Finite Group Databasebeta

← #44560031

Subgroups — #44560031

Order 32128.

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

normal subgroups only

This group has 484 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×#22548721C211yesmaximal, characteristic, complemented
2#30632265C212yesmaximal, complemented
3#30631930C214yesmaximal, complemented
4#30631925C212yesmaximal, complemented
5#30631920C212yesmaximal, complemented
6C2×#22548649C211yesmaximal, characteristic, complemented
7#30631688C212yesmaximal, complemented
8C2×#135×C251C211yesmaximal, characteristic, complemented
9C2×#16860810C2211yescharacteristic
10#22548721—24no—
11C2×#16860794C2211yescharacteristic
12C2×#16860793C2212yes—
13C2×C8×D251C2212yes—
14#22548694—22no—
15#22548691C2214yes—
16#22548689—22no—
17#22548687—24no—
18#22548686C2214yes—
19C2×C4⋊Dic251C2212yes—
20#22548649C2212yes—
21#22548649C2212yes—
22#22548639C2214yes—
23#22548636C2214yes—
24C2×#41×C251C2212yes—
25#135×C251—24no—
26C2×Q16×C251C2211yescharacteristic
27C2×SD16×C251C2212yes—
28C2×D8×C251C2211yescharacteristic
29C22×C8×C251C2211yescharacteristic
30#16860810—22no—
31#16860810—22no—
32C2×Dic502D411yescharacteristic
33C2×D1004D411yescharacteristic
34C2×C22⋊D251—24no—
35C2×C22⋊D251D412yes—
36C22×Dic251—22no—
37C2×C4×D251C2312yes—
38#16860794—22no—
39#16860793—24no—
40C2×C251⋊C8C2312yes—
41C8×D251—24no—
42C4⋊Dic251—22no—
43C4⋊Dic251C2314yes—
44Dic251⋊C4—24no—
45C22⋊Dic251—24no—
46(C2×D251)⋊C4—24no—
47#41×C251—24no—
48#41×C251—24no—
49C2×Q8×C251C2312yes—
50C2×D4×C251—22no—
51C2×D4×C251C2312yes—
52C22×C4×C251—22no—
53C22×C4×C251C2311yescharacteristic
54Q16×C251—22no—
55SD16×C251—24no—
56D8×C251—22no—
57C2×C8×C251D414yescomplemented
58C2×C8×C251C2311yescharacteristic
59C2×C8×C251C2311yescharacteristic
60C22×D251C2×D412yes—
61Dic502—22no—
62D1004—22no—
63C22⋊D251—24no—
64C22⋊D251—84no—
65C2×Dic251—24no—
66C2×Dic251—82no—
67C2×Dic251C2×D412yes—
68C4×D251—22no—
69C4×D251—22no—
70C251⋊C8—22no—
71C23×C251—22no—
72C23×C251C2×D411yescharacteristic
73Q8×C251—22no—
74Q8×C251—22no—
75D4×C251—22no—
76D4×C251—22no—
77D4×C251—44no—
78C2×C4×C251—22no—
79C2×C4×C251—44no—
80C2×C4×C251C2×D411yescharacteristic
81C2×C4×C251C2×D412yes—
82C2×C4×C251#4112yes—
83C2×C4×C251C2411yescharacteristic
84C8×C251C2×D412yes—
85C8×C251C2×D412yes—
86C22×C251—21no—
87C22×C251—21no—
88C22×C251—42no—
89C22×C251—42no—
90C22×C251—42no—
91C22×C251#12112yes—
92C22×C251C22×D411yescharacteristic
93C2×D251—22no—
94C2×D251—22no—
95C2×D251—22no—
96C4×C251—42no—
97C4×C251#12112yes—
98C4×C251C22×D411yescharacteristic
99C4×C251C2×#4111yescharacteristic
100Dic251—22no—
101Dic251—44no—
102C2×C251—22no—
103C2×C251—82no—
104C2×C251C2×#12111yescharacteristic
105C2×C251#57611yescharacteristic
106C2×C251#57811yescharacteristic
107D251—42no—
108C251#306111yescharacteristic, complemented, Sylow
109#3061—2511nomaximal, Sylow
