Finite Group Databasebeta

← #49122308

Subgroups — #49122308

Order 36224.

370 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 370 conjugacy classes of subgroups — too many to draw. Only its 75 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×#22583640C212yesmaximal, complemented
2C2×#22583537C212yesmaximal, complemented
3#31826370C211yesmaximal, characteristic, complemented
4#31826234C211yesmaximal, characteristic, complemented
5#537×C283C211yesmaximal, characteristic
6C22×Dic566C2211yescharacteristic
7#22583640—24no—
8#22583640—24no—
9C2×#17985382—22no—
10C2×#17985380—22no—
11C2×#17985376C2212yes—
12C2×C4×Dic283C2211yescharacteristic
13#22583537—24no—
14C283⋊#95—22no—
15C2×#41×C283C2212yes—
16C22⋊Q8×C283—22no—
17#121×C283—22no—
18#121×C283—22no—
19C2×C4⋊C4×C283C2211yescharacteristic
20C2×Dic566—22no—
21C2×Dic566—22no—
22C2×Dic566—21no—
23C2×Dic566D412yescomplemented
24C22×Dic283—21no—
25C22×Dic283—21no—
26#17985382—28no—
27#17985380—28no—
28#17985376—22no—
29#17985376D414yescomplemented
30C2×C283⋊C8—22no—
31C4×Dic283—22no—
32#41×C283—22no—
33#41×C283—22no—
34#41×C283—22no—
35#41×C283—22no—
36C2×Q8×C283D412yes—
37C2×D4×C283—22no—
38C2×D4×C283—22no—
39C2×D4×C283D412yes—
40C22×C4×C283—22no—
41C22×C4×C283D412yes—
42C22×C4×C283C2311yescharacteristic
43C4⋊C4×C283—22no—
44C4⋊C4×C283—22no—
45C22⋊C4×C283—24no—
46Dic566—24no—
47Dic566—42no—
48C2×Dic283—22no—
49C2×Dic283—22no—
50C2×Dic283—41no—
51C2×Dic283—41no—
52C2×Dic283—41no—
53C2×Dic283—41no—
54C283⋊C8—24no—
55C23×C283—22no—
56C23×C283C2×D411yescharacteristic
57Q8×C283—22no—
58Q8×C283—22no—
59D4×C283—22no—
60D4×C283—22no—
61D4×C283—42no—
62D4×C283—42no—
63D4×C283—82no—
64C2×C4×C283—22no—
65C2×C4×C283—22no—
66C2×C4×C283—42no—
67C2×C4×C283—42no—
68C2×C4×C283—42no—
69C2×C4×C283C2×D411yescharacteristic
70C2×C4×C283C2×D412yes—
71C2×C4×C283C2×D412yes—
72C2×C4×C283#4111yescharacteristic
73C22×C283—21no—
74C22×C283—21no—
75C22×C283—42no—
76C22×C283—42no—
77C22×C283—42no—
78C22×C283#12011yescharacteristic
79C22×C283#12111yescharacteristic
80C22×C283#12711yescharacteristic
81C4×C283—42no—
82C4×C283—42no—
83C4×C283#12012yes—
84C4×C283#12112yes—
85Dic283—42no—
86Dic283—42no—
87C2×C283—21no—
88C2×C283—21no—
89C2×C283—82no—
90C2×C283#39211yescharacteristic
91C2×C283#45511yescharacteristic
92C2×C283#45611yescharacteristic
93C283#200711yescharacteristic, complemented, Sylow
94#2007—2831nomaximal, Sylow
95C2×#137—2832no—
96#537D28311yescharacteristic
97C2×#104—2832no—
98#391—2831no—
99#343—2831no—
100C2×#41C2×D28312yes—
101C22×Q8—2831no—
102#137—5664no—
103#137—5664no—
104C2×Q16—5662no—
105C2×SD16—5662no—
106C2×M16—2832no—
107C22⋊Q8—22no—
108#121—22no—
109#121—22no—
110C2×C4⋊C4C2×D28311yescharacteristic
111C2×C42—2831no—
112#104—5664no—
113#95—5662no—
114#41—22no—
115#41—22no—
116#41—22no—
117#41—22no—
118C2×Q8—2832no—
119C2×Q8—5661no—
120C2×Q8—5662no—
121C2×Q8—5662no—
122C2×Q8C22⋊D28312yes—
123C2×D4—22no—
124C2×D4—22no—
125C2×D4C22⋊D28312yes—
126C22×C4—22no—
127C22×C4—5661no—
128C22×C4—5661no—
129C22×C4C22⋊D28312yes—
130C22×C4C22×D28311yescharacteristic
131Q16—5668no—
132SD16—5668no—
133M16—2834no—
134M16—5662no—
135C2×C8—5662no—
136C4⋊C4—22no—
137C4⋊C4—22no—
138C22⋊C4—24no—
139C42—5662no—
140C23—22no—
141C23C2×C22⋊D28311yescharacteristic
142Q8—22no—
143Q8—22no—
144Q8—5664no—
145Q8—11322no—
146D4—22no—
147D4—22no—
148D4—42no—
149D4—42no—
150D4—82no—
151C2×C4—22no—
152C2×C4—22no—
153C2×C4—42no—
154C2×C4—42no—
155C2×C4—42no—
156C2×C4—5662no—
157C2×C4—5662no—
158C2×C4—11321no—
159C2×C4—11321no—
160C2×C4—11321no—
161C2×C4—11321no—
162C2×C4D4×D28311yescharacteristic
163C2×C4D4×D28312yes—
164C2×C4C2×C22⋊D28312yes—
165C2×C4#1798539111yescharacteristic
166C8—5664no—
167C22—21no—
168C22—21no—
169C22—42no—
170C22—42no—
171C22—42no—
172C22#2258357911yescharacteristic
173C22#2258358111yescharacteristic
174C22#2258360611yescharacteristic
175C4—42no—
176C4—42no—
177C4—11322no—
178C4—11322no—
179C4#2258357812yes—
180C4#2258358212yes—
181C2—21no—
182C2—21no—
183C2—82no—
184C2#3182637311yescharacteristic
185C2#3182664311yescharacteristic
186C2#3182664811yescharacteristic
subgroupG/H
centerC22#22583579
commutator, cosocleC22×C4×C283C23
FrattiniC22×C4C22×D283
Fitting, prime radical#537×C283C2
socleC22×C283#120
hypercenter#537D283
hypocenterC283#2007
LCSG ▷ C22×C4×C283 ▷ C2×C283 ▷ C283
UCSC1 ◁ C22 ◁ C22×C4 ◁ #537

Complements

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

NH ≅ G/Ntype
1. C2×#22583640C2semidirect
2. C2×#22583537C2semidirect
3. #31826370C2semidirect
4. #31826234C2semidirect
23. C2×Dic566D4semidirect
29. #17985376D4semidirect
93. C283#2007semidirect

Maximal subgroups

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

subgroupindex
C2×#225836402
C2×#225835372
#318263702
#318262342
#537×C2832
#2007283