Finite Group Databasebeta

← #49123214

Subgroups — #49123214

Order 36224.

216 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 49 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#31827265C211yesmaximal, characteristic
2#31826908C212yesmaximal, complemented
3#31826723C211yesmaximal, characteristic, complemented
4#31826276C212yesmaximal, complemented
5#349×C283C211yesmaximal, characteristic, complemented
6C2×#17985392—21no—
7#22583643—22no—
8#22583638—22no—
9#22583632—21no—
10#22583632C2212yes—
11C2×Dic1132C2211yescharacteristic, complemented
12#22583589—21no—
13Dic2264—22no—
14#22583555—22no—
15#22583542C2211yescharacteristic, complemented
16#22583535—21no—
17C2×M16×C283C2211yescharacteristic
18M32×C283C2212yescomplemented
19#17985392—22no—
20#17985392—22no—
21#17985392—41no—
22#17985392—41no—
23C2×Dic566—21no—
24C2×Dic566—21no—
25C2×D1132—21no—
26C2×C22⋊D283—41no—
27C2×C4×D283—41no—
28Dic1132—21no—
29Dic1132—22no—
30Dic1132D412yes—
31#17985381—22no—
32#17985381—22no—
33D2264—21no—
34D2264D412yes—
35C4⋊Dic283—21no—
36Dic283⋊C4—41no—
37C22⋊Dic283—41no—
38C22×C4×C283D411yescharacteristic
39M16×C283—22no—
40C2×C8×C283D411yescharacteristic
41C2×C8×C283C2311yescharacteristic
42C16×C283—22no—
43C22×D283—41no—
44Dic566—22no—
45Dic566—41no—
46Dic566—41no—
47Dic566—41no—
48D1132—22no—
49D1132—41no—
50C22⋊D283—42no—
51C22⋊D283—42no—
52C2×Dic283—41no—
53C2×Dic283—41no—
54C2×Dic283—41no—
55C4×D283—42no—
56C4×D283—42no—
57C23×C283D811yescharacteristic
58C2×C4×C283—21no—
59C2×C4×C283—21no—
60C2×C4×C283D811yescharacteristic
61C2×C4×C283C2×D411yescharacteristic
62C8×C283—21no—
63C8×C283C2×D412yes—
64C22×C283—21no—
65C22×C283—41no—
66C22×C283C2×D811yescharacteristic
67C2×D283—42no—
68C2×D283—82no—
69C4×C283—21no—
70C4×C283#12011yescharacteristic
71C4×C283AGL(1,ℤ/8)11yescharacteristic
72Dic283—42no—
73Dic283—81no—
74Dic283—81no—
75C2×C283—21no—
76C2×C283—41no—
77C2×C283#44711yescharacteristic
78D283—82no—
79C283#219111yescharacteristic, complemented, Sylow
80#2191—2831nomaximal, Sylow
81#575—2831no—
82#510—2832no—
83#470—2831no—
84#360—2832no—
85#349D28311yescharacteristic, complemented
86C2×#41—5661no—
87#137—5662no—
88AGL(1,ℤ/8)—5662no—
89#135—2832no—
90#135—5661no—
91C2×Q16—2831no—
92C2×M16C2×D28311yescharacteristic
93C22⋊Q8—5661no—
94Q32—5662no—
95SD32—5662no—
96M32C2×D28312yescomplemented
97#106—2831no—
98#103—5661no—
99#41—5662no—
100#41—5662no—
101#41—11321no—
102#41—11321no—
103C2×Q8—5661no—
104C2×Q8—5661no—
105C2×D4—5661no—
106C2×D4—11321no—
107C22×C4—11321no—
108C22×C4D113211yescharacteristic
109Q16—2832no—
110Q16—5661no—
111Q16—5662no—
112SD16—5662no—
113SD16—5662no—
114D8—2832no—
115D8—5661no—
116M16—22no—
117C2×C8D113211yescharacteristic
118C2×C8C22×D28311yescharacteristic
119C4⋊C4—5661no—
120C4⋊C4—11321no—
121C22⋊C4—11321no—
122C16—22no—
123C23—11321no—
124C23D226411yescharacteristic
125Q8—5662no—
126Q8—11321no—
127Q8—11321no—
128Q8—11321no—
129D4—5662no—
130D4—11321no—
131D4—11322no—
132D4—11322no—
133C2×C4—21no—
134C2×C4—21no—
135C2×C4—11322no—
136C2×C4—11321no—
137C2×C4—11322no—
138C2×C4—11321no—
139C2×C4—11321no—
140C2×C4D226411yescharacteristic
141C2×C4C2×D113211yescharacteristic
142C8—21no—
143C8D4×D28312yes—
144C22—21no—
145C22—41no—
146C22—11322no—
147C22—22642no—
148C22C2×D226411yescharacteristic
149C4—21no—
150C4—11322no—
151C4—22641no—
152C4—22641no—
153C4#2258358011yescharacteristic
154C4#2258363811yescharacteristic
155C2—21no—
156C2—41no—
157C2—22642no—
158C2#3182660111yescharacteristic
subgroupG/H
centerC2#31826601
commutator, cosocleC2×C8×C283C23
FrattiniC2×C8C22×D283
Fitting#349×C283C2
socleC2×C283#447
prime radical#31827265C2
hypercenter#349D283
hypocenterC283#2191
LCSG ▷ C2×C8×C283 ▷ C4×C283 ▷ C2×C283 ▷ C283
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×M16 ◁ #349

Complements

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

NH ≅ G/Ntype
2. #31826908C2semidirect
3. #31826723C2semidirect
4. #31826276C2semidirect
5. #349×C283C2semidirect
11. C2×Dic1132C22semidirect
15. #22583542C22semidirect
18. M32×C283C22semidirect
79. C283#2191semidirect
85. #349D283semidirect
96. M32C2×D283semidirect

Maximal subgroups

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

subgroupindex
#318272652
#318269082
#318267232
#318262762
#349×C2832
#2191283