Finite Group Databasebeta

← #46820490

Subgroups — #46820490

Order 34432.

500 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 500 conjugacy classes of subgroups — too many to draw. Only its 141 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
1C22×#16864385C211yesmaximal, characteristic, complemented
2#31771112C212yesmaximal, complemented
3#31770860C212yesmaximal, complemented
4#31770855C212yesmaximal, complemented
5#31770848C214yesmaximal, complemented
6#31770643C211yesmaximal, characteristic, complemented
7#31770601C212yesmaximal, complemented
8C2×#135×C269C211yesmaximal, characteristic, complemented
9C22×C4×D269C2211yescharacteristic
10C2×#16864385—24no—
11C2×#16864385C2212yes—
12C2×#16864385C2212yes—
13C22×#14524130C2211yescharacteristic
14#22575434—22no—
15#22575429C2214yes—
16#22575424C2214yes—
17#22575421—22no—
18C4×C22⋊D269—24no—
19#22575412C2212yes—
20#22575387C2212yescomplemented
21#22575385C2212yescomplemented
22#22575377C2214yescomplemented
23#22575374C2214yescomplemented
24C2×#41×C269C2212yes—
25#135×C269—24no—
26C2×Q16×C269C2211yescharacteristic, complemented
27C2×SD16×C269C2212yescomplemented
28C2×D8×C269C2211yescharacteristic, complemented
29C22×C8×C269C2211yescharacteristic
30C23×D269D411yescharacteristic
31C2×C22⋊D269—24no—
32C22×Dic269D411yescharacteristic
33C2×C4×D269—22no—
34C2×C4×D269—22no—
35C2×C4×D269D412yes—
36C2×C4×D269C2312yes—
37#16864385—24no—
38#16864385—24no—
39C2×#14524130—22no—
40C2×#14524130C2312yes—
41C4⋊Dic269C2314yes—
42#16864381—24no—
43C22⋊Dic269—24no—
44C22⋊Dic269—22no—
45#16864379—24no—
46C4×Dic269—22no—
47#41×C269—24no—
48#41×C269—24no—
49C2×Q8×C269C2312yes—
50C2×D4×C269—22no—
51C2×D4×C269C2312yes—
52C22×C4×C269—22no—
53C22×C4×C269C2311yescharacteristic
54Q16×C269—22no—
55SD16×C269—24no—
56D8×C269—22no—
57C2×C8×C269D414yescomplemented
58C2×C8×C269C2311yescharacteristic
59C2×C8×C269C2311yescharacteristic
60C22×D269—22no—
61C22×D269—42no—
62C22×D269C2×D412yes—
63C22⋊D269—48no—
64C2×Dic269—22no—
65C2×Dic269—24no—
66C2×Dic269C2×D412yes—
67C4×D269—24no—
68C4×D269—22no—
69C4×D269—22no—
70#14524130—22no—
71C23×C269—22no—
72C23×C269C2×D411yescharacteristic
73Q8×C269—22no—
74Q8×C269—22no—
75D4×C269—22no—
76D4×C269—22no—
77D4×C269—44no—
78C2×C4×C269—22no—
79C2×C4×C269—44no—
80C2×C4×C269C2×D411yescharacteristic
81C2×C4×C269C2×D412yes—
82C2×C4×C269#4112yes—
83C2×C4×C269C2411yescharacteristic
84C8×C269C2×D412yes—
85C8×C269C2×D412yes—
86C22×C269—22no—
87C22×C269—42no—
88C22×C269—44no—
89C22×C269#12112yes—
90C22×C269C22×D411yescharacteristic
91C2×D269—22no—
92C2×D269—24no—
93C2×D269—44no—
94C4×C269—42no—
95C4×C269#12112yes—
96C4×C269C22×D411yescharacteristic
97C4×C269C2×#4111yescharacteristic
98Dic269—22no—
99Dic269—44no—
100C2×C269—22no—
101C2×C269—82no—
102C2×C269C2×#12111yescharacteristic
103C2×C269#57512yes—
104D269—42no—
105C269#305411yescharacteristic, complemented, Sylow
106#3054—2691nomaximal, Sylow
107C2×#135D26911yescharacteristic, complemented
108C22×M16—2691no—
109#525—2692no—
110#470—2692no—
111#469—2692no—
112#468—2694no—
113#427—2691no—
