Finite Group Databasebeta

← #46812901

Subgroups — #46812901

Order 34432.

248 conjugacy classes of subgroups; only the 41 normal subgroups are shown.

normal subgroups only

This group has 248 conjugacy classes of subgroups — too many to draw. Only its 41 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#31771347C211yesmaximal, characteristic, complemented
2#31770778C212yesmaximal, complemented
3#31770402C211yesmaximal, characteristic, complemented
4#31770401C212yesmaximal, complemented
5#454×C269C211yesmaximal, characteristic, complemented
6#22575478—22no—
7#22575442C2212yes—
8#22575441—21no—
9#22575439—22no—
10#22575423—22no—
11#22575421—21no—
12#22575380—22no—
13#22575372C2212yescomplemented
142−1+4×C269C2211yescharacteristic, complemented
15AGL(1,ℤ/8)×C269—21no—
16#124×C269C2211yescharacteristic
17#104×C269—21no—
18#101×C269C2211yescharacteristic, complemented
19C23×D269—21no—
20C2×Dic538—21no—
21C2×D1076—21no—
22C2×D1076D412yes—
23C2×C22⋊D269—22no—
24#16864389—42no—
25#16864387—42no—
26#16864386—22no—
27C22⋊Dic269—21no—
28#16864379—22no—
29#16864379—22no—
30#16864379—22no—
31C4×Dic269D412yes—
32#41×C269—22no—
33#41×C269—21no—
34#41×C269—41no—
35C2×Q8×C269—41no—
36C2×Q8×C269C2311yescharacteristic
37C2×D4×C269D411yescharacteristic
38SD16×C269—41no—
39D8×C269—41no—
40M16×C269—21no—
41C22⋊C4×C269—22no—
42C42×C269D411yescharacteristic
43C22×D269—22no—
44C22×D269—82no—
45C22×D269—81no—
46Dic538—81no—
47D1076—42no—
48D1076—81no—
49C22⋊D269—82no—
50C2×Dic269—22no—
51#14524130—42no—
52C23×C269—21no—
53Q8×C269—21no—
54Q8×C269—22no—
55Q8×C269—41no—
56D4×C269—21no—
57D4×C269—22no—
58D4×C269—41no—
59D4×C269—41no—
60C2×C4×C269—21no—
61C2×C4×C269—41no—
62C2×C4×C269—42no—
63C2×C4×C269C2×D412yes—
64C2×C4×C269C2×D411yescharacteristic
65C8×C269—41no—
66C22×C269—42no—
67C22×C269—41no—
68C22×C269—41no—
69C22×C269#12011yescharacteristic
70C2×D269—42no—
71C2×D269—42no—
72C2×D269—42no—
73C2×D269—84no—
74C4×C269—21no—
75C4×C269—22no—
76C4×C269—41no—
77C4×C269—41no—
78Dic269—42no—
79C2×C269—21no—
80C2×C269—81no—
81C2×C269—81no—
82C2×C269B(4,𝔽2)11yescharacteristic
83D269—82no—
84C269#220211yescharacteristic, complemented, Sylow
85#2202—2691nomaximal, Sylow
86H(3,𝔽4)—2691no—
87#454—2692no—
88#454D26911yescharacteristic, complemented
89#355—2691no—
90#355—2692no—
912−1+4C2×D26911yescharacteristic, complemented
92AGL(1,ℤ/8)—21no—
93AGL(1,ℤ/8)—5382no—
94#124—2692no—
95#124—5382no—
96#124—5381no—
97#124C2×D26911yescharacteristic
98#120—5382no—
99#120—5381no—
100#104—21no—
101#104—5382no—
102#101—2692no—
103#101C2×D26911yescharacteristic, complemented
104C24—5381no—
105#41—22no—
106#41—21no—
107#41—41no—
108C2×Q8—41no—
109C2×Q8—5381no—
110C2×Q8C22×D26911yescharacteristic
111C2×D4—2692no—
112C2×D4—5381no—
113C2×D4—5382no—
114C2×D4C22⋊D26911yescharacteristic
115SD16—41no—
116SD16—10762no—
117D8—41no—
118D8—10762no—
119M16—21no—
120M16—5382no—
121C22⋊C4—22no—
122C22⋊C4—5382no—
123C22⋊C4—5381no—
124C22⋊C4—5382no—
125C22⋊C4—5382no—
126C42—2692no—
127C42C22⋊D26911yescharacteristic
128C23—21no—
129C23—5382no—
130C23—21522no—
131C23—21521no—
132Q8—21no—
133Q8—22no—
134Q8—41no—
135Q8—21521no—
136D4—21no—
137D4—22no—
138D4—41no—
139D4—41no—
140D4—10762no—
141D4—21521no—
142D4—21522no—
143C2×C4—21no—
144C2×C4—41no—
145C2×C4—42no—
146C2×C4—5382no—
147C2×C4D4×D26912yes—
148C2×C4C2×C22⋊D26911yescharacteristic
149C8—41no—
150C8—10762no—
151C22—42no—
152C22—41no—
153C22—41no—
154C22—10762no—
155C22—10762no—
156C22—10762no—
157C22—21524no—
158C22#2257542111yescharacteristic
159C4—21no—
160C4—22no—
161C4—41no—
162C4—41no—
163C4—10762no—
164C2—21no—
165C2—81no—
166C2—81no—
167C2—21522no—
168C2#3177079111yescharacteristic
subgroupG/H
centerC2#31770791
commutator, cosocleC2×Q8×C269C23
FrattiniC2×Q8C22×D269
Fitting#454×C269C2
socleC2×C269B(4,𝔽2)
hypercenter#454D269
hypocenterC269#2202
LCSG ▷ C2×Q8×C269 ▷ C22×C269 ▷ C2×C269 ▷ C269
UCSC1 ◁ C2 ◁ C22 ◁ C2×Q8 ◁ #454

Complements

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

NH ≅ G/Ntype
1. #31771347C2semidirect
2. #31770778C2semidirect
3. #31770402C2semidirect
4. #31770401C2semidirect
5. #454×C269C2semidirect
13. #22575372C22semidirect
14. 2−1+4×C269C22semidirect
18. #101×C269C22semidirect
84. C269#2202semidirect
88. #454D269semidirect
91. 2−1+4C2×D269semidirect
103. #101C2×D269semidirect

Maximal subgroups

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

subgroupindex
#317713472
#317707782
#317704022
#317704012
#454×C2692
#2202269