Finite Group Databasebeta

← #44559897

Subgroups — #44559897

Order 32128.

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×#16860800C211yesmaximal, characteristic, complemented
2#30632258C211yesmaximal, characteristic
3#30632165C211yesmaximal, characteristic, complemented
4#30631907C214yesmaximal, complemented
5#30631895C214yesmaximal, complemented
6#30631715C211yesmaximal, characteristic, complemented
7#30631697C211yesmaximal, characteristic, complemented
8#30631683C211yesmaximal, characteristic
9C2×#137×C251C211yesmaximal, characteristic
10C22×Dic502C2211yescharacteristic
11C2×#16860800—24no—
12C2×#16860800C2214yes—
13C22×C251⋊C8C2211yescharacteristic
14#22548713C2212yes—
15#22548702C2212yes—
16#22548696—22no—
17#22548694—22no—
18Q8×Dic251—22no—
19Q8×Dic251C2214yes—
20#22548674—22no—
21#22548674C2211yescharacteristic
22#22548674C2211yescharacteristic
23Dic251⋊C8C2214yes—
24#22548640C2214yes—
25#22548639C2212yes—
26#22548636C2212yes—
27C2×#41×C251C2211yescharacteristic
28C22×Q8×C251C2211yescharacteristic
29#137×C251—24no—
30C2×Q16×C251C2212yes—
31C2×SD16×C251C2212yes—
32C2×M16×C251C2211yescharacteristic
33C2×Dic502—21no—
34C2×Dic502—22no—
35C2×Dic502—22no—
36C2×Dic502C2312yes—
37C22×Dic251—21no—
38#16860800—28no—
39C2×C251⋊C8—22no—
40C2×C251⋊C8C2312yes—
41C4⋊Dic251—24no—
42C4⋊Dic251D412yes—
43C4⋊Dic251C2312yes—
44Dic251⋊C4—24no—
45C22⋊Dic251—24no—
46C22⋊Dic251—21no—
47C22⋊Dic251D412yes—
48C4×Dic251—24no—
49C4×Dic251—21no—
50C4×Dic251C2312yes—
51#41×C251—22no—
52#41×C251—22no—
53C2×Q8×C251—21no—
54C2×Q8×C251—22no—
55C2×Q8×C251D414yes—
56C2×Q8×C251C2311yescharacteristic
57C2×Q8×C251C2312yes—
58C2×D4×C251—21no—
59C2×D4×C251C2311yescharacteristic
60C22×C4×C251—21no—
61C22×C4×C251—21no—
62C22×C4×C251C2311yescharacteristic
63Q16×C251—24no—
64SD16×C251—24no—
65M16×C251—22no—
66C2×C8×C251C2312yes—
67Dic502—22no—
68Dic502—22no—
69Dic502—42no—
70C2×Dic251—22no—
71C2×Dic251—22no—
72C2×Dic251—42no—
73C2×Dic251C2×D414yes—
74C251⋊C8—22no—
75C23×C251—21no—
76C23×C251C2×D411yescharacteristic
77Q8×C251—24no—
78Q8×C251—22no—
79Q8×C251—21no—
80Q8×C251—21no—
81Q8×C251C2×D414yes—
82D4×C251—21no—
83D4×C251—21no—
84D4×C251—42no—
85C2×C4×C251—24no—
86C2×C4×C251—22no—
87C2×C4×C251—21no—
88C2×C4×C251—42no—
89C2×C4×C251C2×D411yescharacteristic
90C2×C4×C251C2×D412yes—
91C2×C4×C251#4112yes—
92C2×C4×C251C2411yescharacteristic
93C8×C251—22no—
94C22×C251—22no—
95C22×C251—41no—
96C22×C251—42no—
97C22×C251#12112yes—
98C22×C251C22×D411yescharacteristic
99C4×C251—24no—
100C4×C251—41no—
101C4×C251#12112yes—
102C4×C251C22×D411yescharacteristic
103C4×C251C2×#4111yescharacteristic
104Dic251—24no—
105Dic251—42no—
106Dic251—42no—
107C2×C251—22no—
108C2×C251—81no—
109C2×C251C2×#12111yescharacteristic
110C2×C251#57812yes—
111C251#304411yescharacteristic, complemented, Sylow
