Finite Group Databasebeta

← #46813603

Subgroups — #46813603

Order 34432.

516 conjugacy classes of subgroups; only the 197 normal subgroups are shown.

normal subgroups only

This group has 516 conjugacy classes of subgroups — too many to draw. Only its 197 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
1C2×#22575451C211yesmaximal, characteristic
2C2×Q8×Dic269C211yesmaximal, characteristic
3C2×#22575418C211yesmaximal, characteristic
4#31770511C211yesmaximal, characteristic
5#31770492C214yesmaximal
6#31770487C214yesmaximal
7#31770470C211yesmaximal, characteristic
8#31770451C211yesmaximal, characteristic
9#391×C269C211yesmaximal, characteristic
10C22×Dic538C2211yescharacteristic
11#22575451C418yescomplemented
12Q8×Dic269—24no—
13#22575418—24no—
14C2×C4⋊Dic269C2211yescharacteristic
15C2×C4⋊Dic269C2212yes—
16C2×#16864381C2212yes—
17C2×#16864381C2214yes—
18C2×C4×Dic269C2212yes—
19C2×C4×Dic269C2212yes—
20C2×C4×Dic269C2211yescharacteristic
21C2×C4×Dic269C2211yescharacteristic
22#22575360C2212yes—
23#22575360C2212yes—
24#22575360C2214yes—
25#22575360C2214yes—
26C22×Q8×C269C2211yescharacteristic
27C2×C4⋊C4×C269C2211yescharacteristic
28C2×C42×C269C2211yescharacteristic
29#95×C269C2214yes—
30C2×Dic538—22no—
31C2×Dic538—22no—
32C2×Dic538C2×C414yes—
33C22×Dic269C2312yes—
34C22×Dic269C2312yes—
35C22×Dic269C2312yes—
36C22×Dic269C2311yescharacteristic
37C22×Dic269C2311yescharacteristic
38C4⋊Dic269—24no—
39C4⋊Dic269—22no—
40#16864381—24no—
41#16864381C2×C4116yes—
42C4×Dic269—24no—
43C4×Dic269—24no—
44C4×Dic269C2×C414yes—
45C4×Dic269D414yes—
46C2×Q8×C269—22no—
47C2×Q8×C269—22no—
48C2×Q8×C269C2×C414yes—
49C22×C4×C269C2312yes—
50C22×C4×C269C2312yes—
51C22×C4×C269C2312yes—
52C22×C4×C269C2311yescharacteristic
53C4⋊C4×C269—22no—
54C42×C269—22no—
55Dic538—28no—
56C2×Dic269—22no—
57C2×Dic269—22no—
58C2×Dic269—22no—
59C2×Dic269—22no—
60C2×Dic269—22no—
61C2×Dic269—22no—
62C2×Dic269—22no—
63C2×Dic269—24no—
64C2×Dic269C22×C414yes—
65C2×Dic269C22×C414yes—
66C2×Dic269C2×D412yes—
67C2×Dic269C2×D412yes—
68C23×C269C2411yescharacteristic
69Q8×C269—28no—
70C2×C4×C269—22no—
71C2×C4×C269—22no—
72C2×C4×C269—22no—
73C2×C4×C269—22no—
74C2×C4×C269—24no—
75C2×C4×C269—22no—
76C2×C4×C269—22no—
77C2×C4×C269C22×C414yes—
78C2×C4×C269C22×C412yes—
79C2×C4×C269C2×D412yes—
80C2×C4×C269#4112yes—
81C22×C269C23×C411yescharacteristic
82C22×C269C22×D411yescharacteristic
83C22×C269C2×#4111yescharacteristic
84C22×C2692−1+412yes—
85C22×C2692−1+412yes—
86C4×C269—24no—
87C4×C269—42no—
88C4×C269—42no—
89C4×C269C4×D414yes—
90Dic269—24no—
91Dic269—24no—
92Dic269—22no—
93Dic269—22no—
94Dic269—42no—
95C2×C269C2×C4×D411yescharacteristic
96C2×C269#51912yes—
