Finite Group Databasebeta

← #46810121

Subgroups — #46810121

Order 34432.

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#31771258C211yesmaximal, characteristic
2#31771008C211yesmaximal, characteristic, complemented
3#31770886C211yesmaximal, characteristic, complemented
4#31770811C211yesmaximal, characteristic, complemented
5#31770332C211yesmaximal, characteristic, complemented
6#31770332C211yesmaximal, characteristic, complemented
7#332×C269C211yesmaximal, characteristic, complemented
8C2×#16864403—21no—
9C2×#16864391—21no—
10#22575453C2211yescharacteristic
11#22575438—22no—
12#22575429—21no—
13#22575428—21no—
14C4×D1076C2211yescharacteristic
15C4×D1076C2211yescharacteristic
16C4⋊C4×D269—21no—
17#22575387—21no—
18#22575386—22no—
19#22575378—21no—
20#22575375—21no—
21C4×#14524130C2211yescharacteristic
22C4⋊Q8×C269C2211yescharacteristic
23C4⋊C8×C269C2211yescharacteristic
24C4⋊C8×C269C2211yescharacteristic
25#16864403—42no—
26#16864403—42no—
27C2×Dic538—21no—
28C2×D1076—22no—
29C2×D1076D411yescharacteristic
30C2×D1076D411yescharacteristic
31C2×C4×D269—22no—
32C2×C4×D269—21no—
33C2×C4×D269—21no—
34#16864391—42no—
35C2×#14524130—21no—
36C4⋊Dic269—21no—
37C4⋊Dic269—21no—
38C4⋊Dic269D411yescharacteristic
39C4⋊Dic269D411yescharacteristic
40#16864381—22no—
41#16864379—22no—
42#16864379—21no—
43#16864379—21no—
44C2×Q8×C269—21no—
45C2×C8×C269—21no—
46C2×C8×C269—21no—
47C4⋊C4×C269—21no—
48C4⋊C4×C269D412yes—
49C42×C269C2311yescharacteristic
50C22×D269—21no—
51C22×D269—21no—
52Dic538—41no—
53Dic538—41no—
54D1076—21no—
55D1076—21no—
56D1076—21no—
57D1076—44no—
58D1076D812yes—
59C2×Dic269—21no—
60C2×Dic269—21no—
61C2×Dic269—41no—
62C4×D269—44no—
63C4×D269—41no—
64C4×D269—41no—
65C4×D269—41no—
66C4×D269—41no—
67#14524130—22no—
68Q8×C269—41no—
69Q8×C269—41no—
70C2×C4×C269—22no—
71C2×C4×C269C2×D411yescharacteristic
72C2×C4×C269C2×D411yescharacteristic
73C2×C4×C269C2×D411yescharacteristic
74C8×C269—41no—
75C8×C269—41no—
76C22×C269#12011yescharacteristic
77C2×D269—22no—
78C2×D269—41no—
79C2×D269—41no—
80C2×D269—41no—
81C2×D269—41no—
82C2×D269—41no—
83C4×C269—21no—
84C4×C269—42no—
85C4×C269C2×D811yescharacteristic
86C4×C269#13511yescharacteristic
87C4×C269AGL(1,ℤ/8)11yescharacteristic
88C4×C269#13711yescharacteristic
89Dic269—41no—
90Dic269—41no—
91Dic269—81no—
92C2×C269#44711yescharacteristic
93C2×C269#45211yescharacteristic
94C2×C269#45611yescharacteristic
95D269—42no—
96D269—81no—
97C269#164711yescharacteristic, complemented, Sylow
98#1647—2691nomaximal, Sylow
99#547—2691no—
100#500—2691no—
101#474—2691no—
102#461—2691no—
103#332D26911yescharacteristic, complemented
104D4⋊C8—2691no—
105D4⋊C8—2691no—
106C2×#41—5381no—
107C2×SD16—5381no—
108C4⋊Q8—2691no—
109C4⋊Q8C2×D26911yescharacteristic
110#123—5382no—
111C22⋊Q8—5381no—
112#121—5381no—
113C4×D4—2691no—
114C4×D4—2691no—
115C2×C4⋊C4—5381no—
116#107—5381no—
117#107—5382no—
118C4⋊C8C2×D26911yescharacteristic
119C4⋊C8C2×D26911yescharacteristic
120#103—5381no—
121#102—5381no—
122C4×C8—2691no—
123#41—10762no—
124#41—10762no—
125C2×Q8—21no—
126C2×Q8—5381no—
127C2×D4—2691no—
128C2×D4—2691no—
129C2×D4—5382no—
130C22×C4—5382no—
131C22×C4—5381no—
132C22×C4—5381no—
133SD16—10762no—
134C2×C8—21no—
135C2×C8—21no—
136C2×C8—5381no—
137C4⋊C4—21no—
138C4⋊C4—2691no—
139C4⋊C4—2691no—
140C4⋊C4—5382no—
141C4⋊C4—5381no—
142C4⋊C4—5381no—
143C4⋊C4D107612yes—
144C22⋊C4—5382no—
145C22⋊C4—5381no—
146C22⋊C4—5381no—
147C42C22×D26911yescharacteristic
148C23—5381no—
149C23—5381no—
150Q8—41no—
151Q8—41no—
152Q8—10761no—
153Q8—10761no—
154D4—2692no—
155D4—5381no—
156D4—5381no—
157D4—5381no—
158D4—10764no—
159C2×C4—22no—
160C2×C4—5381no—
161C2×C4—5381no—
162C2×C4—10764no—
163C2×C4—10761no—
164C2×C4—10761no—
165C2×C4—10761no—
166C2×C4—10761no—
167C2×C4—10761no—
168C2×C4D4×D26911yescharacteristic
169C2×C4D4×D26911yescharacteristic
170C2×C4C2×D107611yescharacteristic
171C8—41no—
172C8—41no—
173C8—5382no—
174C22—5382no—
175C22—10761no—
176C22—10761no—
177C22—10761no—
178C22—10761no—
179C22—10761no—
180C22#2257542311yescharacteristic
181C4—21no—
182C4—42no—
183C4—10761no—
184C4—10761no—
185C4—21521no—
186C4D8×D26911yescharacteristic
187C4#2257547411yescharacteristic
188C4#2257548011yescharacteristic
189C4#2257548511yescharacteristic
190C2—10762no—
191C2—21521no—
192C2#3177074011yescharacteristic
193C2#3177077011yescharacteristic
194C2#3177079011yescharacteristic
subgroupG/H
centerC22#22575423
commutator, cosocleC42×C269C23
FrattiniC42C22×D269
Fitting#332×C269C2
socleC22×C269#120
prime radical#31771258C2
hypercenter#332D269
hypocenterC269#1647
LCSG ▷ C42×C269 ▷ C22×C269 ▷ C269
UCSC1 ◁ C22 ◁ C42 ◁ #332

Complements

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

NH ≅ G/Ntype
2. #31771008C2semidirect
3. #31770886C2semidirect
4. #31770811C2semidirect
5. #31770332C2semidirect
6. #31770332C2semidirect
7. #332×C269C2semidirect
97. C269#1647semidirect
103. #332D269semidirect

Maximal subgroups

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

subgroupindex
#317712582
#317710082
#317708862
#317708112
#317703322
#332×C2692
#1647269