112#3044—2511nomaximal, Sylow
113C2×#137D25111yescharacteristic
114C22×Q16—2511no—
115#536—2511no—
116#519—2511no—
117#464—2514no—
118#462—2514no—
119#424—2511no—
120#419—2511no—
121#416—2511no—
122C2×#41C2×D25111yescharacteristic
123C22×Q8—2511no—
124C22×Q8C2×D25111yescharacteristic
125#137—24no—
126C2×Q16—5024no—
127C2×Q16—2514no—
128C2×Q16C2×D25112yes—
129C2×SD16C2×D25112yes—
130C2×M16C2×D25111yescharacteristic
131C22×C8—2511no—
132C4⋊Q8—2512no—
133#124—2512no—
134#123—5022no—
135C22⋊Q8—5022no—
136C4×Q8—5022no—
137C4×Q8—2514no—
138#117—5022no—
139#117—2511no—
140#117—2511no—
141C4⋊C8—2514no—
142#103—2512no—
143#103—2514no—
144#102—2512no—
145#41—22no—
146#41—22no—
147C2×Q8—21no—
148C2×Q8—22no—
149C2×Q8—5022no—
150C2×Q8—5022no—
151C2×Q8—5021no—
152C2×Q8—2512no—
153C2×Q8C22⋊D25114yes—
154C2×Q8C22×D25111yescharacteristic
155C2×Q8C22×D25112yes—
156C2×D4—21no—
157C2×D4C22×D25111yescharacteristic
158C22×C4—21no—
159C22×C4—21no—
160C22×C4—5021no—
161C22×C4C22×D25111yescharacteristic
162Q16—24no—
163Q16—5028no—
164SD16—24no—
165M16—22no—
166C2×C8—5022no—
167C2×C8—2512no—
168C2×C8C22×D25112yes—
169C4⋊C4—5024no—
170C4⋊C4—5024no—
171C4⋊C4—2512no—
172C4⋊C4—2512no—
173C22⋊C4—5021no—
174C22⋊C4—5024no—
175C22⋊C4—2512no—
176C42—5024no—
177C42—5021no—
178C42—2512no—
179C23—21no—
180C23D4×D25111yescharacteristic
181Q8—22no—
182Q8—24no—
183Q8—21no—
184Q8—21no—
185Q8—10042no—
186Q8—5022no—
187Q8—5022no—
188Q8C2×C22⋊D25114yes—
189D4—21no—
190D4—21no—
191D4—42no—
192C2×C4—22no—
193C2×C4—24no—
194C2×C4—21no—
195C2×C4—42no—
196C2×C4—10042no—
197C2×C4—5022no—
198C2×C4—5022no—
199C2×C4—2514no—
200C2×C4D4×D25111yescharacteristic
201C2×C4C2×C22⋊D25112yes—
202C2×C4#1686080912yes—
203C2×C4C23×D25111yescharacteristic
204C8—22no—
205C8—5022no—
206C22—22no—
207C22—42no—
208C22—41no—
209C22#2254868712yes—
210C22C2×D4×D25111yescharacteristic
211C4—24no—
212C4—41no—
213C4—10042no—
214C4—10042no—
215C4—5024no—
216C4#2254868712yes—
217C4C22×C22⋊D25111yescharacteristic
218C4C2×#1686080911yescharacteristic
219C2—22no—
220C2—81no—
221C2C2×#2254868711yescharacteristic
222C2#3063250312yes—
subgroupG/H
centerC22C2×D4×D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
FittingC2×#137×C251C2
socleC22×C251C22×D4
prime radicalC2×#41×C251C22
hypercenterC2×#137D251
hypocenterC251#3044
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#137

Complements

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

NH ≅ G/Ntype
1. C22×#16860800C2semidirect
3. #30632165C2semidirect
4. #30631907C2semidirect
5. #30631895C2semidirect
6. #30631715C2semidirect
7. #30631697C2semidirect
111. C251#3044semidirect

Maximal subgroups

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

subgroupindex
C22×#168608002
#306322582
#306321652
#306319072
#306318952
#306317152
#306316972
#306316832
C2×#137×C2512
#3044251