Finite Group Databasebeta

← #44559896

Subgroups — #44559896

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
1C2×#22548748C211yesmaximal, characteristic
2C22×Dic1004C211yesmaximal, characteristic, complemented
3#30632263C211yesmaximal, characteristic
4#30632164C211yesmaximal, characteristic, complemented
5#30631910C214yesmaximal, complemented
6#30631898C214yesmaximal, complemented
7#30631697C211yesmaximal, characteristic, complemented
8#30631684C211yesmaximal, characteristic
9#424×C251C211yesmaximal, characteristic, complemented
10C2×#16860810C2211yescharacteristic
11C22×Dic502C2211yescharacteristic
12C22×Dic502C2211yescharacteristic
13#22548748—24no—
14C2×Dic1004—24no—
15C2×Dic1004C2214yes—
16C2×Dic1004C2212yes—
17C2×#16860799C2212yes—
18#22548715C2212yes—
19#22548703C2212yes—
20#22548699—22no—
21#22548695—22no—
22C4×Dic502C2214yes—
23#22548680—22no—
24#22548674C2211yescharacteristic
25#22548671—22no—
26#22548640C2212yes—
27#22548640C2214yes—
28#22548637C2212yes—
29C2×M16×C251C2211yescharacteristic
30C22×C8×C251C2211yescharacteristic
31#117×C251C2211yescharacteristic
32C4⋊C8×C251C2214yes—
33#16860810—22no—
34#16860810—22no—
35C2×Dic502—21no—
36C2×Dic502—21no—
37C2×Dic502—22no—
38C2×Dic502—22no—
39C2×Dic502—22no—
40C2×Dic502D414yes—
41C2×Dic502C2312yes—
42C2×Dic502C2311yescharacteristic
43C2×Dic502C2312yes—
44C2×D1004C2311yescharacteristic
45C2×C22⋊D251—21no—
46C22×Dic251—21no—
47C22×Dic251—21no—
48C2×C4×D251—21no—
49Dic1004—24no—
50Dic1004—28no—
51#16860799—24no—
52C4⋊Dic251—24no—
53C4⋊Dic251C2312yes—
54Dic251⋊C4—24no—
55C22⋊Dic251—21no—
56(C2×D251)⋊C4—24no—
57C4×Dic251—24no—
58C4×Dic251—21no—
59C22×C4×C251C2311yescharacteristic
60M16×C251—22no—
61C2×C8×C251—22no—
62C2×C8×C251C2312yes—
63C2×C8×C251C2312yes—
64C4⋊C4×C251D412yes—
65C22⋊C4×C251D412yes—
66C42×C251C2312yes—
67C22×D251—21no—
68Dic502—24no—
69Dic502—22no—
70Dic502—22no—
71Dic502—21no—
72Dic502—22no—
73Dic502—21no—
74Dic502—42no—
75Dic502C2×D414yes—
76D1004—21no—
77D1004—21no—
78C22⋊D251—42no—
79C2×Dic251—24no—
80C2×Dic251—22no—
81C2×Dic251—22no—
82C2×Dic251—22no—
83C2×Dic251—21no—
84C2×Dic251—42no—
85C4×D251—42no—
86C23×C251C2×D411yescharacteristic
87C2×C4×C251C2×D412yes—
88C2×C4×C251C2×D414yes—
89C2×C4×C251C2×D411yescharacteristic
90C2×C4×C251#4112yes—
91C2×C4×C251C2411yescharacteristic
92C8×C251—22no—
93C8×C251—22no—
94C22×C251—22no—
95C22×C251#12112yes—
96C22×C251C22×D411yescharacteristic
97C2×D251—41no—
98C2×D251—42no—
99C4×C251—24no—
100C4×C251#12112yes—
101C4×C251C22×D411yescharacteristic
102C4×C251C2×#4111yescharacteristic
103Dic251—24no—
104Dic251—41no—
105Dic251—42no—
106Dic251—42no—
107C2×C251—22no—
108C2×C251C2×#12111yescharacteristic
109C2×C251#57812yes—
110D251—81no—
111C251#304411yescharacteristic, complemented, Sylow
