Finite Group Databasebeta

← #46820418

Subgroups — #46820418

Order 34432.

458 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 458 conjugacy classes of subgroups — too many to draw. Only its 133 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×#22575483C211yesmaximal, characteristic
2C22×Dic1076C211yesmaximal, characteristic, complemented
3#31771176C211yesmaximal, characteristic
4#31771082C211yesmaximal, characteristic, complemented
5#31770826C214yesmaximal, complemented
6#31770814C214yesmaximal, complemented
7#31770631C211yesmaximal, characteristic, complemented
8#31770613C211yesmaximal, characteristic, complemented
9#416×C269C211yesmaximal, characteristic
10C22×Dic538C2211yescharacteristic
11C22×Dic538C2211yescharacteristic
12#22575483—24no—
13C2×Dic1076—24no—
14C2×Dic1076C2214yes—
15C2×#16864392C2212yes—
16C2×#16864390C2212yes—
17C2×#16864386C2211yescharacteristic
18#22575451C2212yes—
19#22575440C2212yes—
20#22575434—22no—
21#22575432—22no—
22C4×Dic538—22no—
23#22575418C2214yes—
24#22575412C2211yescharacteristic
25#22575409—22no—
26#22575382C2214yes—
27#22575376C2214yes—
28C2×#41×C269C2211yescharacteristic
29C22×C8×C269C2211yescharacteristic
30#117×C269C2211yescharacteristic
31#103×C269C2212yes—
32#102×C269C2212yes—
33C2×Dic538—21no—
34C2×Dic538—21no—
35C2×Dic538—22no—
36C2×Dic538—22no—
37C2×Dic538—22no—
38C2×Dic538D414yes—
39C2×Dic538C2312yes—
40C2×Dic538C2312yes—
41C22×Dic269—21no—
42C22×Dic269—21no—
43Dic1076—28no—
44#16864392—24no—
45#16864390—24no—
46#16864386—22no—
47C2×#14524130C2312yes—
48C4⋊Dic269D412yes—
49#16864381—24no—
50#16864381—24no—
51C22⋊Dic269—24no—
52C22⋊Dic269D412yes—
53C4×Dic269—24no—
54C4×Dic269C2312yes—
55#41×C269—22no—
56#41×C269—22no—
57C2×Q8×C269C2311yescharacteristic
58C2×D4×C269—21no—
59C2×D4×C269C2311yescharacteristic
60C22×C4×C269—21no—
61C22×C4×C269C2311yescharacteristic
62C2×C8×C269—22no—
63C2×C8×C269C2312yes—
64C4⋊C4×C269C2312yes—
65C22⋊C4×C269—21no—
66C42×C269—21no—
67Dic538—24no—
68Dic538—22no—
69Dic538—22no—
70Dic538—22no—
71Dic538—42no—
72Dic538C2×D414yes—
73C2×Dic269—24no—
74C2×Dic269—22no—
75C2×Dic269—22no—
76C2×Dic269—42no—
77C2×Dic269C2×D414yes—
78#14524130—22no—
79C23×C269—21no—
80C23×C269C2×D411yescharacteristic
81Q8×C269—21no—
82Q8×C269—21no—
83D4×C269—21no—
84D4×C269—21no—
85D4×C269—42no—
86C2×C4×C269—22no—
87C2×C4×C269—21no—
88C2×C4×C269—42no—
89C2×C4×C269C2×D411yescharacteristic
90C2×C4×C269C2×D412yes—
91C2×C4×C269#4112yes—
92C2×C4×C269C2411yescharacteristic
93C8×C269—22no—
94C22×C269—22no—
95C22×C269—41no—
96C22×C269—42no—
97C22×C269#12112yes—
98C22×C269C22×D411yescharacteristic
99C4×C269—41no—
100C4×C269—42no—
101C4×C269#12112yes—
102C4×C269C22×D411yescharacteristic
103C4×C269C2×#4111yescharacteristic
104Dic269—24no—
105Dic269—24no—
106Dic269—42no—
107C2×C269—22no—
108C2×C269—81no—
109C2×C269C2×#12111yescharacteristic
110C2×C269#57812yes—