97C2×C269#53611yescharacteristic
98C2×C269#53611yescharacteristic
99C2×C269#54112yes—
100C269#232711yescharacteristic, complemented, Sylow
101#2327—2691nomaximal, Sylow
102C2×C4⋊Q8—2691no—
103C2×C4×Q8—2691no—
104C2×C4×Q8—2691no—
105#391—2691no—
106#391D26911yescharacteristic
107#387—2694no—
108#387—2694no—
109#382—2691no—
110#376—2691no—
111C22×Q8—2691no—
112C22×Q8C2×D26911yescharacteristic
113C4⋊Q8—2698no—
114C4×Q8—5384no—
115C4×Q8—5384no—
116C2×C4⋊C4—2694no—
117C2×C4⋊C4—2692no—
118C2×C4⋊C4—2691no—
119C2×C4⋊C4—2692no—
120C2×C4⋊C4C2×D26911yescharacteristic
121C2×C42—2691no—
122C2×C42—2691no—
123C2×C42—2692no—
124C2×C42—2692no—
125C2×C42C2×D26911yescharacteristic
126#95—2692no—
127#95—2692no—
128#95—2694no—
129#95—2694no—
130#95C2×D26914yes—
131C2×Q8—22no—
132C2×Q8—22no—
133C2×Q8—2694no—
134C2×Q8—5382no—
135C2×Q8—5382no—
136C2×Q8C4×D26914yes—
137C22×C4—2692no—
138C22×C4—2691no—
139C22×C4—2691no—
140C22×C4—2692no—
141C22×C4—2692no—
142C22×C4C22×D26912yes—
143C22×C4C22×D26912yes—
144C22×C4C22×D26912yes—
145C22×C4C22×D26911yescharacteristic
146C4⋊C4—22no—
147C4⋊C4—26916no—
148C4⋊C4—5384no—
149C4⋊C4—5384no—
150C4⋊C4—5382no—
151C42—22no—
152C42—2694no—
153C42—2694no—
154C42—5384no—
155C42—5384no—
156C23C23×D26911yescharacteristic
157Q8—28no—
158Q8—5388no—
159C2×C4—22no—
160C2×C4—22no—
161C2×C4—22no—
162C2×C4—22no—
163C2×C4—24no—
164C2×C4—22no—
165C2×C4—22no—
166C2×C4—2694no—
167C2×C4—2694no—
168C2×C4—2692no—
169C2×C4—2692no—
170C2×C4—5382no—
171C2×C4—5382no—
172C2×C4—5382no—
173C2×C4—5382no—
174C2×C4—5382no—
175C2×C4—5382no—
176C2×C4—5382no—
177C2×C4—5384no—
178C2×C4C2×C4×D26914yes—
179C2×C4C2×C4×D26912yes—
180C2×C4D4×D26912yes—
181C2×C4#1686440112yes—
182C22C22×C4×D26911yescharacteristic
183C22C2×D4×D26911yescharacteristic
184C22C2×#1686440111yescharacteristic
185C22#2257550012yes—
186C22#2257550112yes—
187C4—24no—
188C4—42no—
189C4—42no—
190C4—5382no—
191C4—5382no—
192C4—5384no—
193C4—5384no—
194C4—10762no—
195C4#2257541314yes—
196C2C2×#2257541311yescharacteristic
197C2#3177108112yes—
198C2#3177117511yescharacteristic
199C2#3177118011yescharacteristic
200C2#3177121512yes—
subgroupG/H
center, FrattiniC23C23×D269
commutatorC22×C269C23×C4
Fitting#391×C269C2
socle, prime radical, cosocleC23×C269C24
hypercenter#391D269
hypocenterC269#2327
LCSG ▷ C22×C269 ▷ C269
UCSC1 ◁ C23 ◁ #391

Complements

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

NH ≅ G/Ntype
11. #22575451C4semidirect
100. C269#2327semidirect

Maximal subgroups

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

subgroupindex
C2×#225754512
C2×Q8×Dic2692
C2×#225754182
#317705112
#317704922
#317704872
#317704702
#317704512
#391×C2692
#2327269