Finite Group Databasebeta

← #46820417

Subgroups — #46820417

Order 34432.

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

normal subgroups only

This group has 458 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×#16864392C211yesmaximal, characteristic, complemented
2#31771176C211yesmaximal, characteristic
3#31771083C211yesmaximal, characteristic, complemented
4#31770825C214yesmaximal, complemented
5#31770813C214yesmaximal, complemented
6#31770633C211yesmaximal, characteristic, complemented
7#31770615C211yesmaximal, characteristic, complemented
8#31770601C211yesmaximal, characteristic
9C2×#137×C269C211yesmaximal, characteristic
10C22×Dic538C2211yescharacteristic
11C2×#16864392—24no—
12C2×#16864392C2214yes—
13C22×#14524130C2211yescharacteristic
14#22575451C2212yes—
15#22575440C2212yes—
16#22575434—22no—
17#22575432—22no—
18Q8×Dic269—22no—
19Q8×Dic269C2214yes—
20#22575412—22no—
21#22575412C2211yescharacteristic
22#22575412C2211yescharacteristic
23#22575382C2214yes—
24#22575378C2214yes—
25#22575377C2212yes—
26#22575374C2212yes—
27C2×#41×C269C2211yescharacteristic
28C22×Q8×C269C2211yescharacteristic
29#137×C269—24no—
30C2×Q16×C269C2212yes—
31C2×SD16×C269C2212yes—
32C2×M16×C269C2211yescharacteristic
33C2×Dic538—21no—
34C2×Dic538—22no—
35C2×Dic538—22no—
36C2×Dic538C2312yes—
37C22×Dic269—21no—
38#16864392—28no—
39C2×#14524130—22no—
40C2×#14524130C2312yes—
41C4⋊Dic269—24no—
42C4⋊Dic269D412yes—
43C4⋊Dic269C2312yes—
44#16864381—24no—
45C22⋊Dic269—24no—
46C22⋊Dic269—21no—
47C22⋊Dic269D412yes—
48C4×Dic269—24no—
49C4×Dic269—21no—
50C4×Dic269C2312yes—
51#41×C269—22no—
52#41×C269—22no—
53C2×Q8×C269—21no—
54C2×Q8×C269—22no—
55C2×Q8×C269D414yes—
56C2×Q8×C269C2311yescharacteristic
57C2×Q8×C269C2312yes—
58C2×D4×C269—21no—
59C2×D4×C269C2311yescharacteristic
60C22×C4×C269—21no—
61C22×C4×C269—21no—
62C22×C4×C269C2311yescharacteristic
63Q16×C269—24no—
64SD16×C269—24no—
65M16×C269—22no—
66C2×C8×C269C2312yes—
67Dic538—22no—
68Dic538—22no—
69Dic538—42no—
70C2×Dic269—22no—
71C2×Dic269—22no—
72C2×Dic269—42no—
73C2×Dic269C2×D414yes—
74#14524130—22no—
75C23×C269—21no—
76C23×C269C2×D411yescharacteristic
77Q8×C269—24no—
78Q8×C269—22no—
79Q8×C269—21no—
80Q8×C269—21no—
81Q8×C269C2×D414yes—
82D4×C269—21no—
83D4×C269—21no—
84D4×C269—42no—
85C2×C4×C269—24no—
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—24no—
100C4×C269—41no—
101C4×C269#12112yes—
102C4×C269C22×D411yescharacteristic
103C4×C269C2×#4111yescharacteristic
104Dic269—24no—
105Dic269—42no—
106Dic269—42no—
107C2×C269—22no—
108C2×C269—81no—
109C2×C269C2×#12111yescharacteristic
110C2×C269#57812yes—
111C269#304411yescharacteristic, complemented, Sylow
112#3044—2691nomaximal, Sylow
113C2×#137D26911yescharacteristic
114C22×Q16—2691no—
115#536—2691no—
