Finite Group Databasebeta

← #46809839

Subgroups — #46809839

Order 34432.

308 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 308 conjugacy classes of subgroups — too many to draw. Only its 53 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#31771167C211yesmaximal, characteristic
2#31770746C212yesmaximal, complemented
3#31770517C211yesmaximal, characteristic, complemented
4#31770340C212yesmaximal, complemented
5#323×C269C211yesmaximal, characteristic, complemented
6C22×Dic538C2211yescharacteristic, complemented
7C22×C22⋊D269—21no—
8C2×#16864391—22no—
9#22575437—22no—
10#22575427—21no—
11#22575427C2212yes—
12#22575423—22no—
13C2×C22⋊Dic269—21no—
14#22575378—22no—
15#22575360—21no—
16#22575360C2211yescharacteristic, complemented
17C2×C22⋊C4×C269C2211yescharacteristic
18C22⋊C8×C269C2212yescomplemented
19C23×D269—21no—
20C2×Dic538—22no—
21C2×Dic538—42no—
22C2×D1076—21no—
23C2×D1076D412yes—
24C2×C22⋊D269—22no—
25C2×C22⋊D269—22no—
26C2×C22⋊D269—42no—
27C2×C22⋊D269—42no—
28C22×Dic269—21no—
29C22×Dic269—21no—
30C22×Dic269—21no—
31#16864391—44no—
32C4⋊Dic269—21no—
33C4⋊Dic269D412yes—
34C22⋊Dic269—42no—
35#16864379—22no—
36#16864379—22no—
37C24×C269D411yescharacteristic
38C22×C4×C269D411yescharacteristic
39C22×C4×C269C2311yescharacteristic
40C2×C8×C269—22no—
41C22⋊C4×C269—22no—
42C22×D269—22no—
43C22×D269—42no—
44C22×D269—81no—
45C22×D269—81no—
46Dic538—24no—
47Dic538—42no—
48D1076—42no—
49D1076—81no—
50C22⋊D269—44no—
51C22⋊D269—42no—
52C22⋊D269—42no—
53C2×Dic269—22no—
54C2×Dic269—22no—
55C2×Dic269—42no—
56C2×Dic269—41no—
57C2×Dic269—41no—
58C2×Dic269—41no—
59C2×Dic269—41no—
60C2×Dic269—41no—
61C23×C269—21no—
62C23×C269—21no—
63C23×C269—41no—
64C23×C269—41no—
65C23×C269SD1612yescomplemented
66C23×C269C2×D411yescharacteristic
67C2×C4×C269—21no—
68C2×C4×C269—41no—
69C2×C4×C269—41no—
70C2×C4×C269C2×D412yes—
71C8×C269—42no—
72C22×C269—21no—
73C22×C269—21no—
74C22×C269—22no—
75C22×C269—42no—
76C22×C269—41no—
77C22×C269—41no—
78C22×C269—81no—
79C22×C269#12011yescharacteristic
80C22×C269C2×SD1611yescharacteristic
81C22×C269AGL(1,ℤ/8)11yescharacteristic
82C2×D269—42no—
83C2×D269—42no—
84C2×D269—42no—
85C2×D269—84no—
86C4×C269—22no—
87C4×C269—41no—
88Dic269—42no—
89Dic269—42no—
90Dic269—81no—
91C2×C269—21no—
92C2×C269—21no—
93C2×C269—42no—
94C2×C269#45011yescharacteristic
95C2×C269#45511yescharacteristic
96C2×C269B(4,𝔽2)11yescharacteristic
97D269—82no—
98C269#160111yescharacteristic, complemented, Sylow
99#1601—2691nomaximal, Sylow
100#534—2691no—
101#448—2692no—
102#393—2691no—
103#327—2692no—
104#323D26911yescharacteristic, complemented
105C22×Q8—2691no—
106C22×D4—5381no—
107C2×SD16—5382no—
108#123—5382no—
109#121—2692no—
110#121—5381no—
111#120—5382no—
112C2×C22⋊C4—5381no—
113C2×C22⋊C4C2×D26911yescharacteristic
114#103—5382no—
115C22⋊C8C2×D26912yescomplemented
116#95—2691no—
117#95—5381no—
118C24—5381no—
119C24D107611yescharacteristic
120C2×Q8—5382no—
121C2×Q8—10762no—
122C2×D4—2692no—
123C2×D4—5381no—
124C2×D4—5382no—
125C2×D4—5382no—
126C2×D4—10762no—
127C2×D4—10762no—
128C22×C4—5381no—
129C22×C4—5381no—
130C22×C4—5381no—
131C22×C4D107611yescharacteristic
132C22×C4C22×D26911yescharacteristic
133SD16—10764no—
134C2×C8—22no—
135C4⋊C4—2692no—
136C4⋊C4—5381no—
137C22⋊C4—22no—
138C22⋊C4—5382no—
139C22⋊C4—5382no—
140C22⋊C4—10762no—
141C23—21no—
142C23—21no—
143C23—41no—
144C23—41no—
145C23—5382no—
146C23—10762no—
147C23—21521no—
148C23—21521no—
149C23#1686439112yescomplemented
150C23C2×D107611yescharacteristic
151Q8—5384no—
152Q8—10762no—
153D4—10762no—
154D4—10764no—
155D4—10762no—
156D4—10762no—
157D4—21521no—
158C2×C4—21no—
159C2×C4—41no—
160C2×C4—41no—
161C2×C4—5382no—
162C2×C4—5382no—
163C2×C4—10762no—
164C2×C4—10761no—
165C2×C4—10761no—
166C2×C4—10761no—
167C2×C4—10761no—
168C2×C4—10761no—
169C2×C4D4×D26912yes—
170C8—42no—
171C22—21no—
172C22—21no—
173C22—22no—
174C22—42no—
175C22—41no—
176C22—41no—
177C22—81no—
178C22—10762no—
179C22—10762no—
180C22—10762no—
181C22—21524no—
182C22#2257542311yescharacteristic
183C22C2×#1686439111yescharacteristic
184C22#2257548111yescharacteristic
185C4—22no—
186C4—41no—
187C4—10762no—
188C4—10762no—
189C4—21521no—
190C2—21no—
191C2—21no—
192C2—42no—
193C2—21522no—
194C2#3177075811yescharacteristic
195C2#3177078511yescharacteristic
196C2#3177079311yescharacteristic
subgroupG/H
centerC22#22575423
commutator, cosocleC22×C4×C269C23
FrattiniC22×C4C22×D269
Fitting#323×C269C2
socleC22×C269#120
prime radical#31771167C2
hypercenter#323D269
hypocenterC269#1601
LCSG ▷ C22×C4×C269 ▷ C22×C269 ▷ C269
UCSC1 ◁ C22 ◁ C22×C4 ◁ #323

Complements

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

NH ≅ G/Ntype
2. #31770746C2semidirect
3. #31770517C2semidirect
4. #31770340C2semidirect
5. #323×C269C2semidirect
6. C22×Dic538C22semidirect
16. #22575360C22semidirect
18. C22⋊C8×C269C22semidirect
65. C23×C269SD16semidirect
98. C269#1601semidirect
104. #323D269semidirect
115. C22⋊C8C2×D269semidirect
149. C23#16864391semidirect

Maximal subgroups

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

subgroupindex
#317711672
#317707462
#317705172
#317703402
#323×C2692
#1601269