Finite Group Databasebeta

← #46810013

Subgroups — #46810013

Order 34432.

256 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 256 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#31771275C211yesmaximal, characteristic
2#31770977C211yesmaximal, characteristic, complemented
3#31770901C211yesmaximal, characteristic, complemented
4#31770896C211yesmaximal, characteristic, complemented
5#31770335C211yesmaximal, characteristic, complemented
6#31770332C211yesmaximal, characteristic, complemented
7#331×C269C211yesmaximal, characteristic, complemented
8C2×#16864401—21no—
9C2×#16864390—22no—
10#22575453C2211yescharacteristic
11#22575449—21no—
12#22575425—22no—
13#22575424—21no—
14C4×Dic538C2211yescharacteristic
15C4×D1076C2211yescharacteristic
16D4×Dic269—21no—
17#22575387—21no—
18#22575385—21no—
19#22575378—21no—
20#22575375—21no—
21C4×#14524130C2211yescharacteristic
22#127×C269C2211yescharacteristic
23C4⋊C8×C269C2211yescharacteristic
24C4⋊C8×C269C2211yescharacteristic
25#16864401—42no—
26#16864401—42no—
27C2×Dic538—21no—
28C2×Dic538D411yescharacteristic
29C2×D1076D411yescharacteristic
30C2×C22⋊D269—22no—
31C2×C22⋊D269—22no—
32C22×Dic269—22no—
33C2×C4×D269—21no—
34#16864390—44no—
35C2×#14524130—21no—
36C4⋊Dic269—21no—
37C4⋊Dic269—21no—
38C4⋊Dic269D411yescharacteristic
39C4⋊Dic269D411yescharacteristic
40#16864381—21no—
41C22⋊Dic269—22no—
42#16864379—21no—
43C4×Dic269—21no—
44C2×D4×C269—21no—
45C2×D4×C269—21no—
46C2×D4×C269D412yes—
47C2×C8×C269—21no—
48C2×C8×C269—21no—
49C42×C269C2311yescharacteristic
50C22×D269—21no—
51Dic538—21no—
52Dic538—42no—
53Dic538SD1612yescomplemented
54D1076—21no—
55D1076—21no—
56C22⋊D269—44no—
57C22⋊D269—82no—
58C2×Dic269—21no—
59C2×Dic269—21no—
60C2×Dic269—21no—
61C2×Dic269—44no—
62C2×Dic269—41no—
63C4×D269—41no—
64C4×D269—41no—
65#14524130—22no—
66C23×C269—22no—
67D4×C269—42no—
68D4×C269—41no—
69D4×C269—41no—
70D4×C269—41no—
71D4×C269—41no—
72C2×C4×C269C2×D411yescharacteristic
73C2×C4×C269C2×D411yescharacteristic
74C2×C4×C269C2×D411yescharacteristic
75C8×C269—41no—
76C8×C269—41no—
77C22×C269—42no—
78C22×C269—42no—
79C22×C269—42no—
80C22×C269#12011yescharacteristic
81C2×D269—41no—
82C2×D269—41no—
83C2×D269—41no—
84C4×C269—21no—
85C4×C269C2×SD1611yescharacteristic
86C4×C269#13511yescharacteristic
87C4×C269AGL(1,ℤ/8)11yescharacteristic
88C4×C269#13711yescharacteristic
89Dic269—22no—
90Dic269—41no—
91Dic269—41no—
92Dic269—81no—
93C2×C269—82no—
94C2×C269#45011yescharacteristic
95C2×C269#45211yescharacteristic
96C2×C269#45311yescharacteristic
97D269—81no—
98C269#163111yescharacteristic, complemented, Sylow
99#1631—2691nomaximal, Sylow
100#550—2691no—
101#492—2691no—
102#477—2691no—
103#476—2691no—
104#331D26911yescharacteristic, complemented
105Q8⋊C8—2691no—
106D4⋊C8—2691no—
107C2×#41—5381no—
108C2×SD16—5382no—
109C4⋊Q8—2691no—
110#127—5381no—
111#127C2×D26911yescharacteristic
112#121—5381no—
113#121—5382no—
114C4×Q8—2691no—
115C4×D4—2691no—
116C4×D4—5381no—
117#107—5381no—
118#106—5381no—
119C4⋊C8C2×D26911yescharacteristic
120C4⋊C8C2×D26911yescharacteristic
121#103—5381no—
122#102—5381no—
123C4×C8—2691no—
124#41—10762no—
125#41—10762no—
126C2×Q8—2691no—
127C2×Q8—5381no—
128C2×D4—21no—
129C2×D4—21no—
130C2×D4—2691no—
131C2×D4—5382no—
132C2×D4—5382no—
133C2×D4D107612yes—
134C22×C4—5382no—
135C22×C4—5381no—
136SD16—10764no—
137C2×C8—21no—
138C2×C8—21no—
139C2×C8—5381no—
140C4⋊C4—2691no—
141C4⋊C4—2691no—
142C4⋊C4—5381no—
143C4⋊C4—5381no—
144C4⋊C4—5381no—
145C22⋊C4—5382no—
146C22⋊C4—5381no—
147C42—5381no—
148C42C22×D26911yescharacteristic
149C23—22no—
150C23—5381no—
151Q8—2692no—
152Q8—5381no—
153Q8—10762no—
154D4—42no—
155D4—41no—
156D4—41no—
157D4—41no—
158D4—41no—
159D4—5381no—
160D4—5381no—
161D4—10764no—
162D4—21522no—
163C2×C4—5381no—
164C2×C4—5381no—
165C2×C4—5381no—
166C2×C4—10764no—
167C2×C4—10761no—
168C2×C4—10761no—
169C2×C4—10761no—
170C2×C4D4×D26911yescharacteristic
171C2×C4D4×D26911yescharacteristic
172C2×C4C2×D107611yescharacteristic
173C8—41no—
174C8—41no—
175C8—5382no—
176C22—42no—
177C22—42no—
178C22—42no—
179C22—10761no—
180C22—10761no—
181C22—10761no—
182C22#2257542311yescharacteristic
183C4—21no—
184C4—5382no—
185C4—10761no—
186C4—10761no—
187C4—21521no—
188C4SD16×D26911yescharacteristic
189C4#2257547211yescharacteristic
190C4#2257547711yescharacteristic
191C4#2257548211yescharacteristic
192C2—82no—
193C2—21521no—
194C2#3177075711yescharacteristic
195C2#3177076811yescharacteristic
196C2#3177077511yescharacteristic
subgroupG/H
centerC22#22575423
commutator, cosocleC42×C269C23
FrattiniC42C22×D269
Fitting#331×C269C2
socleC22×C269#120
prime radical#31771275C2
hypercenter#331D269
hypocenterC269#1631
LCSG ▷ C42×C269 ▷ C22×C269 ▷ C269
UCSC1 ◁ C22 ◁ C42 ◁ #331

Complements

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

NH ≅ G/Ntype
2. #31770977C2semidirect
3. #31770901C2semidirect
4. #31770896C2semidirect
5. #31770335C2semidirect
6. #31770332C2semidirect
7. #331×C269C2semidirect
53. Dic538SD16semidirect
98. C269#1631semidirect
104. #331D269semidirect

Maximal subgroups

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

subgroupindex
#317712752
#317709772
#317709012
#317708962
#317703352
#317703322
#331×C2692
#1631269