116#519—2691no—
117#464—2694no—
118#462—2694no—
119#424—2691no—
120#419—2691no—
121#416—2691no—
122C2×#41C2×D26911yescharacteristic
123C22×Q8—2691no—
124C22×Q8C2×D26911yescharacteristic
125#137—24no—
126C2×Q16—2694no—
127C2×Q16—5384no—
128C2×Q16C2×D26912yes—
129C2×SD16C2×D26912yes—
130C2×M16C2×D26911yescharacteristic
131C22×C8—2691no—
132C4⋊Q8—2692no—
133#124—2692no—
134#123—5382no—
135C22⋊Q8—5382no—
136C4×Q8—2694no—
137C4×Q8—5382no—
138#117—2691no—
139#117—2691no—
140#117—5382no—
141C4⋊C8—2694no—
142#103—2692no—
143#103—2694no—
144#102—2692no—
145#41—22no—
146#41—22no—
147C2×Q8—21no—
148C2×Q8—22no—
149C2×Q8—2692no—
150C2×Q8—5381no—
151C2×Q8—5382no—
152C2×Q8—5382no—
153C2×Q8C22⋊D26914yes—
154C2×Q8C22×D26911yescharacteristic
155C2×Q8C22×D26912yes—
156C2×D4—21no—
157C2×D4C22×D26911yescharacteristic
158C22×C4—21no—
159C22×C4—21no—
160C22×C4—5381no—
161C22×C4C22×D26911yescharacteristic
162Q16—24no—
163Q16—5388no—
164SD16—24no—
165M16—22no—
166C2×C8—2692no—
167C2×C8—5382no—
168C2×C8C22×D26912yes—
169C4⋊C4—2692no—
170C4⋊C4—2692no—
171C4⋊C4—5384no—
172C4⋊C4—5384no—
173C22⋊C4—2692no—
174C22⋊C4—5384no—
175C22⋊C4—5381no—
176C42—2692no—
177C42—5384no—
178C42—5381no—
179C23—21no—
180C23D4×D26911yescharacteristic
181Q8—24no—
182Q8—22no—
183Q8—21no—
184Q8—21no—
185Q8—5382no—
186Q8—5382no—
187Q8—10762no—
188Q8C2×C22⋊D26914yes—
189D4—21no—
190D4—21no—
191D4—42no—
192C2×C4—24no—
193C2×C4—22no—
194C2×C4—21no—
195C2×C4—42no—
196C2×C4—2694no—
197C2×C4—5382no—
198C2×C4—5382no—
199C2×C4—10762no—
200C2×C4D4×D26911yescharacteristic
201C2×C4C2×C22⋊D26912yes—
202C2×C4#1686440112yes—
203C2×C4C23×D26911yescharacteristic
204C8—22no—
205C8—5382no—
206C22—22no—
207C22—42no—
208C22—41no—
209C22#2257542512yes—
210C22C2×D4×D26911yescharacteristic
211C4—24no—
212C4—41no—
213C4—5384no—
214C4—10762no—
215C4—10762no—
216C4#2257542512yes—
217C4C22×C22⋊D26911yescharacteristic
218C4C2×#1686440111yescharacteristic
219C2—22no—
220C2—81no—
221C2C2×#2257542511yescharacteristic
222C2#3177142112yes—
subgroupG/H
centerC22C2×D4×D269
commutator, cosocleC2×C4×C269C24
FrattiniC2×C4C23×D269
FittingC2×#137×C269C2
socleC22×C269C22×D4
prime radicalC2×#41×C269C22
hypercenterC2×#137D269
hypocenterC269#3044
LCSG ▷ C2×C4×C269 ▷ C2×C269 ▷ C269
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#137

Complements

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

NH ≅ G/Ntype
1. C22×#16864392C2semidirect
3. #31771083C2semidirect
4. #31770825C2semidirect
5. #31770813C2semidirect
6. #31770633C2semidirect
7. #31770615C2semidirect
111. C269#3044semidirect

Maximal subgroups

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

subgroupindex
C22×#168643922
#317711762
#317710832
#317708252
#317708132
#317706332
#317706152
#317706012
C2×#137×C2692
#3044269