110C2×#135D25111yescharacteristic, complemented
111C2×#131—2511no—
112#537—2512no—
113#468—2514no—
114#467—2512no—
115#466—2512no—
116C2×#107—2511no—
117#417—2512no—
118C2×#41—2511no—
119C2×#41C2×D25112yes—
120#135—24no—
121C2×Q16C2×D25111yescharacteristic
122C2×SD16C2×D25112yes—
123C2×D8C2×D25111yescharacteristic
124#131—5024no—
125C2×M16—2511no—
126C2×M16—2512no—
127C22×C8—2512no—
128C22×C8C2×D25111yescharacteristic
129C22⋊Q8—5022no—
130C22⋊Q8—2514no—
131#121—5022no—
132#121—5024no—
133#121—2514no—
134C2×C4⋊C4—2512no—
135#107—2512no—
136#107—2512no—
137#103—2514no—
138#102—2514no—
139#41—24no—
140#41—24no—
141#41—5022no—
142#41—5022no—
143C2×Q8—2511no—
144C2×Q8C22×D25112yes—
145C2×D4—22no—
146C2×D4—5024no—
147C2×D4—2512no—
148C2×D4—2511no—
149C2×D4C22×D25112yes—
150C22×C4—22no—
151C22×C4—5022no—
152C22×C4—2512no—
153C22×C4C22×D25111yescharacteristic
154Q16—22no—
155SD16—24no—
156D8—22no—
157M16—5022no—
158M16—5024no—
159C2×C8—5024no—
160C2×C8—2512no—
161C2×C8C22⋊D25114yescomplemented
162C2×C8C22×D25111yescharacteristic
163C2×C8C22×D25111yescharacteristic
164C4⋊C4—5024no—
165C4⋊C4—5022no—
166C4⋊C4—2514no—
167C22⋊C4—5024no—
168C22⋊C4—5024no—
169C23—22no—
170C23—2512no—
171C23D4×D25111yescharacteristic
172Q8—22no—
173Q8—22no—
174Q8—5022no—
175D4—22no—
176D4—22no—
177D4—44no—
178D4—20084no—
179D4—5024no—
180D4—5022no—
181C2×C4—22no—
182C2×C4—44no—
183C2×C4—20082no—
184C2×C4—5024no—
185C2×C4—5022no—
186C2×C4—5022no—
187C2×C4—2512no—
188C2×C4D4×D25111yescharacteristic
189C2×C4C2×C22⋊D25112yes—
190C2×C4#1686080912yes—
191C2×C4C23×D25111yescharacteristic
192C8—5022no—
193C8C2×C22⋊D25112yes—
194C8C2×C22⋊D25112yes—
195C22—21no—
196C22—21no—
197C22—42no—
198C22—42no—
199C22—42no—
200C22—5022no—
201C22—5022no—
202C22—5022no—
203C22#2254868612yes—
204C22C2×D4×D25111yescharacteristic
205C4—42no—
206C4—10044no—
207C4—5022no—
208C4#2254868612yes—
209C4C22×C22⋊D25111yescharacteristic
210C4C2×#1686080911yescharacteristic
211C2—22no—
212C2—82no—
213C2—10042no—
214C2C2×#2254868611yescharacteristic
215C2#3063248911yescharacteristic
216C2#3063250111yescharacteristic
subgroupG/H
centerC22C2×D4×D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
FittingC2×#135×C251C2
socleC22×C251C22×D4
hypercenterC2×#135D251
hypocenterC251#3061
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#135

Complements

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

NH ≅ G/Ntype
1. C2×#22548721C2semidirect
2. #30632265C2semidirect
3. #30631930C2semidirect
4. #30631925C2semidirect
5. #30631920C2semidirect
6. C2×#22548649C2semidirect
7. #30631688C2semidirect
8. C2×#135×C251C2semidirect
57. C2×C8×C251D4semidirect
108. C251#3061semidirect
110. C2×#135D251semidirect
161. C2×C8C22⋊D251semidirect

Maximal subgroups

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

subgroupindex
C2×#225487212
#306322652
#306319302
#306319252
#306319202
C2×#225486492
#306316882
C2×#135×C2512
#3061251