112#3044—2511nomaximal, Sylow
113C2×#137—2511no—
114C22×Q16—2511no—
115#536—2511no—
116#519—2511no—
117#464—2514no—
118#462—2514no—
119#424D25111yescharacteristic, complemented
120#419—2511no—
121#416—2511no—
122C2×#41—2511no—
123C22×Q8—2511no—
124C22×Q8—2511no—
125#137—5024no—
126C2×Q16—5024no—
127C2×Q16—2514no—
128C2×Q16—2512no—
129C2×SD16—2512no—
130C2×M16C2×D25111yescharacteristic
131C22×C8C2×D25111yescharacteristic
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#117C2×D25111yescharacteristic
141C4⋊C8C2×D25114yes—
142#103—2512no—
143#103—2514no—
144#102—2512no—
145#41—5022no—
146#41—5022no—
147C2×Q8—5022no—
148C2×Q8—5022no—
149C2×Q8—5021no—
150C2×Q8—5021no—
151C2×Q8—5022no—
152C2×Q8—2514no—
153C2×Q8—2512no—
154C2×Q8—2511no—
155C2×Q8—2512no—
156C2×D4—5021no—
157C2×D4—2511no—
158C22×C4—5021no—
159C22×C4—5021no—
160C22×C4—5021no—
161C22×C4C22×D25111yescharacteristic
162Q16—5028no—
163Q16—5024no—
164SD16—5024no—
165M16—22no—
166C2×C8—22no—
167C2×C8C22×D25112yes—
168C2×C8C22×D25112yes—
169C4⋊C4—5024no—
170C4⋊C4—5024no—
171C4⋊C4—2512no—
172C4⋊C4D100412yes—
173C22⋊C4—5021no—
174C22⋊C4—5024no—
175C22⋊C4D100412yes—
176C42—5024no—
177C42—5021no—
178C42C22×D25112yes—
179C23—5021no—
180C23C2×D100411yescharacteristic
181Q8—10042no—
182Q8—5022no—
183Q8—5021no—
184Q8—5022no—
185Q8—5021no—
186Q8—5022no—
187Q8—5024no—
188Q8—2514no—
189D4—10042no—
190D4—5021no—
191D4—5021no—
192C2×C4—10042no—
193C2×C4—10042no—
194C2×C4—5022no—
195C2×C4—5022no—
196C2×C4—5021no—
197C2×C4—5022no—
198C2×C4—5024no—
199C2×C4D4×D25112yes—
200C2×C4C2×D100414yes—
201C2×C4C2×D100411yescharacteristic
202C2×C4#1686081112yes—
203C2×C4C23×D25111yescharacteristic
204C8—22no—
205C8—22no—
206C22—22no—
207C22—10042no—
208C22—10041no—
209C22#2254869012yes—
210C22C22×D100411yescharacteristic
211C4—24no—
212C4—10042no—
213C4—10041no—
214C4—10042no—
215C4—5024no—
216C4#2254869012yes—
217C4C2×D4×D25111yescharacteristic
218C4C2×#1686081111yescharacteristic
219C2—22no—
220C2—20081no—
221C2C2×#2254869011yescharacteristic
222C2#3063250412yes—
subgroupG/H
centerC22C22×D1004
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
Fitting#424×C251C2
socleC22×C251C22×D4
prime radicalC2×#16860810C22
hypercenter#424D251
hypocenterC251#3044
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ #117 ◁ #424

Complements

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

NH ≅ G/Ntype
2. C22×Dic1004C2semidirect
4. #30632164C2semidirect
5. #30631910C2semidirect
6. #30631898C2semidirect
7. #30631697C2semidirect
9. #424×C251C2semidirect
111. C251#3044semidirect
119. #424D251semidirect

Maximal subgroups

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

subgroupindex
C2×#225487482
C22×Dic10042
#306322632
#306321642
#306319102
#306318982
#306316972
#306316842
#424×C2512
#3044251