114#416—2692no—
115C2×#41C2×D26912yes—
116C23×C4—2691no—
117#135—24no—
118C2×Q16C2×D26911yescharacteristic, complemented
119C2×SD16C2×D26912yescomplemented
120C2×D8C2×D26911yescharacteristic, complemented
121C2×M16—2692no—
122C2×M16—2692no—
123C2×M16—5384no—
124C22×C8—2691no—
125C22×C8C2×D26911yescharacteristic
126#123—5382no—
127C22⋊Q8—2694no—
128#121—2694no—
129#120—5382no—
130C4×D4—5384no—
131#117—2692no—
132#107—2692no—
133#106—2692no—
134#103—2694no—
135#102—2694no—
136C24—2691no—
137#41—24no—
138#41—24no—
139C2×Q8C22×D26912yes—
140C2×D4—22no—
141C2×D4—5384no—
142C2×D4C22×D26912yes—
143C22×C4—22no—
144C22×C4—2691no—
145C22×C4—2692no—
146C22×C4—2692no—
147C22×C4—5382no—
148C22×C4—5382no—
149C22×C4C22×D26911yescharacteristic
150Q16—22no—
151SD16—24no—
152D8—22no—
153M16—5384no—
154M16—5384no—
155C2×C8—2692no—
156C2×C8—5382no—
157C2×C8C22⋊D26914yescomplemented
158C2×C8C22×D26911yescharacteristic
159C2×C8C22×D26911yescharacteristic
160C4⋊C4—2694no—
161C4⋊C4—5384no—
162C22⋊C4—5384no—
163C22⋊C4—5384no—
164C22⋊C4—5382no—
165C42—5382no—
166C23—22no—
167C23—2692no—
168C23—5382no—
169C23—10762no—
170C23D4×D26911yescharacteristic
171Q8—22no—
172Q8—22no—
173D4—22no—
174D4—22no—
175D4—44no—
176D4—10768no—
177C2×C4—22no—
178C2×C4—44no—
179C2×C4—2692no—
180C2×C4—5382no—
181C2×C4—5384no—
182C2×C4—5382no—
183C2×C4—5382no—
184C2×C4—5384no—
185C2×C4D4×D26911yescharacteristic
186C2×C4C2×C22⋊D26912yes—
187C2×C4#1686440112yes—
188C2×C4C23×D26911yescharacteristic
189C8—5382no—
190C8C2×C22⋊D26912yes—
191C8C2×C22⋊D26912yes—
192C22—22no—
193C22—44no—
194C22—42no—
195C22—5384no—
196C22—5382no—
197C22—10764no—
198C22#2257542412yes—
199C22C2×D4×D26911yescharacteristic
200C4—42no—
201C4—5382no—
202C4—10764no—
203C4#2257542412yes—
204C4C22×C22⋊D26911yescharacteristic
205C4C2×#1686440111yescharacteristic
206C2—22no—
207C2—82no—
208C2—10762no—
209C2C2×#2257542411yescharacteristic
210C2#3177140212yes—
subgroupG/H
centerC2×C4D4×D269
commutator, cosocleC2×C4×C269C24
FrattiniC2×C4C23×D269
FittingC2×#135×C269C2
socleC22×C269C22×D4
hypercenterC2×#135D269
hypocenterC269#3054
LCSG ▷ C2×C4×C269 ▷ C2×C269 ▷ C269
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C2×#135

Complements

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

NH ≅ G/Ntype
1. C22×#16864385C2semidirect
2. #31771112C2semidirect
3. #31770860C2semidirect
4. #31770855C2semidirect
5. #31770848C2semidirect
6. #31770643C2semidirect
7. #31770601C2semidirect
8. C2×#135×C269C2semidirect
20. #22575387C22semidirect
21. #22575385C22semidirect
22. #22575377C22semidirect
23. #22575374C22semidirect
26. C2×Q16×C269C22semidirect
27. C2×SD16×C269C22semidirect
28. C2×D8×C269C22semidirect
57. C2×C8×C269D4semidirect
105. C269#3054semidirect
107. C2×#135D269semidirect
118. C2×Q16C2×D269semidirect
119. C2×SD16C2×D269semidirect
120. C2×D8C2×D269semidirect
157. C2×C8C22⋊D269semidirect

Maximal subgroups

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

subgroupindex
C22×#168643852
#317711122
#317708602
#317708552
#317708482
#317706432
#317706012
C2×#135×C2692
#3054269