Finite Group Databasebeta

← #46810168

Subgroups — #46810168

Order 34432.

240 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 240 conjugacy classes of subgroups — too many to draw. Only its 57 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#31771259C211yesmaximal, characteristic
2#31771001C211yesmaximal, characteristic, complemented
3#31770807C211yesmaximal, characteristic, complemented
4#31770352C211yesmaximal, characteristic, complemented
5D4⋊#14524130C211yesmaximal, characteristic, complemented
6#31770332C211yesmaximal, characteristic, complemented
7#474×C269C211yesmaximal, characteristic, complemented
8C2×#16864401—21no—
9C2×#16864389—21no—
10#22575453C2211yescharacteristic
11#22575437—22no—
12#22575433—21no—
13#22575427—21no—
14C4×D1076C2211yescharacteristic
15C2×C4⋊Dic269—21no—
16#22575384—22no—
17C4⋊#14524130C2211yescharacteristic
18#22575377—21no—
19C4×#14524130C2211yescharacteristic
20C4⋊Q8×C269C2211yescharacteristic
21C4×D4×C269C2211yescharacteristic
22#107×C269—21no—
23C4⋊C8×C269C2211yescharacteristic
24#102×C269—21no—
25#16864401—42no—
26#16864401—42no—
27C2×Dic538—21no—
28C2×D1076D411yescharacteristic
29C2×C22⋊D269—22no—
30C22×Dic269—22no—
31C2×C4×D269—21no—
32#16864389—42no—
33C2×#14524130—21no—
34C2×#14524130—21no—
35C4⋊Dic269—22no—
36C4⋊Dic269—21no—
37C4⋊Dic269D411yescharacteristic
38C4⋊Dic269D412yes—
39#16864379—22no—
40#16864379—21no—
41C2×Q8×C269—21no—
42C2×D4×C269D411yescharacteristic
43C22×C4×C269—21no—
44C2×C8×C269—21no—
45C4⋊C4×C269—21no—
46C4⋊C4×C269—21no—
47C4⋊C4×C269D411yescharacteristic
48C22⋊C4×C269—21no—
49C42×C269C2311yescharacteristic
50C22×D269—21no—
51Dic538—41no—
52Dic538—41no—
53D1076—21no—
54D1076—21no—
55C22⋊D269—44no—
56C2×Dic269—21no—
57C2×Dic269—22no—
58C2×Dic269—44no—
59C4×D269—41no—
60C4×D269—41no—
61#14524130—22no—
62#14524130—41no—
63C23×C269—21no—
64Q8×C269—41no—
65Q8×C269—41no—
66D4×C269—21no—
67D4×C269SD1612yescomplemented
68C2×C4×C269—21no—
69C2×C4×C269—41no—
70C2×C4×C269—41no—
71C2×C4×C269—41no—
72C2×C4×C269C2×D411yescharacteristic
73C2×C4×C269C2×D411yescharacteristic
74C2×C4×C269C2×D411yescharacteristic
75C8×C269—41no—
76C22×C269—22no—
77C22×C269—41no—
78C22×C269—41no—
79C22×C269#12011yescharacteristic
80C2×D269—41no—
81C2×D269—41no—
82C2×D269—41no—
83C4×C269—21no—
84C4×C269—41no—
85C4×C269—81no—
86C4×C269C2×SD1611yescharacteristic
87C4×C269#13511yescharacteristic
88C4×C269AGL(1,ℤ/8)11yescharacteristic
89C4×C269AGL(1,ℤ/8)11yescharacteristic
90Dic269—41no—
91Dic269—42no—
92C2×C269—42no—
93C2×C269#44911yescharacteristic
94C2×C269#45011yescharacteristic
95C2×C269#45611yescharacteristic
96D269—81no—
97C269#165411yescharacteristic, complemented, Sylow
98#1654—2691nomaximal, Sylow
99#547—2691no—
100#498—2691no—
101#474D26911yescharacteristic, complemented
102#461—2691no—
103#332—2691no—
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×D4C2×D26911yescharacteristic
115C2×C4⋊C4—5381no—
116#107—21no—
117#106—5382no—
118C4⋊C8—2691no—
119C4⋊C8C2×D26911yescharacteristic
120#103—5381no—
121#102—21no—
122C4×C8—2691no—
123#41—10762no—
124#41—10762no—
125C2×Q8—21no—
126C2×Q8—5381no—
127C2×D4—2691no—
128C2×D4—5382no—
129C2×D4C22⋊D26911yescharacteristic
130C22×C4—21no—
131C22×C4—5382no—
132C22×C4—5381no—
133SD16—10762no—
134C2×C8—21no—
135C2×C8—5381no—
136C2×C8—5381no—
137C4⋊C4—21no—
138C4⋊C4—21no—
139C4⋊C4—2691no—
140C4⋊C4—2692no—
141C4⋊C4—5382no—
142C4⋊C4—5381no—
143C4⋊C4C22⋊D26911yescharacteristic
144C22⋊C4—21no—
145C22⋊C4—5382no—
146C22⋊C4—5381no—
147C42C22×D26911yescharacteristic
148C23—21no—
149C23—5381no—
150Q8—41no—
151Q8—41no—
152Q8—10761no—
153Q8—10761no—
154D4—21no—
155D4—5381no—
156D4—5381no—
157D4—10764no—
158D4#1686438912yescomplemented
159C2×C4—21no—
160C2×C4—41no—
161C2×C4—41no—
162C2×C4—41no—
163C2×C4—5381no—
164C2×C4—5382no—
165C2×C4—10764no—
166C2×C4—10761no—
167C2×C4—10761no—
168C2×C4D4×D26911yescharacteristic
169C2×C4D4×D26911yescharacteristic
170C2×C4C2×C22⋊D26911yescharacteristic
171C8—41no—
172C8—5382no—
173C8—10761no—
174C22—22no—
175C22—41no—
176C22—41no—
177C22—10761no—
178C22—10761no—
179C22—10761no—
180C22#2257542111yescharacteristic
181C4—21no—
182C4—41no—
183C4—81no—
184C4—10761no—
185C4—10762no—
186C4C2×#1686438911yescharacteristic
187C4#2257547211yescharacteristic
188C4#2257547711yescharacteristic
189C4#2257547811yescharacteristic
190C2—42no—
191C2—21521no—
192C2#3177074711yescharacteristic
193C2#3177075511yescharacteristic
194C2#3177078611yescharacteristic
subgroupG/H
centerC22#22575421
commutator, cosocleC42×C269C23
FrattiniC42C22×D269
Fitting#474×C269C2
socleC22×C269#120
prime radical#31771259C2
hypercenter#474D269
hypocenterC269#1654
LCSG ▷ C42×C269 ▷ C22×C269 ▷ C269
UCSC1 ◁ C22 ◁ C42 ◁ #474

Complements

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

NH ≅ G/Ntype
2. #31771001C2semidirect
3. #31770807C2semidirect
4. #31770352C2semidirect
5. D4⋊#14524130C2semidirect
6. #31770332C2semidirect
7. #474×C269C2semidirect
67. D4×C269SD16semidirect
97. C269#1654semidirect
101. #474D269semidirect
158. D4#16864389semidirect

Maximal subgroups

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

subgroupindex
#317712592
#317710012
#317708072
#317703522
D4⋊#145241302
#317703322
#474×C2692
#1654269