111C269#304411yescharacteristic, complemented, Sylow
112#3044—2691nomaximal, Sylow
113C2×#137—2691no—
114C22×Q16—2691no—
115#536—2691no—
116#519—2691no—
117#464—2694no—
118#462—2694no—
119#424—2691no—
120#419—2691no—
121#416D26911yescharacteristic
122C2×#41C2×D26911yescharacteristic
123C22×Q8—2691no—
124C22×Q8—2691no—
125#137—5384no—
126C2×Q16—2692no—
127C2×Q16—2694no—
128C2×Q16—5384no—
129C2×SD16—2692no—
130C2×M16—2691no—
131C22×C8C2×D26911yescharacteristic
132C4⋊Q8—2692no—
133#124—2692no—
134#123—5382no—
135C22⋊Q8—5382no—
136C4×Q8—2694no—
137C4×Q8—5382no—
138#117—2691no—
139#117—5382no—
140#117C2×D26911yescharacteristic
141C4⋊C8—2694no—
142#103—2694no—
143#103C2×D26912yes—
144#102C2×D26912yes—
145#41—22no—
146#41—22no—
147C2×Q8—2694no—
148C2×Q8—2692no—
149C2×Q8—2692no—
150C2×Q8—5381no—
151C2×Q8—5381no—
152C2×Q8—5382no—
153C2×Q8—5382no—
154C2×Q8—5382no—
155C2×Q8C22×D26911yescharacteristic
156C2×D4—21no—
157C2×D4C22×D26911yescharacteristic
158C22×C4—21no—
159C22×C4—5381no—
160C22×C4—5381no—
161C22×C4C22×D26911yescharacteristic
162Q16—5388no—
163Q16—5384no—
164SD16—5384no—
165M16—5382no—
166C2×C8—22no—
167C2×C8—2692no—
168C2×C8C22×D26912yes—
169C4⋊C4—2692no—
170C4⋊C4—5384no—
171C4⋊C4—5384no—
172C4⋊C4C22×D26912yes—
173C22⋊C4—21no—
174C22⋊C4—2692no—
175C22⋊C4—5384no—
176C42—21no—
177C42—2692no—
178C42—5384no—
179C23—21no—
180C23D4×D26911yescharacteristic
181Q8—21no—
182Q8—21no—
183Q8—2694no—
184Q8—5384no—
185Q8—5382no—
186Q8—5382no—
187Q8—5382no—
188Q8—10762no—
189D4—21no—
190D4—21no—
191D4—42no—
192C2×C4—22no—
193C2×C4—21no—
194C2×C4—42no—
195C2×C4—2694no—
196C2×C4—5384no—
197C2×C4—5382no—
198C2×C4—5382no—
199C2×C4—10762no—
200C2×C4D4×D26911yescharacteristic
201C2×C4D4×D26912yes—
202C2×C4#1686440212yes—
203C2×C4C23×D26911yescharacteristic
204C8—22no—
205C8—5382no—
206C22—22no—
207C22—42no—
208C22—41no—
209C22#2257542612yes—
210C22C2×D4×D26911yescharacteristic
211C4—42no—
212C4—41no—
213C4—5384no—
214C4—5384no—
215C4—10762no—
216C4#2257542612yes—
217C4C2×D4×D26911yescharacteristic
218C4C2×#1686440211yescharacteristic
219C2—22no—
220C2—81no—
221C2C2×#2257542611yescharacteristic
222C2#3177141912yes—
subgroupG/H
centerC22C2×D4×D269
commutator, cosocleC2×C4×C269C24
FrattiniC2×C4C23×D269
Fitting#416×C269C2
socleC22×C269C22×D4
prime radicalC2×#41×C269C22
hypercenter#416D269
hypocenterC269#3044
LCSG ▷ C2×C4×C269 ▷ C2×C269 ▷ C269
UCSC1 ◁ C22 ◁ C22×C4 ◁ #416

Complements

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

NH ≅ G/Ntype
2. C22×Dic1076C2semidirect
4. #31771082C2semidirect
5. #31770826C2semidirect
6. #31770814C2semidirect
7. #31770631C2semidirect
8. #31770613C2semidirect
111. C269#3044semidirect

Maximal subgroups

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

subgroupindex
C2×#225754832
C22×Dic10762
#317711762
#317710822
#317708262
#317708142
#317706312
#317706132
#416×C2692
#3044269