Finite Group Databasebeta

← #46812838

Subgroups — #46812838

Order 34432.

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#31771405C211yesmaximal, characteristic
2#31771048C212yesmaximal, complemented
3#31770863C211yesmaximal, characteristic, complemented
4#31770416C212yesmaximal, complemented
5#349×C269C211yesmaximal, characteristic, complemented
6C2×#16864402—21no—
7#22575486—22no—
8#22575481—22no—
9#22575475—21no—
10#22575475C2212yes—
11C2×Dic1076C2211yescharacteristic, complemented
12#22575432—21no—
13Dic2152—22no—
14#22575398—22no—
15#22575385C2211yescharacteristic, complemented
16#22575378—21no—
17C2×M16×C269C2211yescharacteristic
18M32×C269C2212yescomplemented
19#16864402—22no—
20#16864402—22no—
21#16864402—41no—
22#16864402—41no—
23C2×Dic538—21no—
24C2×Dic538—21no—
25C2×D1076—21no—
26C2×C22⋊D269—41no—
27C2×C4×D269—41no—
28Dic1076—21no—
29Dic1076—22no—
30Dic1076D412yes—
31#16864391—22no—
32#16864391—22no—
33D2152—21no—
34D2152D412yes—
35C4⋊Dic269—21no—
36#16864381—41no—
37C22⋊Dic269—41no—
38C22×C4×C269D411yescharacteristic
39M16×C269—22no—
40C2×C8×C269D411yescharacteristic
41C2×C8×C269C2311yescharacteristic
42C16×C269—22no—
43C22×D269—41no—
44Dic538—22no—
45Dic538—41no—
46Dic538—41no—
47Dic538—41no—
48D1076—22no—
49D1076—41no—
50C22⋊D269—42no—
51C22⋊D269—42no—
52C2×Dic269—41no—
53C2×Dic269—41no—
54C2×Dic269—41no—
55C4×D269—42no—
56C4×D269—42no—
57C23×C269D811yescharacteristic
58C2×C4×C269—21no—
59C2×C4×C269—21no—
60C2×C4×C269D811yescharacteristic
61C2×C4×C269C2×D411yescharacteristic
62C8×C269—21no—
63C8×C269C2×D412yes—
64C22×C269—21no—
65C22×C269—41no—
66C22×C269C2×D811yescharacteristic
67C2×D269—42no—
68C2×D269—82no—
69C4×C269—21no—
70C4×C269#12011yescharacteristic
71C4×C269AGL(1,ℤ/8)11yescharacteristic
72Dic269—42no—
73Dic269—81no—
74Dic269—81no—
75C2×C269—21no—
76C2×C269—41no—
77C2×C269#44711yescharacteristic
78D269—82no—
79C269#219111yescharacteristic, complemented, Sylow
80#2191—2691nomaximal, Sylow
81#575—2691no—
82#510—2692no—
83#470—2691no—
84#360—2692no—
85#349D26911yescharacteristic, complemented
86C2×#41—5381no—
87#137—5382no—
88AGL(1,ℤ/8)—5382no—
89#135—2692no—
90#135—5381no—
91C2×Q16—2691no—
92C2×M16C2×D26911yescharacteristic
93C22⋊Q8—5381no—
94Q32—5382no—
95SD32—5382no—
96M32C2×D26912yescomplemented
97#106—2691no—
98#103—5381no—
99#41—5382no—
100#41—5382no—
101#41—10761no—
102#41—10761no—
103C2×Q8—5381no—
104C2×Q8—5381no—
105C2×D4—5381no—
106C2×D4—10761no—
107C22×C4—10761no—
108C22×C4D107611yescharacteristic
109Q16—2692no—
110Q16—5381no—
111Q16—5382no—
112SD16—5382no—
113SD16—5382no—
114D8—2692no—
115D8—5381no—
116M16—22no—
117C2×C8D107611yescharacteristic
118C2×C8C22×D26911yescharacteristic
119C4⋊C4—5381no—
120C4⋊C4—10761no—
121C22⋊C4—10761no—
122C16—22no—
123C23—10761no—
124C23D215211yescharacteristic
125Q8—5382no—
126Q8—10761no—
127Q8—10761no—
128Q8—10761no—
129D4—5382no—
130D4—10761no—
131D4—10762no—
132D4—10762no—
133C2×C4—21no—
134C2×C4—21no—
135C2×C4—10762no—
136C2×C4—10761no—
137C2×C4—10762no—
138C2×C4—10761no—
139C2×C4—10761no—
140C2×C4D215211yescharacteristic
141C2×C4C2×D107611yescharacteristic
142C8—21no—
143C8D4×D26912yes—
144C22—21no—
145C22—41no—
146C22—10762no—
147C22—21522no—
148C22C2×D215211yescharacteristic
149C4—21no—
150C4—10762no—
151C4—21521no—
152C4—21521no—
153C4#2257542311yescharacteristic
154C4#2257548111yescharacteristic
155C2—21no—
156C2—41no—
157C2—21522no—
158C2#3177074111yescharacteristic
subgroupG/H
centerC2#31770741
commutator, cosocleC2×C8×C269C23
FrattiniC2×C8C22×D269
Fitting#349×C269C2
socleC2×C269#447
prime radical#31771405C2
hypercenter#349D269
hypocenterC269#2191
LCSG ▷ C2×C8×C269 ▷ C4×C269 ▷ C2×C269 ▷ C269
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×M16 ◁ #349

Complements

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

NH ≅ G/Ntype
2. #31771048C2semidirect
3. #31770863C2semidirect
4. #31770416C2semidirect
5. #349×C269C2semidirect
11. C2×Dic1076C22semidirect
15. #22575385C22semidirect
18. M32×C269C22semidirect
79. C269#2191semidirect
85. #349D269semidirect
96. M32C2×D269semidirect

Maximal subgroups

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

subgroupindex
#317714052
#317710482
#317708632
#317704162
#349×C2692
#2191269