no—
135C2×C8—5661no—
136C2×C8—5661no—
137C4⋊C4—21no—
138C4⋊C4—21no—
139C4⋊C4—21no—
140C4⋊C4—2831no—
141C4⋊C4—2831no—
142C4⋊C4—5662no—
143C4⋊C4C22⋊D28312yes—
144C22⋊C4—5662no—
145C22⋊C4—5661no—
146C22⋊C4—5661no—
147C42C22×D28311yescharacteristic
148C23—5661no—
149C23—5661no—
150Q8—41no—
151Q8—41no—
152Q8—41no—
153Q8—41no—
154D4—2832no—
155D4—5661no—
156D4—5661no—
157D4—5661no—
158D4—11324no—
159C2×C4—22no—
160C2×C4—41no—
161C2×C4—5661no—
162C2×C4—5661no—
163C2×C4—11324no—
164C2×C4—11321no—
165C2×C4—11321no—
166C2×C4—11321no—
167C2×C4—11321no—
168C2×C4D4×D28311yescharacteristic
169C2×C4D4×D28311yescharacteristic
170C2×C4C2×C22⋊D28311yescharacteristic
171C8—22no—
172C8—11321no—
173C8—11321no—
174C22—5662no—
175C22—11321no—
176C22—11321no—
177C22—11321no—
178C22—11321no—
179C22—11321no—
180C22#2258357811yescharacteristic
181C4—21no—
182C4—42no—
183C4—81no—
184C4—11321no—
185C4—11321no—
186C4D8×D28311yescharacteristic
187C4#2258363311yescharacteristic
188C4#2258363411yescharacteristic
189C4#2258364211yescharacteristic
190C2—11322no—
191C2—22641no—
192C2#3182659711yescharacteristic
193C2#3182662611yescharacteristic
194C2#3182664711yescharacteristic
subgroupG/H
centerC22#22583578
commutator, cosocleC42×C283C23
FrattiniC42C22×D283
Fitting#500×C283C2
socleC22×C283#120
prime radical#31827118C2
hypercenter#500D283
hypocenterC283#1647
LCSG ▷ C42×C283 ▷ C22×C283 ▷ C283
UCSC1 ◁ C22 ◁ C42 ◁ #500

Complements

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

NH ≅ G/Ntype
2. #31826742C2semidirect
3. #31826667C2semidirect
4. #31826211C2semidirect
5. #31826194C2semidirect
6. #31826194C2semidirect
7. #500×C283C2semidirect
97. C283#1647semidirect
100. #500D283semidirect

Maximal subgroups

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

subgroupindex
#318271182
#318267422
#318266672
#318262112
#318261942
#500×C2832
#1647283